MNIST ML
Gary N. Thomas
11/3/2019
Introduction:
We can learn the basics of Keras by walking through a simple example: recognizing handwritten digits from the MNIST dataset. MNIST consists of 28 x 28 grayscale images of handwritten digits like these:
The dataset also includes labels for each image, telling us which digit it is. For example, the labels for the above images are 5, 0, 4, and 1
PREPARING THE DATA
The MNIST dataset is included with Keras and can be accessed using the dataset_mnist() function. Here we load the dataset then create variables for our test and training data:
library(keras)
mnist <- dataset_mnist()## Warning in normalizePath(path.expand(path), winslash, mustWork):
## path[1]="C:\Users\garyn\AppData\Local\conda\conda\envs\rstudio-1.1.456/
## python.exe": The system cannot find the file specifiedx_train <- mnist$train$x
y_train <- mnist$train$y
x_test <- mnist$test$x
y_test <- mnist$test$yThe x data is a 3-d array (images,width,height) of grayscale values . To prepare the data for training we convert the 3-d arrays into matrices by reshaping width and height into a single dimension (28x28 images are flattened into length 784 vectors). Then, we convert the grayscale values from integers ranging between 0 to 255 into floating point values ranging between 0 and 1:
# reshape
x_train <- array_reshape(x_train, c(nrow(x_train), 784))
x_test <- array_reshape(x_test, c(nrow(x_test), 784))
# rescale
x_train <- x_train / 255
x_test <- x_test / 255Note that we use the array_reshape() function rather than the dim<-() function to reshape the array. This is so that the data is re-interpreted using row-major semantics (as opposed to R’s default column-major semantics), which is in turn compatible with the way that the numerical libraries called by Keras interpret array dimensions.
The y data is an integer vector with values ranging from 0 to 9. To prepare this data for training we one-hot encode the vectors into binary class matrices using the Keras to_categorical() function:
y_train <- to_categorical(y_train, 10)
y_test <- to_categorical(y_test, 10)DEFINING THE MODEL
The core data structure of Keras is a model, a way to organize layers. The simplest type of model is the Sequential model, a linear stack of layers.
We begin by creating a sequential model and then adding layers using the pipe (%>%) operator:
model <- keras_model_sequential()
model %>%
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 input_shape argument to the first layer specifies the shape of the input data (a length 784 numeric vector representing a grayscale image). The final layer outputs a length 10 numeric vector (probabilities for each digit) using a softmax activation function.
Use the summary() function to print the details of the model:
summary(model)## Model: "sequential"
## ___________________________________________________________________________
## Layer (type) Output Shape Param #
## ===========================================================================
## dense (Dense) (None, 256) 200960
## ___________________________________________________________________________
## dropout (Dropout) (None, 256) 0
## ___________________________________________________________________________
## dense_1 (Dense) (None, 128) 32896
## ___________________________________________________________________________
## dropout_1 (Dropout) (None, 128) 0
## ___________________________________________________________________________
## dense_2 (Dense) (None, 10) 1290
## ===========================================================================
## Total params: 235,146
## Trainable params: 235,146
## Non-trainable params: 0
## ___________________________________________________________________________Next, compile the model with appropriate loss function, optimizer, and metrics:
model %>% compile(
