Deep Learning

In this section, we show how to fit the examples discussed in the text. We use the keras package, which interfaces to the tensorflow package which in turn links to efficient python code. This code is impressively fast, and the package is well-structured. A good companion is the text Deep Learning with R (F. Chollet and J.J. Allaire, Deep Learning with R (2018), Manning Publications.), and most of our code is adapted from there.

Getting keras up and running on your computer can be a challenge. The book website <www.statlearning.com> gives step-by-step instructions on how to achieve this. (Many thanks to Balasubramanian Narasimhan for preparing the keras installation instructions.) Guidance can also be found at <keras.rstudio.com>.

Since the first printing of this book, the torch package has become available as an alternative to the keras package for deep learning. While torch does not require a python installation, the current implementation appears to be a fair bit slower than keras. A version of this lab that makes use of torch is available on the book website. (Many thanks to Daniel Falbel and Sigrid Keydana for preparing the torch version of this lab.)

A Single Layer Network on the Hitters Data

We start by fitting the models in Section 10.6. We set up the data, and separate out a training and test set.

library(ISLR2)
Gitters <- na.omit(Hitters)
n <- nrow(Gitters)
set.seed(13)
ntest <- trunc(n / 3)
testid <- sample(1:n, ntest)

The linear model should be familiar, but we present it anyway.

lfit <- lm(Salary ~ ., data = Gitters[-testid, ])
lpred <- predict(lfit, Gitters[testid, ])
with(Gitters[testid, ], mean(abs(lpred - Salary)))

## [1] 254.6687

Notice the use of the with() command: the first argument is a dataframe, and the second an expression that can refer to elements of the dataframe by name. In this instance the dataframe corresponds to the test data and the expression computes the mean absolute prediction error on this data.

Next we fit the lasso using glmnet. Since this package does not use formulas, we create x and y first.

x <- scale(model.matrix(Salary ~ . - 1, data = Gitters))
y <- Gitters$Salary

The first line makes a call to model.matrix(), which produces the same matrix that was used by lm() (the -1 omits the intercept). This function automatically converts factors to dummy variables. The scale() function standardizes the matrix so each column has mean zero and variance one.

library(glmnet)

## Loading required package: Matrix

## Loaded glmnet 4.1-8

cvfit <- cv.glmnet(x[-testid, ], y[-testid],
    type.measure = "mae")
cpred <- predict(cvfit, x[testid, ], s = "lambda.min")
mean(abs(y[testid] - cpred))

## [1] 252.2994

To fit the neural network, we first set up a model structure that describes the network.

library(keras)
reticulate::use_condaenv(condaenv = "r-tensorflow")
modnn <- keras_model_sequential() %>%
  layer_dense(units = 50, activation = "relu",
        input_shape = ncol(x)) %>%
   layer_dropout(rate = 0.4) %>%
   layer_dense(units = 1)

We have created a vanilla model object called modnn, and have added details about the successive layers in a sequential manner, using the function keras_model_sequential(). The pipe operator \%>\% passes the previous term as the first argument to the next function, and returns the result. It allows us to specify the layers of a neural network in a readable form.

We illustrate the use of the pipe operator on a simple example. Earlier, we created x using the command

x <- scale(model.matrix(Salary ~ . - 1, data = Gitters))

We first make a matrix, and then we center each of the variables. Compound expressions like this can be difficult to parse. We could have obtained the same result using the pipe operator:

x <- model.matrix(Salary ~ . - 1, data = Gitters) %>% scale()

Using the pipe operator makes it easier to follow the sequence of operations.

We now return to our neural network. The object modnn has a single hidden layer with 50 hidden units, and a ReLU activation function. It then has a dropout layer, in which a random 40% of the 50 activations from the previous layer are set to zero during each iteration of the stochastic gradient descent algorithm. Finally, the output layer has just one unit with no activation function, indicating that the model provides a single quantitative output.

Next we add details to modnn that control the fitting algorithm. Here we have simply followed the examples given in the Keras book. We minimize squared-error loss as in (10.23). The algorithm tracks the mean absolute error on the training data, and on validation data if it is supplied.

modnn %>% compile(loss = "mse",
    optimizer = optimizer_rmsprop(),
    metrics = list("mean_absolute_error")
   )

In the previous line, the pipe operator passes modnn as the first argument to compile(). The compile() function does not actually change the R object modnn, but it does communicate these specifications to the corresponding python instance of this model that has been created along the way.

Now we fit the model. We supply the training data and two fitting parameters, epochs and batch_size. Using 32 for the latter means that at each step of SGD, the algorithm randomly selects 32 training observations for the computation of the gradient. Recall from Sections 10.4 and 10.7 that an epoch amounts to the number of SGD steps required to process \(n\) observations. Since the training set has \(n=176\), an epoch is \(176/32=5.5\) SGD steps. The fit() function has an argument validation_data; these data are not used in the fitting, but can be used to track the progress of the model (in this case reporting the mean absolute error). Here we actually supply the test data so we can see the mean absolute error of both the training data and test data as the epochs proceed. To see more options for fitting, use ?fit.keras.engine.training.Model.

history <- modnn %>% fit(
#    x[-testid, ], y[-testid], epochs = 1500, batch_size = 32,
    x[-testid, ], y[-testid], epochs = 600, batch_size = 32,
    validation_data = list(x[testid, ], y[testid])
  )

