From b95146b3e7172ba164de8bbc2afe0d9cc4dba4f6 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Fri, 16 Oct 2020 11:19:23 +0200 Subject: [PATCH] update week 42 --- doc/pub/week42/html/._week42-bs000.html | 20 +- doc/pub/week42/html/._week42-bs001.html | 20 +- doc/pub/week42/html/._week42-bs002.html | 20 +- doc/pub/week42/html/._week42-bs003.html | 20 +- doc/pub/week42/html/._week42-bs004.html | 20 +- doc/pub/week42/html/._week42-bs005.html | 20 +- doc/pub/week42/html/._week42-bs006.html | 20 +- doc/pub/week42/html/._week42-bs007.html | 20 +- doc/pub/week42/html/._week42-bs008.html | 20 +- doc/pub/week42/html/._week42-bs009.html | 20 +- doc/pub/week42/html/._week42-bs010.html | 20 +- doc/pub/week42/html/._week42-bs011.html | 20 +- doc/pub/week42/html/._week42-bs012.html | 20 +- doc/pub/week42/html/._week42-bs013.html | 20 +- doc/pub/week42/html/._week42-bs014.html | 20 +- doc/pub/week42/html/._week42-bs015.html | 20 +- doc/pub/week42/html/._week42-bs016.html | 20 +- doc/pub/week42/html/._week42-bs017.html | 20 +- doc/pub/week42/html/._week42-bs018.html | 20 +- doc/pub/week42/html/._week42-bs019.html | 20 +- doc/pub/week42/html/._week42-bs020.html | 20 +- doc/pub/week42/html/._week42-bs021.html | 21 +- doc/pub/week42/html/._week42-bs022.html | 22 +- doc/pub/week42/html/._week42-bs023.html | 23 +- doc/pub/week42/html/._week42-bs024.html | 24 +- doc/pub/week42/html/week42-bs.html | 20 +- doc/pub/week42/html/week42-reveal.html | 587 ++++++++++++++++++ doc/pub/week42/html/week42-solarized.html | 595 +++++++++++++++++- doc/pub/week42/html/week42.html | 595 +++++++++++++++++- doc/pub/week42/ipynb/ipynb-week42-src.tar.gz | Bin 87344 -> 87344 bytes doc/pub/week42/ipynb/week42.ipynb | 618 +++++++++++++++++++ doc/src/week42/week42.do.txt | 568 +++++++++++++++++ 32 files changed, 3444 insertions(+), 49 deletions(-) diff --git a/doc/pub/week42/html/._week42-bs000.html b/doc/pub/week42/html/._week42-bs000.html index 80d4c12ae..71ebbd12c 100644 --- a/doc/pub/week42/html/._week42-bs000.html +++ b/doc/pub/week42/html/._week42-bs000.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -214,7 +230,7 @@ MathJax.Hub.Config({
  • 9
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  • ...
    • -
  • 30
    • +
  • 35
  • »
    • diff --git a/doc/pub/week42/html/._week42-bs001.html b/doc/pub/week42/html/._week42-bs001.html index 4925a04a6..7697e4771 100644 --- a/doc/pub/week42/html/._week42-bs001.html +++ b/doc/pub/week42/html/._week42-bs001.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -210,7 +226,7 @@ Reading suggestions for both days: 10
  • 11
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs002.html b/doc/pub/week42/html/._week42-bs002.html index b66be4f85..511791796 100644 --- a/doc/pub/week42/html/._week42-bs002.html +++ b/doc/pub/week42/html/._week42-bs002.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -225,7 +241,7 @@ Another good read is the article here 11
  • 12
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    • -
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs003.html b/doc/pub/week42/html/._week42-bs003.html index 517a0d942..0f6c4b3a0 100644 --- a/doc/pub/week42/html/._week42-bs003.html +++ b/doc/pub/week42/html/._week42-bs003.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -201,7 +217,7 @@ before the transformation.
  • 12
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  • ...
    • -
  • 30
    • +
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs004.html b/doc/pub/week42/html/._week42-bs004.html index f2286bc98..a5614c5a3 100644 --- a/doc/pub/week42/html/._week42-bs004.html +++ b/doc/pub/week42/html/._week42-bs004.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -216,7 +232,7 @@ in the input).
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    • diff --git a/doc/pub/week42/html/._week42-bs005.html b/doc/pub/week42/html/._week42-bs005.html index 04ddd2f77..158a253c3 100644 --- a/doc/pub/week42/html/._week42-bs005.html +++ b/doc/pub/week42/html/._week42-bs005.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -217,7 +233,7 @@ would quickly lead to possible overfitting.
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    • diff --git a/doc/pub/week42/html/._week42-bs006.html b/doc/pub/week42/html/._week42-bs006.html index 444489b28..49436fae3 100644 --- a/doc/pub/week42/html/._week42-bs006.html +++ b/doc/pub/week42/html/._week42-bs006.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -230,7 +246,7 @@ dimension.
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    • diff --git a/doc/pub/week42/html/._week42-bs007.html b/doc/pub/week42/html/._week42-bs007.html index 0cedc966b..01d55f26e 100644 --- a/doc/pub/week42/html/._week42-bs007.html +++ b/doc/pub/week42/html/._week42-bs007.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -214,7 +230,7 @@ A simple CNN for image classification could have the architecture:
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    • diff --git a/doc/pub/week42/html/._week42-bs008.html b/doc/pub/week42/html/._week42-bs008.html index 99dbd52b3..b56f0938d 100644 --- a/doc/pub/week42/html/._week42-bs008.html +++ b/doc/pub/week42/html/._week42-bs008.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -211,7 +227,7 @@ are consistent with the labels in the training set for each image.
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs009.html b/doc/pub/week42/html/._week42-bs009.html index 120bb635f..f2d9f73a6 100644 --- a/doc/pub/week42/html/._week42-bs009.html +++ b/doc/pub/week42/html/._week42-bs009.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -214,7 +230,7 @@ and the slides of 18
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    • diff --git a/doc/pub/week42/html/._week42-bs010.html b/doc/pub/week42/html/._week42-bs010.html index 3379f986c..8eb54ec47 100644 --- a/doc/pub/week42/html/._week42-bs010.html +++ b/doc/pub/week42/html/._week42-bs010.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -210,7 +226,7 @@ matrices, typically 1 for each color dimension (Red, Green, Blue).
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    • diff --git a/doc/pub/week42/html/._week42-bs011.html b/doc/pub/week42/html/._week42-bs011.html index 393cef2c0..dde085d65 100644 --- a/doc/pub/week42/html/._week42-bs011.html +++ b/doc/pub/week42/html/._week42-bs011.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -206,7 +222,7 @@ $$
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs012.html b/doc/pub/week42/html/._week42-bs012.html index 893bc62bb..1243fce94 100644 --- a/doc/pub/week42/html/._week42-bs012.html +++ b/doc/pub/week42/html/._week42-bs012.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -212,7 +228,7 @@ single neuron in the first hidden layer.
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs013.html b/doc/pub/week42/html/._week42-bs013.html index 5005706eb..2e921c3e9 100644 --- a/doc/pub/week42/html/._week42-bs013.html +++ b/doc/pub/week42/html/._week42-bs013.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -212,7 +228,7 @@ fixed, and known as a 22
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs014.html b/doc/pub/week42/html/._week42-bs014.html index e407170a9..5dba37fbe 100644 --- a/doc/pub/week42/html/._week42-bs014.html +++ b/doc/pub/week42/html/._week42-bs014.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -216,7 +232,7 @@ layer.
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    • diff --git a/doc/pub/week42/html/._week42-bs015.html b/doc/pub/week42/html/._week42-bs015.html index bcde70b93..ee7cd1a7e 100644 --- a/doc/pub/week42/html/._week42-bs015.html +++ b/doc/pub/week42/html/._week42-bs015.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -209,7 +225,7 @@ classification.
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    • diff --git a/doc/pub/week42/html/._week42-bs016.html b/doc/pub/week42/html/._week42-bs016.html index 32c817d77..d2421b7e5 100644 --- a/doc/pub/week42/html/._week42-bs016.html +++ b/doc/pub/week42/html/._week42-bs016.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -242,7 +258,7 @@ plt.show()
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    • diff --git a/doc/pub/week42/html/._week42-bs017.html b/doc/pub/week42/html/._week42-bs017.html index 1f66acf6a..f6ee9d522 100644 --- a/doc/pub/week42/html/._week42-bs017.html +++ b/doc/pub/week42/html/._week42-bs017.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -224,7 +240,7 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/week42/html/._week42-bs018.html b/doc/pub/week42/html/._week42-bs018.html index 0fa49ca46..08196da8f 100644 --- a/doc/pub/week42/html/._week42-bs018.html +++ b/doc/pub/week42/html/._week42-bs018.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -229,7 +245,7 @@ lmbd_vals = np.
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    • diff --git a/doc/pub/week42/html/._week42-bs019.html b/doc/pub/week42/html/._week42-bs019.html index 2c99685ce..a4608d53c 100644 --- a/doc/pub/week42/html/._week42-bs019.html +++ b/doc/pub/week42/html/._week42-bs019.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -219,7 +235,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs020.html b/doc/pub/week42/html/._week42-bs020.html index ea8066b4a..6a96e45d6 100644 --- a/doc/pub/week42/html/._week42-bs020.html +++ b/doc/pub/week42/html/._week42-bs020.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -232,6 +248,8 @@ plt.show()
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    • diff --git a/doc/pub/week42/html/._week42-bs021.html b/doc/pub/week42/html/._week42-bs021.html index c3817c327..28e0ce8fe 100644 --- a/doc/pub/week42/html/._week42-bs021.html +++ b/doc/pub/week42/html/._week42-bs021.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -217,6 +233,9 @@ train_images, test_images = train_images 28
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  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -217,6 +233,10 @@ plt.show()
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    • diff --git a/doc/pub/week42/html/._week42-bs023.html b/doc/pub/week42/html/._week42-bs023.html index 7078784da..f45dc6fbd 100644 --- a/doc/pub/week42/html/._week42-bs023.html +++ b/doc/pub/week42/html/._week42-bs023.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -218,6 +234,11 @@ You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tenso
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    • diff --git a/doc/pub/week42/html/._week42-bs024.html b/doc/pub/week42/html/._week42-bs024.html index a04528bc2..a66324221 100644 --- a/doc/pub/week42/html/._week42-bs024.html +++ b/doc/pub/week42/html/._week42-bs024.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -216,6 +232,12 @@ As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (102
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    • diff --git a/doc/pub/week42/html/week42-bs.html b/doc/pub/week42/html/week42-bs.html index 80d4c12ae..71ebbd12c 100644 --- a/doc/pub/week42/html/week42-bs.html +++ b/doc/pub/week42/html/week42-bs.html @@ -89,7 +89,18 @@ Automatically generated HTML file from DocOnce source None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -156,6 +167,11 @@ MathJax.Hub.Config({
  • Recurrent neural networks: Overarching view
  • Set up of an RNN
  • A simple example
    • +
  • An extrapolation example
    • +
  • Formatting the Data
    • +
  • Predicting New Points With A Trained Recurrent Neural Network
    • +
  • Other Things to Try
    • +
  • Other Types of Recurrent Neural Networks
    • @@ -214,7 +230,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/week42-reveal.html b/doc/pub/week42/html/week42-reveal.html index 3e2338e10..abae9c934 100644 --- a/doc/pub/week42/html/week42-reveal.html +++ b/doc/pub/week42/html/week42-reveal.html @@ -939,6 +939,593 @@ plt.show() +
    +