loss = 'categorical_crossentropy',
optimizer = optimizer_rmsprop(),
metrics = c('accuracy')
)TRAINING AND EVALUATION
Use the fit() function to train the model for 30 epochs using batches of 128 images:
history <- model %>% fit(
x_train, y_train,
epochs = 30, batch_size = 128,
validation_split = 0.2
)The history object returned by fit() includes loss and accuracy metrics which we can plot:
plot(history)
Evaluate the model’s performance on the test data:
model %>% evaluate(x_test, y_test)## $loss
## [1] 0.1046164
##
## $accuracy
## [1] 0.982Generate predictions on new data:
model %>% predict_classes(x_test)## [1] 7 2 1 0 4 1 4 9 6 9 0 6 9 0 1 5 9 7 3 4 9 6 6 5 4 0 7 4 0 1 3 1 3 4
## [35] 7 2 7 1 2 1 1 7 4 2 3 5 1 2 4 4 6 3 5 5 6 0 4 1 9 5 7 8 9 3 7 4 6 4
## [69] 3 0 7 0 2 9 1 7 3 2 9 7 7 6 2 7 8 4 7 3 6 1 3 6 9 3 1 4 1 7 6 9 6 0
## [103] 5 4 9 9 2 1 9 4 8 7 3 9 7 4 4 4 9 2 5 4 7 6 7 9 0 5 8 5 6 6 5 7 8 1
## [137] 0 1 6 4 6 7 3 1 7 1 8 2 0 2 9 9 5 5 1 5 6 0 3 4 4 6 5 4 6 5 4 5 1 4
## [171] 4 7 2 3 2 7 1 8 1 8 1 8 5 0 8 9 2 5 0 1 1 1 0 9 0 3 1 6 4 2 3 6 1 1
## [205] 1 3 9 5 2 9 4 5 9 3 9 0 3 6 5 5 7 2 2 7 1 2 8 4 1 7 3 3 8 8 7 9 2 2
## [239] 4 1 5 9 8 7 2 3 0 6 4 2 4 1 9 5 7 7 2 8 2 0 8 5 7 7 9 1 8 1 8 0 3 0
## [273] 1 9 9 4 1 8 2 1 2 9 7 5 9 2 6 4 1 5 8 2 9 2 0 4 0 0 2 8 4 7 1 2 4 0
## [307] 2 7 4 3 3 0 0 3 1 9 6 5 2 5 8 7 9 3 0 4 2 0 7 1 1 2 1 5 3 3 9 7 8 6
## [341] 5 6 1 3 8 1 0 5 1 3 1 5 5 6 1 8 5 1 7 9 4 6 2 2 5 0 6 5 6 3 7 2 0 8
## [375] 8 5 4 1 1 4 0 3 3 7 6 1 6 2 1 9 2 8 6 1 9 5 2 5 4 4 2 8 3 8 2 4 5 0
## [409] 3 1 7 7 5 7 9 7 1 9 2 1 4 2 9 2 0 4 9 1 4 8 1 8 4 5 9 8 8 3 7 6 0 0
## [443] 3 0 2 0 6 9 9 3 3 3 2 3 9 1 2 6 8 0 5 6 6 6 3 8 8 2 7 5 8 9 6 1 8 4
## [477] 1 2 5 9 1 9 7 5 4 0 8 9 9 1 0 5 2 3 7 0 9 4 0 6 3 9 5 2 1 3 1 3 6 5
## [511] 7 4 2 2 6 3 2 6 5 4 8 9 7 1 3 0 3 8 3 1 9 3 4 4 6 4 2 1 8 2 5 4 8 8
## [545] 4 0 0 2 3 2 7 7 0 8 7 4 4 7 9 6 9 0 9 8 0 4 6 0 6 3 5 4 8 3 3 9 3 3
## [579] 3 7 8 0 2 2 1 7 0 6 5 4 3 8 0 9 6 3 8 0 9 9 6 8 6 8 5 7 8 6 0 2 4 0
## [613] 2 2 3 1 9 7 5 8 0 8 4 6 2 6 7 9 3 2 9 8 2 2 9 1 7 3 5 9 1 8 0 2 0 5
## [647] 2 1 3 7 6 7 1 2 5 8 0 3 7 7 4 0 9 1 8 6 7 7 4 3 4 9 1 9 5 1 7 3 9 7
## [681] 6 9 1 3 3 8 3 3 6 7 2 8 5 8 5 1 1 4 4 3 1 0 7 7 0 7 9 4 4 8 5 5 4 0
## [715] 8 2 1 6 8 4 8 0 4 0 6 1 9 3 2 6 7 2 6 9 3 1 4 6 2 5 4 2 0 6 2 1 7 3
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## [1021] 3 7 5 4 4 1 8 1 3 8 1 2 5 8 0 6 2 1 1 7 1 5 3 4 6 9 5 0 9 2 2 4 8 2
## [1055] 1 7 2 4 9 4 4 0 3 9 2 2 3 3 8 3 5 7 3 5 8 1 2 4 4 6 4 9 5 1 0 6 9 5
## [1089] 9 5 9 7 3 8 0 3 7 1 3 6 7 8 5 9 7 9 6 3 6 3 7 4 6 5 3 5 4 7 8 7 8 0
## [1123] 7 6 8 8 7 3 3 1 9 5 2 7 3 5 1 1 2 1 4 7 4 7 5 4 5 4 0 8 3 6 9 6 0 2
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## [1225] 2 8 2 2 9 8 4 0 9 5 8 5 1 2 1 3 1 7 9 5 7 2 0 3 8 8 6 2 5 4 1 9 2 1