## Epoch 1/600
## 6/6 - 1s - loss: 457334.9062 - mean_absolute_error: 533.8674 - val_loss: 556238.4375 - val_mean_absolute_error: 539.7639 - 506ms/epoch - 84ms/step
## Epoch 2/600
## 6/6 - 0s - loss: 456833.5938 - mean_absolute_error: 533.5650 - val_loss: 555896.6250 - val_mean_absolute_error: 539.5463 - 50ms/epoch - 8ms/step
## Epoch 3/600
## 6/6 - 0s - loss: 456592.0000 - mean_absolute_error: 533.4005 - val_loss: 555594.5000 - val_mean_absolute_error: 539.3562 - 29ms/epoch - 5ms/step
## Epoch 4/600
## 6/6 - 0s - loss: 456382.4062 - mean_absolute_error: 533.2407 - val_loss: 555313.8125 - val_mean_absolute_error: 539.1793 - 28ms/epoch - 5ms/step
## Epoch 5/600
## 6/6 - 0s - loss: 456072.0000 - mean_absolute_error: 533.0283 - val_loss: 555009.5625 - val_mean_absolute_error: 538.9904 - 35ms/epoch - 6ms/step
## Epoch 6/600
## 6/6 - 0s - loss: 455756.5938 - mean_absolute_error: 532.8202 - val_loss: 554737.5625 - val_mean_absolute_error: 538.8115 - 28ms/epoch - 5ms/step
## Epoch 7/600
## 6/6 - 0s - loss: 455522.0312 - mean_absolute_error: 532.6313 - val_loss: 554442.6875 - val_mean_absolute_error: 538.6324 - 27ms/epoch - 4ms/step
## Epoch 8/600
## 6/6 - 0s - loss: 455281.5000 - mean_absolute_error: 532.3987 - val_loss: 554147.7500 - val_mean_absolute_error: 538.4508 - 27ms/epoch - 5ms/step
## Epoch 9/600
## 6/6 - 0s - loss: 455031.5312 - mean_absolute_error: 532.2232 - val_loss: 553844.6875 - val_mean_absolute_error: 538.2698 - 27ms/epoch - 4ms/step
## Epoch 10/600
## 6/6 - 0s - loss: 454795.2188 - mean_absolute_error: 532.1102 - val_loss: 553525.2500 - val_mean_absolute_error: 538.0775 - 27ms/epoch - 4ms/step
## Epoch 11/600
## 6/6 - 0s - loss: 454600.8125 - mean_absolute_error: 531.9166 - val_loss: 553196.7500 - val_mean_absolute_error: 537.8855 - 27ms/epoch - 4ms/step
## Epoch 12/600
## 6/6 - 0s - loss: 454052.5000 - mean_absolute_error: 531.5363 - val_loss: 552825.6875 - val_mean_absolute_error: 537.6652 - 27ms/epoch - 5ms/step
## Epoch 13/600
## 6/6 - 0s - loss: 453859.6875 - mean_absolute_error: 531.4360 - val_loss: 552517.1250 - val_mean_absolute_error: 537.4695 - 27ms/epoch - 4ms/step
## Epoch 14/600
## 6/6 - 0s - loss: 453863.5000 - mean_absolute_error: 531.3553 - val_loss: 552172.9375 - val_mean_absolute_error: 537.2648 - 27ms/epoch - 4ms/step
## Epoch 15/600
## 6/6 - 0s - loss: 453250.0000 - mean_absolute_error: 531.0011 - val_loss: 551801.7500 - val_mean_absolute_error: 537.0378 - 27ms/epoch - 4ms/step
## Epoch 16/600
## 6/6 - 0s - loss: 453011.8125 - mean_absolute_error: 530.8214 - val_loss: 551437.4375 - val_mean_absolute_error: 536.8190 - 26ms/epoch - 4ms/step
## Epoch 17/600
## 6/6 - 0s - loss: 452861.4688 - mean_absolute_error: 530.6567 - val_loss: 551043.1875 - val_mean_absolute_error: 536.5903 - 26ms/epoch - 4ms/step
## Epoch 18/600
## 6/6 - 0s - loss: 452115.2188 - mean_absolute_error: 530.1516 - val_loss: 550608.4375 - val_mean_absolute_error: 536.3353 - 26ms/epoch - 4ms/step
## Epoch 19/600
## 6/6 - 0s - loss: 451976.4688 - mean_absolute_error: 530.1111 - val_loss: 550186.6875 - val_mean_absolute_error: 536.0891 - 26ms/epoch - 4ms/step
## Epoch 20/600
## 6/6 - 0s - loss: 451324.9688 - mean_absolute_error: 529.6184 - val_loss: 549744.4375 - val_mean_absolute_error: 535.8237 - 28ms/epoch - 5ms/step
## Epoch 21/600
## 6/6 - 0s - loss: 451435.0312 - mean_absolute_error: 529.5406 - val_loss: 549299.7500 - val_mean_absolute_error: 535.5673 - 27ms/epoch - 4ms/step
## Epoch 22/600
## 6/6 - 0s - loss: 450814.6250 - mean_absolute_error: 529.1381 - val_loss: 548868.9375 - val_mean_absolute_error: 535.3138 - 25ms/epoch - 4ms/step
## Epoch 23/600
## 6/6 - 0s - loss: 450577.5312 - mean_absolute_error: 529.0134 - val_loss: 548387.3125 - val_mean_absolute_error: 535.0267 - 27ms/epoch - 5ms/step
## Epoch 24/600
## 6/6 - 0s - loss: 450144.8125 - mean_absolute_error: 528.7249 - val_loss: 547929.3125 - val_mean_absolute_error: 534.7604 - 28ms/epoch - 5ms/step
## Epoch 25/600
## 6/6 - 0s - loss: 449781.2812 - mean_absolute_error: 528.4890 - val_loss: 547436.2500 - val_mean_absolute_error: 534.4739 - 28ms/epoch - 5ms/step
## Epoch 26/600
## 6/6 - 0s - loss: 449317.0000 - mean_absolute_error: 528.0663 - val_loss: 546951.6250 - val_mean_absolute_error: 534.1855 - 30ms/epoch - 5ms/step
## Epoch 27/600
## 6/6 - 0s - loss: 448741.0000 - mean_absolute_error: 527.7154 - val_loss: 546404.8125 - val_mean_absolute_error: 533.8765 - 28ms/epoch - 5ms/step
## Epoch 28/600
## 6/6 - 0s - loss: 448128.4062 - mean_absolute_error: 527.3689 - val_loss: 545848.8125 - val_mean_absolute_error: 533.5617 - 27ms/epoch - 5ms/step
## Epoch 29/600
## 6/6 - 0s - loss: 447865.7812 - mean_absolute_error: 527.1992 - val_loss: 545261.9375 - val_mean_absolute_error: 533.2348 - 26ms/epoch - 4ms/step
## Epoch 30/600
## 6/6 - 0s - loss: 447602.5000 - mean_absolute_error: 526.9998 - val_loss: 544706.3750 - val_mean_absolute_error: 532.9114 - 26ms/epoch - 4ms/step
## Epoch 31/600
## 6/6 - 0s - loss: 446477.6250 - mean_absolute_error: 526.4104 - val_loss: 544070.5000 - val_mean_absolute_error: 532.5593 - 26ms/epoch - 4ms/step
## Epoch 32/600
## 6/6 - 0s - loss: 446721.5938 - mean_absolute_error: 526.3264 - val_loss: 543493.0625 - val_mean_absolute_error: 532.2207 - 26ms/epoch - 4ms/step
## Epoch 33/600
## 6/6 - 0s - loss: 446038.5938 - mean_absolute_error: 525.8576 - val_loss: 542859.0000 - val_mean_absolute_error: 531.8646 - 26ms/epoch - 4ms/step
## Epoch 34/600
## 6/6 - 0s - loss: 445998.6875 - mean_absolute_error: 525.7407 - val_loss: 542280.3750 - val_mean_absolute_error: 531.5212 - 26ms/epoch - 4ms/step
## Epoch 35/600
## 6/6 - 0s - loss: 445156.9062 - mean_absolute_error: 525.1526 - val_loss: 541600.1875 - val_mean_absolute_error: 531.1434 - 26ms/epoch - 4ms/step
## Epoch 36/600
## 6/6 - 0s - loss: 444795.1875 - mean_absolute_error: 524.9016 - val_loss: 540929.9375 - val_mean_absolute_error: 530.7648 - 26ms/epoch - 4ms/step
## Epoch 37/600
## 6/6 - 0s - loss: 443573.5000 - mean_absolute_error: 524.2570 - val_loss: 540201.7500 - val_mean_absolute_error: 530.3696 - 28ms/epoch - 5ms/step
## Epoch 38/600
## 6/6 - 0s - loss: 442829.1250 - mean_absolute_error: 523.7537 - val_loss: 539491.5625 - val_mean_absolute_error: 529.9716 - 29ms/epoch - 5ms/step
## Epoch 39/600
## 6/6 - 0s - loss: 442595.6250 - mean_absolute_error: 523.3256 - val_loss: 538749.1250 - val_mean_absolute_error: 529.5636 - 27ms/epoch - 5ms/step
## Epoch 40/600
## 6/6 - 0s - loss: 441503.5312 - mean_absolute_error: 522.7891 - val_loss: 537985.8125 - val_mean_absolute_error: 529.1382 - 26ms/epoch - 4ms/step
## Epoch 41/600
## 6/6 - 0s - loss: 441055.6250 - mean_absolute_error: 522.4424 - val_loss: 537181.8750 - val_mean_absolute_error: 528.6934 - 26ms/epoch - 4ms/step
## Epoch 42/600
## 6/6 - 0s - loss: 440775.2812 - mean_absolute_error: 522.1063 - val_loss: 536433.9375 - val_mean_absolute_error: 528.2747 - 26ms/epoch - 4ms/step
## Epoch 43/600
## 6/6 - 0s - loss: 440280.7812 - mean_absolute_error: 521.9980 - val_loss: 535695.1250 - val_mean_absolute_error: 527.8558 - 27ms/epoch - 4ms/step
## Epoch 44/600
## 6/6 - 0s - loss: 440286.2812 - mean_absolute_error: 521.7043 - val_loss: 534923.7500 - val_mean_absolute_error: 527.4075 - 27ms/epoch - 5ms/step
## Epoch 45/600
## 6/6 - 0s - loss: 438514.4688 - mean_absolute_error: 520.4506 - val_loss: 534051.1250 - val_mean_absolute_error: 526.9357 - 25ms/epoch - 4ms/step
## Epoch 46/600
## 6/6 - 0s - loss: 438250.3750 - mean_absolute_error: 520.4128 - val_loss: 533173.6875 - val_mean_absolute_error: 526.4585 - 25ms/epoch - 4ms/step
## Epoch 47/600
## 6/6 - 0s - loss: 437779.8750 - mean_absolute_error: 520.0002 - val_loss: 532346.7500 - val_mean_absolute_error: 526.0046 - 26ms/epoch - 4ms/step
## Epoch 48/600
## 6/6 - 0s - loss: 436727.5000 - mean_absolute_error: 519.4863 - val_loss: 531510.3750 - val_mean_absolute_error: 525.5261 - 28ms/epoch - 5ms/step
## Epoch 49/600
## 6/6 - 0s - loss: 436120.6875 - mean_absolute_error: 519.2165 - val_loss: 530570.8750 - val_mean_absolute_error: 525.0003 - 26ms/epoch - 4ms/step
## Epoch 50/600
## 6/6 - 0s - loss: 435588.4062 - mean_absolute_error: 518.7215 - val_loss: 529738.1875 - val_mean_absolute_error: 524.5460 - 25ms/epoch - 4ms/step
## Epoch 51/600
## 6/6 - 0s - loss: 433610.6875 - mean_absolute_error: 517.5278 - val_loss: 528750.4375 - val_mean_absolute_error: 523.9998 - 29ms/epoch - 5ms/step
## Epoch 52/600
## 6/6 - 0s - loss: 432966.5312 - mean_absolute_error: 517.0633 - val_loss: 527815.7500 - val_mean_absolute_error: 523.4831 - 28ms/epoch - 5ms/step
## Epoch 53/600
## 6/6 - 0s - loss: 432563.1875 - mean_absolute_error: 516.7186 - val_loss: 526852.8125 - val_mean_absolute_error: 522.9677 - 28ms/epoch - 5ms/step
## Epoch 54/600
## 6/6 - 0s - loss: 432462.4688 - mean_absolute_error: 516.1799 - val_loss: 525937.5000 - val_mean_absolute_error: 522.4429 - 27ms/epoch - 4ms/step
## Epoch 55/600
## 6/6 - 0s - loss: 431893.5312 - mean_absolute_error: 515.6415 - val_loss: 524968.9375 - val_mean_absolute_error: 521.9095 - 30ms/epoch - 5ms/step
## Epoch 56/600
## 6/6 - 0s - loss: 430110.1875 - mean_absolute_error: 514.9573 - val_loss: 523952.5938 - val_mean_absolute_error: 521.3519 - 28ms/epoch - 5ms/step
## Epoch 57/600
## 6/6 - 0s - loss: 429932.3125 - mean_absolute_error: 514.7479 - val_loss: 522945.7812 - val_mean_absolute_error: 520.7855 - 27ms/epoch - 4ms/step
## Epoch 58/600
## 6/6 - 0s - loss: 428999.2188 - mean_absolute_error: 514.1412 - val_loss: 521918.5625 - val_mean_absolute_error: 520.2151 - 27ms/epoch - 5ms/step
## Epoch 59/600
## 6/6 - 0s - loss: 427342.0938 - mean_absolute_error: 513.1627 - val_loss: 520844.7500 - val_mean_absolute_error: 519.6105 - 27ms/epoch - 4ms/step
## Epoch 60/600
## 6/6 - 0s - loss: 427830.1875 - mean_absolute_error: 513.4539 - val_loss: 519821.6875 - val_mean_absolute_error: 519.0368 - 26ms/epoch - 4ms/step
## Epoch 61/600
## 6/6 - 0s - loss: 427648.6250 - mean_absolute_error: 512.7138 - val_loss: 518781.9375 - val_mean_absolute_error: 518.4414 - 25ms/epoch - 4ms/step
## Epoch 62/600
## 6/6 - 0s - loss: 425454.7812 - mean_absolute_error: 511.5315 - val_loss: 517630.4062 - val_mean_absolute_error: 517.7946 - 25ms/epoch - 4ms/step
## Epoch 63/600
## 6/6 - 0s - loss: 424744.8125 - mean_absolute_error: 511.1264 - val_loss: 516433.6562 - val_mean_absolute_error: 517.1368 - 25ms/epoch - 4ms/step
## Epoch 64/600
## 6/6 - 0s - loss: 423329.6250 - mean_absolute_error: 510.0350 - val_loss: 515272.1875 - val_mean_absolute_error: 516.4924 - 25ms/epoch - 4ms/step
## Epoch 65/600
## 6/6 - 0s - loss: 423673.5312 - mean_absolute_error: 509.7447 - val_loss: 514188.7812 - val_mean_absolute_error: 515.8770 - 25ms/epoch - 4ms/step
## Epoch 66/600
## 6/6 - 0s - loss: 421225.4688 - mean_absolute_error: 508.5004 - val_loss: 513020.8750 - val_mean_absolute_error: 515.2063 - 25ms/epoch - 4ms/step
## Epoch 67/600
## 6/6 - 0s - loss: 420182.0312 - mean_absolute_error: 507.8695 - val_loss: 511910.4375 - val_mean_absolute_error: 514.5726 - 25ms/epoch - 4ms/step
## Epoch 68/600
## 6/6 - 0s - loss: 418583.7188 - mean_absolute_error: 506.6761 - val_loss: 510671.7188 - val_mean_absolute_error: 513.8807 - 25ms/epoch - 4ms/step
## Epoch 69/600
## 6/6 - 0s - loss: 421019.0938 - mean_absolute_error: 508.0124 - val_loss: 509433.6562 - val_mean_absolute_error: 513.1757 - 26ms/epoch - 4ms/step
## Epoch 70/600
## 6/6 - 0s - loss: 418883.3750 - mean_absolute_error: 506.2811 - val_loss: 508210.2188 - val_mean_absolute_error: 512.4781 - 25ms/epoch - 4ms/step
## Epoch 71/600
## 6/6 - 0s - loss: 415869.5312 - mean_absolute_error: 505.2627 - val_loss: 506840.1875 - val_mean_absolute_error: 511.7153 - 26ms/epoch - 4ms/step
## Epoch 72/600
## 6/6 - 0s - loss: 416814.0312 - mean_absolute_error: 504.7061 - val_loss: 505650.0312 - val_mean_absolute_error: 511.0283 - 25ms/epoch - 4ms/step
## Epoch 73/600
## 6/6 - 0s - loss: 416098.7188 - mean_absolute_error: 504.4624 - val_loss: 504471.3125 - val_mean_absolute_error: 510.3434 - 26ms/epoch - 4ms/step
## Epoch 74/600
## 6/6 - 0s - loss: 413247.5312 - mean_absolute_error: 502.3066 - val_loss: 503099.0312 - val_mean_absolute_error: 509.5471 - 29ms/epoch - 5ms/step
## Epoch 75/600
## 6/6 - 0s - loss: 412697.1875 - mean_absolute_error: 502.4846 - val_loss: 501734.8125 - val_mean_absolute_error: 508.7642 - 27ms/epoch - 4ms/step
## Epoch 76/600
## 6/6 - 0s - loss: 413610.4062 - mean_absolute_error: 502.4651 - val_loss: 500482.4688 - val_mean_absolute_error: 508.0488 - 26ms/epoch - 4ms/step
## Epoch 77/600
## 6/6 - 0s - loss: 411666.5312 - mean_absolute_error: 501.6781 - val_loss: 499203.1250 - val_mean_absolute_error: 507.3081 - 26ms/epoch - 4ms/step
## Epoch 78/600
## 6/6 - 0s - loss: 409110.5312 - mean_absolute_error: 499.6575 - val_loss: 497913.2500 - val_mean_absolute_error: 506.5450 - 25ms/epoch - 4ms/step
## Epoch 79/600
## 6/6 - 0s - loss: 408476.6250 - mean_absolute_error: 499.4651 - val_loss: 496532.7812 - val_mean_absolute_error: 505.7570 - 26ms/epoch - 4ms/step
## Epoch 80/600
## 6/6 - 0s - loss: 407735.0938 - mean_absolute_error: 498.9062 - val_loss: 495205.6875 - val_mean_absolute_error: 504.9855 - 26ms/epoch - 4ms/step
## Epoch 81/600
## 6/6 - 0s - loss: 406066.0938 - mean_absolute_error: 497.3821 - val_loss: 493738.1562 - val_mean_absolute_error: 504.1259 - 26ms/epoch - 4ms/step
## Epoch 82/600
## 6/6 - 0s - loss: 406502.7812 - mean_absolute_error: 498.4032 - val_loss: 492313.6875 - val_mean_absolute_error: 503.3254 - 178ms/epoch - 30ms/step
## Epoch 83/600
## 6/6 - 0s - loss: 404177.1875 - mean_absolute_error: 496.1947 - val_loss: 490863.1250 - val_mean_absolute_error: 502.4976 - 34ms/epoch - 6ms/step
## Epoch 84/600
## 6/6 - 0s - loss: 401088.6250 - mean_absolute_error: 494.5773 - val_loss: 489462.6562 - val_mean_absolute_error: 501.6749 - 27ms/epoch - 5ms/step
## Epoch 85/600
## 6/6 - 0s - loss: 402272.3125 - mean_absolute_error: 494.0710 - val_loss: 487989.4688 - val_mean_absolute_error: 500.7877 - 25ms/epoch - 4ms/step
## Epoch 86/600
## 6/6 - 0s - loss: 400949.0000 - mean_absolute_error: 493.9557 - val_loss: 486578.1562 - val_mean_absolute_error: 499.9749 - 29ms/epoch - 5ms/step
## Epoch 87/600
## 6/6 - 0s - loss: 399558.5938 - mean_absolute_error: 492.1721 - val_loss: 485038.5312 - val_mean_absolute_error: 499.0951 - 28ms/epoch - 5ms/step
## Epoch 88/600
## 6/6 - 0s - loss: 397040.7812 - mean_absolute_error: 491.3809 - val_loss: 483350.4375 - val_mean_absolute_error: 498.1400 - 27ms/epoch - 4ms/step
## Epoch 89/600
## 6/6 - 0s - loss: 397518.2188 - mean_absolute_error: 490.8634 - val_loss: 481932.6875 - val_mean_absolute_error: 497.3062 - 25ms/epoch - 4ms/step
## Epoch 90/600
## 6/6 - 0s - loss: 395334.6875 - mean_absolute_error: 489.5264 - val_loss: 480439.0938 - val_mean_absolute_error: 496.4182 - 25ms/epoch - 4ms/step
## Epoch 91/600
## 6/6 - 0s - loss: 393817.2188 - mean_absolute_error: 488.9073 - val_loss: 478922.6562 - val_mean_absolute_error: 495.5181 - 25ms/epoch - 4ms/step
## Epoch 92/600
## 6/6 - 0s - loss: 393000.2812 - mean_absolute_error: 487.8376 - val_loss: 477421.3750 - val_mean_absolute_error: 494.6064 - 25ms/epoch - 4ms/step
## Epoch 93/600
## 6/6 - 0s - loss: 393586.0000 - mean_absolute_error: 487.6530 - val_loss: 475927.0312 - val_mean_absolute_error: 493.7278 - 25ms/epoch - 4ms/step
## Epoch 94/600
## 6/6 - 0s - loss: 391697.3125 - mean_absolute_error: 486.1693 - val_loss: 474370.6250 - val_mean_absolute_error: 492.8169 - 24ms/epoch - 4ms/step
## Epoch 95/600
## 6/6 - 0s - loss: 389599.1875 - mean_absolute_error: 485.4552 - val_loss: 472705.1875 - val_mean_absolute_error: 491.8261 - 26ms/epoch - 4ms/step
## Epoch 96/600
## 6/6 - 0s - loss: 388615.9688 - mean_absolute_error: 484.9270 - val_loss: 471097.8438 - val_mean_absolute_error: 490.8682 - 25ms/epoch - 4ms/step
## Epoch 97/600
## 6/6 - 0s - loss: 385517.4062 - mean_absolute_error: 482.8735 - val_loss: 469474.5625 - val_mean_absolute_error: 489.8810 - 25ms/epoch - 4ms/step
## Epoch 98/600
## 6/6 - 0s - loss: 383573.8750 - mean_absolute_error: 480.9003 - val_loss: 467837.0625 - val_mean_absolute_error: 488.9008 - 25ms/epoch - 4ms/step
## Epoch 99/600
## 6/6 - 0s - loss: 386086.6875 - mean_absolute_error: 480.6908 - val_loss: 466276.6562 - val_mean_absolute_error: 487.9557 - 25ms/epoch - 4ms/step
## Epoch 100/600
## 6/6 - 0s - loss: 383583.0938 - mean_absolute_error: 481.1078 - val_loss: 464513.4688 - val_mean_absolute_error: 486.9306 - 24ms/epoch - 4ms/step
## Epoch 101/600
## 6/6 - 0s - loss: 384005.1250 - mean_absolute_error: 481.2814 - val_loss: 462969.6562 - val_mean_absolute_error: 485.9801 - 24ms/epoch - 4ms/step
## Epoch 102/600
## 6/6 - 0s - loss: 381616.7188 - mean_absolute_error: 478.2743 - val_loss: 461329.7812 - val_mean_absolute_error: 484.9706 - 25ms/epoch - 4ms/step
## Epoch 103/600
## 6/6 - 0s - loss: 378164.5312 - mean_absolute_error: 476.0782 - val_loss: 459595.9688 - val_mean_absolute_error: 483.9182 - 25ms/epoch - 4ms/step
## Epoch 104/600
## 6/6 - 0s - loss: 376022.3750 - mean_absolute_error: 476.2958 - val_loss: 457903.5000 - val_mean_absolute_error: 482.8835 - 25ms/epoch - 4ms/step
## Epoch 105/600
## 6/6 - 0s - loss: 376507.0312 - mean_absolute_error: 474.3524 - val_loss: 456097.1562 - val_mean_absolute_error: 481.7996 - 31ms/epoch - 5ms/step
## Epoch 106/600
## 6/6 - 0s - loss: 373935.5938 - mean_absolute_error: 474.3067 - val_loss: 454338.9375 - val_mean_absolute_error: 480.7029 - 30ms/epoch - 5ms/step
## Epoch 107/600
## 6/6 - 0s - loss: 371676.0000 - mean_absolute_error: 471.5953 - val_loss: 452662.6250 - val_mean_absolute_error: 479.6789 - 28ms/epoch - 5ms/step
## Epoch 108/600
## 6/6 - 0s - loss: 371367.2500 - mean_absolute_error: 471.3351 - val_loss: 450818.0312 - val_mean_absolute_error: 478.5365 - 27ms/epoch - 5ms/step
## Epoch 109/600
## 6/6 - 0s - loss: 370395.5312 - mean_absolute_error: 471.0870 - val_loss: 449100.1250 - val_mean_absolute_error: 477.4482 - 25ms/epoch - 4ms/step
## Epoch 110/600
## 6/6 - 0s - loss: 373272.7812 - mean_absolute_error: 472.1678 - val_loss: 447342.2500 - val_mean_absolute_error: 476.3732 - 25ms/epoch - 4ms/step
## Epoch 111/600
## 6/6 - 0s - loss: 366213.2812 - mean_absolute_error: 466.9176 - val_loss: 445490.7188 - val_mean_absolute_error: 475.2433 - 26ms/epoch - 4ms/step
## Epoch 112/600
## 6/6 - 0s - loss: 365108.9688 - mean_absolute_error: 467.1488 - val_loss: 443766.4688 - val_mean_absolute_error: 474.1650 - 25ms/epoch - 4ms/step
## Epoch 113/600
## 6/6 - 0s - loss: 360668.2188 - mean_absolute_error: 463.9647 - val_loss: 441979.5000 - val_mean_absolute_error: 473.0699 - 25ms/epoch - 4ms/step
## Epoch 114/600
## 6/6 - 0s - loss: 362347.0312 - mean_absolute_error: 463.8652 - val_loss: 440118.5312 - val_mean_absolute_error: 471.8859 - 25ms/epoch - 4ms/step
## Epoch 115/600
## 6/6 - 0s - loss: 360578.1250 - mean_absolute_error: 464.3354 - val_loss: 438254.6250 - val_mean_absolute_error: 470.7080 - 25ms/epoch - 4ms/step
## Epoch 116/600
## 6/6 - 0s - loss: 363291.2188 - mean_absolute_error: 464.7184 - val_loss: 436483.9688 - val_mean_absolute_error: 469.5871 - 27ms/epoch - 4ms/step
## Epoch 117/600
## 6/6 - 0s - loss: 359985.9688 - mean_absolute_error: 462.5975 - val_loss: 434578.9375 - val_mean_absolute_error: 468.3883 - 33ms/epoch - 5ms/step
## Epoch 118/600
## 6/6 - 0s - loss: 357216.4062 - mean_absolute_error: 461.2295 - val_loss: 432647.6875 - val_mean_absolute_error: 467.1489 - 27ms/epoch - 4ms/step
## Epoch 119/600
## 6/6 - 0s - loss: 357366.8750 - mean_absolute_error: 460.2693 - val_loss: 430880.2812 - val_mean_absolute_error: 466.0241 - 28ms/epoch - 5ms/step
## Epoch 120/600
## 6/6 - 0s - loss: 356053.6875 - mean_absolute_error: 458.3414 - val_loss: 429084.2188 - val_mean_absolute_error: 464.8762 - 29ms/epoch - 5ms/step
## Epoch 121/600
## 6/6 - 0s - loss: 355353.4688 - mean_absolute_error: 457.7813 - val_loss: 427226.8438 - val_mean_absolute_error: 463.7062 - 26ms/epoch - 4ms/step
## Epoch 122/600
## 6/6 - 0s - loss: 351503.5625 - mean_absolute_error: 455.8487 - val_loss: 425238.6250 - val_mean_absolute_error: 462.4673 - 26ms/epoch - 4ms/step
## Epoch 123/600
## 6/6 - 0s - loss: 352051.0938 - mean_absolute_error: 456.3943 - val_loss: 423191.0000 - val_mean_absolute_error: 461.1853 - 28ms/epoch - 5ms/step
## Epoch 124/600
## 6/6 - 0s - loss: 346510.7188 - mean_absolute_error: 453.0446 - val_loss: 421268.9688 - val_mean_absolute_error: 459.9184 - 29ms/epoch - 5ms/step
## Epoch 125/600
## 6/6 - 0s - loss: 351864.4688 - mean_absolute_error: 454.3657 - val_loss: 419287.0000 - val_mean_absolute_error: 458.6414 - 30ms/epoch - 5ms/step
## Epoch 126/600
## 6/6 - 0s - loss: 344354.6250 - mean_absolute_error: 451.2169 - val_loss: 417205.0000 - val_mean_absolute_error: 457.3200 - 27ms/epoch - 5ms/step
## Epoch 127/600
## 6/6 - 0s - loss: 345262.2188 - mean_absolute_error: 451.3086 - val_loss: 415369.0000 - val_mean_absolute_error: 456.1115 - 25ms/epoch - 4ms/step
## Epoch 128/600
## 6/6 - 0s - loss: 341683.2188 - mean_absolute_error: 449.1001 - val_loss: 413431.1875 - val_mean_absolute_error: 454.8399 - 25ms/epoch - 4ms/step
## Epoch 129/600
## 6/6 - 0s - loss: 343375.3750 - mean_absolute_error: 447.1724 - val_loss: 411571.9062 - val_mean_absolute_error: 453.5984 - 29ms/epoch - 5ms/step
## Epoch 130/600
## 6/6 - 0s - loss: 339973.8438 - mean_absolute_error: 445.6505 - val_loss: 409646.5312 - val_mean_absolute_error: 452.3265 - 31ms/epoch - 5ms/step
## Epoch 131/600
## 6/6 - 0s - loss: 341175.3750 - mean_absolute_error: 446.7209 - val_loss: 407721.0938 - val_mean_absolute_error: 451.0374 - 32ms/epoch - 5ms/step
## Epoch 132/600
## 6/6 - 0s - loss: 339907.6875 - mean_absolute_error: 446.2447 - val_loss: 405682.0312 - val_mean_absolute_error: 449.7021 - 27ms/epoch - 5ms/step
## Epoch 133/600
## 6/6 - 0s - loss: 333701.8125 - mean_absolute_error: 441.4829 - val_loss: 403552.9062 - val_mean_absolute_error: 448.3045 - 26ms/epoch - 4ms/step
## Epoch 134/600
## 6/6 - 0s - loss: 333407.5938 - mean_absolute_error: 441.0358 - val_loss: 401546.4688 - val_mean_absolute_error: 446.9675 - 29ms/epoch - 5ms/step
## Epoch 135/600
## 6/6 - 0s - loss: 328983.8125 - mean_absolute_error: 438.8489 - val_loss: 399704.1250 - val_mean_absolute_error: 445.7027 - 28ms/epoch - 5ms/step
## Epoch 136/600
## 6/6 - 0s - loss: 331659.6250 - mean_absolute_error: 439.6227 - val_loss: 397696.6875 - val_mean_absolute_error: 444.3489 - 28ms/epoch - 5ms/step
## Epoch 137/600
## 6/6 - 0s - loss: 329107.0312 - mean_absolute_error: 437.8187 - val_loss: 395563.5000 - val_mean_absolute_error: 442.8992 - 27ms/epoch - 4ms/step
## Epoch 138/600
## 6/6 - 0s - loss: 326950.0000 - mean_absolute_error: 436.8653 - val_loss: 393511.2500 - val_mean_absolute_error: 441.5312 - 25ms/epoch - 4ms/step
## Epoch 139/600
## 6/6 - 0s - loss: 324962.7812 - mean_absolute_error: 434.1416 - val_loss: 391507.9688 - val_mean_absolute_error: 440.1665 - 25ms/epoch - 4ms/step
## Epoch 140/600
## 6/6 - 0s - loss: 326138.9688 - mean_absolute_error: 434.7666 - val_loss: 389596.8125 - val_mean_absolute_error: 438.8579 - 26ms/epoch - 4ms/step
## Epoch 141/600
## 6/6 - 0s - loss: 327028.3125 - mean_absolute_error: 435.9652 - val_loss: 387518.9062 - val_mean_absolute_error: 437.4563 - 30ms/epoch - 5ms/step
## Epoch 142/600
## 6/6 - 0s - loss: 323514.0938 - mean_absolute_error: 432.4516 - val_loss: 385423.6562 - val_mean_absolute_error: 436.0193 - 30ms/epoch - 5ms/step
## Epoch 143/600
## 6/6 - 0s - loss: 318441.4062 - mean_absolute_error: 428.3032 - val_loss: 383280.3750 - val_mean_absolute_error: 434.5535 - 31ms/epoch - 5ms/step
## Epoch 144/600
## 6/6 - 0s - loss: 317589.1250 - mean_absolute_error: 429.4677 - val_loss: 381115.3125 - val_mean_absolute_error: 433.0764 - 31ms/epoch - 5ms/step
## Epoch 145/600
## 6/6 - 0s - loss: 314881.5938 - mean_absolute_error: 425.9915 - val_loss: 379048.7188 - val_mean_absolute_error: 431.6014 - 31ms/epoch - 5ms/step
## Epoch 146/600
## 6/6 - 0s - loss: 313982.7500 - mean_absolute_error: 425.8568 - val_loss: 377022.2500 - val_mean_absolute_error: 430.2033 - 30ms/epoch - 5ms/step
## Epoch 147/600
## 6/6 - 0s - loss: 312139.3125 - mean_absolute_error: 425.3507 - val_loss: 374814.6562 - val_mean_absolute_error: 428.7388 - 31ms/epoch - 5ms/step
## Epoch 148/600
## 6/6 - 0s - loss: 309592.0312 - mean_absolute_error: 420.4062 - val_loss: 372843.9688 - val_mean_absolute_error: 427.5083 - 30ms/epoch - 5ms/step
## Epoch 149/600
## 6/6 - 0s - loss: 308513.1562 - mean_absolute_error: 420.8706 - val_loss: 370753.6875 - val_mean_absolute_error: 426.2332 - 30ms/epoch - 5ms/step
## Epoch 150/600
## 6/6 - 0s - loss: 307534.7500 - mean_absolute_error: 419.3484 - val_loss: 368674.5625 - val_mean_absolute_error: 424.9420 - 30ms/epoch - 5ms/step
## Epoch 151/600
## 6/6 - 0s - loss: 306152.7812 - mean_absolute_error: 418.3391 - val_loss: 366651.8125 - val_mean_absolute_error: 423.6177 - 31ms/epoch - 5ms/step
## Epoch 152/600
## 6/6 - 0s - loss: 301381.6562 - mean_absolute_error: 415.9030 - val_loss: 364679.8125 - val_mean_absolute_error: 422.3855 - 28ms/epoch - 5ms/step
## Epoch 153/600
## 6/6 - 0s - loss: 298051.2188 - mean_absolute_error: 413.3892 - val_loss: 362349.2500 - val_mean_absolute_error: 420.9692 - 26ms/epoch - 4ms/step
## Epoch 154/600
## 6/6 - 0s - loss: 302275.5000 - mean_absolute_error: 415.9080 - val_loss: 360315.6875 - val_mean_absolute_error: 419.6880 - 28ms/epoch - 5ms/step
## Epoch 155/600
## 6/6 - 0s - loss: 301805.5000 - mean_absolute_error: 414.8617 - val_loss: 358355.2188 - val_mean_absolute_error: 418.4425 - 28ms/epoch - 5ms/step
## Epoch 156/600
## 6/6 - 0s - loss: 302794.7812 - mean_absolute_error: 416.0512 - val_loss: 356254.0312 - val_mean_absolute_error: 417.1143 - 26ms/epoch - 4ms/step
## Epoch 157/600
## 6/6 - 0s - loss: 295308.7500 - mean_absolute_error: 410.0387 - val_loss: 354225.7812 - val_mean_absolute_error: 415.8027 - 31ms/epoch - 5ms/step
## Epoch 158/600
## 6/6 - 0s - loss: 300708.2812 - mean_absolute_error: 409.5067 - val_loss: 352154.4375 - val_mean_absolute_error: 414.4698 - 28ms/epoch - 5ms/step
## Epoch 159/600
## 6/6 - 0s - loss: 295294.9062 - mean_absolute_error: 408.7884 - val_loss: 349916.1875 - val_mean_absolute_error: 413.0573 - 26ms/epoch - 4ms/step
## Epoch 160/600
## 6/6 - 0s - loss: 289635.3438 - mean_absolute_error: 406.5879 - val_loss: 347830.8438 - val_mean_absolute_error: 411.6722 - 26ms/epoch - 4ms/step
## Epoch 161/600
## 6/6 - 0s - loss: 288513.1250 - mean_absolute_error: 406.4907 - val_loss: 345522.2500 - val_mean_absolute_error: 410.2192 - 25ms/epoch - 4ms/step
## Epoch 162/600
## 6/6 - 0s - loss: 288938.4688 - mean_absolute_error: 403.8246 - val_loss: 343569.0938 - val_mean_absolute_error: 408.8722 - 27ms/epoch - 4ms/step
## Epoch 163/600
## 6/6 - 0s - loss: 291455.0625 - mean_absolute_error: 404.5919 - val_loss: 341623.0938 - val_mean_absolute_error: 407.6009 - 26ms/epoch - 4ms/step
## Epoch 164/600
## 6/6 - 0s - loss: 284292.9688 - mean_absolute_error: 399.4587 - val_loss: 339578.7500 - val_mean_absolute_error: 406.2690 - 26ms/epoch - 4ms/step
## Epoch 165/600
## 6/6 - 0s - loss: 292673.5938 - mean_absolute_error: 405.6833 - val_loss: 337583.2500 - val_mean_absolute_error: 404.9427 - 27ms/epoch - 5ms/step
## Epoch 166/600
## 6/6 - 0s - loss: 287279.9375 - mean_absolute_error: 401.5246 - val_loss: 335513.2188 - val_mean_absolute_error: 403.5635 - 26ms/epoch - 4ms/step
## Epoch 167/600
## 6/6 - 0s - loss: 286551.8125 - mean_absolute_error: 401.8062 - val_loss: 333418.0938 - val_mean_absolute_error: 402.1772 - 26ms/epoch - 4ms/step
## Epoch 168/600
## 6/6 - 0s - loss: 276774.6562 - mean_absolute_error: 395.0574 - val_loss: 331319.5312 - val_mean_absolute_error: 400.7603 - 27ms/epoch - 4ms/step
## Epoch 169/600
## 6/6 - 0s - loss: 272075.0938 - mean_absolute_error: 389.5170 - val_loss: 329083.1250 - val_mean_absolute_error: 399.2652 - 26ms/epoch - 4ms/step
## Epoch 170/600
## 6/6 - 0s - loss: 271259.2500 - mean_absolute_error: 393.1587 - val_loss: 327030.4062 - val_mean_absolute_error: 397.9098 - 26ms/epoch - 4ms/step
## Epoch 171/600
## 6/6 - 0s - loss: 275360.5312 - mean_absolute_error: 393.1476 - val_loss: 324847.4062 - val_mean_absolute_error: 396.4725 - 27ms/epoch - 4ms/step
## Epoch 172/600
## 6/6 - 0s - loss: 269245.5938 - mean_absolute_error: 390.4897 - val_loss: 322745.8438 - val_mean_absolute_error: 395.0698 - 26ms/epoch - 4ms/step
## Epoch 173/600
## 6/6 - 0s - loss: 275503.0625 - mean_absolute_error: 393.5775 - val_loss: 320743.5000 - val_mean_absolute_error: 393.7159 - 35ms/epoch - 6ms/step
## Epoch 174/600
## 6/6 - 0s - loss: 271772.5312 - mean_absolute_error: 391.0435 - val_loss: 318544.7500 - val_mean_absolute_error: 392.2451 - 30ms/epoch - 5ms/step
## Epoch 175/600
## 6/6 - 0s - loss: 266578.1875 - mean_absolute_error: 385.0287 - val_loss: 316471.9375 - val_mean_absolute_error: 390.8643 - 38ms/epoch - 6ms/step
## Epoch 176/600
## 6/6 - 0s - loss: 273886.6250 - mean_absolute_error: 388.7032 - val_loss: 314454.8750 - val_mean_absolute_error: 389.4836 - 30ms/epoch - 5ms/step
## Epoch 177/600
## 6/6 - 0s - loss: 265836.8750 - mean_absolute_error: 384.4574 - val_loss: 312313.3438 - val_mean_absolute_error: 388.0485 - 28ms/epoch - 5ms/step
## Epoch 178/600
## 6/6 - 0s - loss: 267930.0625 - mean_absolute_error: 390.4769 - val_loss: 310220.8125 - val_mean_absolute_error: 386.6729 - 27ms/epoch - 4ms/step
## Epoch 179/600
## 6/6 - 0s - loss: 265426.0938 - mean_absolute_error: 383.2952 - val_loss: 308224.6250 - val_mean_absolute_error: 385.3666 - 27ms/epoch - 4ms/step
## Epoch 180/600
## 6/6 - 0s - loss: 263044.4375 - mean_absolute_error: 380.1407 - val_loss: 306213.8750 - val_mean_absolute_error: 384.0842 - 26ms/epoch - 4ms/step
## Epoch 181/600
## 6/6 - 0s - loss: 251926.0625 - mean_absolute_error: 375.8667 - val_loss: 304288.5000 - val_mean_absolute_error: 382.8922 - 27ms/epoch - 5ms/step
## Epoch 182/600
## 6/6 - 0s - loss: 264784.8750 - mean_absolute_error: 381.5002 - val_loss: 302245.4062 - val_mean_absolute_error: 381.6260 - 26ms/epoch - 4ms/step
## Epoch 183/600
## 6/6 - 0s - loss: 257672.8594 - mean_absolute_error: 377.2373 - val_loss: 300218.2500 - val_mean_absolute_error: 380.3401 - 27ms/epoch - 4ms/step
## Epoch 184/600
## 6/6 - 0s - loss: 254216.8906 - mean_absolute_error: 374.0303 - val_loss: 297963.5938 - val_mean_absolute_error: 378.9618 - 26ms/epoch - 4ms/step
## Epoch 185/600
## 6/6 - 0s - loss: 251741.9375 - mean_absolute_error: 372.2934 - val_loss: 296099.3750 - val_mean_absolute_error: 377.7774 - 27ms/epoch - 4ms/step
## Epoch 186/600
## 6/6 - 0s - loss: 252463.2031 - mean_absolute_error: 370.7031 - val_loss: 294223.7500 - val_mean_absolute_error: 376.5762 - 27ms/epoch - 5ms/step
## Epoch 187/600
## 6/6 - 0s - loss: 254639.7031 - mean_absolute_error: 372.7122 - val_loss: 292227.1875 - val_mean_absolute_error: 375.2910 - 27ms/epoch - 4ms/step
## Epoch 188/600
## 6/6 - 0s - loss: 248234.1406 - mean_absolute_error: 371.2953 - val_loss: 290104.3125 - val_mean_absolute_error: 373.9243 - 25ms/epoch - 4ms/step
## Epoch 189/600
## 6/6 - 0s - loss: 246469.6094 - mean_absolute_error: 365.6683 - val_loss: 288004.0625 - val_mean_absolute_error: 372.5510 - 27ms/epoch - 4ms/step
## Epoch 190/600
## 6/6 - 0s - loss: 245901.6875 - mean_absolute_error: 366.8048 - val_loss: 286009.5625 - val_mean_absolute_error: 371.2370 - 27ms/epoch - 4ms/step
## Epoch 191/600
## 6/6 - 0s - loss: 247700.1406 - mean_absolute_error: 367.3911 - val_loss: 284012.9688 - val_mean_absolute_error: 369.9144 - 26ms/epoch - 4ms/step
## Epoch 192/600
## 6/6 - 0s - loss: 244886.1562 - mean_absolute_error: 367.6219 - val_loss: 282157.6875 - val_mean_absolute_error: 368.6370 - 30ms/epoch - 5ms/step
## Epoch 193/600
## 6/6 - 0s - loss: 243215.5000 - mean_absolute_error: 362.4976 - val_loss: 280196.0312 - val_mean_absolute_error: 367.2997 - 31ms/epoch - 5ms/step
## Epoch 194/600
## 6/6 - 0s - loss: 242047.2656 - mean_absolute_error: 362.7990 - val_loss: 278179.1250 - val_mean_absolute_error: 365.9651 - 28ms/epoch - 5ms/step
## Epoch 195/600
## 6/6 - 0s - loss: 231093.2500 - mean_absolute_error: 355.3278 - val_loss: 276266.7188 - val_mean_absolute_error: 364.6546 - 27ms/epoch - 5ms/step
## Epoch 196/600
## 6/6 - 0s - loss: 234564.5938 - mean_absolute_error: 357.8267 - val_loss: 274234.6250 - val_mean_absolute_error: 363.2620 - 26ms/epoch - 4ms/step
## Epoch 197/600
## 6/6 - 0s - loss: 239610.7969 - mean_absolute_error: 362.7503 - val_loss: 272150.1250 - val_mean_absolute_error: 361.8377 - 27ms/epoch - 4ms/step
## Epoch 198/600
## 6/6 - 0s - loss: 234867.0625 - mean_absolute_error: 354.8141 - val_loss: 270327.8750 - val_mean_absolute_error: 360.5737 - 27ms/epoch - 4ms/step
## Epoch 199/600
## 6/6 - 0s - loss: 239995.0000 - mean_absolute_error: 359.1109 - val_loss: 268471.8750 - val_mean_absolute_error: 359.3359 - 27ms/epoch - 5ms/step
## Epoch 200/600
## 6/6 - 0s - loss: 233912.9375 - mean_absolute_error: 353.6921 - val_loss: 266595.5938 - val_mean_absolute_error: 358.0282 - 26ms/epoch - 4ms/step
## Epoch 201/600
## 6/6 - 0s - loss: 232488.3125 - mean_absolute_error: 356.6218 - val_loss: 264813.6562 - val_mean_absolute_error: 356.7843 - 26ms/epoch - 4ms/step
## Epoch 202/600
## 6/6 - 0s - loss: 228763.6562 - mean_absolute_error: 352.0723 - val_loss: 262855.3750 - val_mean_absolute_error: 355.4591 - 25ms/epoch - 4ms/step
## Epoch 203/600
## 6/6 - 0s - loss: 228862.9844 - mean_absolute_error: 352.7469 - val_loss: 261007.2031 - val_mean_absolute_error: 354.2414 - 25ms/epoch - 4ms/step
## Epoch 204/600
## 6/6 - 0s - loss: 224439.6094 - mean_absolute_error: 347.6244 - val_loss: 259034.1406 - val_mean_absolute_error: 352.9305 - 26ms/epoch - 4ms/step
## Epoch 205/600
## 6/6 - 0s - loss: 223751.4531 - mean_absolute_error: 342.8485 - val_loss: 257335.1719 - val_mean_absolute_error: 351.7973 - 26ms/epoch - 4ms/step
## Epoch 206/600
## 6/6 - 0s - loss: 228703.0938 - mean_absolute_error: 348.3377 - val_loss: 255529.4219 - val_mean_absolute_error: 350.5745 - 25ms/epoch - 4ms/step
## Epoch 207/600
## 6/6 - 0s - loss: 229406.8438 - mean_absolute_error: 351.3371 - val_loss: 253807.9062 - val_mean_absolute_error: 349.4330 - 25ms/epoch - 4ms/step
## Epoch 208/600
## 6/6 - 0s - loss: 222947.2500 - mean_absolute_error: 341.4500 - val_loss: 251909.3750 - val_mean_absolute_error: 348.1523 - 28ms/epoch - 5ms/step
## Epoch 209/600
## 6/6 - 0s - loss: 217274.3438 - mean_absolute_error: 342.3869 - val_loss: 250193.8906 - val_mean_absolute_error: 346.9653 - 27ms/epoch - 4ms/step
## Epoch 210/600
## 6/6 - 0s - loss: 219578.4844 - mean_absolute_error: 344.3673 - val_loss: 248371.6562 - val_mean_absolute_error: 345.7274 - 28ms/epoch - 5ms/step
## Epoch 211/600
## 6/6 - 0s - loss: 216943.3438 - mean_absolute_error: 341.8289 - val_loss: 246617.7500 - val_mean_absolute_error: 344.5426 - 28ms/epoch - 5ms/step
## Epoch 212/600
## 6/6 - 0s - loss: 211199.3125 - mean_absolute_error: 338.9657 - val_loss: 244732.4062 - val_mean_absolute_error: 343.2891 - 26ms/epoch - 4ms/step
## Epoch 213/600
## 6/6 - 0s - loss: 214023.4062 - mean_absolute_error: 340.9576 - val_loss: 242973.6719 - val_mean_absolute_error: 342.2507 - 26ms/epoch - 4ms/step
## Epoch 214/600
## 6/6 - 0s - loss: 222940.3125 - mean_absolute_error: 347.2662 - val_loss: 241304.6250 - val_mean_absolute_error: 341.2769 - 26ms/epoch - 4ms/step
## Epoch 215/600
## 6/6 - 0s - loss: 220123.4062 - mean_absolute_error: 344.4516 - val_loss: 239486.4375 - val_mean_absolute_error: 340.2341 - 26ms/epoch - 4ms/step
## Epoch 216/600
## 6/6 - 0s - loss: 216324.8906 - mean_absolute_error: 341.8695 - val_loss: 237858.2031 - val_mean_absolute_error: 339.2708 - 26ms/epoch - 4ms/step
## Epoch 217/600
## 6/6 - 0s - loss: 219636.6875 - mean_absolute_error: 338.4530 - val_loss: 236343.5938 - val_mean_absolute_error: 338.3815 - 25ms/epoch - 4ms/step
## Epoch 218/600
## 6/6 - 0s - loss: 208233.8594 - mean_absolute_error: 332.5022 - val_loss: 234715.7031 - val_mean_absolute_error: 337.4190 - 25ms/epoch - 4ms/step
## Epoch 219/600
## 6/6 - 0s - loss: 219752.7344 - mean_absolute_error: 340.7679 - val_loss: 233032.6250 - val_mean_absolute_error: 336.4548 - 26ms/epoch - 4ms/step
## Epoch 220/600
## 6/6 - 0s - loss: 209590.4531 - mean_absolute_error: 335.9479 - val_loss: 231626.2031 - val_mean_absolute_error: 335.6566 - 25ms/epoch - 4ms/step
## Epoch 221/600
## 6/6 - 0s - loss: 204073.0469 - mean_absolute_error: 330.7683 - val_loss: 229980.3750 - val_mean_absolute_error: 334.7594 - 25ms/epoch - 4ms/step
## Epoch 222/600
## 6/6 - 0s - loss: 212047.0469 - mean_absolute_error: 334.1384 - val_loss: 228530.8438 - val_mean_absolute_error: 333.9563 - 25ms/epoch - 4ms/step
## Epoch 223/600
## 6/6 - 0s - loss: 207136.6406 - mean_absolute_error: 328.7033 - val_loss: 226940.3750 - val_mean_absolute_error: 333.0653 - 26ms/epoch - 4ms/step
## Epoch 224/600
## 6/6 - 0s - loss: 200280.3125 - mean_absolute_error: 328.7877 - val_loss: 225068.4531 - val_mean_absolute_error: 332.0254 - 25ms/epoch - 4ms/step
## Epoch 225/600
## 6/6 - 0s - loss: 203105.4062 - mean_absolute_error: 332.4039 - val_loss: 223544.7812 - val_mean_absolute_error: 331.1442 - 25ms/epoch - 4ms/step
## Epoch 226/600
## 6/6 - 0s - loss: 203717.1094 - mean_absolute_error: 328.7611 - val_loss: 221876.5781 - val_mean_absolute_error: 330.1771 - 25ms/epoch - 4ms/step
## Epoch 227/600
## 6/6 - 0s - loss: 202249.7031 - mean_absolute_error: 324.1928 - val_loss: 220344.8281 - val_mean_absolute_error: 329.3004 - 26ms/epoch - 4ms/step
## Epoch 228/600
## 6/6 - 0s - loss: 196257.0000 - mean_absolute_error: 321.4007 - val_loss: 218870.9688 - val_mean_absolute_error: 328.4212 - 27ms/epoch - 4ms/step
## Epoch 229/600
## 6/6 - 0s - loss: 204967.3594 - mean_absolute_error: 326.1835 - val_loss: 217337.6094 - val_mean_absolute_error: 327.5039 - 30ms/epoch - 5ms/step
## Epoch 230/600
## 6/6 - 0s - loss: 189913.8125 - mean_absolute_error: 317.6097 - val_loss: 215717.0156 - val_mean_absolute_error: 326.5752 - 28ms/epoch - 5ms/step
## Epoch 231/600
## 6/6 - 0s - loss: 194913.1875 - mean_absolute_error: 324.2822 - val_loss: 214127.7656 - val_mean_absolute_error: 325.7606 - 26ms/epoch - 4ms/step
## Epoch 232/600
## 6/6 - 0s - loss: 201320.4375 - mean_absolute_error: 330.1907 - val_loss: 212952.6875 - val_mean_absolute_error: 325.1514 - 26ms/epoch - 4ms/step
## Epoch 233/600
## 6/6 - 0s - loss: 195879.9844 - mean_absolute_error: 319.6938 - val_loss: 211557.3125 - val_mean_absolute_error: 324.4186 - 25ms/epoch - 4ms/step
## Epoch 234/600
## 6/6 - 0s - loss: 198933.5625 - mean_absolute_error: 328.7068 - val_loss: 210304.3281 - val_mean_absolute_error: 323.7604 - 25ms/epoch - 4ms/step
## Epoch 235/600
## 6/6 - 0s - loss: 185399.9844 - mean_absolute_error: 318.5748 - val_loss: 208824.2969 - val_mean_absolute_error: 322.9792 - 26ms/epoch - 4ms/step
## Epoch 236/600
## 6/6 - 0s - loss: 198684.2344 - mean_absolute_error: 322.3805 - val_loss: 207501.6094 - val_mean_absolute_error: 322.2625 - 26ms/epoch - 4ms/step
## Epoch 237/600
## 6/6 - 0s - loss: 193468.4844 - mean_absolute_error: 317.7323 - val_loss: 206169.7500 - val_mean_absolute_error: 321.5375 - 26ms/epoch - 4ms/step
## Epoch 238/600
## 6/6 - 0s - loss: 190186.9688 - mean_absolute_error: 316.3083 - val_loss: 204727.7188 - val_mean_absolute_error: 320.7454 - 27ms/epoch - 4ms/step
## Epoch 239/600
## 6/6 - 0s - loss: 189161.0156 - mean_absolute_error: 319.3826 - val_loss: 203412.3281 - val_mean_absolute_error: 320.0362 - 25ms/epoch - 4ms/step
## Epoch 240/600
## 6/6 - 0s - loss: 190973.6406 - mean_absolute_error: 319.6608 - val_loss: 201938.7969 - val_mean_absolute_error: 319.2151 - 26ms/epoch - 4ms/step
## Epoch 241/600
## 6/6 - 0s - loss: 188414.3438 - mean_absolute_error: 317.9039 - val_loss: 200615.5000 - val_mean_absolute_error: 318.4688 - 25ms/epoch - 4ms/step
## Epoch 242/600
## 6/6 - 0s - loss: 184340.0469 - mean_absolute_error: 315.2698 - val_loss: 199346.6875 - val_mean_absolute_error: 317.7443 - 27ms/epoch - 4ms/step
## Epoch 243/600
## 6/6 - 0s - loss: 196279.7031 - mean_absolute_error: 325.9991 - val_loss: 198297.6094 - val_mean_absolute_error: 317.1308 - 29ms/epoch - 5ms/step
## Epoch 244/600
## 6/6 - 0s - loss: 188602.0625 - mean_absolute_error: 317.1823 - val_loss: 197218.9688 - val_mean_absolute_error: 316.4943 - 25ms/epoch - 4ms/step
## Epoch 245/600
## 6/6 - 0s - loss: 186846.1094 - mean_absolute_error: 314.6614 - val_loss: 195817.0312 - val_mean_absolute_error: 315.6760 - 26ms/epoch - 4ms/step
## Epoch 246/600
## 6/6 - 0s - loss: 187318.3906 - mean_absolute_error: 314.0659 - val_loss: 194689.5156 - val_mean_absolute_error: 315.0136 - 25ms/epoch - 4ms/step
## Epoch 247/600
## 6/6 - 0s - loss: 182965.9375 - mean_absolute_error: 312.3962 - val_loss: 193614.3438 - val_mean_absolute_error: 314.3657 - 30ms/epoch - 5ms/step
## Epoch 248/600
## 6/6 - 0s - loss: 185667.4531 - mean_absolute_error: 318.0804 - val_loss: 192688.6406 - val_mean_absolute_error: 313.7946 - 30ms/epoch - 5ms/step
## Epoch 249/600
## 6/6 - 0s - loss: 186268.3438 - mean_absolute_error: 318.5378 - val_loss: 191372.5469 - val_mean_absolute_error: 312.9926 - 26ms/epoch - 4ms/step
## Epoch 250/600
## 6/6 - 0s - loss: 179185.9531 - mean_absolute_error: 311.9117 - val_loss: 190376.7656 - val_mean_absolute_error: 312.3776 - 26ms/epoch - 4ms/step
## Epoch 251/600
## 6/6 - 0s - loss: 180610.3750 - mean_absolute_error: 315.0345 - val_loss: 189116.3750 - val_mean_absolute_error: 311.5994 - 26ms/epoch - 4ms/step
## Epoch 252/600
## 6/6 - 0s - loss: 182562.4375 - mean_absolute_error: 309.5172 - val_loss: 188234.6250 - val_mean_absolute_error: 311.0303 - 26ms/epoch - 4ms/step
## Epoch 253/600
## 6/6 - 0s - loss: 176743.7344 - mean_absolute_error: 306.3499 - val_loss: 187362.1094 - val_mean_absolute_error: 310.4661 - 25ms/epoch - 4ms/step
## Epoch 254/600
## 6/6 - 0s - loss: 177555.1406 - mean_absolute_error: 311.5244 - val_loss: 186385.9844 - val_mean_absolute_error: 309.8350 - 25ms/epoch - 4ms/step
## Epoch 255/600
## 6/6 - 0s - loss: 173414.8281 - mean_absolute_error: 305.0019 - val_loss: 185400.7969 - val_mean_absolute_error: 309.2071 - 25ms/epoch - 4ms/step
## Epoch 256/600
## 6/6 - 0s - loss: 175929.4844 - mean_absolute_error: 308.1652 - val_loss: 184251.7656 - val_mean_absolute_error: 308.4698 - 25ms/epoch - 4ms/step
## Epoch 257/600
## 6/6 - 0s - loss: 170827.5469 - mean_absolute_error: 301.8254 - val_loss: 183071.7656 - val_mean_absolute_error: 307.6857 - 25ms/epoch - 4ms/step
## Epoch 258/600
## 6/6 - 0s - loss: 173828.8594 - mean_absolute_error: 314.5431 - val_loss: 182083.7969 - val_mean_absolute_error: 307.0217 - 25ms/epoch - 4ms/step
## Epoch 259/600
## 6/6 - 0s - loss: 177895.6094 - mean_absolute_error: 309.0095 - val_loss: 181222.0625 - val_mean_absolute_error: 306.4349 - 25ms/epoch - 4ms/step
## Epoch 260/600
## 6/6 - 0s - loss: 175635.7188 - mean_absolute_error: 304.6745 - val_loss: 180313.7969 - val_mean_absolute_error: 305.8215 - 25ms/epoch - 4ms/step
## Epoch 261/600
## 6/6 - 0s - loss: 170283.4688 - mean_absolute_error: 304.4434 - val_loss: 179406.9688 - val_mean_absolute_error: 305.1984 - 25ms/epoch - 4ms/step
## Epoch 262/600
## 6/6 - 0s - loss: 170817.3906 - mean_absolute_error: 300.8650 - val_loss: 178408.8281 - val_mean_absolute_error: 304.5121 - 25ms/epoch - 4ms/step
## Epoch 263/600
## 6/6 - 0s - loss: 173272.6875 - mean_absolute_error: 310.5077 - val_loss: 177489.9844 - val_mean_absolute_error: 303.8766 - 25ms/epoch - 4ms/step
## Epoch 264/600
## 6/6 - 0s - loss: 169520.5781 - mean_absolute_error: 306.0300 - val_loss: 176613.5312 - val_mean_absolute_error: 303.2565 - 26ms/epoch - 4ms/step
## Epoch 265/600
## 6/6 - 0s - loss: 181059.4844 - mean_absolute_error: 306.5016 - val_loss: 175852.8281 - val_mean_absolute_error: 302.7489 - 25ms/epoch - 4ms/step
## Epoch 266/600
## 6/6 - 0s - loss: 172759.4375 - mean_absolute_error: 305.5162 - val_loss: 174997.2031 - val_mean_absolute_error: 302.1859 - 28ms/epoch - 5ms/step
## Epoch 267/600
## 6/6 - 0s - loss: 173792.6406 - mean_absolute_error: 303.4190 - val_loss: 174117.9375 - val_mean_absolute_error: 301.6273 - 28ms/epoch - 5ms/step
## Epoch 268/600
## 6/6 - 0s - loss: 169155.2031 - mean_absolute_error: 302.9970 - val_loss: 173345.6719 - val_mean_absolute_error: 301.1217 - 26ms/epoch - 4ms/step
## Epoch 269/600
## 6/6 - 0s - loss: 169001.2656 - mean_absolute_error: 303.2621 - val_loss: 172321.5469 - val_mean_absolute_error: 300.4407 - 26ms/epoch - 4ms/step
## Epoch 270/600
## 6/6 - 0s - loss: 184434.1562 - mean_absolute_error: 314.9073 - val_loss: 171685.2188 - val_mean_absolute_error: 300.0373 - 26ms/epoch - 4ms/step
## Epoch 271/600
## 6/6 - 0s - loss: 170137.9844 - mean_absolute_error: 299.7080 - val_loss: 170751.7031 - val_mean_absolute_error: 299.4109 - 24ms/epoch - 4ms/step
## Epoch 272/600
## 6/6 - 0s - loss: 159502.6406 - mean_absolute_error: 294.1754 - val_loss: 169863.6719 - val_mean_absolute_error: 298.7999 - 25ms/epoch - 4ms/step
## Epoch 273/600
## 6/6 - 0s - loss: 163208.8906 - mean_absolute_error: 294.5403 - val_loss: 169272.4219 - val_mean_absolute_error: 298.4037 - 25ms/epoch - 4ms/step
## Epoch 274/600
## 6/6 - 0s - loss: 163001.4375 - mean_absolute_error: 300.7539 - val_loss: 168438.7188 - val_mean_absolute_error: 297.8254 - 25ms/epoch - 4ms/step
## Epoch 275/600
## 6/6 - 0s - loss: 173622.0625 - mean_absolute_error: 300.1421 - val_loss: 167726.9219 - val_mean_absolute_error: 297.3642 - 25ms/epoch - 4ms/step
## Epoch 276/600
## 6/6 - 0s - loss: 162005.4531 - mean_absolute_error: 296.9312 - val_loss: 166989.4688 - val_mean_absolute_error: 296.8469 - 25ms/epoch - 4ms/step
## Epoch 277/600
## 6/6 - 0s - loss: 168836.8906 - mean_absolute_error: 300.7788 - val_loss: 166259.7969 - val_mean_absolute_error: 296.3679 - 25ms/epoch - 4ms/step
## Epoch 278/600
## 6/6 - 0s - loss: 164424.7656 - mean_absolute_error: 295.0987 - val_loss: 165488.2969 - val_mean_absolute_error: 295.8545 - 25ms/epoch - 4ms/step
## Epoch 279/600
## 6/6 - 0s - loss: 156854.9688 - mean_absolute_error: 293.3288 - val_loss: 164704.4375 - val_mean_absolute_error: 295.3775 - 25ms/epoch - 4ms/step
## Epoch 280/600
## 6/6 - 0s - loss: 175027.7344 - mean_absolute_error: 306.9126 - val_loss: 164186.8125 - val_mean_absolute_error: 295.0701 - 24ms/epoch - 4ms/step
## Epoch 281/600
## 6/6 - 0s - loss: 162289.9375 - mean_absolute_error: 296.5539 - val_loss: 163477.5625 - val_mean_absolute_error: 294.6250 - 26ms/epoch - 4ms/step
## Epoch 282/600
## 6/6 - 0s - loss: 161679.0156 - mean_absolute_error: 294.2578 - val_loss: 162799.3125 - val_mean_absolute_error: 294.2043 - 25ms/epoch - 4ms/step
## Epoch 283/600
## 6/6 - 0s - loss: 158414.2969 - mean_absolute_error: 298.0010 - val_loss: 162119.7656 - val_mean_absolute_error: 293.7825 - 26ms/epoch - 4ms/step
## Epoch 284/600
## 6/6 - 0s - loss: 171057.6562 - mean_absolute_error: 301.8890 - val_loss: 161561.7656 - val_mean_absolute_error: 293.4399 - 25ms/epoch - 4ms/step
## Epoch 285/600
## 6/6 - 0s - loss: 171789.8594 - mean_absolute_error: 306.4397 - val_loss: 161026.4531 - val_mean_absolute_error: 293.0872 - 30ms/epoch - 5ms/step
## Epoch 286/600
## 6/6 - 0s - loss: 157897.3594 - mean_absolute_error: 285.4824 - val_loss: 160377.7500 - val_mean_absolute_error: 292.6837 - 29ms/epoch - 5ms/step
## Epoch 287/600
## 6/6 - 0s - loss: 166581.9531 - mean_absolute_error: 306.3670 - val_loss: 159955.0625 - val_mean_absolute_error: 292.4145 - 25ms/epoch - 4ms/step
## Epoch 288/600
## 6/6 - 0s - loss: 167064.4531 - mean_absolute_error: 297.3846 - val_loss: 159503.6094 - val_mean_absolute_error: 292.1194 - 26ms/epoch - 4ms/step
## Epoch 289/600
## 6/6 - 0s - loss: 153774.9375 - mean_absolute_error: 289.5004 - val_loss: 158829.3750 - val_mean_absolute_error: 291.6718 - 26ms/epoch - 4ms/step
## Epoch 290/600
## 6/6 - 0s - loss: 164849.0156 - mean_absolute_error: 298.8032 - val_loss: 158281.7969 - val_mean_absolute_error: 291.3076 - 26ms/epoch - 4ms/step
## Epoch 291/600
## 6/6 - 0s - loss: 159777.1875 - mean_absolute_error: 296.3013 - val_loss: 157639.2344 - val_mean_absolute_error: 290.8587 - 26ms/epoch - 4ms/step
## Epoch 292/600
## 6/6 - 0s - loss: 156661.0156 - mean_absolute_error: 291.1309 - val_loss: 157257.2812 - val_mean_absolute_error: 290.6170 - 27ms/epoch - 4ms/step
## Epoch 293/600
## 6/6 - 0s - loss: 169319.7188 - mean_absolute_error: 300.8426 - val_loss: 156929.8906 - val_mean_absolute_error: 290.4232 - 25ms/epoch - 4ms/step
## Epoch 294/600
## 6/6 - 0s - loss: 160131.9219 - mean_absolute_error: 293.0064 - val_loss: 156537.5469 - val_mean_absolute_error: 290.1803 - 28ms/epoch - 5ms/step
## Epoch 295/600
## 6/6 - 0s - loss: 157177.7344 - mean_absolute_error: 286.8962 - val_loss: 155904.6250 - val_mean_absolute_error: 289.7768 - 24ms/epoch - 4ms/step
## Epoch 296/600
## 6/6 - 0s - loss: 149745.5938 - mean_absolute_error: 284.7392 - val_loss: 155187.0625 - val_mean_absolute_error: 289.3274 - 28ms/epoch - 5ms/step
## Epoch 297/600
## 6/6 - 0s - loss: 154079.6719 - mean_absolute_error: 285.1041 - val_loss: 154599.4531 - val_mean_absolute_error: 288.9578 - 25ms/epoch - 4ms/step
## Epoch 298/600
## 6/6 - 0s - loss: 146227.9375 - mean_absolute_error: 285.8015 - val_loss: 154013.6562 - val_mean_absolute_error: 288.5718 - 28ms/epoch - 5ms/step
## Epoch 299/600
## 6/6 - 0s - loss: 154029.4531 - mean_absolute_error: 294.8844 - val_loss: 153558.1875 - val_mean_absolute_error: 288.2781 - 26ms/epoch - 4ms/step
## Epoch 300/600
## 6/6 - 0s - loss: 148955.5156 - mean_absolute_error: 286.0232 - val_loss: 153080.2344 - val_mean_absolute_error: 287.9681 - 27ms/epoch - 5ms/step
## Epoch 301/600
## 6/6 - 0s - loss: 159462.6406 - mean_absolute_error: 299.0741 - val_loss: 152668.5938 - val_mean_absolute_error: 287.6901 - 25ms/epoch - 4ms/step
## Epoch 302/600
## 6/6 - 0s - loss: 161148.7344 - mean_absolute_error: 297.3358 - val_loss: 152343.2500 - val_mean_absolute_error: 287.4702 - 27ms/epoch - 5ms/step
## Epoch 303/600
## 6/6 - 0s - loss: 141160.9219 - mean_absolute_error: 276.4745 - val_loss: 151792.5938 - val_mean_absolute_error: 287.0924 - 27ms/epoch - 4ms/step
## Epoch 304/600
## 6/6 - 0s - loss: 146504.5156 - mean_absolute_error: 286.1325 - val_loss: 151210.0156 - val_mean_absolute_error: 286.6979 - 29ms/epoch - 5ms/step
## Epoch 305/600
## 6/6 - 0s - loss: 166877.1094 - mean_absolute_error: 302.1207 - val_loss: 150849.5781 - val_mean_absolute_error: 286.4820 - 28ms/epoch - 5ms/step
## Epoch 306/600
## 6/6 - 0s - loss: 151095.9219 - mean_absolute_error: 291.0655 - val_loss: 150323.0625 - val_mean_absolute_error: 286.1754 - 26ms/epoch - 4ms/step
## Epoch 307/600
## 6/6 - 0s - loss: 150711.9531 - mean_absolute_error: 294.9019 - val_loss: 149912.0469 - val_mean_absolute_error: 285.9234 - 28ms/epoch - 5ms/step
## Epoch 308/600
## 6/6 - 0s - loss: 168872.0000 - mean_absolute_error: 299.4656 - val_loss: 149626.4844 - val_mean_absolute_error: 285.7541 - 25ms/epoch - 4ms/step
## Epoch 309/600
## 6/6 - 0s - loss: 145513.5625 - mean_absolute_error: 285.6674 - val_loss: 149158.6406 - val_mean_absolute_error: 285.5048 - 28ms/epoch - 5ms/step
## Epoch 310/600
## 6/6 - 0s - loss: 146778.0156 - mean_absolute_error: 289.4349 - val_loss: 148734.2500 - val_mean_absolute_error: 285.2399 - 26ms/epoch - 4ms/step
## Epoch 311/600
## 6/6 - 0s - loss: 151213.7344 - mean_absolute_error: 286.0302 - val_loss: 148286.2812 - val_mean_absolute_error: 284.9493 - 26ms/epoch - 4ms/step
## Epoch 312/600
## 6/6 - 0s - loss: 144384.6875 - mean_absolute_error: 283.3375 - val_loss: 147834.6875 - val_mean_absolute_error: 284.6657 - 28ms/epoch - 5ms/step
## Epoch 313/600
## 6/6 - 0s - loss: 155171.6875 - mean_absolute_error: 293.3555 - val_loss: 147504.0000 - val_mean_absolute_error: 284.4341 - 26ms/epoch - 4ms/step
## Epoch 314/600
## 6/6 - 0s - loss: 154110.9531 - mean_absolute_error: 296.0150 - val_loss: 147318.5781 - val_mean_absolute_error: 284.3599 - 28ms/epoch - 5ms/step
## Epoch 315/600
## 6/6 - 0s - loss: 155066.1250 - mean_absolute_error: 291.0930 - val_loss: 146845.2500 - val_mean_absolute_error: 284.0663 - 25ms/epoch - 4ms/step
## Epoch 316/600
## 6/6 - 0s - loss: 142520.2500 - mean_absolute_error: 281.0582 - val_loss: 146357.7812 - val_mean_absolute_error: 283.7374 - 28ms/epoch - 5ms/step
## Epoch 317/600
## 6/6 - 0s - loss: 150364.5156 - mean_absolute_error: 282.7378 - val_loss: 145955.0938 - val_mean_absolute_error: 283.4840 - 24ms/epoch - 4ms/step
## Epoch 318/600
## 6/6 - 0s - loss: 148431.7188 - mean_absolute_error: 287.2853 - val_loss: 145632.8750 - val_mean_absolute_error: 283.2948 - 28ms/epoch - 5ms/step
## Epoch 319/600
## 6/6 - 0s - loss: 150167.1562 - mean_absolute_error: 289.2221 - val_loss: 145336.3281 - val_mean_absolute_error: 283.1039 - 25ms/epoch - 4ms/step
## Epoch 320/600
## 6/6 - 0s - loss: 145063.5469 - mean_absolute_error: 282.4423 - val_loss: 144866.6250 - val_mean_absolute_error: 282.7823 - 27ms/epoch - 4ms/step
## Epoch 321/600
## 6/6 - 0s - loss: 152696.3906 - mean_absolute_error: 291.9814 - val_loss: 144696.7344 - val_mean_absolute_error: 282.6949 - 26ms/epoch - 4ms/step
## Epoch 322/600
## 6/6 - 0s - loss: 149403.7500 - mean_absolute_error: 284.1905 - val_loss: 144565.3281 - val_mean_absolute_error: 282.6128 - 29ms/epoch - 5ms/step
## Epoch 323/600
## 6/6 - 0s - loss: 155205.3281 - mean_absolute_error: 292.7885 - val_loss: 144402.0156 - val_mean_absolute_error: 282.5105 - 28ms/epoch - 5ms/step
## Epoch 324/600
## 6/6 - 0s - loss: 149197.8594 - mean_absolute_error: 290.9965 - val_loss: 144144.1406 - val_mean_absolute_error: 282.3376 - 26ms/epoch - 4ms/step
## Epoch 325/600
## 6/6 - 0s - loss: 143407.8906 - mean_absolute_error: 281.9965 - val_loss: 143947.0312 - val_mean_absolute_error: 282.2053 - 27ms/epoch - 5ms/step
## Epoch 326/600
## 6/6 - 0s - loss: 136803.2812 - mean_absolute_error: 277.4599 - val_loss: 143659.5938 - val_mean_absolute_error: 282.0005 - 24ms/epoch - 4ms/step
## Epoch 327/600
## 6/6 - 0s - loss: 136495.6094 - mean_absolute_error: 279.3854 - val_loss: 143244.1094 - val_mean_absolute_error: 281.7154 - 28ms/epoch - 5ms/step
## Epoch 328/600
## 6/6 - 0s - loss: 157286.7344 - mean_absolute_error: 299.9039 - val_loss: 143231.3281 - val_mean_absolute_error: 281.7016 - 25ms/epoch - 4ms/step
## Epoch 329/600
## 6/6 - 0s - loss: 150274.7812 - mean_absolute_error: 292.2265 - val_loss: 142913.3750 - val_mean_absolute_error: 281.4656 - 26ms/epoch - 4ms/step
## Epoch 330/600
## 6/6 - 0s - loss: 147982.5000 - mean_absolute_error: 287.4987 - val_loss: 142677.7344 - val_mean_absolute_error: 281.3048 - 25ms/epoch - 4ms/step
## Epoch 331/600
## 6/6 - 0s - loss: 153909.1094 - mean_absolute_error: 295.2041 - val_loss: 142489.8906 - val_mean_absolute_error: 281.1685 - 26ms/epoch - 4ms/step
## Epoch 332/600
## 6/6 - 0s - loss: 149013.4219 - mean_absolute_error: 286.0586 - val_loss: 142108.8438 - val_mean_absolute_error: 280.9004 - 26ms/epoch - 4ms/step
## Epoch 333/600
## 6/6 - 0s - loss: 146246.2344 - mean_absolute_error: 286.8116 - val_loss: 141843.3281 - val_mean_absolute_error: 280.7124 - 25ms/epoch - 4ms/step
## Epoch 334/600
## 6/6 - 0s - loss: 155815.6875 - mean_absolute_error: 291.2949 - val_loss: 141745.6562 - val_mean_absolute_error: 280.6975 - 26ms/epoch - 4ms/step
## Epoch 335/600
## 6/6 - 0s - loss: 145381.8125 - mean_absolute_error: 288.3372 - val_loss: 141505.5156 - val_mean_absolute_error: 280.5540 - 25ms/epoch - 4ms/step
## Epoch 336/600
## 6/6 - 0s - loss: 136595.0625 - mean_absolute_error: 270.5172 - val_loss: 141238.1562 - val_mean_absolute_error: 280.4128 - 26ms/epoch - 4ms/step
## Epoch 337/600
## 6/6 - 0s - loss: 147567.9375 - mean_absolute_error: 286.4100 - val_loss: 141050.8438 - val_mean_absolute_error: 280.3098 - 23ms/epoch - 4ms/step
## Epoch 338/600
## 6/6 - 0s - loss: 140326.3125 - mean_absolute_error: 281.5829 - val_loss: 140755.2344 - val_mean_absolute_error: 280.1262 - 27ms/epoch - 5ms/step
## Epoch 339/600
## 6/6 - 0s - loss: 149085.2031 - mean_absolute_error: 290.5926 - val_loss: 140556.3906 - val_mean_absolute_error: 279.9589 - 24ms/epoch - 4ms/step
## Epoch 340/600
## 6/6 - 0s - loss: 151514.7188 - mean_absolute_error: 292.1308 - val_loss: 140494.3906 - val_mean_absolute_error: 279.9482 - 29ms/epoch - 5ms/step
## Epoch 341/600
## 6/6 - 0s - loss: 148481.2188 - mean_absolute_error: 287.4736 - val_loss: 140200.0469 - val_mean_absolute_error: 279.7432 - 29ms/epoch - 5ms/step
## Epoch 342/600
## 6/6 - 0s - loss: 139258.0156 - mean_absolute_error: 278.5034 - val_loss: 139907.7969 - val_mean_absolute_error: 279.5774 - 28ms/epoch - 5ms/step
## Epoch 343/600
## 6/6 - 0s - loss: 140487.7656 - mean_absolute_error: 281.2410 - val_loss: 139548.1562 - val_mean_absolute_error: 279.3187 - 28ms/epoch - 5ms/step
## Epoch 344/600
## 6/6 - 0s - loss: 145564.2031 - mean_absolute_error: 283.6754 - val_loss: 139321.2500 - val_mean_absolute_error: 279.1833 - 26ms/epoch - 4ms/step
## Epoch 345/600
## 6/6 - 0s - loss: 143818.9531 - mean_absolute_error: 285.0037 - val_loss: 139046.2812 - val_mean_absolute_error: 278.9939 - 27ms/epoch - 4ms/step
## Epoch 346/600
## 6/6 - 0s - loss: 140140.3125 - mean_absolute_error: 278.6250 - val_loss: 138863.2031 - val_mean_absolute_error: 278.8638 - 26ms/epoch - 4ms/step
## Epoch 347/600
## 6/6 - 0s - loss: 153236.5156 - mean_absolute_error: 292.7872 - val_loss: 138652.3906 - val_mean_absolute_error: 278.6930 - 29ms/epoch - 5ms/step
## Epoch 348/600
## 6/6 - 0s - loss: 149562.5625 - mean_absolute_error: 291.8971 - val_loss: 138559.8438 - val_mean_absolute_error: 278.6411 - 25ms/epoch - 4ms/step
## Epoch 349/600
## 6/6 - 0s - loss: 157138.6875 - mean_absolute_error: 301.1214 - val_loss: 138566.6875 - val_mean_absolute_error: 278.6397 - 28ms/epoch - 5ms/step
## Epoch 350/600
## 6/6 - 0s - loss: 145506.0625 - mean_absolute_error: 282.5005 - val_loss: 138268.5000 - val_mean_absolute_error: 278.3950 - 26ms/epoch - 4ms/step
## Epoch 351/600
## 6/6 - 0s - loss: 158143.0781 - mean_absolute_error: 298.9182 - val_loss: 138256.3438 - val_mean_absolute_error: 278.4153 - 28ms/epoch - 5ms/step
## Epoch 352/600
## 6/6 - 0s - loss: 147922.1719 - mean_absolute_error: 284.9606 - val_loss: 137940.3281 - val_mean_absolute_error: 278.1440 - 26ms/epoch - 4ms/step
## Epoch 353/600
## 6/6 - 0s - loss: 137926.9844 - mean_absolute_error: 278.6203 - val_loss: 137627.8438 - val_mean_absolute_error: 277.8691 - 26ms/epoch - 4ms/step
## Epoch 354/600
## 6/6 - 0s - loss: 134545.2031 - mean_absolute_error: 276.2552 - val_loss: 137319.4688 - val_mean_absolute_error: 277.6292 - 28ms/epoch - 5ms/step
## Epoch 355/600
## 6/6 - 0s - loss: 145529.6406 - mean_absolute_error: 289.4013 - val_loss: 137146.4531 - val_mean_absolute_error: 277.4931 - 25ms/epoch - 4ms/step
## Epoch 356/600
## 6/6 - 0s - loss: 141693.7969 - mean_absolute_error: 283.5644 - val_loss: 136940.8281 - val_mean_absolute_error: 277.3295 - 27ms/epoch - 5ms/step
## Epoch 357/600
## 6/6 - 0s - loss: 144705.7500 - mean_absolute_error: 288.9565 - val_loss: 136735.8125 - val_mean_absolute_error: 277.1871 - 26ms/epoch - 4ms/step
## Epoch 358/600
## 6/6 - 0s - loss: 143315.1719 - mean_absolute_error: 284.9522 - val_loss: 136667.9844 - val_mean_absolute_error: 277.1407 - 29ms/epoch - 5ms/step
## Epoch 359/600
## 6/6 - 0s - loss: 144464.5000 - mean_absolute_error: 286.9171 - val_loss: 136635.7188 - val_mean_absolute_error: 277.1295 - 29ms/epoch - 5ms/step
## Epoch 360/600
## 6/6 - 0s - loss: 135637.9531 - mean_absolute_error: 276.2336 - val_loss: 136341.1094 - val_mean_absolute_error: 276.9042 - 26ms/epoch - 4ms/step
## Epoch 361/600
## 6/6 - 0s - loss: 144885.8906 - mean_absolute_error: 287.1216 - val_loss: 136091.2031 - val_mean_absolute_error: 276.7258 - 27ms/epoch - 4ms/step
## Epoch 362/600
## 6/6 - 0s - loss: 141664.7188 - mean_absolute_error: 280.5451 - val_loss: 135933.5938 - val_mean_absolute_error: 276.5869 - 26ms/epoch - 4ms/step
## Epoch 363/600
## 6/6 - 0s - loss: 141706.7344 - mean_absolute_error: 282.8755 - val_loss: 135812.7188 - val_mean_absolute_error: 276.5416 - 28ms/epoch - 5ms/step
## Epoch 364/600
## 6/6 - 0s - loss: 147719.1094 - mean_absolute_error: 281.2887 - val_loss: 135716.6875 - val_mean_absolute_error: 276.5006 - 25ms/epoch - 4ms/step
## Epoch 365/600
## 6/6 - 0s - loss: 138949.0938 - mean_absolute_error: 281.0826 - val_loss: 135482.8906 - val_mean_absolute_error: 276.3145 - 27ms/epoch - 5ms/step
## Epoch 366/600
## 6/6 - 0s - loss: 136370.5625 - mean_absolute_error: 280.3179 - val_loss: 135336.4062 - val_mean_absolute_error: 276.2306 - 25ms/epoch - 4ms/step
## Epoch 367/600
## 6/6 - 0s - loss: 137712.2969 - mean_absolute_error: 282.8065 - val_loss: 135089.3594 - val_mean_absolute_error: 276.0478 - 28ms/epoch - 5ms/step
## Epoch 368/600
## 6/6 - 0s - loss: 148172.8438 - mean_absolute_error: 284.6494 - val_loss: 134991.6562 - val_mean_absolute_error: 276.0076 - 24ms/epoch - 4ms/step
## Epoch 369/600
## 6/6 - 0s - loss: 132624.1719 - mean_absolute_error: 277.8628 - val_loss: 134622.4531 - val_mean_absolute_error: 275.6854 - 29ms/epoch - 5ms/step
## Epoch 370/600
## 6/6 - 0s - loss: 130968.1953 - mean_absolute_error: 269.7246 - val_loss: 134343.7188 - val_mean_absolute_error: 275.4729 - 26ms/epoch - 4ms/step
## Epoch 371/600
## 6/6 - 0s - loss: 132765.1406 - mean_absolute_error: 277.4439 - val_loss: 134047.5625 - val_mean_absolute_error: 275.2391 - 27ms/epoch - 4ms/step
## Epoch 372/600
## 6/6 - 0s - loss: 132944.8750 - mean_absolute_error: 275.1205 - val_loss: 133787.8125 - val_mean_absolute_error: 275.0648 - 26ms/epoch - 4ms/step
## Epoch 373/600
## 6/6 - 0s - loss: 140006.6406 - mean_absolute_error: 280.5021 - val_loss: 133747.2812 - val_mean_absolute_error: 275.0915 - 26ms/epoch - 4ms/step
## Epoch 374/600
## 6/6 - 0s - loss: 146879.9531 - mean_absolute_error: 288.1548 - val_loss: 133823.2812 - val_mean_absolute_error: 275.1694 - 27ms/epoch - 4ms/step
## Epoch 375/600
## 6/6 - 0s - loss: 134243.0312 - mean_absolute_error: 283.3741 - val_loss: 133600.1094 - val_mean_absolute_error: 274.9947 - 25ms/epoch - 4ms/step
## Epoch 376/600
## 6/6 - 0s - loss: 136793.5469 - mean_absolute_error: 279.3319 - val_loss: 133504.2812 - val_mean_absolute_error: 274.9716 - 30ms/epoch - 5ms/step
## Epoch 377/600
## 6/6 - 0s - loss: 146253.1094 - mean_absolute_error: 291.4985 - val_loss: 133347.4531 - val_mean_absolute_error: 274.8465 - 29ms/epoch - 5ms/step
## Epoch 378/600
## 6/6 - 0s - loss: 130767.9766 - mean_absolute_error: 272.8818 - val_loss: 133297.7344 - val_mean_absolute_error: 274.8659 - 27ms/epoch - 4ms/step
## Epoch 379/600
## 6/6 - 0s - loss: 130681.8438 - mean_absolute_error: 274.3810 - val_loss: 133110.3281 - val_mean_absolute_error: 274.7425 - 27ms/epoch - 5ms/step
## Epoch 380/600
## 6/6 - 0s - loss: 134060.1719 - mean_absolute_error: 273.1665 - val_loss: 132904.6875 - val_mean_absolute_error: 274.5910 - 26ms/epoch - 4ms/step
## Epoch 381/600
## 6/6 - 0s - loss: 142821.0938 - mean_absolute_error: 287.1338 - val_loss: 132894.5938 - val_mean_absolute_error: 274.5932 - 26ms/epoch - 4ms/step
## Epoch 382/600
## 6/6 - 0s - loss: 127965.0078 - mean_absolute_error: 273.5743 - val_loss: 132691.0469 - val_mean_absolute_error: 274.4850 - 26ms/epoch - 4ms/step
## Epoch 383/600
## 6/6 - 0s - loss: 145372.5000 - mean_absolute_error: 285.5329 - val_loss: 132549.0469 - val_mean_absolute_error: 274.3582 - 26ms/epoch - 4ms/step
## Epoch 384/600
## 6/6 - 0s - loss: 143331.7188 - mean_absolute_error: 289.7511 - val_loss: 132447.6875 - val_mean_absolute_error: 274.2856 - 25ms/epoch - 4ms/step
## Epoch 385/600
## 6/6 - 0s - loss: 135060.0000 - mean_absolute_error: 284.1106 - val_loss: 132402.3906 - val_mean_absolute_error: 274.2467 - 26ms/epoch - 4ms/step
## Epoch 386/600
## 6/6 - 0s - loss: 137463.0312 - mean_absolute_error: 285.6078 - val_loss: 132187.1094 - val_mean_absolute_error: 274.0547 - 25ms/epoch - 4ms/step
## Epoch 387/600
## 6/6 - 0s - loss: 134068.0625 - mean_absolute_error: 274.9309 - val_loss: 132009.8125 - val_mean_absolute_error: 273.9297 - 27ms/epoch - 5ms/step
## Epoch 388/600
## 6/6 - 0s - loss: 136333.7344 - mean_absolute_error: 279.4600 - val_loss: 131856.4219 - val_mean_absolute_error: 273.8391 - 24ms/epoch - 4ms/step
## Epoch 389/600
## 6/6 - 0s - loss: 137622.8594 - mean_absolute_error: 273.7798 - val_loss: 131774.5938 - val_mean_absolute_error: 273.8041 - 28ms/epoch - 5ms/step
## Epoch 390/600
## 6/6 - 0s - loss: 134488.8906 - mean_absolute_error: 283.7192 - val_loss: 131517.1094 - val_mean_absolute_error: 273.5680 - 25ms/epoch - 4ms/step
## Epoch 391/600
## 6/6 - 0s - loss: 128006.3438 - mean_absolute_error: 271.7634 - val_loss: 131424.7656 - val_mean_absolute_error: 273.5204 - 27ms/epoch - 5ms/step
## Epoch 392/600
## 6/6 - 0s - loss: 134391.6250 - mean_absolute_error: 279.3407 - val_loss: 131399.7812 - val_mean_absolute_error: 273.5465 - 26ms/epoch - 4ms/step
## Epoch 393/600
## 6/6 - 0s - loss: 140809.6094 - mean_absolute_error: 284.4016 - val_loss: 131396.0156 - val_mean_absolute_error: 273.5585 - 27ms/epoch - 5ms/step
## Epoch 394/600
## 6/6 - 0s - loss: 134426.1562 - mean_absolute_error: 278.8283 - val_loss: 131151.3594 - val_mean_absolute_error: 273.3451 - 29ms/epoch - 5ms/step
## Epoch 395/600
## 6/6 - 0s - loss: 138568.5938 - mean_absolute_error: 282.4146 - val_loss: 130915.9453 - val_mean_absolute_error: 273.1672 - 29ms/epoch - 5ms/step
## Epoch 396/600
## 6/6 - 0s - loss: 138113.4375 - mean_absolute_error: 276.8884 - val_loss: 130797.9062 - val_mean_absolute_error: 273.0582 - 28ms/epoch - 5ms/step
## Epoch 397/600
## 6/6 - 0s - loss: 138519.9375 - mean_absolute_error: 279.4238 - val_loss: 130798.9297 - val_mean_absolute_error: 273.1422 - 30ms/epoch - 5ms/step
## Epoch 398/600
## 6/6 - 0s - loss: 131709.7344 - mean_absolute_error: 276.6749 - val_loss: 130633.2969 - val_mean_absolute_error: 272.9892 - 30ms/epoch - 5ms/step
## Epoch 399/600
## 6/6 - 0s - loss: 143601.1406 - mean_absolute_error: 289.0594 - val_loss: 130647.1641 - val_mean_absolute_error: 272.9972 - 31ms/epoch - 5ms/step
## Epoch 400/600
## 6/6 - 0s - loss: 143188.2344 - mean_absolute_error: 279.0416 - val_loss: 130673.1016 - val_mean_absolute_error: 273.0354 - 30ms/epoch - 5ms/step
## Epoch 401/600
## 6/6 - 0s - loss: 132211.7031 - mean_absolute_error: 269.6628 - val_loss: 130538.1875 - val_mean_absolute_error: 272.9566 - 28ms/epoch - 5ms/step
## Epoch 402/600
## 6/6 - 0s - loss: 133693.9375 - mean_absolute_error: 275.4669 - val_loss: 130463.8594 - val_mean_absolute_error: 272.9707 - 26ms/epoch - 4ms/step
## Epoch 403/600
## 6/6 - 0s - loss: 135901.2188 - mean_absolute_error: 274.8310 - val_loss: 130401.2188 - val_mean_absolute_error: 272.9510 - 24ms/epoch - 4ms/step
## Epoch 404/600
## 6/6 - 0s - loss: 127646.1797 - mean_absolute_error: 274.2939 - val_loss: 130274.7734 - val_mean_absolute_error: 272.8829 - 30ms/epoch - 5ms/step
## Epoch 405/600
## 6/6 - 0s - loss: 139477.8594 - mean_absolute_error: 278.7483 - val_loss: 130066.2969 - val_mean_absolute_error: 272.7268 - 24ms/epoch - 4ms/step
## Epoch 406/600
## 6/6 - 0s - loss: 141439.1406 - mean_absolute_error: 281.9819 - val_loss: 130050.3906 - val_mean_absolute_error: 272.7266 - 24ms/epoch - 4ms/step
## Epoch 407/600
## 6/6 - 0s - loss: 141724.4844 - mean_absolute_error: 284.3767 - val_loss: 130136.7969 - val_mean_absolute_error: 272.7864 - 26ms/epoch - 4ms/step
## Epoch 408/600
## 6/6 - 0s - loss: 146034.0156 - mean_absolute_error: 283.5807 - val_loss: 130107.6094 - val_mean_absolute_error: 272.8304 - 25ms/epoch - 4ms/step
## Epoch 409/600
## 6/6 - 0s - loss: 134245.6875 - mean_absolute_error: 278.6707 - val_loss: 129977.0703 - val_mean_absolute_error: 272.7412 - 24ms/epoch - 4ms/step
## Epoch 410/600
## 6/6 - 0s - loss: 128059.9062 - mean_absolute_error: 276.9586 - val_loss: 129831.8359 - val_mean_absolute_error: 272.6414 - 25ms/epoch - 4ms/step
## Epoch 411/600
## 6/6 - 0s - loss: 131347.1406 - mean_absolute_error: 275.5037 - val_loss: 129752.8516 - val_mean_absolute_error: 272.5891 - 26ms/epoch - 4ms/step
## Epoch 412/600
## 6/6 - 0s - loss: 134061.8906 - mean_absolute_error: 274.5652 - val_loss: 129707.3594 - val_mean_absolute_error: 272.5933 - 26ms/epoch - 4ms/step
## Epoch 413/600
## 6/6 - 0s - loss: 129769.3828 - mean_absolute_error: 279.0428 - val_loss: 129536.8047 - val_mean_absolute_error: 272.4981 - 30ms/epoch - 5ms/step
## Epoch 414/600
## 6/6 - 0s - loss: 142485.0469 - mean_absolute_error: 286.9203 - val_loss: 129462.2969 - val_mean_absolute_error: 272.4361 - 25ms/epoch - 4ms/step
## Epoch 415/600
## 6/6 - 0s - loss: 135731.0625 - mean_absolute_error: 282.0503 - val_loss: 129500.0938 - val_mean_absolute_error: 272.5366 - 27ms/epoch - 4ms/step
## Epoch 416/600
## 6/6 - 0s - loss: 127807.0078 - mean_absolute_error: 272.8477 - val_loss: 129480.7031 - val_mean_absolute_error: 272.5567 - 26ms/epoch - 4ms/step
## Epoch 417/600
## 6/6 - 0s - loss: 135650.9844 - mean_absolute_error: 276.2612 - val_loss: 129411.5859 - val_mean_absolute_error: 272.5662 - 24ms/epoch - 4ms/step
## Epoch 418/600
## 6/6 - 0s - loss: 130075.0234 - mean_absolute_error: 278.5264 - val_loss: 129411.1953 - val_mean_absolute_error: 272.6328 - 25ms/epoch - 4ms/step
## Epoch 419/600
## 6/6 - 0s - loss: 133243.0938 - mean_absolute_error: 280.0276 - val_loss: 129266.8750 - val_mean_absolute_error: 272.5526 - 25ms/epoch - 4ms/step
## Epoch 420/600
## 6/6 - 0s - loss: 128448.6172 - mean_absolute_error: 272.8798 - val_loss: 129248.5312 - val_mean_absolute_error: 272.5927 - 26ms/epoch - 4ms/step
## Epoch 421/600
## 6/6 - 0s - loss: 131571.7344 - mean_absolute_error: 284.2420 - val_loss: 129266.4141 - val_mean_absolute_error: 272.6355 - 25ms/epoch - 4ms/step
## Epoch 422/600
## 6/6 - 0s - loss: 136612.3906 - mean_absolute_error: 279.0088 - val_loss: 129184.5156 - val_mean_absolute_error: 272.6097 - 24ms/epoch - 4ms/step
## Epoch 423/600
## 6/6 - 0s - loss: 130256.3203 - mean_absolute_error: 273.9220 - val_loss: 129083.8594 - val_mean_absolute_error: 272.5705 - 24ms/epoch - 4ms/step
## Epoch 424/600
## 6/6 - 0s - loss: 124155.7031 - mean_absolute_error: 271.0119 - val_loss: 128864.4688 - val_mean_absolute_error: 272.3846 - 24ms/epoch - 4ms/step
## Epoch 425/600
## 6/6 - 0s - loss: 130586.5938 - mean_absolute_error: 282.6036 - val_loss: 128903.3672 - val_mean_absolute_error: 272.4448 - 24ms/epoch - 4ms/step
## Epoch 426/600
## 6/6 - 0s - loss: 132786.0625 - mean_absolute_error: 282.8328 - val_loss: 128855.6328 - val_mean_absolute_error: 272.4478 - 24ms/epoch - 4ms/step
## Epoch 427/600
## 6/6 - 0s - loss: 138025.4844 - mean_absolute_error: 281.9857 - val_loss: 128876.5781 - val_mean_absolute_error: 272.5634 - 25ms/epoch - 4ms/step
## Epoch 428/600
## 6/6 - 0s - loss: 130660.5781 - mean_absolute_error: 275.1533 - val_loss: 128852.5312 - val_mean_absolute_error: 272.5992 - 25ms/epoch - 4ms/step
## Epoch 429/600
## 6/6 - 0s - loss: 135501.7656 - mean_absolute_error: 270.6758 - val_loss: 128764.2969 - val_mean_absolute_error: 272.5835 - 25ms/epoch - 4ms/step
## Epoch 430/600
## 6/6 - 0s - loss: 127933.7578 - mean_absolute_error: 274.7909 - val_loss: 128587.8594 - val_mean_absolute_error: 272.4873 - 25ms/epoch - 4ms/step
## Epoch 431/600
## 6/6 - 0s - loss: 140171.7031 - mean_absolute_error: 279.1570 - val_loss: 128521.7812 - val_mean_absolute_error: 272.4694 - 26ms/epoch - 4ms/step
## Epoch 432/600
## 6/6 - 0s - loss: 146013.4375 - mean_absolute_error: 281.2237 - val_loss: 128531.8750 - val_mean_absolute_error: 272.5148 - 30ms/epoch - 5ms/step
## Epoch 433/600
## 6/6 - 0s - loss: 124987.1328 - mean_absolute_error: 272.3760 - val_loss: 128406.4219 - val_mean_absolute_error: 272.4001 - 26ms/epoch - 4ms/step
## Epoch 434/600
## 6/6 - 0s - loss: 129938.9062 - mean_absolute_error: 273.1972 - val_loss: 128413.2891 - val_mean_absolute_error: 272.4775 - 25ms/epoch - 4ms/step
## Epoch 435/600
## 6/6 - 0s - loss: 126414.3984 - mean_absolute_error: 273.2758 - val_loss: 128198.6094 - val_mean_absolute_error: 272.3320 - 26ms/epoch - 4ms/step
## Epoch 436/600
## 6/6 - 0s - loss: 134308.2344 - mean_absolute_error: 275.2979 - val_loss: 128059.6562 - val_mean_absolute_error: 272.2328 - 25ms/epoch - 4ms/step
## Epoch 437/600
## 6/6 - 0s - loss: 128416.9766 - mean_absolute_error: 268.5712 - val_loss: 128039.3438 - val_mean_absolute_error: 272.2149 - 25ms/epoch - 4ms/step
## Epoch 438/600
## 6/6 - 0s - loss: 122220.2578 - mean_absolute_error: 268.9008 - val_loss: 127855.1719 - val_mean_absolute_error: 272.1157 - 25ms/epoch - 4ms/step
## Epoch 439/600
## 6/6 - 0s - loss: 138002.5469 - mean_absolute_error: 283.3926 - val_loss: 127682.0703 - val_mean_absolute_error: 271.9575 - 25ms/epoch - 4ms/step
## Epoch 440/600
## 6/6 - 0s - loss: 131544.0156 - mean_absolute_error: 276.5154 - val_loss: 127528.9922 - val_mean_absolute_error: 271.8620 - 26ms/epoch - 4ms/step
## Epoch 441/600
## 6/6 - 0s - loss: 139610.8594 - mean_absolute_error: 284.4831 - val_loss: 127439.6641 - val_mean_absolute_error: 271.8123 - 25ms/epoch - 4ms/step
## Epoch 442/600
## 6/6 - 0s - loss: 118489.2812 - mean_absolute_error: 265.2676 - val_loss: 127275.8125 - val_mean_absolute_error: 271.7209 - 24ms/epoch - 4ms/step
## Epoch 443/600
## 6/6 - 0s - loss: 129391.9297 - mean_absolute_error: 270.6159 - val_loss: 127126.0781 - val_mean_absolute_error: 271.5981 - 24ms/epoch - 4ms/step
## Epoch 444/600
## 6/6 - 0s - loss: 130452.0312 - mean_absolute_error: 275.4501 - val_loss: 127125.8359 - val_mean_absolute_error: 271.6345 - 25ms/epoch - 4ms/step
## Epoch 445/600
## 6/6 - 0s - loss: 134128.3281 - mean_absolute_error: 273.7907 - val_loss: 127067.2188 - val_mean_absolute_error: 271.6133 - 25ms/epoch - 4ms/step
## Epoch 446/600
## 6/6 - 0s - loss: 122105.5938 - mean_absolute_error: 262.3990 - val_loss: 126946.1016 - val_mean_absolute_error: 271.5554 - 24ms/epoch - 4ms/step
## Epoch 447/600
## 6/6 - 0s - loss: 121408.1562 - mean_absolute_error: 268.5659 - val_loss: 126759.3828 - val_mean_absolute_error: 271.3973 - 24ms/epoch - 4ms/step
## Epoch 448/600
## 6/6 - 0s - loss: 127403.6719 - mean_absolute_error: 272.0333 - val_loss: 126683.6016 - val_mean_absolute_error: 271.3319 - 25ms/epoch - 4ms/step
## Epoch 449/600
## 6/6 - 0s - loss: 132290.7500 - mean_absolute_error: 272.4439 - val_loss: 126481.1250 - val_mean_absolute_error: 271.1199 - 25ms/epoch - 4ms/step
## Epoch 450/600
## 6/6 - 0s - loss: 120202.1328 - mean_absolute_error: 263.1240 - val_loss: 126398.6875 - val_mean_absolute_error: 271.0589 - 25ms/epoch - 4ms/step
## Epoch 451/600
## 6/6 - 