    An extrapolation example

    + +

    +The following code provides an example of how recurrent neural +networks can be used to extrapolate to unknown values of physics data +sets. Specifically, the data sets used in this program come from +a quantum mechanical many-body calculation of energies as functions of the number of particles. + +

    + + +

    # For matrices and calculations
+import numpy as np
+# For machine learning (backend for keras)
+import tensorflow as tf
+# User-friendly machine learning library
+# Front end for TensorFlow
+import tensorflow.keras
+# Different methods from Keras needed to create an RNN
+# This is not necessary but it shortened function calls 
+# that need to be used in the code.
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras import regularizers
+from tensorflow.keras.models import Model, Sequential
+from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
+# For timing the code
+from timeit import default_timer as timer
+# For plotting
+import matplotlib.pyplot as plt
+
+
+# The data set
+datatype='VaryDimension'
+X_tot = np.arange(2, 42, 2)
+y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,
+	-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, 
+	-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])
+
    +
    + + +
    +

    Formatting the Data

    + +

    +The way the recurrent neural networks are trained in this program +differs from how machine learning algorithms are usually trained. +Typically a machine learning algorithm is trained by learning the +relationship between the x data and the y data. In this program, the +recurrent neural network will be trained to recognize the relationship +in a sequence of y values. This is type of data formatting is +typically used time series forcasting, but it can also be used in any +extrapolation (time series forecasting is just a specific type of +extrapolation along the time axis). This method of data formatting +does not use the x data and assumes that the y data are evenly spaced. + +

    +For a standard machine learning algorithm, the training data has the +form of (x,y) so the machine learning algorithm learns to assiciate a +y value with a given x value. This is useful when the test data has x +values within the same range as the training data. However, for this +application, the x values of the test data are outside of the x values +of the training data and the traditional method of training a machine +learning algorithm does not work as well. For this reason, the +recurrent neural network is trained on sequences of y values of the +form ((y1, y2), y3), so that the network is concerned with learning +the pattern of the y data and not the relation between the x and y +data. As long as the pattern of y data outside of the training region +stays relatively stable compared to what was inside the training +region, this method of training can produce accurate extrapolations to +y values far removed from the training data set. + +

    + + + + + + + +

    + + +

    # FORMAT_DATA
+def format_data(data, length_of_sequence = 2):  
+    """
+        Inputs:
+            data(a numpy array): the data that will be the inputs to the recurrent neural
+                network
+            length_of_sequence (an int): the number of elements in one iteration of the
+                sequence patter.  For a function approximator use length_of_sequence = 2.
+        Returns:
+            rnn_input (a 3D numpy array): the input data for the recurrent neural network.  Its
+                dimensions are length of data - length of sequence, length of sequence, 
+                dimnsion of data
+            rnn_output (a numpy array): the training data for the neural network
+        Formats data to be used in a recurrent neural network.
+    """
+
+    X, Y = [], []
+    for i in range(len(data)-length_of_sequence):
+        # Get the next length_of_sequence elements
+        a = data[i:i+length_of_sequence]
+        # Get the element that immediately follows that
+        b = data[i+length_of_sequence]
+        # Reshape so that each data point is contained in its own array
+        a = np.reshape (a, (len(a), 1))
+        X.append(a)
+        Y.append(b)
+    rnn_input = np.array(X)
+    rnn_output = np.array(Y)
+
+    return rnn_input, rnn_output
+
+
+# ## Defining the Recurrent Neural Network Using Keras
+# 
+# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.
+
+def rnn(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with one hidden layer and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer
+    hidden_neurons = 200
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to 
+    # the network immediately after the input layer
+    rnn = SimpleRNN(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN")(inp)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
    +
    + + +
    +

    Predicting New Points With A Trained Recurrent Neural Network

    + +

    + + +

    def test_rnn (x1, y_test, plot_min, plot_max):
+    """
+        Inputs:
+            x1 (a list or numpy array): The complete x component of the data set
+            y_test (a list or numpy array): The complete y component of the data set
+            plot_min (an int or float): the smallest x value used in the training data
+            plot_max (an int or float): the largest x valye used in the training data
+        Returns:
+            None.
+        Uses a trained recurrent neural network model to predict future points in the 
+        series.  Computes the MSE of the predicted data set from the true data set, saves
+        the predicted data set to a csv file, and plots the predicted and true data sets w
+        while also displaying the data range used for training.
+    """
+    # Add the training data as the first dim points in the predicted data array as these
+    # are known values.
+    y_pred = y_test[:dim].tolist()
+    # Generate the first input to the trained recurrent neural network using the last two 
+    # points of the training data.  Based on how the network was trained this means that it
+    # will predict the first point in the data set after the training data.  All of the 
+    # brackets are necessary for Tensorflow.
+    next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])
+    # Save the very last point in the training data set.  This will be used later.
+    last = [y_test[dim-1]]
+
+    # Iterate until the complete data set is created.
+    for i in range (dim, len(y_test)):
+        # Predict the next point in the data set using the previous two points.
+        next = model.predict(next_input)
+        # Append just the number of the predicted data set
+        y_pred.append(next[0][0])
+        # Create the input that will be used to predict the next data point in the data set.
+        next_input = np.array([[last, next[0]]], dtype=np.float64)
+        last = next
+
+    # Print the mean squared error between the known data set and the predicted data set.
+    print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())
+    # Save the predicted data set as a csv file for later use
+    name = datatype + 'Predicted'+str(dim)+'.csv'
+    np.savetxt(name, y_pred, delimiter=',')
+    # Plot the known data set and the predicted data set.  The red box represents the region that was used
+    # for the training data.
+    fig, ax = plt.subplots()
+    ax.plot(x1, y_test, label="true", linewidth=3)
+    ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4)
+    ax.legend()
+    # Created a red region to represent the points used in the training data.
+    ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')
+    plt.show()
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn(length_of_sequences = rnn_input.shape[1])
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    + + +
    +

    Other Things to Try

    + +

    +Changing the size of the recurrent neural network and its parameters +can drastically change the results you get from the model. The below +code takes the simple recurrent neural network from above and adds a +second hidden layer, changes the number of neurons in the hidden +layer, and explicitly declares the activation function of the hidden +layers to be a sigmoid function. The loss function and optimizer can +also be changed but are kept the same as the above network. These +parameters can be tuned to provide the optimal result from the +network. For some ideas on how to improve the performance of a +recurrent neural network. + +