## [1259] 5 8 1 0 2 4 4 3 6 8 8 2 4 0 5 0 4 4 7 9 3 4 1 5 9 7 3 5 8 8 0 5 3 3
## [1293] 6 6 0 1 6 0 3 5 4 4 1 2 9 1 4 6 9 9 3 9 8 4 4 3 1 3 1 3 8 7 9 4 8 8
## [1327] 7 9 9 1 4 5 6 0 5 2 2 2 1 5 5 2 4 9 6 2 7 7 2 2 1 1 2 8 3 7 2 4 1 7
## [1361] 1 7 6 7 8 2 7 3 1 7 5 8 2 6 2 2 5 6 5 0 9 2 4 3 3 9 7 6 6 8 0 4 1 3
## [1395] 8 2 9 1 8 0 6 7 2 1 0 5 5 2 0 2 2 0 2 4 9 8 0 9 9 4 6 5 4 9 1 8 3 4
## [1429] 9 9 1 2 2 8 1 9 6 4 0 9 4 8 3 8 6 0 2 5 1 9 6 2 9 4 0 9 6 0 6 2 5 4
## [1463] 2 3 8 4 5 5 0 3 8 5 3 5 8 6 5 7 6 3 3 9 6 1 1 2 9 0 4 3 3 6 9 5 7 3
## [1497] 7 7 7 8 3 9 8 3 0 7 2 7 9 4 5 4 9 3 2 1 4 0 2 3 7 5 9 8 8 5 0 6 1 4
## [1531] 7 3 9 0 0 0 6 6 2 3 7 8 4 7 7 9 2 4 1 6 5 2 4 9 9 1 8 4 0 9 8 4 8 7
## [1565] 7 0 7 8 8 6 0 4 8 8 2 4 7 6 6 6 4 7 1 8 8 2 3 6 3 0 0 3 7 6 9 7 9 9
## [1599] 5 4 3 3 6 1 2 3 7 3 3 2 0 3 3 8 4 3 6 3 5 0 2 6 9 0 7 4 6 9 3 5 1 9
## [1633] 6 1 4 5 4 5 0 5 9 5 2 1 2 9 1 9 9 4 0 8 4 5 2 9 2 1 2 1 7 3 6 8 8 4
## [1667] 9 1 9 8 5 7 5 1 1 8 6 5 0 4 4 7 2 3 5 6 8 8 6 2 3 1 0 5 8 9 2 9 6 7
## [1701] 0 4 8 7 1 7 4 1 0 3 7 2 0 0 9 1 7 0 7 8 4 7 2 0 4 6 0 3 1 1 3 3 9 6
## [1735] 7 4 1 5 3 0 8 7 3 9 6 9 3 5 0 2 7 4 5 1 1 5 8 0 8 8 1 5 0 3 0 3 1 4
## [1769] 0 3 7 2 7 1 8 0 7 0 4 3 1 9 8 7 7 1 4 9 9 3 8 1 7 9 0 2 0 3 3 7 6 9
## [1803] 2 3 3 7 7 0 0 7 5 2 9 8 7 4 4 2 6 6 1 9 6 8 2 9 0 8 3 1 1 6 3 5 1 1
## [1837] 1 3 1 2 3 0 2 0 1 3 5 5 7 4 8 9 6 9 6 8 3 6 6 8 5 1 4 2 4 4 5 1 1 9
## [1871] 0 2 4 9 5 7 1 8 8 5 6 9 8 7 1 1 6 7 6 3 2 2 0 8 9 2 5 1 0 8 1 4 5 7
## [1905] 9 6 9 0 6 1 5 5 8 3 8 2 6 5 0 7 4 6 1 3 4 7 3 2 3 4 2 5 2 7 1 7 2 6
## [1939] 4 1 5 2 8 6 0 1 8 2 5 7 7 6 9 3 5 8 4 2 4 0 8 8 3 4 9 2 7 5 8 6 5 6
## [1973] 0 8 6 7 3 6 4 9 4 6 6 3 2 4 1 0 1 4 6 2 9 1 1 0 6 3 9 5 6 5 6 5 9 4
## [2007] 6 4 3 9 1 3 4 1 9 1 7 1 1 9 3 5 4 0 7 3 6 1 7 5 5 3 3 0 1 5 7 5 8 6
## [2041] 5 1 0 8 2 3 4 6 7 9 8 1 8 9 9 2 8 6 2 7 0 0 6 7 5 8 6 0 9 3 7 1 3 5
## [2075] 4 3 3 5 5 6 3 0 2 3 4 2 3 0 9 9 4 7 2 1 4 7 0 6 0 8 5 2 8 5 7 3 0 8
## [2109] 2 1 2 8 2 5 5 7 6 4 6 8 4 8 2 7 4 5 2 0 3 9 9 6 7 2 5 1 1 1 2 3 6 7
## [2143] 8 7 6 4 8 9 4 8 6 3 8 3 1 0 6 2 2 5 6 9 5 8 1 4 1 7 8 4 6 1 8 4 3 1
## [2177] 2 8 0 8 5 9 1 4 2 8 2 7 0 9 0 2 5 7 6 7 9 4 2 6 2 4 4 8 0 4 4 5 8 0
## [2211] 6 8 9 8 5 6 9 0 4 8 7 1 3 4 5 8 0 9 1 3 3 6 9 8 7 1 0 5 7 1 7 5 2 7
## [2245] 9 1 8 5 2 4 9 4 7 2 2 3 4 9 1 9 2 1 7 9 4 4 1 6 7 2 7 8 0 1 9 7 1 1
## [2279] 7 5 3 3 5 1 3 7 6 1 3 8 7 5 9 6 0 0 2 8 8 2 3 7 1 3 0 3 4 4 3 8 9 2
## [2313] 3 9 7 1 1 7 0 4 9 6 5 9 1 7 0 2 0 2 4 6 7 0 7 1 4 6 4 5 4 9 9 1 7 9
## [2347] 5 3 3 8 2 3 6 2 2 1 1 1 1 1 6 9 8 4 3 7 1 6 4 8 0 4 7 4 2 4 0 7 0 1
## [2381] 9 8 8 6 0 0 4 1 6 8 2 2 3 8 4 8 2 2 1 7 5 4 4 0 4 3 4 7 3 1 0 1 2 5
## [2415] 9 2 1 0 1 8 9 1 4 8 3 8 9 3 6 2 8 3 2 2 1 0 4 2 9 2 4 3 7 9 1 5 2 4
## [2449] 9 0 3 8 5 3 6 0 9 4 6 2 5 0 0 7 4 6 6 8 6 6 8 6 9 1 7 2 5 9 9 0 7 2
## [2483] 7 6 7 0 6 5 4 4 7 2 0 9 9 2 2 9 4 4 2 3 3 2 1 7 0 7 6 4 1 3 8 7 4 5
## [2517] 9 2 5 1 8 7 3 7 1 5 5 0 9 1 4 0 6 3 3 6 0 4 9 7 5 1 6 8 9 5 5 7 9 3
## [2551] 8 3 8 1 5 3 5 0 5 5 3 8 6 7 7 7 3 7 0 5 9 0 2 5 5 3 1 7 7 8 6 5 5 3
## [2585] 8 9 5 3 7 9 1 7 0 0 3 7 2 3 8 1 8 6 2 9 5 7 5 7 8 6 2 5 1 4 8 4 5 8
## [2619] 3 0 6 2 7 3 3 2 1 0 7 3 4 0 3 9 3 2 8 9 0 3 8 0 7 6 5 4 7 3 5 0 8 6
## [2653] 2 5 1 1 0 0 4 4 0 1 2 3 2 7 7 8 5 2 5 7 6 9 1 4 1 6 4 2 4 3 5 4 3 9
## [2687] 5 0 1 5 3 8 9 1 9 7 9 5 5 2 7 4 6 0 1 1 1 0 4 4 7 6 3 0 0 4 3 0 6 1
## [2721] 9 6 1 3 8 1 2 5 6 2 7 3 6 0 1 9 7 6 6 8 9 2 9 5 8 3 1 0 0 7 6 6 2 1
## [2755] 6 9 3 1 8 6 9 0 6 0 0 0 6 3 5 9 3 4 5 5 8 5 3 0 4 0 2 9 6 8 2 3 1 2
## [2789] 1 1 5 6 9 8 0 6 6 5 5 3 8 6 2 1 4 5 4 3 7 8 3 0 9 3 5 1 1 0 4 4 7 0
## [2823] 1 7 0 1 6 1 4 5 6 6 5 7 8 4 4 7 2 5 3 7 0 7 7 9 6 4 2 8 5 7 8 3 9 5
## [2857] 8 9 9 8 6 2 8 9 2 3 6 1 1 