0s - loss: 126697.8438 - mean_absolute_error: 273.4726 - val_loss: 126256.3438 - val_mean_absolute_error: 270.8771 - 31ms/epoch - 5ms/step
## Epoch 452/600
## 6/6 - 0s - loss: 129642.3516 - mean_absolute_error: 273.2444 - val_loss: 126270.0234 - val_mean_absolute_error: 270.8922 - 27ms/epoch - 5ms/step
## Epoch 453/600
## 6/6 - 0s - loss: 123063.4766 - mean_absolute_error: 268.6275 - val_loss: 126145.3203 - val_mean_absolute_error: 270.8247 - 24ms/epoch - 4ms/step
## Epoch 454/600
## 6/6 - 0s - loss: 122446.4297 - mean_absolute_error: 264.5496 - val_loss: 126112.6016 - val_mean_absolute_error: 270.8298 - 26ms/epoch - 4ms/step
## Epoch 455/600
## 6/6 - 0s - loss: 129712.0938 - mean_absolute_error: 275.1669 - val_loss: 125976.4922 - val_mean_absolute_error: 270.6800 - 24ms/epoch - 4ms/step
## Epoch 456/600
## 6/6 - 0s - loss: 129757.6016 - mean_absolute_error: 268.2074 - val_loss: 125982.9766 - val_mean_absolute_error: 270.7341 - 25ms/epoch - 4ms/step
## Epoch 457/600
## 6/6 - 0s - loss: 125467.0469 - mean_absolute_error: 273.2118 - val_loss: 125895.3594 - val_mean_absolute_error: 270.6867 - 24ms/epoch - 4ms/step
## Epoch 458/600
## 6/6 - 0s - loss: 120753.7188 - mean_absolute_error: 270.7338 - val_loss: 125930.9219 - val_mean_absolute_error: 270.7680 - 24ms/epoch - 4ms/step
## Epoch 459/600
## 6/6 - 0s - loss: 125822.9297 - mean_absolute_error: 271.6499 - val_loss: 125902.1641 - val_mean_absolute_error: 270.7639 - 24ms/epoch - 4ms/step
## Epoch 460/600
## 6/6 - 0s - loss: 126451.8203 - mean_absolute_error: 274.3952 - val_loss: 125817.0312 - val_mean_absolute_error: 270.6894 - 25ms/epoch - 4ms/step
## Epoch 461/600
## 6/6 - 0s - loss: 122585.1953 - mean_absolute_error: 265.1931 - val_loss: 125812.0703 - val_mean_absolute_error: 270.7606 - 25ms/epoch - 4ms/step
## Epoch 462/600
## 6/6 - 0s - loss: 129631.5547 - mean_absolute_error: 275.5483 - val_loss: 125959.1719 - val_mean_absolute_error: 270.9298 - 25ms/epoch - 4ms/step
## Epoch 463/600
## 6/6 - 0s - loss: 124061.4297 - mean_absolute_error: 274.2781 - val_loss: 126007.5625 - val_mean_absolute_error: 271.0061 - 24ms/epoch - 4ms/step
## Epoch 464/600
## 6/6 - 0s - loss: 132418.6719 - mean_absolute_error: 274.7596 - val_loss: 126039.3672 - val_mean_absolute_error: 271.0685 - 25ms/epoch - 4ms/step
## Epoch 465/600
## 6/6 - 0s - loss: 129855.2500 - mean_absolute_error: 273.1653 - val_loss: 126026.3828 - val_mean_absolute_error: 271.0923 - 25ms/epoch - 4ms/step
## Epoch 466/600
## 6/6 - 0s - loss: 134875.6406 - mean_absolute_error: 274.0345 - val_loss: 125940.3828 - val_mean_absolute_error: 271.0277 - 24ms/epoch - 4ms/step
## Epoch 467/600
## 6/6 - 0s - loss: 125197.5781 - mean_absolute_error: 266.5958 - val_loss: 125890.4609 - val_mean_absolute_error: 270.9965 - 25ms/epoch - 4ms/step
## Epoch 468/600
## 6/6 - 0s - loss: 123747.2969 - mean_absolute_error: 265.0471 - val_loss: 125803.0469 - val_mean_absolute_error: 270.9416 - 24ms/epoch - 4ms/step
## Epoch 469/600
## 6/6 - 0s - loss: 117140.9453 - mean_absolute_error: 261.6905 - val_loss: 125728.8984 - val_mean_absolute_error: 270.8763 - 25ms/epoch - 4ms/step
## Epoch 470/600
## 6/6 - 0s - loss: 127551.1016 - mean_absolute_error: 269.8280 - val_loss: 125727.0703 - val_mean_absolute_error: 270.8956 - 30ms/epoch - 5ms/step
## Epoch 471/600
## 6/6 - 0s - loss: 127681.1250 - mean_absolute_error: 272.5217 - val_loss: 125615.3438 - val_mean_absolute_error: 270.7754 - 28ms/epoch - 5ms/step
## Epoch 472/600
## 6/6 - 0s - loss: 138772.3750 - mean_absolute_error: 281.4794 - val_loss: 125684.8047 - val_mean_absolute_error: 270.8595 - 25ms/epoch - 4ms/step
## Epoch 473/600
## 6/6 - 0s - loss: 115402.9766 - mean_absolute_error: 254.2523 - val_loss: 125671.9688 - val_mean_absolute_error: 270.8958 - 26ms/epoch - 4ms/step
## Epoch 474/600
## 6/6 - 0s - loss: 129244.0703 - mean_absolute_error: 274.3181 - val_loss: 125739.8594 - val_mean_absolute_error: 270.9915 - 25ms/epoch - 4ms/step
## Epoch 475/600
## 6/6 - 0s - loss: 124304.1719 - mean_absolute_error: 266.8078 - val_loss: 125671.3203 - val_mean_absolute_error: 270.9478 - 26ms/epoch - 4ms/step
## Epoch 476/600
## 6/6 - 0s - loss: 120913.7422 - mean_absolute_error: 264.4384 - val_loss: 125578.4609 - val_mean_absolute_error: 270.8657 - 25ms/epoch - 4ms/step
## Epoch 477/600
## 6/6 - 0s - loss: 122407.8438 - mean_absolute_error: 261.7377 - val_loss: 125413.5312 - val_mean_absolute_error: 270.7080 - 24ms/epoch - 4ms/step
## Epoch 478/600
## 6/6 - 0s - loss: 126155.5312 - mean_absolute_error: 272.9483 - val_loss: 125408.3438 - val_mean_absolute_error: 270.6929 - 25ms/epoch - 4ms/step
## Epoch 479/600
## 6/6 - 0s - loss: 126995.1562 - mean_absolute_error: 271.0168 - val_loss: 125415.3438 - val_mean_absolute_error: 270.7007 - 25ms/epoch - 4ms/step
## Epoch 480/600
## 6/6 - 0s - loss: 124730.8984 - mean_absolute_error: 263.1997 - val_loss: 125340.5625 - val_mean_absolute_error: 270.6673 - 24ms/epoch - 4ms/step
## Epoch 481/600
## 6/6 - 0s - loss: 134893.2031 - mean_absolute_error: 278.0999 - val_loss: 125407.8281 - val_mean_absolute_error: 270.7942 - 25ms/epoch - 4ms/step
## Epoch 482/600
## 6/6 - 0s - loss: 125300.8203 - mean_absolute_error: 267.3581 - val_loss: 125294.9688 - val_mean_absolute_error: 270.6966 - 25ms/epoch - 4ms/step
## Epoch 483/600
## 6/6 - 0s - loss: 115409.8516 - mean_absolute_error: 258.3402 - val_loss: 125255.2734 - val_mean_absolute_error: 270.7158 - 24ms/epoch - 4ms/step
## Epoch 484/600
## 6/6 - 0s - loss: 123789.1953 - mean_absolute_error: 260.9417 - val_loss: 125128.6875 - val_mean_absolute_error: 270.6131 - 25ms/epoch - 4ms/step
## Epoch 485/600
## 6/6 - 0s - loss: 124182.6250 - mean_absolute_error: 271.2110 - val_loss: 125051.2188 - val_mean_absolute_error: 270.5628 - 25ms/epoch - 4ms/step
## Epoch 486/600
## 6/6 - 0s - loss: 125092.0312 - mean_absolute_error: 263.3574 - val_loss: 124997.0469 - val_mean_absolute_error: 270.5287 - 24ms/epoch - 4ms/step
## Epoch 487/600
## 6/6 - 0s - loss: 114333.5078 - mean_absolute_error: 262.0895 - val_loss: 124833.0000 - val_mean_absolute_error: 270.3665 - 25ms/epoch - 4ms/step
## Epoch 488/600
## 6/6 - 0s - loss: 127480.6172 - mean_absolute_error: 267.3306 - val_loss: 124635.4922 - val_mean_absolute_error: 270.1313 - 26ms/epoch - 4ms/step
## Epoch 489/600
## 6/6 - 0s - loss: 131497.6250 - mean_absolute_error: 272.9057 - val_loss: 124569.6641 - val_mean_absolute_error: 270.0952 - 26ms/epoch - 4ms/step
## Epoch 490/600
## 6/6 - 0s - loss: 136215.0156 - mean_absolute_error: 279.4718 - val_loss: 124518.1484 - val_mean_absolute_error: 270.0175 - 30ms/epoch - 5ms/step
## Epoch 491/600
## 6/6 - 0s - loss: 124557.4453 - mean_absolute_error: 270.2972 - val_loss: 124460.9922 - val_mean_absolute_error: 269.9780 - 26ms/epoch - 4ms/step
## Epoch 492/600
## 6/6 - 0s - loss: 123097.5938 - mean_absolute_error: 263.6534 - val_loss: 124401.8594 - val_mean_absolute_error: 269.8936 - 26ms/epoch - 4ms/step
## Epoch 493/600
## 6/6 - 0s - loss: 121928.1797 - mean_absolute_error: 264.7653 - val_loss: 124321.8047 - val_mean_absolute_error: 269.8488 - 25ms/epoch - 4ms/step
## Epoch 494/600
## 6/6 - 0s - loss: 139030.3906 - mean_absolute_error: 284.2428 - val_loss: 124330.7109 - val_mean_absolute_error: 269.8958 - 25ms/epoch - 4ms/step
## Epoch 495/600
## 6/6 - 0s - loss: 127830.4688 - mean_absolute_error: 276.7632 - val_loss: 124320.4688 - val_mean_absolute_error: 269.9070 - 25ms/epoch - 4ms/step
## Epoch 496/600
## 6/6 - 0s - loss: 118391.5000 - mean_absolute_error: 262.0421 - val_loss: 124243.2188 - val_mean_absolute_error: 269.8849 - 25ms/epoch - 4ms/step
## Epoch 497/600
## 6/6 - 0s - loss: 122453.8281 - mean_absolute_error: 266.5708 - val_loss: 124188.8516 - val_mean_absolute_error: 269.8651 - 24ms/epoch - 4ms/step
## Epoch 498/600
## 6/6 - 0s - loss: 125327.0781 - mean_absolute_error: 272.8235 - val_loss: 124111.2031 - val_mean_absolute_error: 269.8194 - 25ms/epoch - 4ms/step
## Epoch 499/600
## 6/6 - 0s - loss: 123786.5234 - mean_absolute_error: 264.2314 - val_loss: 124044.1406 - val_mean_absolute_error: 269.7620 - 24ms/epoch - 4ms/step
## Epoch 500/600
## 6/6 - 0s - loss: 123856.1562 - mean_absolute_error: 265.4048 - val_loss: 124042.3203 - val_mean_absolute_error: 269.7883 - 25ms/epoch - 4ms/step
## Epoch 501/600
## 6/6 - 0s - loss: 121102.1250 - mean_absolute_error: 265.8095 - val_loss: 123926.6406 - val_mean_absolute_error: 269.6776 - 24ms/epoch - 4ms/step
## Epoch 502/600
## 6/6 - 0s - loss: 126423.5781 - mean_absolute_error: 269.5818 - val_loss: 123910.6641 - val_mean_absolute_error: 269.6590 - 26ms/epoch - 4ms/step
## Epoch 503/600
## 6/6 - 0s - loss: 114935.9688 - mean_absolute_error: 260.2321 - val_loss: 123863.7266 - val_mean_absolute_error: 269.6180 - 24ms/epoch - 4ms/step
## Epoch 504/600
## 6/6 - 0s - loss: 119288.6719 - mean_absolute_error: 266.4413 - val_loss: 123774.8750 - val_mean_absolute_error: 269.5045 - 25ms/epoch - 4ms/step
## Epoch 505/600
## 6/6 - 0s - loss: 116334.2031 - mean_absolute_error: 255.5049 - val_loss: 123632.2734 - val_mean_absolute_error: 269.3542 - 25ms/epoch - 4ms/step
## Epoch 506/600
## 6/6 - 0s - loss: 118820.7188 - mean_absolute_error: 267.4717 - val_loss: 123549.0312 - val_mean_absolute_error: 269.2683 - 25ms/epoch - 4ms/step
## Epoch 507/600
## 6/6 - 0s - loss: 122845.1172 - mean_absolute_error: 263.7417 - val_loss: 123525.3203 - val_mean_absolute_error: 269.2698 - 25ms/epoch - 4ms/step
## Epoch 508/600
## 6/6 - 0s - loss: 125952.1719 - mean_absolute_error: 263.6483 - val_loss: 123446.1406 - val_mean_absolute_error: 269.1967 - 26ms/epoch - 4ms/step
## Epoch 509/600
## 6/6 - 0s - loss: 117856.2031 - mean_absolute_error: 264.9116 - val_loss: 123519.7578 - val_mean_absolute_error: 269.2638 - 29ms/epoch - 5ms/step
## Epoch 510/600
## 6/6 - 0s - loss: 133888.8594 - mean_absolute_error: 274.6753 - val_loss: 123550.3906 - val_mean_absolute_error: 269.3254 - 26ms/epoch - 4ms/step
## Epoch 511/600
## 6/6 - 0s - loss: 125351.9062 - mean_absolute_error: 273.2261 - val_loss: 123507.9062 - val_mean_absolute_error: 269.2733 - 25ms/epoch - 4ms/step
## Epoch 512/600
## 6/6 - 0s - loss: 123330.8047 - mean_absolute_error: 266.7308 - val_loss: 123486.2266 - val_mean_absolute_error: 269.2920 - 25ms/epoch - 4ms/step
## Epoch 513/600
## 6/6 - 0s - loss: 122188.8203 - mean_absolute_error: 266.1061 - val_loss: 123380.3906 - val_mean_absolute_error: 269.1962 - 25ms/epoch - 4ms/step
## Epoch 514/600
## 6/6 - 0s - loss: 127314.2734 - mean_absolute_error: 271.4080 - val_loss: 123391.0000 - val_mean_absolute_error: 269.2334 - 25ms/epoch - 4ms/step
## Epoch 515/600
## 6/6 - 0s - loss: 123076.7500 - mean_absolute_error: 271.6270 - val_loss: 123403.9062 - val_mean_absolute_error: 269.2808 - 25ms/epoch - 4ms/step
## Epoch 516/600
## 6/6 - 0s - loss: 120080.0312 - mean_absolute_error: 266.0625 - val_loss: 123382.4609 - val_mean_absolute_error: 269.2851 - 26ms/epoch - 4ms/step
## Epoch 517/600
## 6/6 - 0s - loss: 110620.0234 - mean_absolute_error: 253.3100 - val_loss: 123264.0703 - val_mean_absolute_error: 269.1763 - 26ms/epoch - 4ms/step
## Epoch 518/600
## 6/6 - 0s - loss: 112909.0938 - mean_absolute_error: 255.9296 - val_loss: 123214.6641 - val_mean_absolute_error: 269.1393 - 24ms/epoch - 4ms/step
## Epoch 519/600
## 6/6 - 0s - loss: 113431.2266 - mean_absolute_error: 257.0557 - val_loss: 123209.3359 - val_mean_absolute_error: 269.1310 - 24ms/epoch - 4ms/step
## Epoch 520/600
## 6/6 - 0s - loss: 121216.9062 - mean_absolute_error: 265.4352 - val_loss: 123207.1953 - val_mean_absolute_error: 269.1487 - 24ms/epoch - 4ms/step
## Epoch 521/600
## 6/6 - 0s - loss: 123890.8047 - mean_absolute_error: 262.3613 - val_loss: 123190.7344 - val_mean_absolute_error: 269.1819 - 25ms/epoch - 4ms/step
## Epoch 522/600
## 6/6 - 0s - loss: 118304.5312 - mean_absolute_error: 254.3886 - val_loss: 123076.1641 - val_mean_absolute_error: 269.0361 - 25ms/epoch - 4ms/step
## Epoch 523/600
## 6/6 - 0s - loss: 127682.9062 - mean_absolute_error: 271.1933 - val_loss: 123026.3125 - val_mean_absolute_error: 268.9743 - 26ms/epoch - 4ms/step
## Epoch 524/600
## 6/6 - 0s - loss: 125802.5547 - mean_absolute_error: 266.0329 - val_loss: 122968.0469 - val_mean_absolute_error: 268.9134 - 25ms/epoch - 4ms/step
## Epoch 525/600
## 6/6 - 0s - loss: 118518.1562 - mean_absolute_error: 264.1398 - val_loss: 122879.7734 - val_mean_absolute_error: 268.8054 - 25ms/epoch - 4ms/step
## Epoch 526/600
## 6/6 - 0s - loss: 123735.5312 - mean_absolute_error: 264.5235 - val_loss: 122878.9297 - val_mean_absolute_error: 268.8024 - 25ms/epoch - 4ms/step
## Epoch 527/600
## 6/6 - 0s - loss: 119204.0781 - mean_absolute_error: 261.2545 - val_loss: 122792.0938 - val_mean_absolute_error: 268.7585 - 26ms/epoch - 4ms/step
## Epoch 528/600
## 6/6 - 0s - loss: 123895.9297 - mean_absolute_error: 270.3891 - val_loss: 122879.2266 - val_mean_absolute_error: 268.8412 - 28ms/epoch - 5ms/step
## Epoch 529/600
## 6/6 - 0s - loss: 125404.9062 - mean_absolute_error: 264.7766 - val_loss: 122886.7109 - val_mean_absolute_error: 268.8465 - 26ms/epoch - 4ms/step
## Epoch 530/600
## 6/6 - 0s - loss: 116067.9766 - mean_absolute_error: 260.2440 - val_loss: 122815.3125 - val_mean_absolute_error: 268.7909 - 25ms/epoch - 4ms/step
## Epoch 531/600
## 6/6 - 0s - loss: 121524.5078 - mean_absolute_error: 259.0569 - val_loss: 122775.1719 - val_mean_absolute_error: 268.7478 - 26ms/epoch - 4ms/step
## Epoch 532/600
## 6/6 - 0s - loss: 120758.0234 - mean_absolute_error: 263.5826 - val_loss: 122752.7578 - val_mean_absolute_error: 268.7306 - 25ms/epoch - 4ms/step
## Epoch 533/600
## 6/6 - 0s - loss: 111301.4922 - mean_absolute_error: 256.1684 - val_loss: 122623.5078 - val_mean_absolute_error: 268.5923 - 25ms/epoch - 4ms/step
## Epoch 534/600
## 6/6 - 0s - loss: 122223.1172 - mean_absolute_error: 266.1873 - val_loss: 122595.1953 - val_mean_absolute_error: 268.5775 - 24ms/epoch - 4ms/step
## Epoch 535/600
## 6/6 - 0s - loss: 117949.9531 - mean_absolute_error: 258.1115 - val_loss: 122671.0000 - val_mean_absolute_error: 268.7118 - 25ms/epoch - 4ms/step
## Epoch 536/600
## 6/6 - 0s - loss: 121147.6016 - mean_absolute_error: 265.5214 - val_loss: 122658.6875 - val_mean_absolute_error: 268.7058 - 25ms/epoch - 4ms/step
## Epoch 537/600
## 6/6 - 0s - loss: 113948.6172 - mean_absolute_error: 260.4852 - val_loss: 122479.2969 - val_mean_absolute_error: 268.5077 - 25ms/epoch - 4ms/step
## Epoch 538/600
## 6/6 - 0s - loss: 129004.4219 - mean_absolute_error: 272.0306 - val_loss: 122506.5781 - val_mean_absolute_error: 268.5179 - 24ms/epoch - 4ms/step
## Epoch 539/600
## 6/6 - 0s - loss: 122267.5547 - mean_absolute_error: 257.2210 - val_loss: 122572.1250 - val_mean_absolute_error: 268.6521 - 24ms/epoch - 4ms/step
## Epoch 540/600
## 6/6 - 0s - loss: 120368.1250 - mean_absolute_error: 260.4856 - val_loss: 122629.1250 - val_mean_absolute_error: 268.7570 - 24ms/epoch - 4ms/step
## Epoch 541/600
## 6/6 - 0s - loss: 116620.4922 - mean_absolute_error: 258.5653 - val_loss: 122631.7500 - val_mean_absolute_error: 268.7354 - 25ms/epoch - 4ms/step
## Epoch 542/600
## 6/6 - 0s - loss: 130324.7500 - mean_absolute_error: 269.6649 - val_loss: 122470.4688 - val_mean_absolute_error: 268.5107 - 24ms/epoch - 4ms/step
## Epoch 543/600
## 6/6 - 0s - loss: 125239.2266 - mean_absolute_error: 269.4655 - val_loss: 122366.6562 - val_mean_absolute_error: 268.3950 - 24ms/epoch - 4ms/step
## Epoch 544/600
## 6/6 - 0s - loss: 123395.6953 - mean_absolute_error: 267.1058 - val_loss: 122270.5781 - val_mean_absolute_error: 268.2826 - 24ms/epoch - 4ms/step
## Epoch 545/600
## 6/6 - 0s - loss: 121083.1484 - mean_absolute_error: 259.9524 - val_loss: 122267.4453 - val_mean_absolute_error: 268.2845 - 25ms/epoch - 4ms/step
## Epoch 546/600
## 6/6 - 0s - loss: 109899.3984 - mean_absolute_error: 250.6523 - val_loss: 122191.7969 - val_mean_absolute_error: 268.2017 - 25ms/epoch - 4ms/step
## Epoch 547/600
## 6/6 - 0s - loss: 120956.2812 - mean_absolute_error: 265.7276 - val_loss: 122016.1172 - val_mean_absolute_error: 267.9405 - 30ms/epoch - 5ms/step
## Epoch 548/600
## 6/6 - 0s - loss: 117950.4453 - mean_absolute_error: 263.9824 - val_loss: 122006.3438 - val_mean_absolute_error: 267.9271 - 29ms/epoch - 5ms/step
## Epoch 549/600
## 6/6 - 0s - loss: 119000.3672 - mean_absolute_error: 263.0527 - val_loss: 121896.8828 - val_mean_absolute_error: 267.7984 - 25ms/epoch - 4ms/step
## Epoch 550/600
## 6/6 - 0s - loss: 115317.7031 - mean_absolute_error: 257.9549 - val_loss: 121846.2031 - val_mean_absolute_error: 267.7318 - 27ms/epoch - 4ms/step
## Epoch 551/600
## 6/6 - 0s - loss: 123795.6484 - mean_absolute_error: 266.4873 - val_loss: 121845.4453 - val_mean_absolute_error: 267.7596 - 25ms/epoch - 4ms/step
## Epoch 552/600
## 6/6 - 0s - loss: 121847.4531 - mean_absolute_error: 266.0386 - val_loss: 121866.1016 - val_mean_absolute_error: 267.7720 - 25ms/epoch - 4ms/step
## Epoch 553/600
## 6/6 - 0s - loss: 127348.2969 - mean_absolute_error: 262.3014 - val_loss: 121826.7578 - val_mean_absolute_error: 267.7024 - 25ms/epoch - 4ms/step
## Epoch 554/600
## 6/6 - 0s - loss: 130819.2031 - mean_absolute_error: 269.4442 - val_loss: 121986.8516 - val_mean_absolute_error: 267.8794 - 24ms/epoch - 4ms/step
## Epoch 555/600
## 6/6 - 0s - loss: 117471.5703 - mean_absolute_error: 260.7635 - val_loss: 121959.9297 - val_mean_absolute_error: 267.8761 - 25ms/epoch - 4ms/step
## Epoch 556/600
## 6/6 - 0s - loss: 128533.9922 - mean_absolute_error: 268.0547 - val_loss: 121969.5156 - val_mean_absolute_error: 267.8889 - 24ms/epoch - 4ms/step
## Epoch 557/600
## 6/6 - 0s - loss: 113930.8828 - mean_absolute_error: 258.7264 - val_loss: 121959.1953 - val_mean_absolute_error: 267.9321 - 24ms/epoch - 4ms/step
## Epoch 558/600
## 6/6 - 0s - loss: 120506.8438 - mean_absolute_error: 257.3590 - val_loss: 121852.0781 - val_mean_absolute_error: 267.7893 - 25ms/epoch - 4ms/step
## Epoch 559/600
## 6/6 - 0s - loss: 115616.7578 - mean_absolute_error: 259.5575 - val_loss: 121871.7266 - val_mean_absolute_error: 267.7897 - 24ms/epoch - 4ms/step
## Epoch 560/600
## 6/6 - 0s - loss: 120822.3516 - mean_absolute_error: 263.5886 - val_loss: 121809.7031 - val_mean_absolute_error: 267.7443 - 24ms/epoch - 4ms/step
## Epoch 561/600
## 6/6 - 0s - loss: 117378.1953 - mean_absolute_error: 256.6993 - val_loss: 121867.6406 - val_mean_absolute_error: 267.8291 - 24ms/epoch - 4ms/step
## Epoch 562/600
## 6/6 - 0s - loss: 118268.9531 - mean_absolute_error: 268.6290 - val_loss: 121839.3984 - val_mean_absolute_error: 267.7766 - 24ms/epoch - 4ms/step
## Epoch 563/600
## 6/6 - 0s - loss: 119346.9766 - mean_absolute_error: 263.3104 - val_loss: 121709.5781 - val_mean_absolute_error: 267.6098 - 24ms/epoch - 4ms/step
## Epoch 564/600
## 6/6 - 0s - loss: 114669.9922 - mean_absolute_error: 251.4128 - val_loss: 121635.3906 - val_mean_absolute_error: 267.5048 - 29ms/epoch - 5ms/step
## Epoch 565/600
## 6/6 - 0s - loss: 111953.2812 - mean_absolute_error: 259.7254 - val_loss: 121592.8281 - val_mean_absolute_error: 267.4333 - 26ms/epoch - 4ms/step
## Epoch 566/600
## 6/6 - 0s - loss: 122088.6797 - mean_absolute_error: 262.5455 - val_loss: 121713.7109 - val_mean_absolute_error: 267.6042 - 29ms/epoch - 5ms/step
## Epoch 567/600
## 6/6 - 0s - loss: 117125.1953 - mean_absolute_error: 256.7731 - val_loss: 121672.9062 - val_mean_absolute_error: 267.5598 - 27ms/epoch - 5ms/step
## Epoch 568/600
## 6/6 - 0s - loss: 111641.6562 - mean_absolute_error: 248.6690 - val_loss: 121551.1719 - val_mean_absolute_error: 267.4403 - 25ms/epoch - 4ms/step
## Epoch 569/600
## 6/6 - 0s - loss: 127001.9922 - mean_absolute_error: 268.6897 - val_loss: 121624.9688 - val_mean_absolute_error: 267.5192 - 26ms/epoch - 4ms/step
## Epoch 570/600
## 6/6 - 0s - loss: 118997.2969 - mean_absolute_error: 260.4953 - val_loss: 121530.7266 - val_mean_absolute_error: 267.3910 - 25ms/epoch - 4ms/step
## Epoch 571/600
## 6/6 - 0s - loss: 119538.2500 - mean_absolute_error: 259.0325 - val_loss: 121570.7578 - val_mean_absolute_error: 267.4254 - 25ms/epoch - 4ms/step
## Epoch 572/600
## 6/6 - 0s - loss: 118318.8516 - mean_absolute_error: 258.9564 - val_loss: 121447.9062 - val_mean_absolute_error: 267.2573 - 24ms/epoch - 4ms/step
## Epoch 573/600
## 6/6 - 0s - loss: 120034.2578 - mean_absolute_error: 261.7945 - val_loss: 121497.2266 - val_mean_absolute_error: 267.3278 - 24ms/epoch - 4ms/step
## Epoch 574/600
## 6/6 - 0s - loss: 109306.2266 - mean_absolute_error: 250.0794 - val_loss: 121385.4609 - val_mean_absolute_error: 267.1909 - 25ms/epoch - 4ms/step
## Epoch 575/600
## 6/6 - 0s - loss: 115742.7266 - mean_absolute_error: 259.8546 - val_loss: 121346.1250 - val_mean_absolute_error: 267.1386 - 25ms/epoch - 4ms/step
## Epoch 576/600
## 6/6 - 0s - loss: 111234.0469 - mean_absolute_error: 254.5857 - val_loss: 121309.5625 - val_mean_absolute_error: 267.0995 - 24ms/epoch - 4ms/step
## Epoch 577/600
## 6/6 - 0s - loss: 119244.1953 - mean_absolute_error: 267.4169 - val_loss: 121154.0000 - val_mean_absolute_error: 266.9035 - 24ms/epoch - 4ms/step
## Epoch 578/600
## 6/6 - 0s - loss: 120340.7969 - mean_absolute_error: 259.0182 - val_loss: 121177.1719 - val_mean_absolute_error: 266.9163 - 24ms/epoch - 4ms/step
## Epoch 579/600
## 6/6 - 0s - loss: 117331.1562 - mean_absolute_error: 254.8001 - val_loss: 121135.6797 - val_mean_absolute_error: 266.8169 - 24ms/epoch - 4ms/step
## Epoch 580/600
## 6/6 - 0s - loss: 116016.0312 - mean_absolute_error: 260.8574 - val_loss: 121116.7266 - val_mean_absolute_error: 266.8182 - 25ms/epoch - 4ms/step
## Epoch 581/600
## 6/6 - 0s - loss: 107189.9219 - mean_absolute_error: 247.1023 - val_loss: 121121.7266 - val_mean_absolute_error: 266.8371 - 24ms/epoch - 4ms/step
## Epoch 582/600
## 6/6 - 0s - loss: 113008.2422 - mean_absolute_error: 250.3622 - val_loss: 121016.0938 - val_mean_absolute_error: 266.7039 - 25ms/epoch - 4ms/step
## Epoch 583/600
## 6/6 - 0s - loss: 117231.8672 - mean_absolute_error: 253.2794 - val_loss: 120877.0000 - val_mean_absolute_error: 266.5075 - 26ms/epoch - 4ms/step
## Epoch 584/600
## 6/6 - 0s - loss: 120413.5703 - mean_absolute_error: 261.7800 - val_loss: 120841.5156 - val_mean_absolute_error: 266.4577 - 24ms/epoch - 4ms/step
## Epoch 585/600
## 6/6 - 0s - loss: 116286.7734 - mean_absolute_error: 260.7272 - val_loss: 120890.5547 - val_mean_absolute_error: 266.5221 - 27ms/epoch - 5ms/step
## Epoch 586/600
## 6/6 - 0s - loss: 118615.7500 - mean_absolute_error: 257.3634 - val_loss: 120873.4688 - val_mean_absolute_error: 266.4990 - 29ms/epoch - 5ms/step
## Epoch 587/600
## 6/6 - 0s - loss: 116101.8984 - mean_absolute_error: 259.4239 - val_loss: 120831.3359 - val_mean_absolute_error: 266.4591 - 25ms/epoch - 4ms/step
## Epoch 588/600
## 6/6 - 0s - loss: 107355.9766 - mean_absolute_error: 251.5270 - val_loss: 120737.6406 - val_mean_absolute_error: 266.3634 - 25ms/epoch - 4ms/step
## Epoch 589/600
## 6/6 - 0s - loss: 112207.6797 - mean_absolute_error: 257.8389 - val_loss: 120636.5625 - val_mean_absolute_error: 266.2362 - 27ms/epoch - 4ms/step
## Epoch 590/600
## 6/6 - 0s - loss: 107000.4922 - mean_absolute_error: 249.7298 - val_loss: 120650.5547 - val_mean_absolute_error: 266.2586 - 25ms/epoch - 4ms/step
## Epoch 591/600
## 6/6 - 0s - loss: 120649.1562 - mean_absolute_error: 258.8677 - val_loss: 120630.1406 - val_mean_absolute_error: 266.2668 - 25ms/epoch - 4ms/step
## Epoch 592/600
## 6/6 - 0s - loss: 107273.9922 - mean_absolute_error: 245.1542 - val_loss: 120595.0078 - val_mean_absolute_error: 266.2388 - 24ms/epoch - 4ms/step
## Epoch 593/600
## 6/6 - 0s - loss: 125474.2422 - mean_absolute_error: 269.6733 - val_loss: 120575.5547 - val_mean_absolute_error: 266.2202 - 25ms/epoch - 4ms/step
## Epoch 594/600
## 6/6 - 0s - loss: 110512.5703 - mean_absolute_error: 258.0105 - val_loss: 120421.9922 - val_mean_absolute_error: 265.9999 - 25ms/epoch - 4ms/step
## Epoch 595/600
## 6/6 - 0s - loss: 115844.3281 - mean_absolute_error: 259.0305 - val_loss: 120314.4844 - val_mean_absolute_error: 265.8346 - 25ms/epoch - 4ms/step
## Epoch 596/600
## 6/6 - 0s - loss: 114330.1172 - mean_absolute_error: 258.9388 - val_loss: 120382.2031 - val_mean_absolute_error: 265.8875 - 24ms/epoch - 4ms/step
## Epoch 597/600
## 6/6 - 0s - loss: 126110.6562 - mean_absolute_error: 261.7068 - val_loss: 120407.9766 - val_mean_absolute_error: 265.8789 - 24ms/epoch - 4ms/step
## Epoch 598/600
## 6/6 - 0s - loss: 120084.8438 - mean_absolute_error: 265.3672 - val_loss: 120433.3125 - val_mean_absolute_error: 265.9001 - 24ms/epoch - 4ms/step
## Epoch 599/600
## 6/6 - 0s - loss: 106686.8203 - mean_absolute_error: 248.4153 - val_loss: 120431.4219 - val_mean_absolute_error: 265.9124 - 24ms/epoch - 4ms/step
## Epoch 600/600
## 6/6 - 0s - loss: 120953.7812 - mean_absolute_error: 265.0257 - val_loss: 120386.9219 - val_mean_absolute_error: 265.8430 - 24ms/epoch - 4ms/step