    + + +

    def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two hidden layers and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer, increased from the first network
+    hidden_neurons = 500
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Create two hidden layers instead of one hidden layer.  Explicitly set the activation
+    # function to be the sigmoid function (the default value is hyperbolic tangent)
+    rnn1 = SimpleRNN(hidden_neurons, 
+                    return_sequences=True,  # This needs to be True if another hidden layer is to follow
+                    stateful = stateful, activation = 'sigmoid',
+                    name="RNN1")(inp)
+    rnn2 = SimpleRNN(hidden_neurons, 
+                    return_sequences=False, activation = 'sigmoid',
+                    stateful = stateful,
+                    name="RNN2")(rnn1)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn2)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn_2layers(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    + + +
    +

    Other Types of Recurrent Neural Networks

    + +

    +Besides a simple recurrent neural network layer, there are two other +commonly used types of recurrent neural network layers: Long Short +Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +introduction to these layers see https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b +and https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b. + +

    +The first network created below is similar to the previous network, +but it replaces the SimpleRNN layers with LSTM layers. The second +network below has two hidden layers made up of GRUs, which are +preceeded by two dense (feeddorward) neural network layers. These +dense layers "preprocess" the data before it reaches the recurrent +layers. This architecture has been shown to improve the performance +of recurrent neural networks (see the link above and also +https://arxiv.org/pdf/1807.02857.pdf. + +

    + + +

    def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.
+    """
+    # Number of neurons on the input/output layer and the number of neurons in the hidden layer
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input Layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)
+    rnn= LSTM(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN", use_bias=True, activation='tanh')(inp)
+    rnn1 = LSTM(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True, activation='tanh')(rnn)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn1)
+    # Define the midel
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the model
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with four hidden layers (two dense followed by
+        two GRU layers) and returns the model.
+    """    
+    # Number of neurons on the input/output layers and hidden layers
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden Dense (feedforward) layers
+    dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)
+    dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)
+    # Hidden GRU layers
+    rnn1 = GRU(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True)(dnn1)
+    rnn = GRU(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN", use_bias=True)(rnn1)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Define the model
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the mdoel
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Change the method name to reflect which network you want to use
+model = dnn2_gru2(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
+
+# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)
+# 
+# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+# Reshape the data for Keras specifications
+X_train = X_train.reshape((dim, 1))
+y_train = y_train.reshape((dim, 1))
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Set the sequence length to 1 for regular data formatting 
+model = rnn(length_of_sequences = 1)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(X_train, y_train, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict the remaining data points
+X_pred = X_tot[dim:]
+X_pred = X_pred.reshape((len(X_pred), 1))
+y_model = model.predict(X_pred)
+y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))
+
+# Plot the known data set and the predicted data set.  The red box represents the region that was used
+# for the training data.
+fig, ax = plt.subplots()
+ax.plot(X_tot, y_tot, label="true", linewidth=3)
+ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4)
+ax.legend()
+# Created a red region to represent the points used in the training data.
+ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')
+plt.show()
+
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +
    + + diff --git a/doc/pub/week42/html/week42-solarized.html b/doc/pub/week42/html/week42-solarized.html index 15acaea3a..d00badbb0 100644 --- a/doc/pub/week42/html/week42-solarized.html +++ b/doc/pub/week42/html/week42-solarized.html @@ -109,7 +109,18 @@ div { text-align: justify; text-justify: inter-word; } None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -921,6 +932,588 @@ plt.axvline(df.index[Tp], c="r") plt.show()

    +









    + +

    An extrapolation example

    + +

    +The following code provides an example of how recurrent neural +networks can be used to extrapolate to unknown values of physics data +sets. Specifically, the data sets used in this program come from +a quantum mechanical many-body calculation of energies as functions of the number of particles. + +

    + + +

    # For matrices and calculations
+import numpy as np
+# For machine learning (backend for keras)
+import tensorflow as tf
+# User-friendly machine learning library
+# Front end for TensorFlow
+import tensorflow.keras
+# Different methods from Keras needed to create an RNN
+# This is not necessary but it shortened function calls 
+# that need to be used in the code.
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras import regularizers
+from tensorflow.keras.models import Model, Sequential
+from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
+# For timing the code
+from timeit import default_timer as timer
+# For plotting
+import matplotlib.pyplot as plt
+
+
+# The data set
+datatype='VaryDimension'
+X_tot = np.arange(2, 42, 2)
+y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,
+	-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, 
+	-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])
+
    +

    +









    + +

    Formatting the Data

    + +

    +The way the recurrent neural networks are trained in this program +differs from how machine learning algorithms are usually trained. +Typically a machine learning algorithm is trained by learning the +relationship between the x data and the y data. In this program, the +recurrent neural network will be trained to recognize the relationship +in a sequence of y values. This is type of data formatting is +typically used time series forcasting, but it can also be used in any +extrapolation (time series forecasting is just a specific type of +extrapolation along the time axis). This method of data formatting +does not use the x data and assumes that the y data are evenly spaced. + +

    +For a standard machine learning algorithm, the training data has the +form of (x,y) so the machine learning algorithm learns to assiciate a +y value with a given x value. This is useful when the test data has x +values within the same range as the training data. However, for this +application, the x values of the test data are outside of the x values +of the training data and the traditional method of training a machine +learning algorithm does not work as well. For this reason, the +recurrent neural network is trained on sequences of y values of the +form ((y1, y2), y3), so that the network is concerned with learning +the pattern of the y data and not the relation between the x and y +data. As long as the pattern of y data outside of the training region +stays relatively stable compared to what was inside the training +region, this method of training can produce accurate extrapolations to +y values far removed from the training data set. + +

    + + + + + + + +

    + + +

    # FORMAT_DATA
+def format_data(data, length_of_sequence = 2):  
+    """
+        Inputs:
+            data(a numpy array): the data that will be the inputs to the recurrent neural
+                network
+            length_of_sequence (an int): the number of elements in one iteration of the
+                sequence patter.  For a function approximator use length_of_sequence = 2.
+        Returns:
+            rnn_input (a 3D numpy array): the input data for the recurrent neural network.  Its
+                dimensions are length of data - length of sequence, length of sequence, 
+                dimnsion of data
+            rnn_output (a numpy array): the training data for the neural network
+        Formats data to be used in a recurrent neural network.
+    """
+
+    X, Y = [], []
+    for i in range(len(data)-length_of_sequence):
+        # Get the next length_of_sequence elements
+        a = data[i:i+length_of_sequence]
+        # Get the element that immediately follows that
+        b = data[i+length_of_sequence]
+        # Reshape so that each data point is contained in its own array
+        a = np.reshape (a, (len(a), 1))
+        X.append(a)
+        Y.append(b)
+    rnn_input = np.array(X)
+    rnn_output = np.array(Y)
+
+    return rnn_input, rnn_output
+
+
+# ## Defining the Recurrent Neural Network Using Keras
+# 
+# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.
+
+def rnn(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with one hidden layer and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer
+    hidden_neurons = 200
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to 
+    # the network immediately after the input layer
+    rnn = SimpleRNN(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN")(inp)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
    +

    +









    + +

    Predicting New Points With A Trained Recurrent Neural Network

    + +

    + + +

    def test_rnn (x1, y_test, plot_min, plot_max):
+    """
+        Inputs:
+            x1 (a list or numpy array): The complete x component of the data set
+            y_test (a list or numpy array): The complete y component of the data set
+            plot_min (an int or float): the smallest x value used in the training data
+            plot_max (an int or float): the largest x valye used in the training data
+        Returns:
+            None.
+        Uses a trained recurrent neural network model to predict future points in the 
+        series.  Computes the MSE of the predicted data set from the true data set, saves
+        the predicted data set to a csv file, and plots the predicted and true data sets w
+        while also displaying the data range used for training.
+    """
+    # Add the training data as the first dim points in the predicted data array as these
+    # are known values.
+    y_pred = y_test[:dim].tolist()
+    # Generate the first input to the trained recurrent neural network using the last two 
+    # points of the training data.  Based on how the network was trained this means that it
+    # will predict the first point in the data set after the training data.  All of the 
+    # brackets are necessary for Tensorflow.
+    next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])
+    # Save the very last point in the training data set.  This will be used later.
+    last = [y_test[dim-1]]
+
+    # Iterate until the complete data set is created.
+    for i in range (dim, len(y_test)):
+        # Predict the next point in the data set using the previous two points.
+        next = model.predict(next_input)
+        # Append just the number of the predicted data set
+        y_pred.append(next[0][0])
+        # Create the input that will be used to predict the next data point in the data set.
+        next_input = np.array([[last, next[0]]], dtype=np.float64)
+        last = next
+
+    # Print the mean squared error between the known data set and the predicted data set.
+    print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())
+    # Save the predicted data set as a csv file for later use
+    name = datatype + 'Predicted'+str(dim)+'.csv'
+    np.savetxt(name, y_pred, delimiter=',')
+    # Plot the known data set and the predicted data set.  The red box represents the region that was used
+    # for the training data.
+    fig, ax = plt.subplots()
+    ax.plot(x1, y_test, label="true", linewidth=3)
+    ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4)
+    ax.legend()
+    # Created a red region to represent the points used in the training data.
+    ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')
+    plt.show()
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn(length_of_sequences = rnn_input.shape[1])
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    +