8 9 3 4 0 7 9 6 4 1 4 1 3 4 9 3 1 4 7 7 4
## [2891] 7 2 9 3 0 8 8 8 4 0 4 4 1 5 2 8 3 4 9 5 2 8 1 5 3 7 9 4 2 5 6 3 5 9
## [2925] 3 5 9 2 1 9 5 3 0 6 9 8 4 0 4 7 2 9 0 1 0 3 1 6 5 8 1 5 5 5 0 3 5 5
## [2959] 9 2 8 7 0 4 9 1 9 7 7 5 3 2 0 9 1 8 6 2 3 9 6 2 1 9 1 3 5 5 0 3 8 3
## [2993] 3 7 6 6 0 1 4 0 6 9 8 1 2 9 9 5 9 7 3 7 8 0 1 3 0 4 6 1 0 2 5 8 4 4
## [3027] 1 1 5 4 8 6 0 6 9 2 6 2 7 1 7 9 4 0 0 3 8 2 2 3 1 6 0 5 7 7 9 2 6 7
## [3061] 9 7 8 6 8 8 4 6 8 4 1 2 8 2 3 9 4 0 3 7 3 2 3 3 7 3 4 0 6 2 0 8 1 5
## [3095] 3 5 4 1 7 1 5 7 5 7 3 2 2 7 3 7 3 7 8 5 4 5 2 9 6 5 3 6 7 4 1 7 1 5
## [3129] 2 3 6 3 1 4 2 6 7 4 3 8 0 6 2 1 6 5 3 9 1 9 3 2 1 8 4 4 6 5 8 6 9 7
## [3163] 7 8 6 9 7 3 9 4 0 5 4 6 4 1 2 3 0 0 2 6 6 5 7 0 8 6 4 7 9 0 7 3 4 2
## [3197] 1 8 8 5 9 2 7 1 8 8 8 2 7 6 0 1 2 7 1 0 8 3 6 0 5 3 6 2 8 7 0 1 4 2
## [3231] 1 1 4 4 4 4 7 1 6 2 9 9 0 0 1 8 8 4 3 4 2 0 6 1 6 1 2 2 2 1 2 3 7 8
## [3265] 1 0 0 2 1 6 6 0 1 6 2 5 1 7 4 8 2 1 4 3 8 3 9 9 4 8 3 4 7 2 7 5 7 0
## [3299] 4 3 3 2 6 7 6 0 0 6 7 7 0 5 5 8 1 0 7 0 2 8 1 5 0 8 8 0 3 2 7 7 2 6
## [3333] 4 7 5 5 7 2 9 2 8 4 6 8 6 5 0 0 8 7 6 1 7 1 1 2 7 4 0 0 7 7 6 3 8 6
## [3367] 4 2 0 9 4 0 5 7 8 2 7 4 7 1 1 3 6 6 2 9 1 9 4 8 3 6 9 5 9 6 2 4 6 7
## [3401] 7 0 6 6 9 4 8 3 5 3 4 9 0 0 5 2 5 0 7 1 1 1 0 7 6 7 9 6 6 4 1 4 3 1
## [3435] 1 2 2 4 1 0 8 7 6 3 4 0 0 6 3 3 0 9 1 7 1 1 3 1 0 9 9 7 5 4 1 4 8 9
## [3469] 5 3 5 1 9 8 2 7 3 9 9 0 1 0 2 9 3 9 3 3 6 2 4 9 8 3 7 4 0 4 7 8 4 9
## [3503] 8 1 9 7 5 9 2 8 2 2 0 2 2 3 8 4 6 8 4 8 2 4 6 7 9 3 3 9 4 3 1 4 4 7
## [3537] 0 5 9 6 0 4 4 4 4 6 1 2 3 3 6 4 5 9 6 8 5 6 0 5 6 4 1 8 6 5 2 5 4 5
## [3571] 5 4 7 7 0 7 8 2 2 3 7 0 1 8 0 7 1 9 8 7 5 5 9 1 7 5 4 3 1 2 2 1 6 6
## [3605] 7 1 1 4 0 7 4 2 4 0 6 4 7 6 9 5 3 4 6 5 0 1 8 8 2 8 3 5 7 8 0 8 5 7
## [3639] 1 1 0 1 3 7 8 5 0 7 1 1 0 1 1 4 5 2 7 6 2 3 0 2 8 5 9 6 9 7 2 1 3 6
## [3673] 4 1 8 2 4 0 5 1 0 3 2 6 4 4 3 9 6 1 6 5 7 9 2 0 2 6 0 1 4 3 5 2 8 8
## [3707] 0 8 8 9 0 9 6 7 6 3 9 3 4 7 7 7 4 9 0 6 4 4 4 2 7 2 8 1 0 0 7 8 3 3
## [3741] 3 1 3 7 6 1 3 1 6 4 5 2 4 7 5 9 5 8 4 9 9 1 6 5 0 1 3 7 0 3 4 8 2 2
## [3775] 0 2 8 1 5 1 6 8 8 9 1 2 1 3 5 1 0 9 4 4 8 3 2 5 9 7 6 6 2 0 0 0 5 8
## [3809] 3 1 5 2 3 8 5 1 8 2 0 4 9 9 6 2 3 3 5 6 4 8 0 9 2 8 3 6 7 5 7 2 9 4
## [3843] 9 1 2 8 6 0 7 0 9 1 1 5 7 5 9 9 1 9 5 9 2 5 0 4 1 0 8 9 0 8 9 8 9 4
## [3877] 2 5 7 9 8 9 8 0 9 9 6 8 9 9 5 9 8 6 1 0 3 3 5 2 1 6 5 0 2 8 2 5 6 2
## [3911] 3 0 2 2 6 4 3 5 5 1 7 2 1 6 9 1 9 9 5 5 1 6 2 2 8 6 7 1 4 6 0 2 0 5
## [3945] 3 2 2 3 6 8 9 8 5 3 8 5 4 5 2 0 5 6 3 2 8 3 9 9 5 7 9 4 6 7 1 3 7 3
## [3979] 6 6 0 9 0 1 9 4 2 8 8 0 1 6 9 7 5 3 4 7 4 9 9 4 3 6 3 1 1 7 6 9 1 8
## [4013] 4 1 1 9 9 4 3 6 8 1 6 0 4 1 3 7 7 4 9 5 1 0 0 1 1 6 2 1 9 8 4 0 3 6
## [4047] 4 9 0 7 1 6 5 7 5 2 5 1 8 5 4 7 0 6 7 3 2 5 8 1 0 4 5 7 1 3 5 1 9 0
## [4081] 0 6 0 7 3 1 8 3 9 7 0 0 8 9 5 9 8 3 2 7 2 9 7 2 1 1 3 7 5 3 1 9 8 2
## [4115] 2 2 8 8 5 7 3 8 9 8 8 6 8 2 3 9 7 5 6 2 9 2 8 8 1 6 2 8 7 9 1 8 0 1
## [4149] 7 2 0 7 5 1 9 0 2 0 9 8 6 2 3 9 3 8 0 2 1 1 1 1 4 2 9 7 7 5 1 1 2 1
## [4183] 9 9 9 1 0 2 0 2 1 1 4 6 4 1 5 4 9 9 7 1 5 6 2 1 2 8 0 6 9 6 1 9 7 7
## [4217] 1 4 8 5 3 4 3 4 7 7 5 0 7 4 8 8 1 5 3 9 5 9 7 6 9 0 3 6 3 9 8 2 8 1
## [4251] 2 8 6 8 5 3 3 9 4 9 2 5 1 5 1 1 4 1 4 4 3 5 9 1 2 2 3 3 0 2 9 0 0 9
## [4285] 9 6 0 9 3 7 8 4 1 9 9 7 2 7 9 9 5 9 5 1 1 8 3 5 1 9 5 3 5 4 9 5 9 3
## [4319] 1 9 0 9 7 5 4 9 2 0 1 0 5 1 4 9 3 3 6 1 5 2 5 2 2 0 9 2 6 6 0 1 2 0
## [4353] 3 0 2 5 5 7 9 5 5 0 8 9 5 0 3 2 5 9 0 8 