(Here and elsewhere we have reduced the number of epochs to make runtimes manageable; users can of course change back)

We can plot the history to display the mean absolute error for the training and test data. For the best aesthetics, install the ggplot2 package before calling the plot() function. If you have not installed ggplot2, then the code below will still run, but the plot will be less attractive.

plot(history)

It is worth noting that if you run the fit() command a second time in the same session, then the fitting process will pick up where it left off. Try re-running the fit() command, and then the plot() command, to see!

Finally, we predict from the final model, and evaluate its performance on the test data. Due to the use of SGD, the results vary slightly with each fit. Unfortunately the set.seed() function does not ensure identical results (since the fitting is done in python), so your results will differ slightly.

npred <- predict(modnn, x[testid, ])

## 3/3 - 0s - 53ms/epoch - 18ms/step

mean(abs(y[testid] - npred))

## [1] 265.843

A Multilayer Network on the MNIST Digit Data

The keras package comes with a number of example datasets, including the MNIST digit data. Our first step is to load the MNIST data. The dataset_mnist() function is provided for this purpose.

mnist <- dataset_mnist()
x_train <- mnist$train$x
g_train <- mnist$train$y
x_test <- mnist$test$x
g_test <- mnist$test$y
dim(x_train)

## [1] 60000    28    28

dim(x_test)

## [1] 10000    28    28

There are 60,000 images in the training data and 10,000 in the test data. The images are \(28\times 28\), and stored as a three-dimensional array, so we need to reshape them into a matrix. Also, we need to “one-hot” encode the class label. Luckily keras has a lot of built-in functions that do this for us.

x_train <- array_reshape(x_train, c(nrow(x_train), 784))
x_test <- array_reshape(x_test, c(nrow(x_test), 784))
y_train <- to_categorical(g_train, 10)
y_test <- to_categorical(g_test, 10)

Neural networks are somewhat sensitive to the scale of the inputs. For example, ridge and lasso regularization are affected by scaling. Here the inputs are eight-bit grayscale values between 0 and 255, so we rescale to the unit interval. (Eight bits means \(2^8\), which equals 256. Since the convention is to start at \(0\), the possible values range from \(0\) to \(255\).)

x_train <- x_train / 255
x_test <- x_test / 255

Now we are ready to fit our neural network.

modelnn <- keras_model_sequential()
modelnn %>%
  layer_dense(units = 256, activation = "relu",
       input_shape = c(784)) %>%
  layer_dropout(rate = 0.4) %>%
  layer_dense(units = 128, activation = "relu") %>%
  layer_dropout(rate = 0.3) %>%
  layer_dense(units = 10, activation = "softmax")

The first layer goes from \(28\times28=784\) input units to a hidden layer of \(256\) units, which uses the ReLU activation function. This is specified by a call to layer_dense(), which takes as input a modelnn object, and returns a modified modelnn object. This is then piped through layer_dropout() to perform dropout regularization. The second hidden layer comes next, with \(128\) hidden units, followed by a dropout layer. The final layer is the output layer, with activation "softmax" (10.13) for the 10-class classification problem, which defines the map from the second hidden layer to class probabilities. Finally, we use summary() to summarize the model, and to make sure we got it all right.

summary(modelnn)

## Model: "sequential_1"
## ________________________________________________________________________________
##  Layer (type)                       Output Shape                    Param #     
## ================================================================================
##  dense_4 (Dense)                    (None, 256)                     200960      
##  dropout_2 (Dropout)                (None, 256)                     0           
##  dense_3 (Dense)                    (None, 128)                     32896       
##  dropout_1 (Dropout)                (None, 128)                     0           
##  dense_2 (Dense)                    (None, 10)                      1290        
## ================================================================================
## Total params: 235146 (918.54 KB)
## Trainable params: 235146 (918.54 KB)
## Non-trainable params: 0 (0.00 Byte)
## ________________________________________________________________________________

The parameters for each layer include a bias term, which results in a parameter count of 235,146. For example, the first hidden layer involves \((784+1)\times256=200{,}960\) parameters.

Notice that the layer names such as dropout_1 and dense_2 have subscripts. These may appear somewhat random; in fact, if you fit the same model again, these will change. They are of no consequence: they vary because the model specification code is run in python, and these subscripts are incremented every time keras_model_sequential() is called.

Next, we add details to the model to specify the fitting algorithm. We fit the model by minimizing the cross-entropy function given by (10.14).

modelnn %>% compile(loss = "categorical_crossentropy",
    optimizer = optimizer_rmsprop(), metrics = c("accuracy")
  )

Now we are ready to go. The final step is to supply training data, and fit the model.

system.time(
  history <- modelnn %>%
#     fit(x_train, y_train, epochs = 30, batch_size = 128,
      fit(x_train, y_train, epochs = 15, batch_size = 128,
        validation_split = 0.2)
)

## Epoch 1/15
## 375/375 - 2s - loss: 0.4405 - accuracy: 0.8673 - val_loss: 0.1631 - val_accuracy: 0.9507 - 2s/epoch - 5ms/step
## Epoch 2/15
## 375/375 - 1s - loss: 0.2042 - accuracy: 0.9385 - val_loss: 0.1201 - val_accuracy: 0.9645 - 1s/epoch - 3ms/step
## Epoch 3/15
## 375/375 - 1s - loss: 0.1597 - accuracy: 0.9513 - val_loss: 0.1062 - val_accuracy: 0.9676 - 1s/epoch - 3ms/step
## Epoch 4/15
## 375/375 - 1s - loss: 0.1307 - accuracy: 0.9613 - val_loss: 0.0999 - val_accuracy: 0.9712 - 1s/epoch - 3ms/step
## Epoch 5/15
## 375/375 - 1s - loss: 0.1158 - accuracy: 0.9657 - val_loss: 0.0886 - val_accuracy: 0.9753 - 1s/epoch - 4ms/step
## Epoch 6/15
## 375/375 - 1s - loss: 0.1035 - accuracy: 0.9680 - val_loss: 0.0933 - val_accuracy: 0.9745 - 1s/epoch - 4ms/step
## Epoch 7/15
## 375/375 - 2s - loss: 0.0940 - accuracy: 0.9719 - val_loss: 0.0909 - val_accuracy: 0.9760 - 2s/epoch - 4ms/step
## Epoch 8/15
## 375/375 - 2s - loss: 0.0877 - accuracy: 0.9737 - val_loss: 0.0903 - val_accuracy: 0.9758 - 2s/epoch - 4ms/step
## Epoch 9/15
## 375/375 - 2s - loss: 0.0858 - accuracy: 0.9741 - val_loss: 0.0864 - val_accuracy: 0.9772 - 2s/epoch - 4ms/step
## Epoch 10/15
## 375/375 - 1s - loss: 0.0764 - accuracy: 0.9763 - val_loss: 0.0846 - val_accuracy: 0.9772 - 1s/epoch - 4ms/step
## Epoch 11/15
## 375/375 - 1s - loss: 0.0737 - accuracy: 0.9780 - val_loss: 0.0859 - val_accuracy: 0.9784 - 1s/epoch - 4ms/step
## Epoch 12/15
## 375/375 - 2s - loss: 0.0697 - accuracy: 0.9789 - val_loss: 0.0906 - val_accuracy: 0.9790 - 2s/epoch - 5ms/step
## Epoch 13/15
## 375/375 - 1s - loss: 0.0673 - accuracy: 0.9802 - val_loss: 0.0841 - val_accuracy: 0.9788 - 1s/epoch - 4ms/step
## Epoch 14/15
## 375/375 - 1s - loss: 0.0624 - accuracy: 0.9810 - val_loss: 0.0870 - val_accuracy: 0.9786 - 1s/epoch - 4ms/step
## Epoch 15/15
## 375/375 - 1s - loss: 0.0617 - accuracy: 0.9813 - val_loss: 0.0874 - val_accuracy: 0.9783 - 1s/epoch - 4ms/step

##    user  system elapsed 
##  49.815  20.194  22.050

plot(history, smooth = FALSE)

We have suppressed the output here, which is a progress report on the fitting of the model, grouped by epoch. This is very useful, since on large datasets fitting can take time. Fitting this model took 144 seconds on a 2.9 GHz MacBook Pro with 4 cores and 32 GB of RAM. Here we specified a validation split of 20%, so the training is actually performed on 80% of the 60,000 observations in the training set. This is an alternative to actually supplying validation data, like we did in Section 10.9.1. See ?fit.keras.engine.training.Model for all the optional fitting arguments. SGD uses batches of 128 observations in computing the gradient, and doing the arithmetic, we see that an epoch corresponds to 375 gradient steps. The last plot() command produces a figure similar to Figure 10.18.

To obtain the test error in Table 10.1, we first write a simple function accuracy() that compares predicted and true class labels, and then use it to evaluate our predictions.

accuracy <- function(pred, truth)
  mean(drop(as.numeric(pred)) == drop(truth))
modelnn %>% predict(x_test) %>% k_argmax() %>% accuracy(g_test)

## 313/313 - 0s - 322ms/epoch - 1ms/step

## [1] 0.981

The table also reports LDA (Chapter 4) and multiclass logistic regression. Although packages such as glmnet can handle multiclass logistic regression, they are quite slow on this large dataset. It is much faster and quite easy to fit such a model using the keras software. We just have an input layer and output layer, and omit the hidden layers!

modellr <- keras_model_sequential() %>%
  layer_dense(input_shape = 784, units = 10,
       activation = "softmax")
summary(modellr)

## Model: "sequential_2"
## ________________________________________________________________________________
##  Layer (type)                       Output Shape                    Param #     
## ================================================================================
##  dense_5 (Dense)                    (None, 10)                      7850        
## ================================================================================
## Total params: 7850 (30.66 KB)
## Trainable params: 7850 (30.66 KB)
## Non-trainable params: 0 (0.00 Byte)
## ________________________________________________________________________________

We fit the model just as before.

modellr %>% compile(loss = "categorical_crossentropy",
     optimizer = optimizer_rmsprop(), metrics = c("accuracy"))
modellr %>% fit(x_train, y_train, epochs = 30,
      batch_size = 128, validation_split = 0.2)

## Epoch 1/30
## 375/375 - 1s - loss: 0.6681 - accuracy: 0.8352 - val_loss: 0.3613 - val_accuracy: 0.9012 - 1s/epoch - 3ms/step
## Epoch 2/30
## 375/375 - 1s - loss: 0.3541 - accuracy: 0.9024 - val_loss: 0.3097 - val_accuracy: 0.9153 - 519ms/epoch - 1ms/step
## Epoch 3/30
## 375/375 - 1s - loss: 0.3184 - accuracy: 0.9113 - val_loss: 0.2936 - val_accuracy: 0.9173 - 515ms/epoch - 1ms/step
## Epoch 4/30
## 375/375 - 1s - loss: 0.3021 - accuracy: 0.9156 - val_loss: 0.2853 - val_accuracy: 0.9212 - 504ms/epoch - 1ms/step
## Epoch 5/30
## 375/375 - 1s - loss: 0.2927 - accuracy: 0.9180 - val_loss: 0.2787 - val_accuracy: 0.9235 - 508ms/epoch - 1ms/step
## Epoch 6/30
## 375/375 - 1s - loss: 0.2857 - accuracy: 0.9205 - val_loss: 0.2758 - val_accuracy: 0.9231 - 520ms/epoch - 1ms/step
## Epoch 7/30
## 375/375 - 1s - loss: 0.2811 - accuracy: 0.9218 - val_loss: 0.2730 - val_accuracy: 0.9246 - 544ms/epoch - 1ms/step
## Epoch 8/30
## 375/375 - 1s - loss: 0.2768 - accuracy: 0.9230 - val_loss: 0.2718 - val_accuracy: 0.9243 - 529ms/epoch - 1ms/step
## Epoch 9/30
## 375/375 - 1s - loss: 0.2739 - accuracy: 0.9235 - val_loss: 0.2700 - val_accuracy: 0.9258 - 523ms/epoch - 1ms/step
## Epoch 10/30
## 375/375 - 1s - loss: 0.2710 - accuracy: 0.9244 - val_loss: 0.2676 - val_accuracy: 0.9270 - 514ms/epoch - 1ms/step
## Epoch 11/30
## 375/375 - 1s - loss: 0.2685 - accuracy: 0.9255 - val_loss: 0.2663 - val_accuracy: 0.9277 - 524ms/epoch - 1ms/step
## Epoch 12/30
## 375/375 - 1s - loss: 0.2668 - accuracy: 0.9257 - val_loss: 0.2661 - val_accuracy: 0.9280 - 513ms/epoch - 1ms/step
## Epoch 13/30
## 375/375 - 1s - loss: 0.2649 - accuracy: 0.9273 - val_loss: 0.2644 - val_accuracy: 0.9277 - 515ms/epoch - 1ms/step
## Epoch 14/30
## 375/375 - 1s - loss: 0.2635 - accuracy: 0.9274 - val_loss: 0.2651 - val_accuracy: 0.9277 - 505ms/epoch - 1ms/step
## Epoch 15/30
## 375/375 - 1s - loss: 0.2617 - accuracy: 0.9273 - val_loss: 0.2636 - val_accuracy: 0.9293 - 554ms/epoch - 1ms/step
## Epoch 16/30
## 375/375 - 1s - loss: 0.2608 - accuracy: 0.9285 - val_loss: 0.2629 - val_accuracy: 0.9296 - 558ms/epoch - 1ms/step
## Epoch 17/30
## 375/375 - 1s - loss: 0.2592 - accuracy: 0.9286 - val_loss: 0.2634 - val_accuracy: 0.9293 - 511ms/epoch - 1ms/step
## Epoch 18/30
## 375/375 - 1s - loss: 0.2582 - accuracy: 0.9289 - val_loss: 0.2634 - val_accuracy: 0.9293 - 506ms/epoch - 1ms/step
## Epoch 19/30
## 375/375 - 1s - loss: 0.2571 - accuracy: 0.9290 - val_loss: 0.2629 - val_accuracy: 0.9307 - 508ms/epoch - 1ms/step
## Epoch 20/30
## 375/375 - 1s - loss: 0.2564 - accuracy: 0.9297 - val_loss: 0.2624 - val_accuracy: 0.9291 - 535ms/epoch - 1ms/step
## Epoch 21/30
## 375/375 - 1s - loss: 0.2554 - accuracy: 0.9298 - val_loss: 0.2621 - val_accuracy: 0.9306 - 518ms/epoch - 1ms/step
## Epoch 22/30
## 375/375 - 1s - loss: 0.2546 - accuracy: 0.9301 - val_loss: 0.2627 - val_accuracy: 0.9306 - 523ms/epoch - 1ms/step
## Epoch 23/30
## 375/375 - 1s - loss: 0.2538 - accuracy: 0.9302 - val_loss: 0.2624 - val_accuracy: 0.9307 - 526ms/epoch - 1ms/step
## Epoch 24/30
## 375/375 - 1s - loss: 0.2531 - accuracy: 0.9307 - val_loss: 0.2624 - val_accuracy: 0.9307 - 516ms/epoch - 1ms/step
## Epoch 25/30
## 375/375 - 1s - loss: 0.2525 - accuracy: 0.9307 - val_loss: 0.2617 - val_accuracy: 0.9312 - 523ms/epoch - 1ms/step
## Epoch 26/30
## 375/375 - 1s - loss: 0.2518 - accuracy: 0.9313 - val_loss: 0.2632 - val_accuracy: 0.9296 - 507ms/epoch - 1ms/step
## Epoch 27/30
## 375/375 - 1s - loss: 0.2514 - accuracy: 0.9314 - val_loss: 0.2629 - val_accuracy: 0.9316 - 504ms/epoch - 1ms/step
## Epoch 28/30
## 375/375 - 1s - loss: 0.2503 - accuracy: 0.9316 - val_loss: 0.2623 - val_accuracy: 0.9308 - 513ms/epoch - 1ms/step
## Epoch 29/30
## 375/375 - 0s - loss: 0.2500 - accuracy: 0.9320 - val_loss: 0.2627 - val_accuracy: 0.9298 - 493ms/epoch - 1ms/step
## Epoch 30/30
## 375/375 - 0s - loss: 0.2494 - accuracy: 0.9321 - val_loss: 0.2610 - val_accuracy: 0.9319 - 491ms/epoch - 1ms/step

modellr %>% predict(x_test) %>% k_argmax() %>% accuracy(g_test)

## 313/313 - 0s - 192ms/epoch - 612us/step

## [1] 0.9277

Convolutional Neural Networks

In this section we fit a CNN to the CIFAR data, which is available in the keras package. It is arranged in a similar fashion as the MNIST data.

cifar100 <- dataset_cifar100()
names(cifar100)

## [1] "train" "test"

x_train <- cifar100$train$x
g_train <- cifar100$train$y
x_test <- cifar100$test$x
g_test <- cifar100$test$y
dim(x_train)

## [1] 50000    32    32     3

range(x_train[1,,, 1])

## [1]  13 255

The array of 50,000 training images has four dimensions: each three-color image is represented as a set of three channels, each of which consists of \(32\times 32\) eight-bit pixels. We standardize as we did for the digits, but keep the array structure. We one-hot encode the response factors to produce a 100-column binary matrix.

x_train <- x_train / 255
x_test <- x_test / 255
y_train <- to_categorical(g_train, 100)
dim(y_train)

## [1] 50000   100

Before we start, we look at some of the training images using the jpeg package; similar code produced Figure 10.5 on page 411.

library(jpeg)
par(mar = c(0, 0, 0, 0), mfrow = c(5, 5))
index <- sample(seq(50000), 25)
for (i in index) plot(as.raster(x_train[i,,, ]))

The as.raster() function converts the feature map so that it can be plotted as a color image.

Here we specify a moderately-sized CNN for demonstration purposes, similar in structure to Figure 10.8.

model <- keras_model_sequential() %>%
  layer_conv_2d(filters = 32, kernel_size = c(3, 3),
      padding = "same", activation = "relu",
      input_shape = c(32, 32, 3)) %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_conv_2d(filters = 64, kernel_size = c(3, 3),
      padding = "same", activation = "relu") %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_conv_2d(filters = 128, kernel_size = c(3, 3),
      padding = "same", activation = "relu") %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_conv_2d(filters = 256, kernel_size = c(3, 3),
      padding = "same", activation = "relu") %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_flatten() %>%
  layer_dropout(rate = 0.5) %>%
  layer_dense(units = 512, activation = "relu") %>%
  layer_dense(units = 100, activation = "softmax")
summary(model)

## Model: "sequential_3"
## ________________________________________________________________________________
##  Layer (type)                       Output Shape                    Param #     
## ================================================================================
##  conv2d_3 (Conv2D)                  (None, 32, 32, 32)              896         
##  max_pooling2d_3 (MaxPooling2D)     (None, 16, 16, 32)              0           
##  conv2d_2 (Conv2D)                  (None, 16, 16, 64)              18496       
##  max_pooling2d_2 (MaxPooling2D)     (None, 8, 8, 64)                0           
##  conv2d_1 (Conv2D)                  (None, 8, 8, 128)               73856       
##  max_pooling2d_1 (MaxPooling2D)     (None, 4, 4, 128)               0           
##  conv2d (Conv2D)                    (None, 4, 4, 256)               295168      
##  max_pooling2d (MaxPooling2D)       (None, 2, 2, 256)               0           
##  flatten (Flatten)                  (None, 1024)                    0           
##  dropout_3 (Dropout)                (None, 1024)                    0           
##  dense_7 (Dense)                    (None, 512)                     524800      
##  dense_6 (Dense)                    (None, 100)                     51300       
## ================================================================================
## Total params: 964516 (3.68 MB)
## Trainable params: 964516 (3.68 MB)
## Non-trainable params: 0 (0.00 Byte)
## ________________________________________________________________________________

Notice that we used the padding = "same" argument to layer_conv_2D(), which ensures that the output channels have the same dimension as the input channels. There are 32 channels in the first hidden layer, in contrast to the three channels in the input layer. We use a \(3\times 3\) convolution filter for each channel in all the layers. Each convolution is followed by a max-pooling layer over \(2\times2\) blocks. By studying the summary, we can see that the channels halve in both dimensions after each of these max-pooling operations. After the last of these we have a layer with 256 channels of dimension \(2\times 2\). These are then flattened to a dense layer of size 1,024: in other words, each of the \(2\times 2\) matrices is turned into a \(4\)-vector, and put side-by-side in one layer. This is followed by a dropout regularization layer, then another dense layer of size 512, which finally reaches the softmax output layer.

Finally, we specify the fitting algorithm, and fit the model.

model %>% compile(loss = "categorical_crossentropy",
    optimizer = optimizer_rmsprop(), metrics = c("accuracy"))
#history <- model %>% fit(x_train, y_train, epochs = 30,
history <- model %>% fit(x_train, y_train, epochs = 10,
    batch_size = 128, validation_split = 0.2)

## Epoch 1/10
## 313/313 - 24s - loss: 4.2199 - accuracy: 0.0521 - val_loss: 3.9362 - val_accuracy: 0.0994 - 24s/epoch - 75ms/step
## Epoch 2/10
## 313/313 - 25s - loss: 3.6554 - accuracy: 0.1376 - val_loss: 3.4648 - val_accuracy: 0.1699 - 25s/epoch - 80ms/step
## Epoch 3/10
## 313/313 - 24s - loss: 3.3191 - accuracy: 0.1978 - val_loss: 3.1971 - val_accuracy: 0.2227 - 24s/epoch - 78ms/step
## Epoch 4/10
## 313/313 - 25s - loss: 3.0683 - accuracy: 0.2450 - val_loss: 3.1209 - val_accuracy: 0.2374 - 25s/epoch - 79ms/step
## Epoch 5/10
## 313/313 - 25s - loss: 2.8678 - accuracy: 0.2816 - val_loss: 2.8395 - val_accuracy: 0.2906 - 25s/epoch - 81ms/step
## Epoch 6/10
## 313/313 - 25s - loss: 2.6943 - accuracy: 0.3174 - val_loss: 2.7189 - val_accuracy: 0.3121 - 25s/epoch - 81ms/step
## Epoch 7/10
## 313/313 - 26s - loss: 2.5368 - accuracy: 0.3488 - val_loss: 2.6692 - val_accuracy: 0.3260 - 26s/epoch - 81ms/step
## Epoch 8/10
## 313/313 - 26s - loss: 2.3936 - accuracy: 0.3771 - val_loss: 2.5566 - val_accuracy: 0.3497 - 26s/epoch - 82ms/step
## Epoch 9/10
## 313/313 - 26s - loss: 2.2717 - accuracy: 0.4054 - val_loss: 2.3643 - val_accuracy: 0.3965 - 26s/epoch - 84ms/step
## Epoch 10/10
## 313/313 - 27s - loss: 2.1426 - accuracy: 0.4324 - val_loss: 2.3883 - val_accuracy: 0.3886 - 27s/epoch - 85ms/step

model %>% predict(x_test) %>% k_argmax() %>% accuracy(g_test)

## 313/313 - 5s - 5s/epoch - 17ms/step

## [1] 0.3938

This model takes 10 minutes to run and achieves 46% accuracy on the test data. Although this is not terrible for 100-class data (a random classifier gets 1% accuracy), searching the web we see results around 75%. Typically it takes a lot of architecture carpentry, fiddling with regularization, and time to achieve such results.

Using Pretrained CNN Models

We now show how to use a CNN pretrained on the imagenet database to classify natural images, and demonstrate how we produced Figure 10.10. We copied six jpeg images from a digital photo album into the directory book_images. (These images are available from the data section of <www.statlearning.com>, the ISL book website. Download book_images.zip; when clicked it creates the book_images directory.) We first read in the images, and convert them into the array format expected by the keras software to match the specifications in imagenet. Make sure that your working directory in R is set to the folder in which the images are stored.

img_dir <- "book_images"
image_names <- list.files(img_dir)
num_images <- length(image_names)
x <- array(dim = c(num_images, 224, 224, 3))
for (i in 1:num_images) {
  img_path <- paste(img_dir, image_names[i], sep = "/")
  img <- image_load(img_path, target_size = c(224, 224))
  x[i,,, ] <- image_to_array(img)
}
x <- imagenet_preprocess_input(x)

We then load the trained network. The model has 50 layers, with a fair bit of complexity.

model <- application_resnet50(weights = "imagenet")
summary(model)