    + +

    Other Things to Try

    + +

    +Changing the size of the recurrent neural network and its parameters +can drastically change the results you get from the model. The below +code takes the simple recurrent neural network from above and adds a +second hidden layer, changes the number of neurons in the hidden +layer, and explicitly declares the activation function of the hidden +layers to be a sigmoid function. The loss function and optimizer can +also be changed but are kept the same as the above network. These +parameters can be tuned to provide the optimal result from the +network. For some ideas on how to improve the performance of a +recurrent neural network. + +

    + + +

    def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two hidden layers and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer, increased from the first network
+    hidden_neurons = 500
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Create two hidden layers instead of one hidden layer.  Explicitly set the activation
+    # function to be the sigmoid function (the default value is hyperbolic tangent)
+    rnn1 = SimpleRNN(hidden_neurons, 
+                    return_sequences=True,  # This needs to be True if another hidden layer is to follow
+                    stateful = stateful, activation = 'sigmoid',
+                    name="RNN1")(inp)
+    rnn2 = SimpleRNN(hidden_neurons, 
+                    return_sequences=False, activation = 'sigmoid',
+                    stateful = stateful,
+                    name="RNN2")(rnn1)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn2)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn_2layers(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    +









    + +

    Other Types of Recurrent Neural Networks

    + +

    +Besides a simple recurrent neural network layer, there are two other +commonly used types of recurrent neural network layers: Long Short +Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +introduction to these layers see https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b +and https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b. + +

    +The first network created below is similar to the previous network, +but it replaces the SimpleRNN layers with LSTM layers. The second +network below has two hidden layers made up of GRUs, which are +preceeded by two dense (feeddorward) neural network layers. These +dense layers "preprocess" the data before it reaches the recurrent +layers. This architecture has been shown to improve the performance +of recurrent neural networks (see the link above and also +https://arxiv.org/pdf/1807.02857.pdf. + +

    + + +

    def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.
+    """
+    # Number of neurons on the input/output layer and the number of neurons in the hidden layer
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input Layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)
+    rnn= LSTM(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN", use_bias=True, activation='tanh')(inp)
+    rnn1 = LSTM(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True, activation='tanh')(rnn)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn1)
+    # Define the midel
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the model
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with four hidden layers (two dense followed by
+        two GRU layers) and returns the model.
+    """    
+    # Number of neurons on the input/output layers and hidden layers
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden Dense (feedforward) layers
+    dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)
+    dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)
+    # Hidden GRU layers
+    rnn1 = GRU(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True)(dnn1)
+    rnn = GRU(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN", use_bias=True)(rnn1)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Define the model
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the mdoel
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Change the method name to reflect which network you want to use
+model = dnn2_gru2(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
+
+# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)
+# 
+# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+# Reshape the data for Keras specifications
+X_train = X_train.reshape((dim, 1))
+y_train = y_train.reshape((dim, 1))
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Set the sequence length to 1 for regular data formatting 
+model = rnn(length_of_sequences = 1)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(X_train, y_train, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict the remaining data points
+X_pred = X_tot[dim:]
+X_pred = X_pred.reshape((len(X_pred), 1))
+y_model = model.predict(X_pred)
+y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))
+
+# Plot the known data set and the predicted data set.  The red box represents the region that was used
+# for the training data.
+fig, ax = plt.subplots()
+ax.plot(X_tot, y_tot, label="true", linewidth=3)
+ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4)
+ax.legend()
+# Created a red region to represent the points used in the training data.
+ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')
+plt.show()
+
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    diff --git a/doc/pub/week42/html/week42.html b/doc/pub/week42/html/week42.html index ff3687cb1..8c8c688c6 100644 --- a/doc/pub/week42/html/week42.html +++ b/doc/pub/week42/html/week42.html @@ -114,7 +114,18 @@ div { text-align: justify; text-justify: inter-word; } None, '___sec26'), ('Set up of an RNN', 2, None, '___sec27'), - ('A simple example', 2, None, '___sec28')]} + ('A simple example', 2, None, '___sec28'), + ('An extrapolation example', 2, None, '___sec29'), + ('Formatting the Data', 2, None, '___sec30'), + ('Predicting New Points With A Trained Recurrent Neural Network', + 2, + None, + '___sec31'), + ('Other Things to Try', 2, None, '___sec32'), + ('Other Types of Recurrent Neural Networks', + 2, + None, + '___sec33')]} end of tocinfo --> @@ -926,6 +937,588 @@ plt.axvline(df. plt.show()

    +









    + +

    An extrapolation example

    + +

    +The following code provides an example of how recurrent neural +networks can be used to extrapolate to unknown values of physics data +sets. Specifically, the data sets used in this program come from +a quantum mechanical many-body calculation of energies as functions of the number of particles. + +

    + + +

    # For matrices and calculations
+import numpy as np
+# For machine learning (backend for keras)
+import tensorflow as tf
+# User-friendly machine learning library
+# Front end for TensorFlow
+import tensorflow.keras
+# Different methods from Keras needed to create an RNN
+# This is not necessary but it shortened function calls 
+# that need to be used in the code.
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras import regularizers
+from tensorflow.keras.models import Model, Sequential
+from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
+# For timing the code
+from timeit import default_timer as timer
+# For plotting
+import matplotlib.pyplot as plt
+
+
+# The data set
+datatype='VaryDimension'
+X_tot = np.arange(2, 42, 2)
+y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,
+	-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, 
+	-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])
+
    +

    +









    + +

    Formatting the Data

    + +

    +The way the recurrent neural networks are trained in this program +differs from how machine learning algorithms are usually trained. +Typically a machine learning algorithm is trained by learning the +relationship between the x data and the y data. In this program, the +recurrent neural network will be trained to recognize the relationship +in a sequence of y values. This is type of data formatting is +typically used time series forcasting, but it can also be used in any +extrapolation (time series forecasting is just a specific type of +extrapolation along the time axis). This method of data formatting +does not use the x data and assumes that the y data are evenly spaced. + +

    +For a standard machine learning algorithm, the training data has the +form of (x,y) so the machine learning algorithm learns to assiciate a +y value with a given x value. This is useful when the test data has x +values within the same range as the training data. However, for this +application, the x values of the test data are outside of the x values +of the training data and the traditional method of training a machine +learning algorithm does not work as well. For this reason, the +recurrent neural network is trained on sequences of y values of the +form ((y1, y2), y3), so that the network is concerned with learning +the pattern of the y data and not the relation between the x and y +data. As long as the pattern of y data outside of the training region +stays relatively stable compared to what was inside the training +region, this method of training can produce accurate extrapolations to +y values far removed from the training data set. + +

    + + + + + + + +

    + + +

    # FORMAT_DATA
+def format_data(data, length_of_sequence = 2):  
+    """
+        Inputs:
+            data(a numpy array): the data that will be the inputs to the recurrent neural
+                network
+            length_of_sequence (an int): the number of elements in one iteration of the
+                sequence patter.  For a function approximator use length_of_sequence = 2.
+        Returns:
+            rnn_input (a 3D numpy array): the input data for the recurrent neural network.  Its
+                dimensions are length of data - length of sequence, length of sequence, 
+                dimnsion of data
+            rnn_output (a numpy array): the training data for the neural network
+        Formats data to be used in a recurrent neural network.
+    """
+
+    X, Y = [], []
+    for i in range(len(data)-length_of_sequence):
+        # Get the next length_of_sequence elements
+        a = data[i:i+length_of_sequence]
+        # Get the element that immediately follows that
+        b = data[i+length_of_sequence]
+        # Reshape so that each data point is contained in its own array
+        a = np.reshape (a, (len(a), 1))
+        X.append(a)
+        Y.append(b)
+    rnn_input = np.array(X)
+    rnn_output = np.array(Y)
+
+    return rnn_input, rnn_output
+
+
+# ## Defining the Recurrent Neural Network Using Keras
+# 
+# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.
+
+def rnn(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with one hidden layer and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer
+    hidden_neurons = 200
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to 
+    # the network immediately after the input layer
+    rnn = SimpleRNN(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN")(inp)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
    +