8 4 5 8 8 4 5 4 8 5 4 9 3 2
## [4387] 1 2 6 8 8 7 0 3 6 6 4 3 8 8 7 2 2 0 0 4 3 9 9 1 9 8 6 6 4 2 6 9 2 8
## [4421] 5 4 5 7 9 9 9 2 1 8 3 4 0 7 8 7 9 3 4 6 5 6 2 3 9 2 6 0 0 6 1 2 8 7
## [4455] 9 8 2 0 4 7 7 5 0 5 6 4 6 7 4 3 0 7 5 0 7 4 2 0 8 9 9 4 2 4 6 7 8 7
## [4489] 6 9 4 1 3 7 3 0 8 7 7 6 9 3 9 2 2 9 2 1 8 3 2 9 6 8 4 0 1 2 8 4 5 2
## [4523] 7 8 1 1 3 0 3 5 7 0 3 1 9 3 5 3 1 7 7 3 0 8 4 8 2 6 5 2 9 7 3 9 0 9
## [4557] 9 6 4 2 9 7 2 1 1 6 7 4 7 5 9 6 8 2 1 4 4 5 7 6 1 3 2 5 9 9 3 6 1 1
## [4591] 4 6 9 7 2 1 5 1 4 6 3 4 1 1 0 3 1 6 8 4 9 0 7 3 0 2 9 0 6 6 6 3 6 7
## [4625] 7 2 8 6 0 8 3 0 2 9 8 3 2 5 3 8 8 0 0 1 9 5 1 3 9 6 0 1 4 1 7 1 2 3
## [4659] 7 9 7 4 9 9 3 9 2 8 2 7 1 8 0 9 1 0 1 7 7 9 6 9 9 9 2 1 6 1 3 5 7 1
## [4693] 9 7 6 4 5 7 6 6 9 9 6 3 6 2 9 8 1 2 2 5 5 2 3 7 2 1 0 1 0 4 5 2 8 2
## [4727] 8 3 5 1 7 8 1 1 2 9 7 8 4 0 3 0 7 8 8 4 7 7 8 5 8 4 9 8 1 3 8 0 3 1
## [4761] 7 1 5 5 1 6 5 7 4 9 3 5 4 7 1 2 0 8 1 6 0 7 3 4 7 3 9 6 0 8 6 4 8 7
## [4795] 7 9 3 8 6 9 7 2 3 4 0 2 1 0 3 5 5 7 2 4 4 7 2 8 3 0 8 7 8 9 0 8 4 4
## [4829] 5 8 5 6 6 3 0 9 3 7 6 8 9 3 4 9 5 8 9 1 2 8 8 6 8 1 3 7 9 0 1 1 9 7
## [4863] 0 8 1 7 4 5 7 1 2 1 1 3 5 6 2 1 2 8 8 7 6 6 9 3 7 0 5 2 8 0 5 4 3 8
## [4897] 4 6 6 2 7 9 5 1 3 2 4 3 6 1 9 4 4 7 6 5 4 1 9 9 2 7 8 0 1 3 6 1 3 4
## [4931] 1 1 1 5 6 0 7 0 7 2 3 2 5 2 2 9 4 9 8 1 2 1 6 1 2 7 4 0 0 0 8 2 2 9
## [4965] 2 2 1 9 9 2 7 5 1 3 4 9 4 1 8 5 6 2 8 3 1 2 8 4 9 9 3 7 0 7 7 2 3 2
## [4999] 4 0 3 9 9 8 4 1 0 6 0 9 6 8 6 1 1 9 8 9 2 3 5 5 9 4 2 1 9 4 3 9 6 0
## [5033] 4 0 6 0 1 2 3 4 7 8 9 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 9 8 3 4 7 8
## [5067] 6 3 4 0 9 7 1 9 3 8 4 7 5 0 9 1 4 5 4 6 2 0 6 2 1 1 1 1 7 2 4 7 5 2
## [5101] 9 4 5 8 4 2 9 7 0 0 7 5 1 1 7 6 6 6 8 2 2 7 7 4 0 2 4 2 1 8 9 6 1 0
## [5135] 5 9 6 9 8 0 3 0 8 3 9 6 3 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 0 1 2
## [5169] 3 4 5 6 7 8 5 4 8 7 4 7 7 3 9 8 8 3 1 5 8 2 7 4 2 1 5 4 5 5 8 6 4 4
## [5203] 4 1 8 7 5 5 1 8 9 1 3 6 3 3 2 2 6 9 9 6 5 5 3 3 8 1 6 5 6 8 1 9 7 6
## [5237] 8 3 7 4 7 0 9 0 0 3 7 9 3 0 2 0 1 0 1 0 4 0 1 0 4 7 9 6 2 6 2 2 9 9
## [5271] 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 8 0 5 6
## [5305] 6 0 8 0 2 3 7 9 4 7 1 9 1 7 1 4 0 0 4 1 7 5 7 1 3 3 3 6 6 9 7 4 3 0
## [5339] 2 5 2 6 0 8 9 4 3 5 4 8 1 5 9 0 6 4 3 6 3 3 8 1 4 7 5 7 2 2 0 0 1 7
## [5373] 7 9 5 9 8 9 6 8 8 2 3 6 1 2 9 8 9 5 2 6 2 4 8 4 6 5 0 1 5 6 7 8 9 0
## [5407] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 7 4 2 0 9 0 1 5 8 8 0 2 7 8 4
## [5441] 4 6 1 0 4 5 3 9 4 2 0 5 0 1 3 2 9 1 6 0 1 1 8 0 4 7 7 6 3 6 0 7 3 5
## [5475] 4 2 4 1 8 3 5 6 7 0 6 7 1 2 5 8 1 9 3 8 2 8 7 6 7 1 4 6 2 9 3 0 1 2
## [5509] 3 4 5 6 7 0 1 2 3 4 5 0 1 2 8 9 1 4 0 9 5 0 8 0 7 7 1 1 2 9 3 6 7 2
## [5543] 3 8 1 2 9 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2 7 4 9 1 0 6 8 5 5 5 3 5 9
## [5577] 7 4 8 5 9 6 9 3 0 3 8 9 1 8 1 6 0 0 1 2 3 4 5 6 9 8 9 0 1 2 3 4 5 6
## [5611] 7 8 9 0 1 2 3 4 5 6 7 8 9 3 5 3 2 9 3 2 1 4 5 5 3 3 2 1 3 9 7 2 8 2
## [5645] 8 9 1 8 8 7 8 1 0 0 7 7 8 7 5 0 6 1 5 7 4 6 1 2 5 0 7 9 9 0 3 8 1 4
## [5679] 8 1 8 6 5 9 0 0 0 3 7 1 6 4 2 6 6 0 4 5 4 1 3 8 6 3 9 9 5 9 3 7 8 5
## [5713] 6 4 7 6 2 2 0 9 4 0 1 2 3 4 5 6 7 8 9 0 1 2 7 5 6 0 1 2 3 4 5 6 8 7
## [5747] 1 3 2 8 0 7 5 9 9 6 0 9 4 1 3 2 1 2 3 8 3 2 6 5 6 8 2 7 4 8 1 8 0 5
## [5781] 3 9 4 1 9 2 1 9 6 7 9 0 4 6 1 7 3 8 7 2 9 6 5 8 3 9 0 5 7 1 6 1 0 9
## [5815] 3 3 4 4 0 6 2 5 4 2 3 4 6 0 0 2 0 1 