Keras package

Introduction to keras package

• Keras has the following key features:
• Allows the same code to run on CPU or on GPU, seamlessly. *.User-friendly API which makes it easy to quickly prototype deep learning models.
• Built-in support for convolutional networks (for computer vision), recurrent networks (for sequence processing), and any combination of both.
• Supports arbitrary network architectures: multi-input or multi-output models, layer sharing, model sharing, etc. This means that Keras is appropriate for building essentially any deep learning model, from a memory network to a neural Turing machine.
• Is capable of running on top of multiple back-ends including TensorFlow, CNTK, or Theano.
• The good news about Keras and TensorFlow is that you don’t need to choose between them! The default backend for Keras is TensorFlow and Keras can be integrated seamlessly with TensorFlow workflows. There is also a pure-TensorFlow implementation of Keras with deeper integration on the roadmap for later this year.
• Keras and TensorFlow are the state of the art in deep learning tools and with the keras package you can now access both with a fluent R interface.

Dataset Information:

#Getting Started
#Installation
#To begin, install the keras R package from CRAN as follows:
#The Keras R interface uses the TensorFlow backend engine by default. To install both the core Keras library as well as the TensorFlow backend use the install_keras() function:
library(keras)
#install_keras()
#This will provide you with default CPU-based installations of Keras and TensorFlow. If you want a more customized installation, e.g. if you want to take advantage of NVIDIA GPUs, see the documentation for install_keras().

#MNIST Example
#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()
x_train <- mnist$train$x
y_train <- mnist$train$y
x_test <- mnist$test$x
y_test <-  mnist$test$y

# The 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
dim(x_train) <- c(nrow(x_train),784)
dim(x_test) <- c(nrow(x_test),784)

# rescale

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

# 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)

## ___________________________________________________________________________
## Layer (type)                     Output Shape                  Param #     
## ===========================================================================
## dense_1 (Dense)                  (None, 256)                   200960      
## ___________________________________________________________________________
## dropout_1 (Dropout)              (None, 256)                   0           
## ___________________________________________________________________________
## dense_2 (Dense)                  (None, 128)                   32896       
## ___________________________________________________________________________
## dropout_2 (Dropout)              (None, 128)                   0           
## ___________________________________________________________________________
## dense_3 (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

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,verbose = 0)