    +









    + +

    Predicting New Points With A Trained Recurrent Neural Network

    + +

    + + +

    def test_rnn (x1, y_test, plot_min, plot_max):
+    """
+        Inputs:
+            x1 (a list or numpy array): The complete x component of the data set
+            y_test (a list or numpy array): The complete y component of the data set
+            plot_min (an int or float): the smallest x value used in the training data
+            plot_max (an int or float): the largest x valye used in the training data
+        Returns:
+            None.
+        Uses a trained recurrent neural network model to predict future points in the 
+        series.  Computes the MSE of the predicted data set from the true data set, saves
+        the predicted data set to a csv file, and plots the predicted and true data sets w
+        while also displaying the data range used for training.
+    """
+    # Add the training data as the first dim points in the predicted data array as these
+    # are known values.
+    y_pred = y_test[:dim].tolist()
+    # Generate the first input to the trained recurrent neural network using the last two 
+    # points of the training data.  Based on how the network was trained this means that it
+    # will predict the first point in the data set after the training data.  All of the 
+    # brackets are necessary for Tensorflow.
+    next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])
+    # Save the very last point in the training data set.  This will be used later.
+    last = [y_test[dim-1]]
+
+    # Iterate until the complete data set is created.
+    for i in range (dim, len(y_test)):
+        # Predict the next point in the data set using the previous two points.
+        next = model.predict(next_input)
+        # Append just the number of the predicted data set
+        y_pred.append(next[0][0])
+        # Create the input that will be used to predict the next data point in the data set.
+        next_input = np.array([[last, next[0]]], dtype=np.float64)
+        last = next
+
+    # Print the mean squared error between the known data set and the predicted data set.
+    print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())
+    # Save the predicted data set as a csv file for later use
+    name = datatype + 'Predicted'+str(dim)+'.csv'
+    np.savetxt(name, y_pred, delimiter=',')
+    # Plot the known data set and the predicted data set.  The red box represents the region that was used
+    # for the training data.
+    fig, ax = plt.subplots()
+    ax.plot(x1, y_test, label="true", linewidth=3)
+    ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4)
+    ax.legend()
+    # Created a red region to represent the points used in the training data.
+    ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')
+    plt.show()
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn(length_of_sequences = rnn_input.shape[1])
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    +









    + +

    Other Things to Try

    + +

    +Changing the size of the recurrent neural network and its parameters +can drastically change the results you get from the model. The below +code takes the simple recurrent neural network from above and adds a +second hidden layer, changes the number of neurons in the hidden +layer, and explicitly declares the activation function of the hidden +layers to be a sigmoid function. The loss function and optimizer can +also be changed but are kept the same as the above network. These +parameters can be tuned to provide the optimal result from the +network. For some ideas on how to improve the performance of a +recurrent neural network. + +

    + + +

    def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two hidden layers and returns the model.
+    """
+    # Number of neurons in the input and output layers
+    in_out_neurons = 1
+    # Number of neurons in the hidden layer, increased from the first network
+    hidden_neurons = 500
+    # Define the input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons))  
+    # Create two hidden layers instead of one hidden layer.  Explicitly set the activation
+    # function to be the sigmoid function (the default value is hyperbolic tangent)
+    rnn1 = SimpleRNN(hidden_neurons, 
+                    return_sequences=True,  # This needs to be True if another hidden layer is to follow
+                    stateful = stateful, activation = 'sigmoid',
+                    name="RNN1")(inp)
+    rnn2 = SimpleRNN(hidden_neurons, 
+                    return_sequences=False, activation = 'sigmoid',
+                    stateful = stateful,
+                    name="RNN2")(rnn1)
+    # Define the output layer as a dense neural network layer (standard neural network layer)
+    #and add it to the network immediately after the hidden layer.
+    dens = Dense(in_out_neurons,name="dense")(rnn2)
+    # Create the machine learning model starting with the input layer and ending with the 
+    # output layer
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the machine learning model using the mean squared error function as the loss 
+    # function and an Adams optimizer.
+    model.compile(loss="mean_squared_error", optimizer="adam")  
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+model = rnn_2layers(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    +









    + +

    Other Types of Recurrent Neural Networks

    + +

    +Besides a simple recurrent neural network layer, there are two other +commonly used types of recurrent neural network layers: Long Short +Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +introduction to these layers see https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b +and https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b. + +

    +The first network created below is similar to the previous network, +but it replaces the SimpleRNN layers with LSTM layers. The second +network below has two hidden layers made up of GRUs, which are +preceeded by two dense (feeddorward) neural network layers. These +dense layers "preprocess" the data before it reaches the recurrent +layers. This architecture has been shown to improve the performance +of recurrent neural networks (see the link above and also +https://arxiv.org/pdf/1807.02857.pdf. + +

    + + +

    def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.
+    """
+    # Number of neurons on the input/output layer and the number of neurons in the hidden layer
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input Layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)
+    rnn= LSTM(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN", use_bias=True, activation='tanh')(inp)
+    rnn1 = LSTM(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True, activation='tanh')(rnn)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn1)
+    # Define the midel
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the model
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):
+    """
+        Inputs:
+            length_of_sequences (an int): the number of y values in "x data".  This is determined
+                when the data is formatted
+            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
+            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
+        Returns:
+            model (a Keras model): The recurrent neural network that is built and compiled by this
+                method
+        Builds and compiles a recurrent neural network with four hidden layers (two dense followed by
+        two GRU layers) and returns the model.
+    """    
+    # Number of neurons on the input/output layers and hidden layers
+    in_out_neurons = 1
+    hidden_neurons = 250
+    # Input layer
+    inp = Input(batch_shape=(batch_size, 
+                length_of_sequences, 
+                in_out_neurons)) 
+    # Hidden Dense (feedforward) layers
+    dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)
+    dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)
+    # Hidden GRU layers
+    rnn1 = GRU(hidden_neurons, 
+                    return_sequences=True,
+                    stateful = stateful,
+                    name="RNN1", use_bias=True)(dnn1)
+    rnn = GRU(hidden_neurons, 
+                    return_sequences=False,
+                    stateful = stateful,
+                    name="RNN", use_bias=True)(rnn1)
+    # Output layer
+    dens = Dense(in_out_neurons,name="dense")(rnn)
+    # Define the model
+    model = Model(inputs=[inp],outputs=[dens])
+    # Compile the mdoel
+    model.compile(loss='mean_squared_error', optimizer='adam')  
+    # Return the model
+    return model
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+
+# Generate the training data for the RNN, using a sequence of 2
+rnn_input, rnn_training = format_data(y_train, 2)
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Change the method name to reflect which network you want to use
+model = dnn2_gru2(length_of_sequences = 2)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict more points of the data set
+test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
+
+# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)
+# 
+# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.
+
+# Check to make sure the data set is complete
+assert len(X_tot) == len(y_tot)
+
+# This is the number of points that will be used in as the training data
+dim=12
+
+# Separate the training data from the whole data set
+X_train = X_tot[:dim]
+y_train = y_tot[:dim]
+
+# Reshape the data for Keras specifications
+X_train = X_train.reshape((dim, 1))
+y_train = y_train.reshape((dim, 1))
+
+
+# Create a recurrent neural network in Keras and produce a summary of the 
+# machine learning model
+# Set the sequence length to 1 for regular data formatting 
+model = rnn(length_of_sequences = 1)
+model.summary()
+
+# Start the timer.  Want to time training+testing
+start = timer()
+# Fit the model using the training data genenerated above using 150 training iterations and a 5%
+# validation split.  Setting verbose to True prints information about each training iteration.
+hist = model.fit(X_train, y_train, batch_size=None, epochs=150, 
+                 verbose=True,validation_split=0.05)
+
+
+# This section plots the training loss and the validation loss as a function of training iteration.
+# This is not required for analyzing the couple cluster data but can help determine if the network is
+# being overtrained.
+for label in ["loss","val_loss"]:
+    plt.plot(hist.history[label],label=label)
+
+plt.ylabel("loss")
+plt.xlabel("epoch")
+plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
+plt.legend()
+plt.show()
+
+# Use the trained neural network to predict the remaining data points
+X_pred = X_tot[dim:]
+X_pred = X_pred.reshape((len(X_pred), 1))
+y_model = model.predict(X_pred)
+y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))
+
+# Plot the known data set and the predicted data set.  The red box represents the region that was used
+# for the training data.
+fig, ax = plt.subplots()
+ax.plot(X_tot, y_tot, label="true", linewidth=3)
+ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4)
+ax.legend()
+# Created a red region to represent the points used in the training data.
+ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')
+plt.show()
+
+# Stop the timer and calculate the total time needed.
+end = timer()
+print('Time: ', end-start)
+
    +