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0
## [5849] 1 2 3 4 5 6 7 8 9 8 7 1 3 7 5 2 8 0 7 5 9 9 0 9 1 1 5 8 8 6 3 2 1 8
## [5883] 3 2 6 5 6 7 4 1 0 5 3 1 9 2 1 9 6 0 4 6 1 7 3 8 7 2 9 6 5 8 3 5 7 1
## [5917] 6 1 0 9 6 2 5 4 2 3 4 4 6 0 0 2 0 1 2 3 9 3 6 7 8 9 0 1 2 3 4 5 6 7
## [5951] 8 9 0 1 2 8 4 5 6 7 8 9 8 6 5 0 6 8 9 4 1 9 5 8 0 4 8 9 1 4 0 5 5 2
## [5985] 1 5 4 0 7 6 0 1 7 0 6 8 9 5 1 7 9 8 6 0 8 1 7 7 1 3 2 5 1 4 2 0 0 7
## [6019] 8 4 6 4 9 3 8 4 7 2 5 6 3 6 9 6 3 2 2 4 6 9 0 2 5 5 1 3 3 9 7 8 7 2
## [6053] 2 5 7 9 8 2 1 3 1 3 0 1 2 8 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3
## [6087] 4 5 6 7 8 9 1 2 6 5 3 0 7 0 4 1 4 3 6 7 2 3 1 2 1 2 9 6 0 1 3 0 2 7
## [6121] 5 7 6 2 9 1 9 0 6 0 6 0 2 0 6 1 5 8 4 3 0 1 5 4 4 8 5 7 5 7 8 3 4 8
## [6155] 8 5 2 9 7 1 3 8 1 0 7 5 5 6 9 4 7 7 9 9 3 4 4 3 8 6 2 0 1 2 3 4 5 6
## [6189] 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 8 3 9 5 5 2 6 8 4 9
## [6223] 1 7 1 2 3 5 9 6 9 1 1 1 2 9 5 6 8 1 2 0 7 7 5 8 2 9 8 9 0 4 6 7 1 3
## [6257] 4 5 6 0 3 6 8 7 0 4 2 7 4 7 5 4 3 4 2 8 1 5 1 2 0 2 5 6 4 3 0 0 0 3
## [6291] 3 5 7 0 6 4 8 8 6 3 4 6 9 9 8 2 7 7 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
## [6325] 5 6 7 8 0 1 2 3 4 5 6 7 8 2 1 7 2 5 0 8 0 2 7 8 8 3 6 0 2 7 6 6 1 2
## [6359] 8 8 7 7 4 7 7 3 7 4 5 4 3 3 8 4 1 1 9 7 4 3 7 3 3 0 2 5 5 6 6 3 5 2
## [6393] 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1 1 9 8 9 2 3 5 5 9 4 2 1 9 3 9 2 0 6
## [6427] 0 4 0 0 1 2 3 4 7 8 9 0 1 2 3 7 8 9 0 1 2 3 4 7 8 9 7 3 0 3 1 8 7 6
## [6461] 4 0 2 6 8 3 2 8 1 2 0 7 1 0 4 4 5 8 0 6 2 3 1 5 1 8 5 9 4 0 7 5 8 8
## [6495] 3 8 9 2 6 2 5 3 1 7 3 9 1 9 9 6 0 3 9 2 8 1 4 3 5 2 9 2 5 8 9 5 0 1
## [6529] 2 4 5 6 0 1 2 3 4 5 6 7 1 2 3 4 5 1 0 4 5 6 6 3 4 4 2 9 1 0 6 4 9 7
## [6563] 2 3 3 9 2 0 4 3 3 7 1 5 6 3 1 2 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 8
## [6597] 6 7 6 7 5 1 0 8 1 6 7 2 9 7 9 5 8 6 2 6 2 8 1 7 5 0 1 1 3 1 4 9 1 8
## [6631] 6 8 5 0 1 2 3 4 5 6 7 5 9 0 1 2 3 4 7 8 9 5 1 7 8 9 9 8 9 8 4 1 7 7
## [6665] 3 3 7 6 6 6 1 9 0 1 7 6 3 2 1 7 1 3 9 1 7 6 8 4 1 4 3 6 9 6 1 4 4 7
## [6699] 2 4 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 2 3 4 7 8 1 3 5 1 7 7
## [6733] 2 1 4 8 3 4 4 3 9 7 4 1 2 3 5 9 1 6 0 1 0 0 2 9 7 1 1 4 0 4 7 3 6 8
## [6767] 0 3 7 4 0 6 9 2 6 5 8 6 9 0 4 0 6 6 9 2 0 9 5 1 3 7 6 9 3 0 2 2 0 1
## [6801] 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2 1 7 2 5 0
## [6835] 8 0 2 7 8 8 3 0 6 0 2 7 6 6 1 2 8 8 7 7 4 7 7 3 7 4 5 4 3 3 8 4 5 4
## [6869] 1 1 9 7 4 3 7 3 3 0 2 5 5 6 3 1 5 2 5 9 9 8 4 1 0 6 0 9 6 8 8 5 6 1
## [6903] 1 9 8 9 2 3 5 5 9 4 2 1 9 4 9 1 3 9 2 0 6 0 4 0 6 0 1 2 3 4 5 6 7 8
## [6937] 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3 8 0 7 1 0 7 5 5 6 9 0 1
## [6971] 0 0 8 3 4 3 1 5 0 0 9 5 3 4 9 3 7 6 9 2 4 5 7 2 6 4 9 4 9 4 1 2 2 5
## [7005] 8 1 3 2 9 4 3 8 2 2 1 2 8 6 5 1 6 7 2 1 3 9 3 8 7 5 7 0 7 4 8 8 5 0
## [7039] 6 6 3 7 6 9 9 4 8 4 1 0 6 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
## [7073] 0 1 2 3 4 5 6 7 8 9 7 4 0 4 0 1 7 9 5 1 4 2 8 9 4 3 7 8 2 4 4 3 3 6
## [7107] 9 9 5 8 6 7 0 6 8 2 6 3 9 3 2 8 6 1 7 4 8 8 9 0 3 3 9 0 5 2 9 4 1 0
## [7141] 3 7 5 8 7 7 8 2 9 7 1 2 6 4 2 5 2 3 6 6 5 0 0 2 8 1 6 1 0 4 3 1 6 1
## [7175] 9 0 1 4 5 6 7 8 9 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 8 4 0 0 7 2 4 3
## [7209] 8 6 6 3 2 6 3 3 3 1 4 7 8 0 3 1 9 0 1 9 1 2 7 0 1 3 8 2 9 2 7 6 5 5
## [7243] 9 9 8 2 9 1 3 2 3 4 3 1 9 0 9 3 6 8 7 0 1 0 5 8 2 7 7 0 1 2 3 4 5 6