## Model: "resnet50"
## ________________________________________________________________________________
##  Layer (type)       Output Shape         Para   Connected to         Trainable  
##                                          m #                                    
## ================================================================================
##  input_1 (InputLay  [(None, 224, 224,    0      []                   Y          
##  er)                3)]                                                         
##  conv1_pad (ZeroPa  (None, 230, 230, 3   0      ['input_1[0][0]']    Y          
##  dding2D)           )                                                           
##  conv1_conv (Conv2  (None, 112, 112, 6   9472   ['conv1_pad[0][0]'   Y          
##  D)                 4)                          ]                               
##  conv1_bn (BatchNo  (None, 112, 112, 6   256    ['conv1_conv[0][0]   Y          
##  rmalization)       4)                          ']                              
##  conv1_relu (Activ  (None, 112, 112, 6   0      ['conv1_bn[0][0]']   Y          
##  ation)             4)                                                          
##  pool1_pad (ZeroPa  (None, 114, 114, 6   0      ['conv1_relu[0][0]   Y          
##  dding2D)           4)                          ']                              
##  pool1_pool (MaxPo  (None, 56, 56, 64)   0      ['pool1_pad[0][0]'   Y          
##  oling2D)                                       ]                               
##  conv2_block1_1_co  (None, 56, 56, 64)   4160   ['pool1_pool[0][0]   Y          
##  nv (Conv2D)                                    ']                              
##  conv2_block1_1_bn  (None, 56, 56, 64)   256    ['conv2_block1_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block1_1_re  (None, 56, 56, 64)   0      ['conv2_block1_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block1_2_co  (None, 56, 56, 64)   3692   ['conv2_block1_1_r   Y          
##  nv (Conv2D)                             8      elu[0][0]']                     
##  conv2_block1_2_bn  (None, 56, 56, 64)   256    ['conv2_block1_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block1_2_re  (None, 56, 56, 64)   0      ['conv2_block1_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block1_0_co  (None, 56, 56, 256   1664   ['pool1_pool[0][0]   Y          
##  nv (Conv2D)        )                    0      ']                              
##  conv2_block1_3_co  (None, 56, 56, 256   1664   ['conv2_block1_2_r   Y          
##  nv (Conv2D)        )                    0      elu[0][0]']                     
##  conv2_block1_0_bn  (None, 56, 56, 256   1024   ['conv2_block1_0_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv2_block1_3_bn  (None, 56, 56, 256   1024   ['conv2_block1_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv2_block1_add   (None, 56, 56, 256   0      ['conv2_block1_0_b   Y          
##  (Add)              )                           n[0][0]',                       
##                                                  'conv2_block1_3_b              
##                                                 n[0][0]']                       
##  conv2_block1_out   (None, 56, 56, 256   0      ['conv2_block1_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv2_block2_1_co  (None, 56, 56, 64)   1644   ['conv2_block1_out   Y          
##  nv (Conv2D)                             8      [0][0]']                        
##  conv2_block2_1_bn  (None, 56, 56, 64)   256    ['conv2_block2_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block2_1_re  (None, 56, 56, 64)   0      ['conv2_block2_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block2_2_co  (None, 56, 56, 64)   3692   ['conv2_block2_1_r   Y          
##  nv (Conv2D)                             8      elu[0][0]']                     
##  conv2_block2_2_bn  (None, 56, 56, 64)   256    ['conv2_block2_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block2_2_re  (None, 56, 56, 64)   0      ['conv2_block2_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block2_3_co  (None, 56, 56, 256   1664   ['conv2_block2_2_r   Y          
##  nv (Conv2D)        )                    0      elu[0][0]']                     
##  conv2_block2_3_bn  (None, 56, 56, 256   1024   ['conv2_block2_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv2_block2_add   (None, 56, 56, 256   0      ['conv2_block1_out   Y          
##  (Add)              )                           [0][0]',                        
##                                                  'conv2_block2_3_b              
##                                                 n[0][0]']                       
##  conv2_block2_out   (None, 56, 56, 256   0      ['conv2_block2_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv2_block3_1_co  (None, 56, 56, 64)   1644   ['conv2_block2_out   Y          
##  nv (Conv2D)                             8      [0][0]']                        
##  conv2_block3_1_bn  (None, 56, 56, 64)   256    ['conv2_block3_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block3_1_re  (None, 56, 56, 64)   0      ['conv2_block3_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block3_2_co  (None, 56, 56, 64)   3692   ['conv2_block3_1_r   Y          
##  nv (Conv2D)                             8      elu[0][0]']                     
##  conv2_block3_2_bn  (None, 56, 56, 64)   256    ['conv2_block3_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv2_block3_2_re  (None, 56, 56, 64)   0      ['conv2_block3_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv2_block3_3_co  (None, 56, 56, 256   1664   ['conv2_block3_2_r   Y          
##  nv (Conv2D)        )                    0      elu[0][0]']                     
##  conv2_block3_3_bn  (None, 56, 56, 256   1024   ['conv2_block3_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv2_block3_add   (None, 56, 56, 256   0      ['conv2_block2_out   Y          
##  (Add)              )                           [0][0]',                        
##                                                  'conv2_block3_3_b              
##                                                 n[0][0]']                       
##  conv2_block3_out   (None, 56, 56, 256   0      ['conv2_block3_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv3_block1_1_co  (None, 28, 28, 128   3289   ['conv2_block3_out   Y          
##  nv (Conv2D)        )                    6      [0][0]']                        
##  conv3_block1_1_bn  (None, 28, 28, 128   512    ['conv3_block1_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block1_1_re  (None, 28, 28, 128   0      ['conv3_block1_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block1_2_co  (None, 28, 28, 128   1475   ['conv3_block1_1_r   Y          
##  nv (Conv2D)        )                    84     elu[0][0]']                     
##  conv3_block1_2_bn  (None, 28, 28, 128   512    ['conv3_block1_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block1_2_re  (None, 28, 28, 128   0      ['conv3_block1_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block1_0_co  (None, 28, 28, 512   1315   ['conv2_block3_out   Y          
##  nv (Conv2D)        )                    84     [0][0]']                        
##  conv3_block1_3_co  (None, 28, 28, 512   6604   ['conv3_block1_2_r   Y          
##  nv (Conv2D)        )                    8      elu[0][0]']                     
##  conv3_block1_0_bn  (None, 28, 28, 512   2048   ['conv3_block1_0_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block1_3_bn  (None, 28, 28, 512   2048   ['conv3_block1_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block1_add   (None, 28, 28, 512   0      ['conv3_block1_0_b   Y          
##  (Add)              )                           n[0][0]',                       
##                                                  'conv3_block1_3_b              
##                                                 n[0][0]']                       
##  conv3_block1_out   (None, 28, 28, 512   0      ['conv3_block1_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv3_block2_1_co  (None, 28, 28, 128   6566   ['conv3_block1_out   Y          
##  nv (Conv2D)        )                    4      [0][0]']                        
##  conv3_block2_1_bn  (None, 28, 28, 128   512    ['conv3_block2_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block2_1_re  (None, 28, 28, 128   0      ['conv3_block2_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block2_2_co  (None, 28, 28, 128   1475   ['conv3_block2_1_r   Y          
##  nv (Conv2D)        )                    84     elu[0][0]']                     
##  conv3_block2_2_bn  (None, 28, 28, 128   512    ['conv3_block2_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block2_2_re  (None, 28, 28, 128   0      ['conv3_block2_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block2_3_co  (None, 28, 28, 512   6604   ['conv3_block2_2_r   Y          
##  nv (Conv2D)        )                    8      elu[0][0]']                     
##  conv3_block2_3_bn  (None, 28, 28, 512   2048   ['conv3_block2_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block2_add   (None, 28, 28, 512   0      ['conv3_block1_out   Y          
##  (Add)              )                           [0][0]',                        
##                                                  'conv3_block2_3_b              
##                                                 n[0][0]']                       
##  conv3_block2_out   (None, 28, 28, 512   0      ['conv3_block2_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv3_block3_1_co  (None, 28, 28, 128   6566   ['conv3_block2_out   Y          
##  nv (Conv2D)        )                    4      [0][0]']                        
##  conv3_block3_1_bn  (None, 28, 28, 128   512    ['conv3_block3_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block3_1_re  (None, 28, 28, 128   0      ['conv3_block3_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block3_2_co  (None, 28, 28, 128   1475   ['conv3_block3_1_r   Y          
##  nv (Conv2D)        )                    84     elu[0][0]']                     
##  conv3_block3_2_bn  (None, 28, 28, 128   512    ['conv3_block3_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block3_2_re  (None, 28, 28, 128   0      ['conv3_block3_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block3_3_co  (None, 28, 28, 512   6604   ['conv3_block3_2_r   Y          
##  nv (Conv2D)        )                    8      elu[0][0]']                     
##  conv3_block3_3_bn  (None, 28, 28, 512   2048   ['conv3_block3_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block3_add   (None, 28, 28, 512   0      ['conv3_block2_out   Y          
##  (Add)              )                           [0][0]',                        
##                                                  'conv3_block3_3_b              
##                                                 n[0][0]']                       
##  conv3_block3_out   (None, 28, 28, 512   0      ['conv3_block3_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv3_block4_1_co  (None, 28, 28, 128   6566   ['conv3_block3_out   Y          
##  nv (Conv2D)        )                    4      [0][0]']                        
##  conv3_block4_1_bn  (None, 28, 28, 128   512    ['conv3_block4_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block4_1_re  (None, 28, 28, 128   0      ['conv3_block4_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block4_2_co  (None, 28, 28, 128   1475   ['conv3_block4_1_r   Y          
##  nv (Conv2D)        )                    84     elu[0][0]']                     
##  conv3_block4_2_bn  (None, 28, 28, 128   512    ['conv3_block4_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block4_2_re  (None, 28, 28, 128   0      ['conv3_block4_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv3_block4_3_co  (None, 28, 28, 512   6604   ['conv3_block4_2_r   Y          
##  nv (Conv2D)        )                    8      elu[0][0]']                     
##  conv3_block4_3_bn  (None, 28, 28, 512   2048   ['conv3_block4_3_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv3_block4_add   (None, 28, 28, 512   0      ['conv3_block3_out   Y          
##  (Add)              )                           [0][0]',                        
##                                                  'conv3_block4_3_b              
##                                                 n[0][0]']                       
##  conv3_block4_out   (None, 28, 28, 512   0      ['conv3_block4_add   Y          
##  (Activation)       )                           [0][0]']                        
##  conv4_block1_1_co  (None, 14, 14, 256   1313   ['conv3_block4_out   Y          
##  nv (Conv2D)        )                    28     [0][0]']                        
##  conv4_block1_1_bn  (None, 14, 14, 256   1024   ['conv4_block1_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block1_1_re  (None, 14, 14, 256   0      ['conv4_block1_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block1_2_co  (None, 14, 14, 256   5900   ['conv4_block1_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block1_2_bn  (None, 14, 14, 256   1024   ['conv4_block1_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block1_2_re  (None, 14, 14, 256   0      ['conv4_block1_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block1_0_co  (None, 14, 14, 102   5253   ['conv3_block4_out   Y          
##  nv (Conv2D)        4)                   12     [0][0]']                        
##  conv4_block1_3_co  (None, 14, 14, 102   2631   ['conv4_block1_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block1_0_bn  (None, 14, 14, 102   4096   ['conv4_block1_0_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block1_3_bn  (None, 14, 14, 102   4096   ['conv4_block1_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block1_add   (None, 14, 14, 102   0      ['conv4_block1_0_b   Y          
##  (Add)              4)                          n[0][0]',                       
##                                                  'conv4_block1_3_b              
##                                                 n[0][0]']                       
##  conv4_block1_out   (None, 14, 14, 102   0      ['conv4_block1_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv4_block2_1_co  (None, 14, 14, 256   2624   ['conv4_block1_out   Y          
##  nv (Conv2D)        )                    00     [0][0]']                        
##  conv4_block2_1_bn  (None, 14, 14, 256   1024   ['conv4_block2_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block2_1_re  (None, 14, 14, 256   0      ['conv4_block2_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block2_2_co  (None, 14, 14, 256   5900   ['conv4_block2_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block2_2_bn  (None, 14, 14, 256   1024   ['conv4_block2_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block2_2_re  (None, 14, 14, 256   0      ['conv4_block2_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block2_3_co  (None, 14, 14, 102   2631   ['conv4_block2_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block2_3_bn  (None, 14, 14, 102   4096   ['conv4_block2_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block2_add   (None, 14, 14, 102   0      ['conv4_block1_out   Y          
##  (Add)              4)                          [0][0]',                        
##                                                  'conv4_block2_3_b              
##                                                 n[0][0]']                       
##  conv4_block2_out   (None, 14, 14, 102   0      ['conv4_block2_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv4_block3_1_co  (None, 14, 14, 256   2624   ['conv4_block2_out   Y          
##  nv (Conv2D)        )                    00     [0][0]']                        
##  conv4_block3_1_bn  (None, 14, 14, 256   1024   ['conv4_block3_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block3_1_re  (None, 14, 14, 256   0      ['conv4_block3_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block3_2_co  (None, 14, 14, 256   5900   ['conv4_block3_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block3_2_bn  (None, 14, 14, 256   1024   ['conv4_block3_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block3_2_re  (None, 14, 14, 256   0      ['conv4_block3_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block3_3_co  (None, 14, 14, 102   2631   ['conv4_block3_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block3_3_bn  (None, 14, 14, 102   4096   ['conv4_block3_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block3_add   (None, 14, 14, 102   0      ['conv4_block2_out   Y          
##  (Add)              4)                          [0][0]',                        
##                                                  'conv4_block3_3_b              
##                                                 n[0][0]']                       
##  conv4_block3_out   (None, 14, 14, 102   0      ['conv4_block3_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv4_block4_1_co  (None, 14, 14, 256   2624   ['conv4_block3_out   Y          
##  nv (Conv2D)        )                    00     [0][0]']                        
##  conv4_block4_1_bn  (None, 14, 14, 256   1024   ['conv4_block4_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block4_1_re  (None, 14, 14, 256   0      ['conv4_block4_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block4_2_co  (None, 14, 14, 256   5900   ['conv4_block4_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block4_2_bn  (None, 14, 14, 256   1024   ['conv4_block4_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block4_2_re  (None, 14, 14, 256   0      ['conv4_block4_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block4_3_co  (None, 14, 14, 102   2631   ['conv4_block4_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block4_3_bn  (None, 14, 14, 102   4096   ['conv4_block4_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block4_add   (None, 14, 14, 102   0      ['conv4_block3_out   Y          
##  (Add)              4)                          [0][0]',                        
##                                                  'conv4_block4_3_b              
##                                                 n[0][0]']                       
##  conv4_block4_out   (None, 14, 14, 102   0      ['conv4_block4_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv4_block5_1_co  (None, 14, 14, 256   2624   ['conv4_block4_out   Y          
##  nv (Conv2D)        )                    00     [0][0]']                        
##  conv4_block5_1_bn  (None, 14, 14, 256   1024   ['conv4_block5_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block5_1_re  (None, 14, 14, 256   0      ['conv4_block5_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block5_2_co  (None, 14, 14, 256   5900   ['conv4_block5_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block5_2_bn  (None, 14, 14, 256   1024   ['conv4_block5_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block5_2_re  (None, 14, 14, 256   0      ['conv4_block5_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block5_3_co  (None, 14, 14, 102   2631   ['conv4_block5_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block5_3_bn  (None, 14, 14, 102   4096   ['conv4_block5_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block5_add   (None, 14, 14, 102   0      ['conv4_block4_out   Y          
##  (Add)              4)                          [0][0]',                        
##                                                  'conv4_block5_3_b              
##                                                 n[0][0]']                       
##  conv4_block5_out   (None, 14, 14, 102   0      ['conv4_block5_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv4_block6_1_co  (None, 14, 14, 256   2624   ['conv4_block5_out   Y          
##  nv (Conv2D)        )                    00     [0][0]']                        
##  conv4_block6_1_bn  (None, 14, 14, 256   1024   ['conv4_block6_1_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block6_1_re  (None, 14, 14, 256   0      ['conv4_block6_1_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block6_2_co  (None, 14, 14, 256   5900   ['conv4_block6_1_r   Y          
##  nv (Conv2D)        )                    80     elu[0][0]']                     
##  conv4_block6_2_bn  (None, 14, 14, 256   1024   ['conv4_block6_2_c   Y          
##   (BatchNormalizat  )                           onv[0][0]']                     
##  ion)                                                                           
##  conv4_block6_2_re  (None, 14, 14, 256   0      ['conv4_block6_2_b   Y          
##  lu (Activation)    )                           n[0][0]']                       
##  conv4_block6_3_co  (None, 14, 14, 102   2631   ['conv4_block6_2_r   Y          
##  nv (Conv2D)        4)                   68     elu[0][0]']                     
##  conv4_block6_3_bn  (None, 14, 14, 102   4096   ['conv4_block6_3_c   Y          
##   (BatchNormalizat  4)                          onv[0][0]']                     
##  ion)                                                                           
##  conv4_block6_add   (None, 14, 14, 102   0      ['conv4_block5_out   Y          
##  (Add)              4)                          [0][0]',                        
##                                                  'conv4_block6_3_b              
##                                                 n[0][0]']                       
##  conv4_block6_out   (None, 14, 14, 102   0      ['conv4_block6_add   Y          
##  (Activation)       4)                          [0][0]']                        
##  conv5_block1_1_co  (None, 7, 7, 512)    5248   ['conv4_block6_out   Y          
##  nv (Conv2D)                             00     [0][0]']                        
##  conv5_block1_1_bn  (None, 7, 7, 512)    2048   ['conv5_block1_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block1_1_re  (None, 7, 7, 512)    0      ['conv5_block1_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block1_2_co  (None, 7, 7, 512)    2359   ['conv5_block1_1_r   Y          
##  nv (Conv2D)                             808    elu[0][0]']                     
##  conv5_block1_2_bn  (None, 7, 7, 512)    2048   ['conv5_block1_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block1_2_re  (None, 7, 7, 512)    0      ['conv5_block1_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block1_0_co  (None, 7, 7, 2048)   2099   ['conv4_block6_out   Y          
##  nv (Conv2D)                             200    [0][0]']                        
##  conv5_block1_3_co  (None, 7, 7, 2048)   1050   ['conv5_block1_2_r   Y          
##  nv (Conv2D)                             624    elu[0][0]']                     
##  conv5_block1_0_bn  (None, 7, 7, 2048)   8192   ['conv5_block1_0_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block1_3_bn  (None, 7, 7, 2048)   8192   ['conv5_block1_3_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block1_add   (None, 7, 7, 2048)   0      ['conv5_block1_0_b   Y          
##  (Add)                                          n[0][0]',                       
##                                                  'conv5_block1_3_b              
##                                                 n[0][0]']                       
##  conv5_block1_out   (None, 7, 7, 2048)   0      ['conv5_block1_add   Y          
##  (Activation)                                   [0][0]']                        
##  conv5_block2_1_co  (None, 7, 7, 512)    1049   ['conv5_block1_out   Y          
##  nv (Conv2D)                             088    [0][0]']                        
##  conv5_block2_1_bn  (None, 7, 7, 512)    2048   ['conv5_block2_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block2_1_re  (None, 7, 7, 512)    0      ['conv5_block2_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block2_2_co  (None, 7, 7, 512)    2359   ['conv5_block2_1_r   Y          
##  nv (Conv2D)                             808    elu[0][0]']                     
##  conv5_block2_2_bn  (None, 7, 7, 512)    2048   ['conv5_block2_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block2_2_re  (None, 7, 7, 512)    0      ['conv5_block2_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block2_3_co  (None, 7, 7, 2048)   1050   ['conv5_block2_2_r   Y          
##  nv (Conv2D)                             624    elu[0][0]']                     
##  conv5_block2_3_bn  (None, 7, 7, 2048)   8192   ['conv5_block2_3_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block2_add   (None, 7, 7, 2048)   0      ['conv5_block1_out   Y          
##  (Add)                                          [0][0]',                        
##                                                  'conv5_block2_3_b              
##                                                 n[0][0]']                       
##  conv5_block2_out   (None, 7, 7, 2048)   0      ['conv5_block2_add   Y          
##  (Activation)                                   [0][0]']                        
##  conv5_block3_1_co  (None, 7, 7, 512)    1049   ['conv5_block2_out   Y          
##  nv (Conv2D)                             088    [0][0]']                        
##  conv5_block3_1_bn  (None, 7, 7, 512)    2048   ['conv5_block3_1_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block3_1_re  (None, 7, 7, 512)    0      ['conv5_block3_1_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block3_2_co  (None, 7, 7, 512)    2359   ['conv5_block3_1_r   Y          
##  nv (Conv2D)                             808    elu[0][0]']                     
##  conv5_block3_2_bn  (None, 7, 7, 512)    2048   ['conv5_block3_2_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block3_2_re  (None, 7, 7, 512)    0      ['conv5_block3_2_b   Y          
##  lu (Activation)                                n[0][0]']                       
##  conv5_block3_3_co  (None, 7, 7, 2048)   1050   ['conv5_block3_2_r   Y          
##  nv (Conv2D)                             624    elu[0][0]']                     
##  conv5_block3_3_bn  (None, 7, 7, 2048)   8192   ['conv5_block3_3_c   Y          
##   (BatchNormalizat                              onv[0][0]']                     
##  ion)                                                                           
##  conv5_block3_add   (None, 7, 7, 2048)   0      ['conv5_block2_out   Y          
##  (Add)                                          [0][0]',                        
##                                                  'conv5_block3_3_b              
##                                                 n[0][0]']                       
##  conv5_block3_out   (None, 7, 7, 2048)   0      ['conv5_block3_add   Y          
##  (Activation)                                   [0][0]']                        
##  avg_pool (GlobalA  (None, 2048)         0      ['conv5_block3_out   Y          
##  veragePooling2D)                               [0][0]']                        
##  predictions (Dens  (None, 1000)         2049   ['avg_pool[0][0]']   Y          
##  e)                                      000                                    
## ================================================================================
## Total params: 25636712 (97.80 MB)
## Trainable params: 25583592 (97.59 MB)
## Non-trainable params: 53120 (207.50 KB)
## ________________________________________________________________________________

Finally, we classify our six images, and return the top three class choices in terms of predicted probability for each.

pred6 <- model %>% predict(x) %>%
  imagenet_decode_predictions(top = 3)

## 1/1 - 1s - 1s/epoch - 1s/step

names(pred6) <- image_names
print(pred6)

## $flamingo.jpg
##   class_name class_description       score
## 1  n02007558          flamingo 0.926349699
## 2  n02006656         spoonbill 0.071699418
## 3  n02002556       white_stork 0.001228212
## 
## $hawk_cropped.jpeg
##   class_name class_description      score
## 1  n01608432              kite 0.72270888
## 2  n01622779    great_grey_owl 0.08182584
## 3  n01532829       house_finch 0.04218876
## 
## $hawk.jpg
##   class_name class_description     score
## 1  n03388043          fountain 0.2788654
## 2  n03532672              hook 0.1785542
## 3  n03804744              nail 0.1080729
## 
## $huey.jpg
##   class_name           class_description      score
## 1  n02097474             Tibetan_terrier 0.50929689
## 2  n02098413                       Lhasa 0.42209876
## 3  n02098105 soft-coated_wheaten_terrier 0.01695859
## 
## $kitty.jpg
##   class_name    class_description      score
## 1  n02105641 Old_English_sheepdog 0.83265996
## 2  n02086240             Shih-Tzu 0.04513887
## 3  n03223299              doormat 0.03299766
## 
## $weaver.jpg
##   class_name class_description      score
## 1  n01843065           jacamar 0.49795479
## 2  n01818515             macaw 0.22193292
## 3  n02494079   squirrel_monkey 0.04287853

IMDb Document Classification

Now we perform document classification (Section 10.4) on the IMDB dataset, which is available as part of the keras package. We limit the dictionary size to the 10,000 most frequently-used words and tokens.

max_features <- 10000
imdb <- dataset_imdb(num_words = max_features)
c(c(x_train, y_train), c(x_test, y_test)) %<-% imdb

The third line is a shortcut for unpacking the list of lists. Each element of x_train is a vector of numbers between 0 and 9999 (the document), referring to the words found in the dictionary. For example, the first training document is the positive review on page 419. The indices of the first 12 words are given below.

x_train[[1]][1:12]

##  [1]    1   14   22   16   43  530  973 1622 1385   65  458 4468

To see the words, we create a function, decode_review(), that provides a simple interface to the dictionary.

word_index <- dataset_imdb_word_index()
decode_review <- function(text, word_index) {
  word <- names(word_index)
  idx <- unlist(word_index, use.names = FALSE)
  word <- c("<PAD>", "<START>", "<UNK>", "<UNUSED>", word)
  idx <- c(0:3, idx + 3)
  words <- word[match(text, idx, 2)]
  paste(words, collapse = " ")
}
decode_review(x_train[[1]][1:12], word_index)

## [1] "<START> this film was just brilliant casting location scenery story direction everyone's"

Next we write a function to “one-hot” encode each document in a list of documents, and return a binary matrix in sparse-matrix format.

library(Matrix)
one_hot <- function(sequences, dimension) {
  seqlen <- sapply(sequences, length)
  n <- length(seqlen)
  rowind <- rep(1:n, seqlen)
  colind <- unlist(sequences)
  sparseMatrix(i = rowind, j = colind,
      dims = c(n, dimension))
}

To construct the sparse matrix, one supplies just the entries that are nonzero. In the last line we call the function sparseMatrix() and supply the row indices corresponding to each document and the column indices corresponding to the words in each document, since we omit the values they are taken to be all ones. Words that appear more than once in any given document still get recorded as a one.

x_train_1h <- one_hot(x_train, 10000)
x_test_1h <- one_hot(x_test, 10000)
dim(x_train_1h)

## [1] 25000 10000

nnzero(x_train_1h) / (25000 * 10000)

## [1] 0.01316987

Only 1.3% of the entries are nonzero, so this amounts to considerable savings in memory. We create a validation set of size 2,000, leaving 23,000 for training.

set.seed(3)
ival <- sample(seq(along = y_train), 2000)

First we fit a lasso logistic regression model using glmnet() on the training data, and evaluate its performance on the validation data. Finally, we plot the accuracy, acclmv, as a function of the shrinkage parameter, \(\lambda\). Similar expressions compute the performance on the test data, and were used to produce the left plot in Figure 10.11. The code takes advantage of the sparse-matrix format of x_train_1h, and runs in about 5 seconds; in the usual dense format it would take about 5 minutes.

library(glmnet)
fitlm <- glmnet(x_train_1h[-ival, ], y_train[-ival],
    family = "binomial", standardize = FALSE)
classlmv <- predict(fitlm, x_train_1h[ival, ]) > 0
acclmv <- apply(classlmv, 2, accuracy,  y_train[ival] > 0)

We applied the accuracy() function that we wrote in Lab 10.9.2 to every column of the prediction matrix classlmv, and since this is a logical matrix of TRUE/FALSE values, we supply the second argument truth as a logical vector as well.

Before making a plot, we adjust the plotting window.

par(mar = c(4, 4, 4, 4), mfrow = c(1, 1))
plot(-log(fitlm$lambda), acclmv)

Next we fit a fully-connected neural network with two hidden layers, each with 16 units and ReLU activation.

model <- keras_model_sequential() %>%
  layer_dense(units = 16, activation = "relu",
      input_shape = c(10000)) %>%
  layer_dense(units = 16, activation = "relu") %>%
  layer_dense(units = 1, activation = "sigmoid")
model %>% compile(optimizer = "rmsprop",
    loss = "binary_crossentropy", metrics = c("accuracy"))
history <- model %>% fit(x_train_1h[-ival, ], y_train[-ival],
    epochs = 20, batch_size = 512,
    validation_data = list(x_train_1h[ival, ], y_train[ival]))

## Epoch 1/20
## 45/45 - 1s - loss: 0.4862 - accuracy: 0.7980 - val_loss: 0.3643 - val_accuracy: 0.8680 - 1s/epoch - 25ms/step
## Epoch 2/20
## 45/45 - 1s - loss: 0.2897 - accuracy: 0.9015 - val_loss: 0.2989 - val_accuracy: 0.8840 - 522ms/epoch - 12ms/step
## Epoch 3/20
## 45/45 - 1s - loss: 0.2218 - accuracy: 0.9220 - val_loss: 0.2802 - val_accuracy: 0.8910 - 526ms/epoch - 12ms/step
## Epoch 4/20
## 45/45 - 1s - loss: 0.1850 - accuracy: 0.9360 - val_loss: 0.2786 - val_accuracy: 0.8900 - 531ms/epoch - 12ms/step
## Epoch 5/20
## 45/45 - 1s - loss: 0.1624 - accuracy: 0.9434 - val_loss: 0.2814 - val_accuracy: 0.8900 - 534ms/epoch - 12ms/step
## Epoch 6/20
## 45/45 - 0s - loss: 0.1414 - accuracy: 0.9518 - val_loss: 0.2938 - val_accuracy: 0.8885 - 490ms/epoch - 11ms/step
## Epoch 7/20
## 45/45 - 1s - loss: 0.1265 - accuracy: 0.9571 - val_loss: 0.3108 - val_accuracy: 0.8850 - 553ms/epoch - 12ms/step
## Epoch 8/20
## 45/45 - 1s - loss: 0.1130 - accuracy: 0.9627 - val_loss: 0.3463 - val_accuracy: 0.8725 - 512ms/epoch - 11ms/step
## Epoch 9/20
## 45/45 - 1s - loss: 0.1006 - accuracy: 0.9667 - val_loss: 0.3569 - val_accuracy: 0.8690 - 766ms/epoch - 17ms/step
## Epoch 10/20
## 45/45 - 1s - loss: 0.0876 - accuracy: 0.9727 - val_loss: 0.3534 - val_accuracy: 0.8730 - 572ms/epoch - 13ms/step
## Epoch 11/20
## 45/45 - 1s - loss: 0.0795 - accuracy: 0.9750 - val_loss: 0.3818 - val_accuracy: 0.8715 - 542ms/epoch - 12ms/step
## Epoch 12/20
## 45/45 - 1s - loss: 0.0669 - accuracy: 0.9811 - val_loss: 0.3982 - val_accuracy: 0.8710 - 540ms/epoch - 12ms/step
## Epoch 13/20
## 45/45 - 1s - loss: 0.0617 - accuracy: 0.9822 - val_loss: 0.4072 - val_accuracy: 0.8700 - 528ms/epoch - 12ms/step
## Epoch 14/20
## 45/45 - 1s - loss: 0.0536 - accuracy: 0.9847 - val_loss: 0.4259 - val_accuracy: 0.8705 - 519ms/epoch - 12ms/step
## Epoch 15/20
## 45/45 - 1s - loss: 0.0446 - accuracy: 0.9893 - val_loss: 0.4526 - val_accuracy: 0.8655 - 528ms/epoch - 12ms/step
## Epoch 16/20
## 45/45 - 1s - loss: 0.0407 - accuracy: 0.9897 - val_loss: 0.4717 - val_accuracy: 0.8640 - 526ms/epoch - 12ms/step
## Epoch 17/20
## 45/45 - 1s - loss: 0.0342 - accuracy: 0.9917 - val_loss: 0.5014 - val_accuracy: 0.8665 - 515ms/epoch - 11ms/step
## Epoch 18/20
## 45/45 - 1s - loss: 0.0279 - accuracy: 0.9940 - val_loss: 0.5499 - val_accuracy: 0.8605 - 539ms/epoch - 12ms/step
## Epoch 19/20
## 45/45 - 1s - loss: 0.0243 - accuracy: 0.9953 - val_loss: 0.5458 - val_accuracy: 0.8650 - 535ms/epoch - 12ms/step
## Epoch 20/20
## 45/45 - 1s - loss: 0.0225 - accuracy: 0.9951 - val_loss: 0.5760 - val_accuracy: 0.8625 - 521ms/epoch - 12ms/step

The history object has a metrics component that records both the training and validation accuracy at each epoch. Figure 10.11 includes test accuracy at each epoch as well. To compute the test accuracy, we rerun the entire sequence above, replacing the last line with

history <- model %>% fit(
    x_train_1h[-ival, ], y_train[-ival], epochs = 20,
    batch_size = 512, validation_data = list(x_test_1h, y_test)
  )

## Epoch 1/20
## 45/45 - 1s - loss: 0.0178 - accuracy: 0.9961 - val_loss: 0.6489 - val_accuracy: 0.8568 - 979ms/epoch - 22ms/step
## Epoch 2/20
## 45/45 - 1s - loss: 0.0139 - accuracy: 0.9976 - val_loss: 0.6815 - val_accuracy: 0.8555 - 828ms/epoch - 18ms/step
## Epoch 3/20
## 45/45 - 1s - loss: 0.0116 - accuracy: 0.9978 - val_loss: 0.7016 - val_accuracy: 0.8554 - 852ms/epoch - 19ms/step
## Epoch 4/20
## 45/45 - 1s - loss: 0.0114 - accuracy: 0.9975 - val_loss: 0.7383 - val_accuracy: 0.8536 - 843ms/epoch - 19ms/step
## Epoch 5/20
## 45/45 - 1s - loss: 0.0084 - accuracy: 0.9990 - val_loss: 0.7505 - val_accuracy: 0.8556 - 854ms/epoch - 19ms/step
## Epoch 6/20
## 45/45 - 1s - loss: 0.0106 - accuracy: 0.9973 - val_loss: 0.7837 - val_accuracy: 0.8542 - 835ms/epoch - 19ms/step
## Epoch 7/20
## 45/45 - 1s - loss: 0.0077 - accuracy: 0.9980 - val_loss: 0.8139 - val_accuracy: 0.8543 - 827ms/epoch - 18ms/step
## Epoch 8/20
## 45/45 - 1s - loss: 0.0032 - accuracy: 0.9999 - val_loss: 0.8426 - val_accuracy: 0.8538 - 842ms/epoch - 19ms/step
## Epoch 9/20
## 45/45 - 1s - loss: 0.0065 - accuracy: 0.9984 - val_loss: 0.8605 - val_accuracy: 0.8537 - 802ms/epoch - 18ms/step
## Epoch 10/20
## 45/45 - 1s - loss: 0.0064 - accuracy: 0.9984 - val_loss: 0.8852 - val_accuracy: 0.8528 - 818ms/epoch - 18ms/step
## Epoch 11/20
## 45/45 - 1s - loss: 0.0060 - accuracy: 0.9982 - val_loss: 0.8942 - val_accuracy: 0.8550 - 877ms/epoch - 19ms/step
## Epoch 12/20
## 45/45 - 1s - loss: 0.0016 - accuracy: 1.0000 - val_loss: 0.9180 - val_accuracy: 0.8546 - 874ms/epoch - 19ms/step
## Epoch 13/20
## 45/45 - 1s - loss: 0.0074 - accuracy: 0.9980 - val_loss: 0.9463 - val_accuracy: 0.8532 - 886ms/epoch - 20ms/step
## Epoch 14/20
## 45/45 - 1s - loss: 0.0011 - accuracy: 1.0000 - val_loss: 0.9766 - val_accuracy: 0.8532 - 893ms/epoch - 20ms/step
## Epoch 15/20
## 45/45 - 1s - loss: 0.0043 - accuracy: 0.9989 - val_loss: 0.9838 - val_accuracy: 0.8534 - 869ms/epoch - 19ms/step
## Epoch 16/20
## 45/45 - 1s - loss: 0.0040 - accuracy: 0.9989 - val_loss: 1.3050 - val_accuracy: 0.8320 - 812ms/epoch - 18ms/step
## Epoch 17/20
## 45/45 - 1s - loss: 0.0016 - accuracy: 0.9997 - val_loss: 1.0239 - val_accuracy: 0.8540 - 803ms/epoch - 18ms/step
## Epoch 18/20
## 45/45 - 1s - loss: 0.0069 - accuracy: 0.9979 - val_loss: 1.0323 - val_accuracy: 0.8548 - 855ms/epoch - 19ms/step
## Epoch 19/20
## 45/45 - 1s - loss: 6.8344e-04 - accuracy: 1.0000 - val_loss: 1.0539 - val_accuracy: 0.8533 - 816ms/epoch - 18ms/step
## Epoch 20/20
## 45/45 - 1s - loss: 0.0068 - accuracy: 0.9976 - val_loss: 1.0830 - val_accuracy: 0.8528 - 893ms/epoch - 20ms/step

Recurrent Neural Networks

In this lab we fit the models illustrated in Section 10.5. ### Sequential Models for Document Classification Here we fit a simple LSTM RNN for sentiment analysis with the IMDb movie-review data, as discussed in Section 10.5.1. We showed how to input the data in 10.9.5, so we will not repeat that here.