## $loss
## [1] 0.1147017
## 
## $acc
## [1] 0.981

# Generate predictions on new data:
 model %>% predict_classes(x_test)

##     [1] 7 2 1 0 4 1 4 9 5 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 9 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 2 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
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##   [341] 3 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 7 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
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##   [443] 3 0 2 0 6 4 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 2 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
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##   [681] 6 9 1 3 3 8 3 3 6 7 2 4 5 8 5 1 1 4 4 3 1 0 7 7 0 7 9 9 4 8 5 5 4 0
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##   [783] 3 1 4 5 6 8 9 4 1 5 3 8 0 3 2 5 1 2 8 3 4 4 0 8 8 3 3 1 7 3 5 9 6 3
##   [817] 2 6 1 3 6 0 7 2 1 7 1 4 2 4 2 1 7 9 6 1 1 2 4 8 1 7 7 4 7 0 7 3 1 3
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##   [885] 0 6 6 3 2 1 3 2 2 9 3 0 0 5 7 8 3 4 4 6 0 2 9 1 4 7 4 7 3 9 8 8 4 7
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##   [953] 6 4 9 5 2 3 3 4 7 8 9 1 1 5 9 1 4 4 5 4 0 6 2 2 3 1 5 1 2 0 3 8 1 2
##   [987] 6 7 1 6 2 3 9 0 1 2 2 0 8 9 9 0 2 5 1 9 7 8 1 0 4 1 7 9 5 4 2 6 8 1
##  [1021] 3 7 5 4 4 1 8 1 3 8 1 2 5 8 0 6 2 1 1 2 1 5 3 4 8 9 5 0 9 2 2 4 8 2
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##  [1089] 9 5 9 7 3 8 0 3 7 1 3 6 7 8 5 9 7 9 6 9 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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##  [1191] 5 0 9 1 7 0 6 0 8 6 8 1 8 0 3 3 7 2 3 6 2 1 6 1 1 3 7 9 0 8 0 5 4 0
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##  [1259] 5 8 7 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 9 3 3
##  [1293] 6 6 0 1 6 0 3 7 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 7 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 7 9 8 3 0 7 2 7 9 4 5 4 9 3 2 1 4 0 2 3 7 5 7 8 8 5 0 1 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 3 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 3 0 3 3 8 4 3 6 3 5 0 2 0 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 2 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 9 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 2 5 1 2 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 7 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 2 2 9 0 8 7 1 1 6 3 5 1 1
##  [1837] 1 3 1 2 3 0 2 0 1 3 5 5 7 4 3 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 7 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 2 1 2 9 3 5 4 0 7 3 6 1 7 5 5 3 3 0 1 3 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 2 8 5 2 8 5 7 3 0 8
##  [2109] 2 7 2 8 2 5 5 7 6 4 6 8 4 8 2 7 4 9 2 0 3 9 4 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 8 8 1 4 1 7 8 4 6 1 8 4 3 1
##  [2177] 2 8 0 8 5 9 1 4 2 0 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 8 1 9 7 1 1
##  [2279] 7 5 3 3 5 1 3 7 6 1 3 8 7 5 9 0 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 7 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 9 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 5 0 9 4 6 2 5 0 2 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 7 3
##  [2585] 8 9 5 3 7 9 1 7 0 0 3 7 2 5 1 1 8 6 2 9 5 7 5 1 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 0 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] 4 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 5 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 4 2 3 6 1 1 8 9 3 4 0 7 9 6 7 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 2 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 3 5 0 3 5 5
##  [2959] 9 2 8 7 0 4 9 1 9 7 7 5 7 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 3 2 7 6 0 1 2 7 1 0 8 3 6 0 5 3 6 2 8 7 0 1 4 2
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##  [3265] 1 0 0 2 1 6 6 0 1 6 2 5 1 7 4 8 2 1 4 3 2 3 9 9 4 9 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 5 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 7 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 2 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 9 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
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##  [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 5 3 9 4 9 2 5 1 5 1 4 4 1 4 4 3 3 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 7 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 7 3 0 8 9 5 0 3 2 5 4 0 8 8 4 5 8 8 4 5 4 8 5 4 9 2 2
##  [4387] 1 2 6 8 8 7 0 3 6 6 4 3 8 8 7 2 2 0 0 9 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 3 9 2 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 1 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 8 8 2 1 9 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 4 9 0 6 6 6 3 6 7
##  [4625] 7 2 8 6 0 8 3 0 2 9 8 3 2 5 3 9 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 3 1 1 2 9 7 8 4 0 5 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 3 3 5 5 7 2 4 6 7 2 8 3 0 8 7 8 4 0 8 4 4
##  [4829] 5 8 5 6 6 2 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 4 7
##  [4863] 0 8 1 7 4 5 7 1 2 1 1 3 9 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 7 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 3 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 5 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 7 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 2 3 2 1 3 9 7 2 5 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 2 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 3 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 0 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 9 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 3 4 5 6 7 8 3 0 1 2 3 4 5 6 7 8 9 0 1 2 3
##  [6087] 4 5 6 7 8 9 1 8 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 3 6 3 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 9 3 3 7 1 5 6 3 1 7 8 4 0 2 4 0 2 4 7 8 0 7 0 6 9 3 2 8
##  [6597] 6 7 5 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 7 4 9 1 8
##  [6631] 6 8 9 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 7 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 4 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 4 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 8 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 0
##  [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 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 9 0 1 3 1 5 1 8
##  [8093] 4 9 2 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 9 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 9 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 9 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 8 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 7 3 5 4 8 1 5 9 0 6 3 3 8 1 4 7 5 2 0 0 1
##  [9691] 7 8 7 6 8 8 2 3 6 1 2 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 5 3 2 2 0 5 5 6 0
##  [9759] 3 8 1 0 3 0 4 7 4 9 0 9 0 7 1 7 1 6 6 5 6 2 8 7 6 4 9 9 5 3 7 4 3 0
##  [9793] 7 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 6

# Keras 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.

Code Reference : Keras for R - by - JJ Allaire

blog.rstudio.com