## [7277] 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 7 4 8 1 5 6 5 7 2 8
## [7311] 6 3 3 8 6 5 4 0 9 1 7 2 9 1 5 1 3 2 2 3 0 6 4 3 7 6 9 0 4 8 1 4 0 6
## [7345] 1 2 6 9 2 2 3 5 5 1 0 7 7 9 6 2 9 4 7 0 2 3 4 0 0 8 8 8 5 1 3 7 4 9
## [7379] 8 8 9 0 9 8 9 0 2 6 5 6 7 4 7 5 4 1 3 5 3 1 2 3 4 5 6 1 2 3 4 6 0 1
## [7413] 2 4 5 6 7 8 1 7 2 4 1 4 1 4 9 6 8 4 5 3 7 8 8 3 3 5 6 7 0 6 1 6 8 7
## [7447] 0 1 5 0 8 5 0 1 5 8 4 2 3 9 7 6 9 1 9 0 6 7 1 2 3 9 2 4 5 5 3 7 5 3
## [7481] 1 5 2 2 3 0 2 9 4 9 7 0 2 7 4 9 9 2 5 9 8 3 8 6 7 0 0 1 2 3 4 5 6 7
## [7515] 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 7 2 6 5 5 3 7 8 6 6
## [7549] 6 6 4 3 8 8 3 0 1 9 0 5 4 1 9 1 2 7 0 1 3 8 2 9 2 7 4 2 6 5 5 9 9 1
## [7583] 1 5 7 6 8 2 9 4 3 1 9 0 9 3 6 8 7 0 1 0 5 8 2 7 7 0 1 2 3 4 5 6 7 8
## [7617] 9 0 1 2 3 4 5 8 9 0 1 2 3 4 5 6 7 8 9 2 1 2 1 3 9 9 8 5 3 7 0 7 7 5
## [7651] 7 9 9 4 7 0 3 4 1 5 8 1 4 8 4 1 8 6 6 4 6 0 5 5 3 3 5 7 2 5 9 6 9 2
## [7685] 6 2 1 2 0 8 3 8 3 0 8 7 4 9 5 0 9 7 0 0 4 6 0 9 1 6 2 7 6 8 3 5 2 1
## [7719] 8 3 8 6 1 0 2 1 4 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
## [7753] 5 6 7 8 9 7 6 4 7 6 2 3 4 8 7 8 6 9 8 3 2 2 8 4 8 5 6 5 0 2 0 1 1 2
## [7787] 9 6 8 2 1 0 6 5 2 9 7 5 3 9 3 7 1 8 3 8 1 9 5 5 0 1 1 9 8 2 6 0 4 5
## [7821] 0 3 1 8 6 7 5 9 9 3 0 3 1 4 4 0 4 9 0 1 2 3 5 6 7 8 0 1 2 3 5 6 7 8
## [7855] 9 0 1 2 3 5 6 7 8 9 9 7 0 9 0 1 5 8 8 0 9 3 2 7 8 4 6 1 0 4 9 4 2 0
## [7889] 5 0 1 6 9 3 2 9 1 6 0 1 1 8 7 7 6 3 6 0 7 2 4 1 7 0 6 7 1 2 5 8 1 8
## [7923] 2 8 7 6 8 7 1 6 2 9 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
## [7957] 3 4 5 6 7 8 9 8 9 5 7 0 3 1 6 8 4 1 5 6 4 2 7 8 1 3 4 3 4 7 2 0 5 0
## [7991] 1 9 2 3 2 3 5 5 7 8 4 9 9 7 1 1 9 0 7 8 3 4 8 6 3 8 0 9 6 2 1 0 1 0
## [8025] 6 2 3 8 9 0 7 2 3 4 5 5 2 8 5 4 6 6 6 7 9 1 8 2 1 5 3 4 7 9 4 0 0 0
## [8059] 1 1 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 9 0 1 3 1 5 1 2
## [8093] 4 9 8 4 6 8 0 1 1 9 2 6 6 8 7 4 2 9 7 0 2 1 0 3 6 0 1 2 3 4 5 6 7 8
## [8127] 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 8 6 5 9 7 0 2 3 4 3 8 5 1
## [8161] 5 2 3 0 1 2 1 3 2 6 5 3 0 7 2 7 4 6 4 0 5 9 9 8 9 5 3 1 7 4 7 6 5 4
## [8195] 0 0 6 6 2 0 6 3 7 7 4 4 3 9 2 8 9 6 0 9 5 3 8 8 7 1 4 0 4 8 5 2 3 9
## [8229] 0 1 9 1 5 1 7 4 8 6 2 1 6 8 8 0 1 2 3 4 7 8 9 0 1 2 3 4 6 7 8 9 0 1
## [8263] 2 3 4 7 8 9 1 4 5 3 3 0 9 5 4 3 0 8 4 6 7 0 7 7 1 6 9 1 3 6 2 3 8 2
## [8297] 3 8 9 5 8 8 7 1 7 1 1 0 3 4 2 6 4 7 4 2 7 4 2 9 2 7 9 2 1 6 6 5 3 4
## [8331] 8 5 9 6 9 0 6 3 0 8 1 6 0 0 1 2 3 4 5 6 7 0 1 2 3 4 7 8 9 0 1 2 3 4
## [8365] 7 2 5 1 6 4 3 9 9 0 9 7 1 6 4 3 6 2 0 9 8 6 5 7 0 0 1 7 4 3 2 4 1 3
## [8399] 7 6 4 7 7 7 9 8 4 3 8 2 8 3 5 8 0 5 4 7 1 3 1 7 9 6 2 0 9 1 7 3 3 9
## [8433] 1 6 4 3 9 8 2 1 8 6 4 1 5 5 6 5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
## [8467] 8 9 0 1 2 3 4 5 6 7 8 9 6 9 7 0 2 3 4 3 8 5 1 3 0 1 2 1 3 2 0 7 2 6
## [8501] 4 0 5 9 9 8 9 5 3 1 7 4 7 0 0 6 6 6 3 7 4 2 8 9 8 7 1 9 0 4 8 5 2 3
## [8535] 9 0 1 9 1 5 1 7 6 1 2 1 6 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 0
## [8569] 1 2 3 5 6 7 8 1 0 4 5 6 6 3 4 4 2 8 1 0 6 4 9 7 2 9 2 0 9 3 3 9 1 5
## [8603] 2 3 1 6 7 3 7 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 4 8 6 0 5 7 5 1 0 8
## [8637] 1 6 7 2 9 7 9 5 6 5 2 6 2 8 1 7 5 5 7 3 5 0 1 1 3 8 4 9 4 5 1 8 6 8
## [8671] 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3 5 3
## [8705] 2 9 3 2 1 4 5 5 2 3 2 1 3 9 7 2 1 2 8 9 1 8 8 7 8 1 0 0 6 7 7 8 7 5
## [8739] 0 