    diff --git a/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz b/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz index cc34d78c98a3f132895f2050bed70cbe8542ef56..7f61649e51dee3554ee33944a5a9264c574c5378 100644 GIT binary patch delta 20 bcmdn6igm*(RyO%=4u+|zjci-l7_~wHN{0qB delta 20 bcmdn6igm*(RyO%=4u%4qMz*bNj9Q@pM{)(# diff --git a/doc/pub/week42/ipynb/week42.ipynb b/doc/pub/week42/ipynb/week42.ipynb index 0bd948b33..6b3758969 100644 --- a/doc/pub/week42/ipynb/week42.ipynb +++ b/doc/pub/week42/ipynb/week42.ipynb @@ -836,6 +836,624 @@ "plt.axvline(df.index[Tp], c=\"r\")\n", "plt.show()" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## An extrapolation example\n", + "\n", + "The following code provides an example of how recurrent neural\n", + "networks can be used to extrapolate to unknown values of physics data\n", + "sets. Specifically, the data sets used in this program come from\n", + "a quantum mechanical many-body calculation of energies as functions of the number of particles." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "\n", + "# For matrices and calculations\n", + "import numpy as np\n", + "# For machine learning (backend for keras)\n", + "import tensorflow as tf\n", + "# User-friendly machine learning library\n", + "# Front end for TensorFlow\n", + "import tensorflow.keras\n", + "# Different methods from Keras needed to create an RNN\n", + "# This is not necessary but it shortened function calls \n", + "# that need to be used in the code.\n", + "from tensorflow.keras import datasets, layers, models\n", + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras import regularizers\n", + "from tensorflow.keras.models import Model, Sequential\n", + "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", + "# For timing the code\n", + "from timeit import default_timer as timer\n", + "# For plotting\n", + "import matplotlib.pyplot as plt\n", + "\n", + "\n", + "# The data set\n", + "datatype='VaryDimension'\n", + "X_tot = np.arange(2, 42, 2)\n", + "y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,\n", + "\t-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, \n", + "\t-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Formatting the Data\n", + "\n", + "The way the recurrent neural networks are trained in this program\n", + "differs from how machine learning algorithms are usually trained.\n", + "Typically a machine learning algorithm is trained by learning the\n", + "relationship between the x data and the y data. In this program, the\n", + "recurrent neural network will be trained to recognize the relationship\n", + "in a sequence of y values. This is type of data formatting is\n", + "typically used time series forcasting, but it can also be used in any\n", + "extrapolation (time series forecasting is just a specific type of\n", + "extrapolation along the time axis). This method of data formatting\n", + "does not use the x data and assumes that the y data are evenly spaced.\n", + "\n", + "For a standard machine learning algorithm, the training data has the\n", + "form of (x,y) so the machine learning algorithm learns to assiciate a\n", + "y value with a given x value. This is useful when the test data has x\n", + "values within the same range as the training data. However, for this\n", + "application, the x values of the test data are outside of the x values\n", + "of the training data and the traditional method of training a machine\n", + "learning algorithm does not work as well. For this reason, the\n", + "recurrent neural network is trained on sequences of y values of the\n", + "form ((y1, y2), y3), so that the network is concerned with learning\n", + "the pattern of the y data and not the relation between the x and y\n", + "data. As long as the pattern of y data outside of the training region\n", + "stays relatively stable compared to what was inside the training\n", + "region, this method of training can produce accurate extrapolations to\n", + "y values far removed from the training data set.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# FORMAT_DATA\n", + "def format_data(data, length_of_sequence = 2): \n", + " \"\"\"\n", + " Inputs:\n", + " data(a numpy array): the data that will be the inputs to the recurrent neural\n", + " network\n", + " length_of_sequence (an int): the number of elements in one iteration of the\n", + " sequence patter. For a function approximator use length_of_sequence = 2.\n", + " Returns:\n", + " rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its\n", + " dimensions are length of data - length of sequence, length of sequence, \n", + " dimnsion of data\n", + " rnn_output (a numpy array): the training data for the neural network\n", + " Formats data to be used in a recurrent neural network.\n", + " \"\"\"\n", + "\n", + " X, Y = [], []\n", + " for i in range(len(data)-length_of_sequence):\n", + " # Get the next length_of_sequence elements\n", + " a = data[i:i+length_of_sequence]\n", + " # Get the element that immediately follows that\n", + " b = data[i+length_of_sequence]\n", + " # Reshape so that each data point is contained in its own array\n", + " a = np.reshape (a, (len(a), 1))\n", + " X.append(a)\n", + " Y.append(b)\n", + " rnn_input = np.array(X)\n", + " rnn_output = np.array(Y)\n", + "\n", + " return rnn_input, rnn_output\n", + "\n", + "\n", + "# ## Defining the Recurrent Neural Network Using Keras\n", + "# \n", + "# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.\n", + "\n", + "def rnn(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with one hidden layer and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer\n", + " hidden_neurons = 200\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to \n", + " # the network immediately after the input layer\n", + " rnn = SimpleRNN(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\")(inp)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Predicting New Points With A Trained Recurrent Neural Network" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def test_rnn (x1, y_test, plot_min, plot_max):\n", + " \"\"\"\n", + " Inputs:\n", + " x1 (a list or numpy array): The complete x component of the data set\n", + " y_test (a list or numpy array): The complete y component of the data set\n", + " plot_min (an int or float): the smallest x value used in the training data\n", + " plot_max (an int or float): the largest x valye used in the training data\n", + " Returns:\n", + " None.\n", + " Uses a trained recurrent neural network model to predict future points in the \n", + " series. Computes the MSE of the predicted data set from the true data set, saves\n", + " the predicted data set to a csv file, and plots the predicted and true data sets w\n", + " while also displaying the data range used for training.\n", + " \"\"\"\n", + " # Add the training data as the first dim points in the predicted data array as these\n", + " # are known values.\n", + " y_pred = y_test[:dim].tolist()\n", + " # Generate the first input to the trained recurrent neural network using the last two \n", + " # points of the training data. Based on how the network was trained this means that it\n", + " # will predict the first point in the data set after the training data. All of the \n", + " # brackets are necessary for Tensorflow.\n", + " next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])\n", + " # Save the very last point in the training data set. This will be used later.\n", + " last = [y_test[dim-1]]\n", + "\n", + " # Iterate until the complete data set is created.\n", + " for i in range (dim, len(y_test)):\n", + " # Predict the next point in the data set using the previous two points.\n", + " next = model.predict(next_input)\n", + " # Append just the number of the predicted data set\n", + " y_pred.append(next[0][0])\n", + " # Create the input that will be used to predict the next data point in the data set.\n", + " next_input = np.array([[last, next[0]]], dtype=np.float64)\n", + " last = next\n", + "\n", + " # Print the mean squared error between the known data set and the predicted data set.\n", + " print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())\n", + " # Save the predicted data set as a csv file for later use\n", + " name = datatype + 'Predicted'+str(dim)+'.csv'\n", + " np.savetxt(name, y_pred, delimiter=',')\n", + " # Plot the known data set and the predicted data set. The red box represents the region that was used\n", + " # for the training data.\n", + " fig, ax = plt.subplots()\n", + " ax.plot(x1, y_test, label=\"true\", linewidth=3)\n", + " ax.plot(x1, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + " ax.legend()\n", + " # Created a red region to represent the points used in the training data.\n", + " ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')\n", + " plt.show()\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn(length_of_sequences = rnn_input.shape[1])\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Other Things to Try\n", + "\n", + "\n", + "Changing the size of the recurrent neural network and its parameters\n", + "can drastically change the results you get from the model. The below\n", + "code takes the simple recurrent neural network from above and adds a\n", + "second hidden layer, changes the number of neurons in the hidden\n", + "layer, and explicitly declares the activation function of the hidden\n", + "layers to be a sigmoid function. The loss function and optimizer can\n", + "also be changed but are kept the same as the above network. These\n", + "parameters can be tuned to provide the optimal result from the\n", + "network. For some ideas on how to improve the performance of a\n", + "[recurrent neural network](https://danijar.com/tips-for-training-recurrent-neural-networks)." + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons in the input and output layers\n", + " in_out_neurons = 1\n", + " # Number of neurons in the hidden layer, increased from the first network\n", + " hidden_neurons = 500\n", + " # Define the input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Create two hidden layers instead of one hidden layer. Explicitly set the activation\n", + " # function to be the sigmoid function (the default value is hyperbolic tangent)\n", + " rnn1 = SimpleRNN(hidden_neurons, \n", + " return_sequences=True, # This needs to be True if another hidden layer is to follow\n", + " stateful = stateful, activation = 'sigmoid',\n", + " name=\"RNN1\")(inp)\n", + " rnn2 = SimpleRNN(hidden_neurons, \n", + " return_sequences=False, activation = 'sigmoid',\n", + " stateful = stateful,\n", + " name=\"RNN2\")(rnn1)\n", + " # Define the output layer as a dense neural network layer (standard neural network layer)\n", + " #and add it to the network immediately after the hidden layer.\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn2)\n", + " # Create the machine learning model starting with the input layer and ending with the \n", + " # output layer\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the machine learning model using the mean squared error function as the loss \n", + " # function and an Adams optimizer.