We first calculate the lengths of the documents.

wc <- sapply(x_train, length)
median(wc)

## [1] 178

sum(wc <= 500) / length(wc)

## [1] 0.91568

We see that over 91% of the documents have fewer than 500 words. Our RNN requires all the document sequences to have the same length. We hence restrict the document lengths to the last \(L=500\) words, and pad the beginning of the shorter ones with blanks.

maxlen <- 500
x_train <- pad_sequences(x_train, maxlen = maxlen)
x_test <- pad_sequences(x_test, maxlen = maxlen)
dim(x_train)

## [1] 25000   500

dim(x_test)

## [1] 25000   500

x_train[1, 490:500]

##  [1]   16 4472  113  103   32   15   16 5345   19  178   32

The last expression shows the last few words in the first document. At this stage, each of the 500 words in the document is represented using an integer corresponding to the location of that word in the 10,000-word dictionary. The first layer of the RNN is an embedding layer of size 32, which will be learned during training. This layer one-hot encodes each document as a matrix of dimension \(500 \times 10,000\), and then maps these \(10,000\) dimensions down to \(32\).

model <- keras_model_sequential() %>%
  layer_embedding(input_dim = 10000, output_dim = 32) %>%
  layer_lstm(units = 32) %>%
  layer_dense(units = 1, activation = "sigmoid")

The second layer is an LSTM with 32 units, and the output layer is a single sigmoid for the binary classification task.

The rest is now similar to other networks we have fit. We track the test performance as the network is fit, and see that it attains 87% accuracy.

model %>% compile(optimizer = "rmsprop",
    loss = "binary_crossentropy", metrics = c("acc"))
#history <- model %>% fit(x_train, y_train, epochs = 10,
history <- model %>% fit(x_train, y_train, epochs = 3,
    batch_size = 128, validation_data = list(x_test, y_test))

## Epoch 1/3
## 196/196 - 37s - loss: 0.5798 - acc: 0.6840 - val_loss: 0.4883 - val_acc: 0.7926 - 37s/epoch - 188ms/step
## Epoch 2/3
## 196/196 - 33s - loss: 0.3344 - acc: 0.8640 - val_loss: 0.6256 - val_acc: 0.7999 - 33s/epoch - 171ms/step
## Epoch 3/3
## 196/196 - 33s - loss: 0.2629 - acc: 0.8986 - val_loss: 0.3298 - val_acc: 0.8618 - 33s/epoch - 168ms/step

plot(history)

predy <- predict(model, x_test) > 0.5

## 782/782 - 17s - 17s/epoch - 22ms/step

mean(abs(y_test == as.numeric(predy)))

## [1] 0.86176

Time Series Prediction

We now show how to fit the models in Section 10.5.2 for time series prediction. We first set up the data, and standardize each of the variables.

library(ISLR2)
xdata <- data.matrix(
    NYSE[, c("DJ_return", "log_volume","log_volatility")]
  )
istrain <- NYSE[, "train"]
xdata <- scale(xdata)

The variable istrain contains a TRUE for each year that is in the training set, and a FALSE for each year in the test set.

We first write functions to create lagged versions of the three time series. We start with a function that takes as input a data matrix and a lag \(L\), and returns a lagged version of the matrix. It simply inserts \(L\) rows of NA at the top, and truncates the bottom.

lagm <- function(x, k = 1) {
  n <- nrow(x)
  pad <- matrix(NA, k, ncol(x))
  rbind(pad, x[1:(n - k), ])
}

We now use this function to create a data frame with all the required lags, as well as the response variable.

arframe <- data.frame(log_volume = xdata[, "log_volume"],
   L1 = lagm(xdata, 1), L2 = lagm(xdata, 2),
   L3 = lagm(xdata, 3), L4 = lagm(xdata, 4),
   L5 = lagm(xdata, 5)
 )

If we look at the first five rows of this frame, we will see some missing values in the lagged variables (due to the construction above). We remove these rows, and adjust istrain accordingly.

arframe <- arframe[-(1:5), ]
istrain <- istrain[-(1:5)]

We now fit the linear AR model to the training data using lm(), and predict on the test data.

arfit <- lm(log_volume ~ ., data = arframe[istrain, ])
arpred <- predict(arfit, arframe[!istrain, ])
V0 <- var(arframe[!istrain, "log_volume"])
1 - mean((arpred - arframe[!istrain, "log_volume"])^2) / V0

## [1] 0.413223

The last two lines compute the \(R^2\) on the test data, as defined in (3.17).

We refit this model, including the factor variable day_of_week.

arframed <-
    data.frame(day = NYSE[-(1:5), "day_of_week"], arframe)
arfitd <- lm(log_volume ~ ., data = arframed[istrain, ])
arpredd <- predict(arfitd, arframed[!istrain, ])
1 - mean((arpredd - arframe[!istrain, "log_volume"])^2) / V0

## [1] 0.4598616

To fit the RNN, we need to reshape these data, since it expects a sequence of \(L=5\) feature vectors \(X={X_\ell}_1^L\) for each observation, as in (10.20) on page 428. These are lagged versions of the time series going back \(L\) time points.

n <- nrow(arframe)
xrnn <- data.matrix(arframe[, -1])
xrnn <- array(xrnn, c(n, 3, 5))
xrnn <- xrnn[,, 5:1]
xrnn <- aperm(xrnn, c(1, 3, 2))
dim(xrnn)

## [1] 6046    5    3

We have done this in four steps. The first simply extracts the \(n\times 15\) matrix of lagged versions of the three predictor variables from arframe. The second converts this matrix to an \(n\times 3\times 5\) array. We can do this by simply changing the dimension attribute, since the new array is filled column wise. The third step reverses the order of lagged variables, so that index \(1\) is furthest back in time, and index \(5\) closest. The final step rearranges the coordinates of the array (like a partial transpose) into the format that the RNN module in keras expects.

Now we are ready to proceed with the RNN, which uses 12 hidden units.

model <- keras_model_sequential() %>%
  layer_simple_rnn(units = 12,
      input_shape = list(5, 3),
      dropout = 0.1, recurrent_dropout = 0.1) %>%
  layer_dense(units = 1)
model %>% compile(optimizer = optimizer_rmsprop(),
    loss = "mse")

We specify two forms of dropout for the units feeding into the hidden layer. The first is for the input sequence feeding into this layer, and the second is for the previous hidden units feeding into the layer. The output layer has a single unit for the response.

We fit the model in a similar fashion to previous networks. We supply the fit function with test data as validation data, so that when we monitor its progress and plot the history function we can see the progress on the test data. Of course we should not use this as a basis for early stopping, since then the test performance would be biased.

history <- model %>% fit(
    xrnn[istrain,, ], arframe[istrain, "log_volume"],
#    batch_size = 64, epochs = 200,
    batch_size = 64, epochs = 75,
    validation_data =
      list(xrnn[!istrain,, ], arframe[!istrain, "log_volume"])
  )

## Epoch 1/75
## 67/67 - 1s - loss: 0.9548 - val_loss: 0.8558 - 1s/epoch - 17ms/step
## Epoch 2/75
## 67/67 - 0s - loss: 0.6267 - val_loss: 0.7254 - 154ms/epoch - 2ms/step
## Epoch 3/75
## 67/67 - 0s - loss: 0.5414 - val_loss: 0.6830 - 146ms/epoch - 2ms/step
## Epoch 4/75
## 67/67 - 0s - loss: 0.5197 - val_loss: 0.6704 - 143ms/epoch - 2ms/step
## Epoch 5/75
## 67/67 - 0s - loss: 0.5033 - val_loss: 0.6638 - 139ms/epoch - 2ms/step
## Epoch 6/75
## 67/67 - 0s - loss: 0.4974 - val_loss: 0.6583 - 150ms/epoch - 2ms/step
## Epoch 7/75
## 67/67 - 0s - loss: 0.4865 - val_loss: 0.6599 - 150ms/epoch - 2ms/step
## Epoch 8/75
## 67/67 - 0s - loss: 0.4835 - val_loss: 0.6537 - 138ms/epoch - 2ms/step
## Epoch 9/75
## 67/67 - 0s - loss: 0.4831 - val_loss: 0.6524 - 158ms/epoch - 2ms/step
## Epoch 10/75
## 67/67 - 0s - loss: 0.4794 - val_loss: 0.6488 - 145ms/epoch - 2ms/step
## Epoch 11/75
## 67/67 - 0s - loss: 0.4712 - val_loss: 0.6483 - 148ms/epoch - 2ms/step
## Epoch 12/75
## 67/67 - 0s - loss: 0.4786 - val_loss: 0.6492 - 145ms/epoch - 2ms/step
## Epoch 13/75
## 67/67 - 0s - loss: 0.4746 - val_loss: 0.6533 - 145ms/epoch - 2ms/step
## Epoch 14/75
## 67/67 - 0s - loss: 0.4746 - val_loss: 0.6452 - 146ms/epoch - 2ms/step
## Epoch 15/75
## 67/67 - 0s - loss: 0.4728 - val_loss: 0.6441 - 143ms/epoch - 2ms/step
## Epoch 16/75
## 67/67 - 0s - loss: 0.4670 - val_loss: 0.6515 - 158ms/epoch - 2ms/step
## Epoch 17/75
## 67/67 - 0s - loss: 0.4704 - val_loss: 0.6454 - 147ms/epoch - 2ms/step
## Epoch 18/75
## 67/67 - 0s - loss: 0.4628 - val_loss: 0.6427 - 142ms/epoch - 2ms/step
## Epoch 19/75
## 67/67 - 0s - loss: 0.4602 - val_loss: 0.6455 - 150ms/epoch - 2ms/step
## Epoch 20/75
## 67/67 - 0s - loss: 0.4542 - val_loss: 0.6386 - 151ms/epoch - 2ms/step
## Epoch 21/75
## 67/67 - 0s - loss: 0.4649 - val_loss: 0.6387 - 147ms/epoch - 2ms/step
## Epoch 22/75
## 67/67 - 0s - loss: 0.4659 - val_loss: 0.6381 - 144ms/epoch - 2ms/step
## Epoch 23/75
## 67/67 - 0s - loss: 0.4751 - val_loss: 0.6405 - 151ms/epoch - 2ms/step
## Epoch 24/75
## 67/67 - 0s - loss: 0.4529 - val_loss: 0.6413 - 409ms/epoch - 6ms/step
## Epoch 25/75
## 67/67 - 0s - loss: 0.4680 - val_loss: 0.6364 - 145ms/epoch - 2ms/step
## Epoch 26/75
## 67/67 - 0s - loss: 0.4599 - val_loss: 0.6389 - 144ms/epoch - 2ms/step
## Epoch 27/75
## 67/67 - 0s - loss: 0.4633 - val_loss: 0.6377 - 147ms/epoch - 2ms/step
## Epoch 28/75
## 67/67 - 0s - loss: 0.4587 - val_loss: 0.6367 - 146ms/epoch - 2ms/step
## Epoch 29/75
## 67/67 - 0s - loss: 0.4579 - val_loss: 0.6381 - 144ms/epoch - 2ms/step
## Epoch 30/75
## 67/67 - 0s - loss: 0.4588 - val_loss: 0.6354 - 140ms/epoch - 2ms/step
## Epoch 31/75
## 67/67 - 0s - loss: 0.4625 - val_loss: 0.6344 - 151ms/epoch - 2ms/step
## Epoch 32/75
## 67/67 - 0s - loss: 0.4640 - val_loss: 0.6402 - 152ms/epoch - 2ms/step
## Epoch 33/75
## 67/67 - 0s - loss: 0.4580 - val_loss: 0.6331 - 138ms/epoch - 2ms/step
## Epoch 34/75
## 67/67 - 0s - loss: 0.4539 - val_loss: 0.6360 - 145ms/epoch - 2ms/step
## Epoch 35/75
## 67/67 - 0s - loss: 0.4531 - val_loss: 0.6431 - 147ms/epoch - 2ms/step
## Epoch 36/75
## 67/67 - 0s - loss: 0.4544 - val_loss: 0.6375 - 143ms/epoch - 2ms/step
## Epoch 37/75
## 67/67 - 0s - loss: 0.4587 - val_loss: 0.6358 - 142ms/epoch - 2ms/step
## Epoch 38/75
## 67/67 - 0s - loss: 0.4582 - val_loss: 0.6419 - 153ms/epoch - 2ms/step
## Epoch 39/75
## 67/67 - 0s - loss: 0.4539 - val_loss: 0.6341 - 141ms/epoch - 2ms/step
## Epoch 40/75
## 67/67 - 0s - loss: 0.4463 - val_loss: 0.6319 - 134ms/epoch - 2ms/step
## Epoch 41/75
## 67/67 - 0s - loss: 0.4533 - val_loss: 0.6345 - 147ms/epoch - 2ms/step
## Epoch 42/75
## 67/67 - 0s - loss: 0.4567 - val_loss: 0.6306 - 145ms/epoch - 2ms/step
## Epoch 43/75
## 67/67 - 0s - loss: 0.4517 - val_loss: 0.6303 - 143ms/epoch - 2ms/step
## Epoch 44/75
## 67/67 - 0s - loss: 0.4538 - val_loss: 0.6376 - 140ms/epoch - 2ms/step
## Epoch 45/75
## 67/67 - 0s - loss: 0.4469 - val_loss: 0.6306 - 152ms/epoch - 2ms/step
## Epoch 46/75
## 67/67 - 0s - loss: 0.4425 - val_loss: 0.6348 - 146ms/epoch - 2ms/step
## Epoch 47/75
## 67/67 - 0s - loss: 0.4541 - val_loss: 0.6270 - 135ms/epoch - 2ms/step
## Epoch 48/75
## 67/67 - 0s - loss: 0.4386 - val_loss: 0.6277 - 150ms/epoch - 2ms/step
## Epoch 49/75
## 67/67 - 0s - loss: 0.4523 - val_loss: 0.6355 - 147ms/epoch - 2ms/step
## Epoch 50/75
## 67/67 - 0s - loss: 0.4520 - val_loss: 0.6285 - 150ms/epoch - 2ms/step
## Epoch 51/75
## 67/67 - 0s - loss: 0.4475 - val_loss: 0.6288 - 143ms/epoch - 2ms/step
## Epoch 52/75
## 67/67 - 0s - loss: 0.4448 - val_loss: 0.6257 - 149ms/epoch - 2ms/step
## Epoch 53/75
## 67/67 - 0s - loss: 0.4432 - val_loss: 0.6244 - 142ms/epoch - 2ms/step
## Epoch 54/75
## 67/67 - 0s - loss: 0.4394 - val_loss: 0.6298 - 139ms/epoch - 2ms/step
## Epoch 55/75
## 67/67 - 0s - loss: 0.4541 - val_loss: 0.6280 - 149ms/epoch - 2ms/step
## Epoch 56/75
## 67/67 - 0s - loss: 0.4555 - val_loss: 0.6317 - 142ms/epoch - 2ms/step
## Epoch 57/75
## 67/67 - 0s - loss: 0.4526 - val_loss: 0.6280 - 141ms/epoch - 2ms/step
## Epoch 58/75
## 67/67 - 0s - loss: 0.4447 - val_loss: 0.6260 - 143ms/epoch - 2ms/step
## Epoch 59/75
## 67/67 - 0s - loss: 0.4486 - val_loss: 0.6347 - 148ms/epoch - 2ms/step
## Epoch 60/75
## 67/67 - 0s - loss: 0.4530 - val_loss: 0.6329 - 142ms/epoch - 2ms/step
## Epoch 61/75
## 67/67 - 0s - loss: 0.4487 - val_loss: 0.6315 - 136ms/epoch - 2ms/step
## Epoch 62/75
## 67/67 - 0s - loss: 0.4433 - val_loss: 0.6282 - 152ms/epoch - 2ms/step
## Epoch 63/75
## 67/67 - 0s - loss: 0.4501 - val_loss: 0.6291 - 147ms/epoch - 2ms/step
## Epoch 64/75
## 67/67 - 0s - loss: 0.4471 - val_loss: 0.6337 - 138ms/epoch - 2ms/step
## Epoch 65/75
## 67/67 - 0s - loss: 0.4505 - val_loss: 0.6290 - 144ms/epoch - 2ms/step
## Epoch 66/75
## 67/67 - 0s - loss: 0.4541 - val_loss: 0.6323 - 151ms/epoch - 2ms/step
## Epoch 67/75
## 67/67 - 0s - loss: 0.4455 - val_loss: 0.6248 - 143ms/epoch - 2ms/step
## Epoch 68/75
## 67/67 - 0s - loss: 0.4462 - val_loss: 0.6259 - 147ms/epoch - 2ms/step
## Epoch 69/75
## 67/67 - 0s - loss: 0.4553 - val_loss: 0.6296 - 160ms/epoch - 2ms/step
## Epoch 70/75
## 67/67 - 0s - loss: 0.4386 - val_loss: 0.6252 - 147ms/epoch - 2ms/step
## Epoch 71/75
## 67/67 - 0s - loss: 0.4425 - val_loss: 0.6339 - 144ms/epoch - 2ms/step
## Epoch 72/75
## 67/67 - 0s - loss: 0.4480 - val_loss: 0.6274 - 156ms/epoch - 2ms/step
## Epoch 73/75
## 67/67 - 0s - loss: 0.4461 - val_loss: 0.6279 - 147ms/epoch - 2ms/step
## Epoch 74/75
## 67/67 - 0s - loss: 0.4511 - val_loss: 0.6259 - 145ms/epoch - 2ms/step
## Epoch 75/75
## 67/67 - 0s - loss: 0.4466 - val_loss: 0.6276 - 151ms/epoch - 2ms/step

kpred <- predict(model, xrnn[!istrain,, ])

## 56/56 - 0s - 167ms/epoch - 3ms/step

1 - mean((kpred - arframe[!istrain, "log_volume"])^2) / V0

## [1] 0.4046528

This model takes about one minute to train.

We could replace the keras_model_sequential() command above with the following command:

model <- keras_model_sequential() %>%
  layer_flatten(input_shape = c(5, 3)) %>%
  layer_dense(units = 1)

Here, layer_flatten() simply takes the input sequence and turns it into a long vector of predictors. This results in a linear AR model. To fit a nonlinear AR model, we could add in a hidden layer.

However, since we already have the matrix of lagged variables from the AR model that we fit earlier using the lm() command, we can actually fit a nonlinear AR model without needing to perform flattening. We extract the model matrix x from arframed, which includes the day_of_week variable.

x <- model.matrix(log_volume ~ . - 1, data = arframed)
colnames(x)

##  [1] "dayfri"            "daymon"            "daythur"          
##  [4] "daytues"           "daywed"            "L1.DJ_return"     
##  [7] "L1.log_volume"     "L1.log_volatility" "L2.DJ_return"     
## [10] "L2.log_volume"     "L2.log_volatility" "L3.DJ_return"     
## [13] "L3.log_volume"     "L3.log_volatility" "L4.DJ_return"     
## [16] "L4.log_volume"     "L4.log_volatility" "L5.DJ_return"     
## [19] "L5.log_volume"     "L5.log_volatility"

The -1 in the formula avoids the creation of a column of ones for the intercept. The variable day_of_week is a five-level factor (there are five trading days), and the -1 results in five rather than four dummy variables.

The rest of the steps to fit a nonlinear AR model should by now be familiar.

arnnd <- keras_model_sequential() %>%
  layer_dense(units = 32, activation = 'relu',
      input_shape = ncol(x)) %>%
  layer_dropout(rate = 0.5) %>%
  layer_dense(units = 1)
arnnd %>% compile(loss = "mse",
    optimizer = optimizer_rmsprop())
history <- arnnd %>% fit(
#    x[istrain, ], arframe[istrain, "log_volume"], epochs = 100, 
    x[istrain, ], arframe[istrain, "log_volume"], epochs = 30, 
    batch_size = 32, validation_data =
      list(x[!istrain, ], arframe[!istrain, "log_volume"])
  )

## Epoch 1/30
## 134/134 - 1s - loss: 1.0899 - val_loss: 0.6986 - 550ms/epoch - 4ms/step
## Epoch 2/30
## 134/134 - 0s - loss: 0.6487 - val_loss: 0.6266 - 182ms/epoch - 1ms/step
## Epoch 3/30
## 134/134 - 0s - loss: 0.5526 - val_loss: 0.6046 - 174ms/epoch - 1ms/step
## Epoch 4/30
## 134/134 - 0s - loss: 0.5012 - val_loss: 0.5913 - 171ms/epoch - 1ms/step
## Epoch 5/30
## 134/134 - 0s - loss: 0.4764 - val_loss: 0.5857 - 181ms/epoch - 1ms/step
## Epoch 6/30
## 134/134 - 0s - loss: 0.4724 - val_loss: 0.5790 - 174ms/epoch - 1ms/step
## Epoch 7/30
## 134/134 - 0s - loss: 0.4600 - val_loss: 0.5785 - 173ms/epoch - 1ms/step
## Epoch 8/30
## 134/134 - 0s - loss: 0.4476 - val_loss: 0.5735 - 177ms/epoch - 1ms/step
## Epoch 9/30
## 134/134 - 0s - loss: 0.4557 - val_loss: 0.5754 - 174ms/epoch - 1ms/step
## Epoch 10/30
## 134/134 - 0s - loss: 0.4442 - val_loss: 0.5715 - 181ms/epoch - 1ms/step
## Epoch 11/30
## 134/134 - 0s - loss: 0.4495 - val_loss: 0.5723 - 166ms/epoch - 1ms/step
## Epoch 12/30
## 134/134 - 0s - loss: 0.4405 - val_loss: 0.5672 - 179ms/epoch - 1ms/step
## Epoch 13/30
## 134/134 - 0s - loss: 0.4355 - val_loss: 0.5670 - 170ms/epoch - 1ms/step
## Epoch 14/30
## 134/134 - 0s - loss: 0.4437 - val_loss: 0.5666 - 174ms/epoch - 1ms/step
## Epoch 15/30
## 134/134 - 0s - loss: 0.4214 - val_loss: 0.5645 - 169ms/epoch - 1ms/step
## Epoch 16/30
## 134/134 - 0s - loss: 0.4377 - val_loss: 0.5665 - 182ms/epoch - 1ms/step
## Epoch 17/30
## 134/134 - 0s - loss: 0.4419 - val_loss: 0.5664 - 178ms/epoch - 1ms/step
## Epoch 18/30
## 134/134 - 0s - loss: 0.4198 - val_loss: 0.5657 - 167ms/epoch - 1ms/step
## Epoch 19/30
## 134/134 - 0s - loss: 0.4249 - val_loss: 0.5643 - 172ms/epoch - 1ms/step
## Epoch 20/30
## 134/134 - 0s - loss: 0.4277 - val_loss: 0.5648 - 171ms/epoch - 1ms/step
## Epoch 21/30
## 134/134 - 0s - loss: 0.4175 - val_loss: 0.5655 - 168ms/epoch - 1ms/step
## Epoch 22/30
## 134/134 - 0s - loss: 0.4314 - val_loss: 0.5666 - 193ms/epoch - 1ms/step
## Epoch 23/30
## 134/134 - 0s - loss: 0.4213 - val_loss: 0.5638 - 167ms/epoch - 1ms/step
## Epoch 24/30
## 134/134 - 0s - loss: 0.4261 - val_loss: 0.5662 - 168ms/epoch - 1ms/step
## Epoch 25/30
## 134/134 - 0s - loss: 0.4237 - val_loss: 0.5661 - 176ms/epoch - 1ms/step
## Epoch 26/30
## 134/134 - 0s - loss: 0.4271 - val_loss: 0.5655 - 181ms/epoch - 1ms/step
## Epoch 27/30
## 134/134 - 0s - loss: 0.4271 - val_loss: 0.5646 - 179ms/epoch - 1ms/step
## Epoch 28/30
## 134/134 - 0s - loss: 0.4277 - val_loss: 0.5664 - 185ms/epoch - 1ms/step
## Epoch 29/30
## 134/134 - 0s - loss: 0.4213 - val_loss: 0.5637 - 170ms/epoch - 1ms/step
## Epoch 30/30
## 134/134 - 0s - loss: 0.4189 - val_loss: 0.5659 - 193ms/epoch - 1ms/step

plot(history)

npred <- predict(arnnd, x[!istrain, ])

## 56/56 - 0s - 65ms/epoch - 1ms/step

1 - mean((arframe[!istrain, "log_volume"] - npred)^2) / V0

## [1] 0.4631082