6 1 5 7 4 6 1 2 5 0 7 9 9 0 3 4 4 8 4 1 8 6 5 9 0 0 0 3 7 1 6 4 6
## [8773] 0 4 5 4 1 3 8 6 3 9 9 5 9 3 7 8 5 6 4 7 6 2 2 0 9 4 0 1 2 3 4 5 6 7
## [8807] 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 6 4 2 6 4 7 5 5 4 7 2 9
## [8841] 3 9 3 8 2 0 9 5 6 0 1 0 6 5 3 5 3 8 0 0 3 4 1 5 3 0 8 3 0 6 2 7 8 1
## [8875] 7 1 3 8 5 4 2 0 9 7 6 7 4 1 6 2 6 7 1 9 8 0 6 9 4 9 9 6 2 3 7 1 9 2
## [8909] 2 5 3 7 8 0 1 2 3 4 7 8 9 0 1 2 3 4 7 8 9 0 1 7 8 9 8 9 2 6 1 3 5 4
## [8943] 8 2 6 4 3 4 5 9 2 0 3 9 4 9 7 3 8 7 4 4 9 8 5 8 2 6 6 2 3 1 3 2 7 3
## [8977] 1 9 0 1 1 3 5 0 7 8 1 5 1 4 6 0 0 4 9 1 6 6 9 0 7 6 1 1 0 1 2 3 4 2
## [9011] 2 3 4 5 6 2 0 1 2 2 8 6 3 9 2 1 9 3 9 6 1 7 2 4 4 5 7 0 0 1 6 6 8 2
## [9045] 7 7 2 4 2 1 6 1 0 6 9 8 3 9 6 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
## [9079] 8 9 0 1 2 3 4 5 6 7 8 9 1 6 8 9 9 0 1 2 4 4 3 7 4 4 4 0 3 8 7 5 8 2
## [9113] 1 7 5 3 8 5 2 5 1 1 6 2 1 3 8 6 4 2 6 2 5 5 0 2 8 0 6 8 1 7 9 1 9 2
## [9147] 6 7 6 6 8 7 4 9 2 1 3 3 0 5 5 8 0 3 7 9 7 0 2 7 9 1 7 8 0 3 5 3 6 0
## [9181] 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 7 8 9 6 4 2 6 4 7 8
## [9215] 9 2 9 3 9 3 0 0 1 0 4 2 6 3 5 3 0 3 4 1 5 3 0 8 3 0 6 1 7 8 0 9 2 6
## [9249] 7 1 9 6 9 4 9 9 6 7 1 2 5 3 7 8 0 1 2 4 5 6 7 8 9 0 1 3 4 5 6 7 5 0
## [9283] 1 3 4 7 8 9 7 5 5 1 9 9 7 1 0 0 5 9 7 1 7 2 2 3 6 8 3 2 0 0 6 1 7 5
## [9317] 8 6 2 9 4 8 8 7 1 0 8 7 7 5 8 5 3 4 6 1 1 5 5 0 7 2 3 6 4 1 2 4 1 5
## [9351] 4 2 0 4 8 6 1 9 0 2 5 6 9 3 6 3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6
## [9385] 7 8 9 0 1 2 3 5 6 7 8 1 0 9 5 7 5 1 8 6 9 0 4 1 9 3 8 4 4 7 0 1 9 2
## [9419] 8 7 8 2 5 9 6 0 6 5 5 3 3 3 9 8 1 1 0 6 1 0 0 6 2 1 1 3 2 7 7 8 8 7
## [9453] 8 4 6 0 2 0 7 0 3 6 8 7 1 5 9 9 3 7 2 4 9 4 3 6 2 2 5 3 2 5 5 9 4 1
## [9487] 7 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 0
## [9521] 1 2 7 5 3 4 4 0 0 6 9 6 6 5 7 2 3 4 4 9 1 4 0 7 9 5 7 2 3 1 4 4 0 9
## [9555] 9 6 1 8 3 3 7 3 9 8 8 4 7 7 6 2 1 9 8 7 8 8 7 2 2 3 9 3 3 5 5 0 7 4
## [9589] 5 6 5 1 4 1 1 2 8 2 6 1 5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
## [9623] 1 2 3 4 5 6 7 8 8 0 6 0 1 2 3 7 9 4 7 1 9 1 7 1 4 0 0 1 7 5 7 1 3 3
## [9657] 3 1 6 9 7 1 3 0 7 6 0 8 9 4 3 5 4 8 1 5 9 0 6 5 3 8 1 4 7 5 2 0 0 1
## [9691] 7 8 9 6 8 8 2 3 6 1 8 9 5 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
## [9725] 0 1 2 3 4 6 6 7 8 9 7 4 6 1 4 0 9 9 3 7 8 0 7 5 8 6 3 2 2 0 5 8 6 0
## [9759] 3 8 1 0 3 0 4 7 4 9 8 9 0 7 1 7 1 6 6 5 6 2 8 7 6 4 9 9 5 3 7 4 3 0
## [9793] 1 6 6 1 1 3 2 1 0 0 1 2 3 4 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9
## [9827] 0 8 3 9 5 5 2 6 8 4 1 7 1 7 3 5 6 9 1 1 1 2 1 2 0 7 7 5 8 2 9 8 6 7
## [9861] 3 4 6 8 7 0 4 2 7 7 5 4 3 4 2 8 1 5 1 0 2 3 3 5 7 0 6 8 6 3 9 9 8 2
## [9895] 7 7 1 0 1 7 8 9 0 1 2 3 4 5 6 7 8 0 1 2 3 4 7 8 9 7 8 6 4 1 9 3 8 4
## [9929] 4 7 0 1 9 2 8 7 8 2 6 0 6 5 3 3 3 9 1 4 0 6 1 0 0 6 2 1 1 7 7 8 4 6
## [9963] 0 7 0 3 6 8 7 1 5 2 4 9 4 3 6 4 1 7 2 6 6 0 1 2 3 4 5 6 7 8 9 0 1 2
## [9997] 3 4 5 6Keras provides a vocabulary for building deep learning models that is simple, elegant, and intuitive. Building a question answering system, an image classification model, a neural Turing machine, or any other model is just as straightforward.
Of course, it takes more work than that. It assumes you have some degree of familiarity with R and/or Python, Keras, TensorFlow etc.
It is differently possible with some work; ok, a good bit of elbow grease.
Gary Thomas, Data Analyst