\n", + " model.compile(loss=\"mean_squared_error\", optimizer=\"adam\") \n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "model = rnn_2layers(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Other Types of Recurrent Neural Networks\n", + "\n", + "Besides a simple recurrent neural network layer, there are two other\n", + "commonly used types of recurrent neural network layers: Long Short\n", + "Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short\n", + "introduction to these layers see \n", + "and .\n", + "\n", + "The first network created below is similar to the previous network,\n", + "but it replaces the SimpleRNN layers with LSTM layers. The second\n", + "network below has two hidden layers made up of GRUs, which are\n", + "preceeded by two dense (feeddorward) neural network layers. These\n", + "dense layers \"preprocess\" the data before it reaches the recurrent\n", + "layers. This architecture has been shown to improve the performance\n", + "of recurrent neural networks (see the link above and also\n", + "." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.\n", + " \"\"\"\n", + " # Number of neurons on the input/output layer and the number of neurons in the hidden layer\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input Layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)\n", + " rnn= LSTM(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True, activation='tanh')(inp)\n", + " rnn1 = LSTM(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True, activation='tanh')(rnn)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn1)\n", + " # Define the midel\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the model\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):\n", + " \"\"\"\n", + " Inputs:\n", + " length_of_sequences (an int): the number of y values in \"x data\". This is determined\n", + " when the data is formatted\n", + " batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.\n", + " stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.\n", + " Returns:\n", + " model (a Keras model): The recurrent neural network that is built and compiled by this\n", + " method\n", + " Builds and compiles a recurrent neural network with four hidden layers (two dense followed by\n", + " two GRU layers) and returns the model.\n", + " \"\"\" \n", + " # Number of neurons on the input/output layers and hidden layers\n", + " in_out_neurons = 1\n", + " hidden_neurons = 250\n", + " # Input layer\n", + " inp = Input(batch_shape=(batch_size, \n", + " length_of_sequences, \n", + " in_out_neurons)) \n", + " # Hidden Dense (feedforward) layers\n", + " dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)\n", + " dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)\n", + " # Hidden GRU layers\n", + " rnn1 = GRU(hidden_neurons, \n", + " return_sequences=True,\n", + " stateful = stateful,\n", + " name=\"RNN1\", use_bias=True)(dnn1)\n", + " rnn = GRU(hidden_neurons, \n", + " return_sequences=False,\n", + " stateful = stateful,\n", + " name=\"RNN\", use_bias=True)(rnn1)\n", + " # Output layer\n", + " dens = Dense(in_out_neurons,name=\"dense\")(rnn)\n", + " # Define the model\n", + " model = Model(inputs=[inp],outputs=[dens])\n", + " # Compile the mdoel\n", + " model.compile(loss='mean_squared_error', optimizer='adam') \n", + " # Return the model\n", + " return model\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "\n", + "# Generate the training data for the RNN, using a sequence of 2\n", + "rnn_input, rnn_training = format_data(y_train, 2)\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Change the method name to reflect which network you want to use\n", + "model = dnn2_gru2(length_of_sequences = 2)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict more points of the data set\n", + "test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)\n", + "\n", + "\n", + "# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)\n", + "# \n", + "# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.\n", + "\n", + "# Check to make sure the data set is complete\n", + "assert len(X_tot) == len(y_tot)\n", + "\n", + "# This is the number of points that will be used in as the training data\n", + "dim=12\n", + "\n", + "# Separate the training data from the whole data set\n", + "X_train = X_tot[:dim]\n", + "y_train = y_tot[:dim]\n", + "\n", + "# Reshape the data for Keras specifications\n", + "X_train = X_train.reshape((dim, 1))\n", + "y_train = y_train.reshape((dim, 1))\n", + "\n", + "\n", + "# Create a recurrent neural network in Keras and produce a summary of the \n", + "# machine learning model\n", + "# Set the sequence length to 1 for regular data formatting \n", + "model = rnn(length_of_sequences = 1)\n", + "model.summary()\n", + "\n", + "# Start the timer. Want to time training+testing\n", + "start = timer()\n", + "# Fit the model using the training data genenerated above using 150 training iterations and a 5%\n", + "# validation split. Setting verbose to True prints information about each training iteration.\n", + "hist = model.fit(X_train, y_train, batch_size=None, epochs=150, \n", + " verbose=True,validation_split=0.05)\n", + "\n", + "\n", + "# This section plots the training loss and the validation loss as a function of training iteration.\n", + "# This is not required for analyzing the couple cluster data but can help determine if the network is\n", + "# being overtrained.\n", + "for label in [\"loss\",\"val_loss\"]:\n", + " plt.plot(hist.history[label],label=label)\n", + "\n", + "plt.ylabel(\"loss\")\n", + "plt.xlabel(\"epoch\")\n", + "plt.title(\"The final validation loss: {}\".format(hist.history[\"val_loss\"][-1]))\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "# Use the trained neural network to predict the remaining data points\n", + "X_pred = X_tot[dim:]\n", + "X_pred = X_pred.reshape((len(X_pred), 1))\n", + "y_model = model.predict(X_pred)\n", + "y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))\n", + "\n", + "# Plot the known data set and the predicted data set. The red box represents the region that was used\n", + "# for the training data.\n", + "fig, ax = plt.subplots()\n", + "ax.plot(X_tot, y_tot, label=\"true\", linewidth=3)\n", + "ax.plot(X_tot, y_pred, 'g-.',label=\"predicted\", linewidth=4)\n", + "ax.legend()\n", + "# Created a red region to represent the points used in the training data.\n", + "ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')\n", + "plt.show()\n", + "\n", + "# Stop the timer and calculate the total time needed.\n", + "end = timer()\n", + "print('Time: ', end-start)" + ] } ], "metadata": {}, diff --git a/doc/src/week42/week42.do.txt b/doc/src/week42/week42.do.txt index 56bb83f89..2db6df215 100644 --- a/doc/src/week42/week42.do.txt +++ b/doc/src/week42/week42.do.txt @@ -679,3 +679,571 @@ plt.show() !ec +!split +===== An extrapolation example ===== + +The following code provides an example of how recurrent neural +networks can be used to extrapolate to unknown values of physics data +sets. Specifically, the data sets used in this program come from +a quantum mechanical many-body calculation of energies as functions of the number of particles. + + +!bc pycod + +# For matrices and calculations +import numpy as np +# For machine learning (backend for keras) +import tensorflow as tf +# User-friendly machine learning library +# Front end for TensorFlow +import tensorflow.keras +# Different methods from Keras needed to create an RNN +# This is not necessary but it shortened function calls +# that need to be used in the code. +from tensorflow.keras import datasets, layers, models +from tensorflow.keras.layers import Input +from tensorflow.keras import regularizers +from tensorflow.keras.models import Model, Sequential +from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU +# For timing the code +from timeit import default_timer as timer +# For plotting +import matplotlib.pyplot as plt + + +# The data set +datatype='VaryDimension' +X_tot = np.arange(2, 42, 2) +y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846, + -0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, + -1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767]) + +!ec + +!split +===== Formatting the Data ===== + +The way the recurrent neural networks are trained in this program +differs from how machine learning algorithms are usually trained. +Typically a machine learning algorithm is trained by learning the +relationship between the x data and the y data. In this program, the +recurrent neural network will be trained to recognize the relationship +in a sequence of y values. This is type of data formatting is +typically used time series forcasting, but it can also be used in any +extrapolation (time series forecasting is just a specific type of +extrapolation along the time axis). This method of data formatting +does not use the x data and assumes that the y data are evenly spaced. + +For a standard machine learning algorithm, the training data has the +form of (x,y) so the machine learning algorithm learns to assiciate a +y value with a given x value. This is useful when the test data has x +values within the same range as the training data. However, for this +application, the x values of the test data are outside of the x values +of the training data and the traditional method of training a machine +learning algorithm does not work as well. For this reason, the +recurrent neural network is trained on sequences of y values of the +form ((y1, y2), y3), so that the network is concerned with learning +the pattern of the y data and not the relation between the x and y +data. As long as the pattern of y data outside of the training region +stays relatively stable compared to what was inside the training +region, this method of training can produce accurate extrapolations to +y values far removed from the training data set. + + +# +# The idea behind formatting the data in this way comes from [this resource](https://machinelearningmastery.com/time-series-prediction-lstm-recurrent-neural-networks-python-keras/) and [this one](https://fairyonice.github.io/Understand-Keras%27s-RNN-behind-the-scenes-with-a-sin-wave-example.html). +# +# The following method takes in a y data set and formats it so the "x data" are of the form (y1, y2) and the "y data" are of the form y3, with extra brackets added in to make the resulting arrays compatable with both Keras and Tensorflow. +# +# Note: Using a sequence length of two is not required for time series forecasting so any lenght of sequence could be used (for example instead of ((y1, y2) y3) you could change the length of sequence to be 4 and the resulting data points would have the form ((y1, y2, y3, y4), y5)). While the following method can be used to create a data set of any sequence length, the remainder of the code expects the length of sequence to be 2. This is because the data sets are very small and the higher the lenght of the sequence the less resulting data points. + +!bc pycod +# FORMAT_DATA +def format_data(data, length_of_sequence = 2): + """ + Inputs: + data(a numpy array): the data that will be the inputs to the recurrent neural + network + length_of_sequence (an int): the number of elements in one iteration of the + sequence patter. For a function approximator use length_of_sequence = 2. + Returns: + rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its + dimensions are length of data - length of sequence, length of sequence, + dimnsion of data + rnn_output (a numpy array): the training data for the neural network + Formats data to be used in a recurrent neural network. + """ + + X, Y = [], [] + for i in range(len(data)-length_of_sequence): + # Get the next length_of_sequence elements + a = data[i:i+length_of_sequence] + # Get the element that immediately follows that + b = data[i+length_of_sequence] + # Reshape so that each data point is contained in its own array + a = np.reshape (a, (len(a), 1)) + X.append(a) + Y.append(b) + rnn_input = np.array(X) + rnn_output = np.array(Y) + + return rnn_input, rnn_output + + +# ## Defining the Recurrent Neural Network Using Keras +# +# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer. + +def rnn(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with one hidden layer and returns the model. + """ + # Number of neurons in the input and output layers + in_out_neurons = 1 + # Number of neurons in the hidden layer + hidden_neurons = 200 + # Define the input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to + # the network immediately after the input layer + rnn = SimpleRNN(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN")(inp) + # Define the output layer as a dense neural network layer (standard neural network layer) + #and add it to the network immediately after the hidden layer. + dens = Dense(in_out_neurons,name="dense")(rnn) + # Create the machine learning model starting with the input layer and ending with the + # output layer + model = Model(inputs=[inp],outputs=[dens]) + # Compile the machine learning model using the mean squared error function as the loss + # function and an Adams optimizer. + model.compile(loss="mean_squared_error", optimizer="adam") + return model + +!ec + +!split +===== Predicting New Points With A Trained Recurrent Neural Network ===== + +!bc pycod +def test_rnn (x1, y_test, plot_min, plot_max): + """ + Inputs: + x1 (a list or numpy array): The complete x component of the data set + y_test (a list or numpy array): The complete y component of the data set + plot_min (an int or float): the smallest x value used in the training data + plot_max (an int or float): the largest x valye used in the training data + Returns: + None. + Uses a trained recurrent neural network model to predict future points in the + series. Computes the MSE of the predicted data set from the true data set, saves + the predicted data set to a csv file, and plots the predicted and true data sets w + while also displaying the data range used for training. + """ + # Add the training data as the first dim points in the predicted data array as these + # are known values. + y_pred = y_test[:dim].tolist() + # Generate the first input to the trained recurrent neural network using the last two + # points of the training data. Based on how the network was trained this means that it + # will predict the first point in the data set after the training data. All of the + # brackets are necessary for Tensorflow. + next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]]) + # Save the very last point in the training data set. This will be used later. + last = [y_test[dim-1]] + + # Iterate until the complete data set is created. + for i in range (dim, len(y_test)): + # Predict the next point in the data set using the previous two points. + next = model.predict(next_input) + # Append just the number of the predicted data set + y_pred.append(next[0][0]) + # Create the input that will be used to predict the next data point in the data set. + next_input = np.array([[last, next[0]]], dtype=np.float64) + last = next + + # Print the mean squared error between the known data set and the predicted data set. + print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean()) + # Save the predicted data set as a csv file for later use + name = datatype + 'Predicted'+str(dim)+'.csv' + np.savetxt(name, y_pred, delimiter=',') + # Plot the known data set and the predicted data set. The red box represents the region that was used + # for the training data. + fig, ax = plt.subplots() + ax.plot(x1, y_test, label="true", linewidth=3) + ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4) + ax.legend() + # Created a red region to represent the points used in the training data. + ax.axvspan(plot_min, plot_max, alpha=0.25, color='red') + plt.show() + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +model = rnn(length_of_sequences = rnn_input.shape[1]) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) +!ec + +!split +===== Other Things to Try ===== + + +Changing the size of the recurrent neural network and its parameters +can drastically change the results you get from the model. The below +code takes the simple recurrent neural network from above and adds a +second hidden layer, changes the number of neurons in the hidden +layer, and explicitly declares the activation function of the hidden +layers to be a sigmoid function. The loss function and optimizer can +also be changed but are kept the same as the above network. These +parameters can be tuned to provide the optimal result from the +network. For some ideas on how to improve the performance of a +"recurrent neural network":"https://danijar.com/tips-for-training-recurrent-neural-networks". + +!bc pycod +def rnn_2layers(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with two hidden layers and returns the model. + """ + # Number of neurons in the input and output layers + in_out_neurons = 1 + # Number of neurons in the hidden layer, increased from the first network + hidden_neurons = 500 + # Define the input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Create two hidden layers instead of one hidden layer. Explicitly set the activation + # function to be the sigmoid function (the default value is hyperbolic tangent) + rnn1 = SimpleRNN(hidden_neurons, + return_sequences=True, # This needs to be True if another hidden layer is to follow + stateful = stateful, activation = 'sigmoid', + name="RNN1")(inp) + rnn2 = SimpleRNN(hidden_neurons, + return_sequences=False, activation = 'sigmoid', + stateful = stateful, + name="RNN2")(rnn1) + # Define the output layer as a dense neural network layer (standard neural network layer) + #and add it to the network immediately after the hidden layer. + dens = Dense(in_out_neurons,name="dense")(rnn2) + # Create the machine learning model starting with the input layer and ending with the + # output layer + model = Model(inputs=[inp],outputs=[dens]) + # Compile the machine learning model using the mean squared error function as the loss + # function and an Adams optimizer. + model.compile(loss="mean_squared_error", optimizer="adam") + return model + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +model = rnn_2layers(length_of_sequences = 2) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) +!ec + +!split +===== Other Types of Recurrent Neural Networks ===== + +Besides a simple recurrent neural network layer, there are two other +commonly used types of recurrent neural network layers: Long Short +Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short +introduction to these layers see URL:"https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b" +and URL:"https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b". + +The first network created below is similar to the previous network, +but it replaces the SimpleRNN layers with LSTM layers. The second +network below has two hidden layers made up of GRUs, which are +preceeded by two dense (feeddorward) neural network layers. These +dense layers "preprocess" the data before it reaches the recurrent +layers. This architecture has been shown to improve the performance +of recurrent neural networks (see the link above and also +URL:"https://arxiv.org/pdf/1807.02857.pdf". + +!bc pycod +def lstm_2layers(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model. + """ + # Number of neurons on the input/output layer and the number of neurons in the hidden layer + in_out_neurons = 1 + hidden_neurons = 250 + # Input Layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers) + rnn= LSTM(hidden_neurons, + return_sequences=True, + stateful = stateful, + name="RNN", use_bias=True, activation='tanh')(inp) + rnn1 = LSTM(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN1", use_bias=True, activation='tanh')(rnn) + # Output layer + dens = Dense(in_out_neurons,name="dense")(rnn1) + # Define the midel + model = Model(inputs=[inp],outputs=[dens]) + # Compile the model + model.compile(loss='mean_squared_error', optimizer='adam') + # Return the model + return model + +def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False): + """ + Inputs: + length_of_sequences (an int): the number of y values in "x data". This is determined + when the data is formatted + batch_size (an int): Default value is None. See Keras documentation of SimpleRNN. + stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN. + Returns: + model (a Keras model): The recurrent neural network that is built and compiled by this + method + Builds and compiles a recurrent neural network with four hidden layers (two dense followed by + two GRU layers) and returns the model. + """ + # Number of neurons on the input/output layers and hidden layers + in_out_neurons = 1 + hidden_neurons = 250 + # Input layer + inp = Input(batch_shape=(batch_size, + length_of_sequences, + in_out_neurons)) + # Hidden Dense (feedforward) layers + dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp) + dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn) + # Hidden GRU layers + rnn1 = GRU(hidden_neurons, + return_sequences=True, + stateful = stateful, + name="RNN1", use_bias=True)(dnn1) + rnn = GRU(hidden_neurons, + return_sequences=False, + stateful = stateful, + name="RNN", use_bias=True)(rnn1) + # Output layer + dens = Dense(in_out_neurons,name="dense")(rnn) + # Define the model + model = Model(inputs=[inp],outputs=[dens]) + # Compile the mdoel + model.compile(loss='mean_squared_error', optimizer='adam') + # Return the model + return model + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + + +# Generate the training data for the RNN, using a sequence of 2 +rnn_input, rnn_training = format_data(y_train, 2) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +# Change the method name to reflect which network you want to use +model = dnn2_gru2(length_of_sequences = 2) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict more points of the data set +test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1]) +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + + +# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data) +# +# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation. + +# Check to make sure the data set is complete +assert len(X_tot) == len(y_tot) + +# This is the number of points that will be used in as the training data +dim=12 + +# Separate the training data from the whole data set +X_train = X_tot[:dim] +y_train = y_tot[:dim] + +# Reshape the data for Keras specifications +X_train = X_train.reshape((dim, 1)) +y_train = y_train.reshape((dim, 1)) + + +# Create a recurrent neural network in Keras and produce a summary of the +# machine learning model +# Set the sequence length to 1 for regular data formatting +model = rnn(length_of_sequences = 1) +model.summary() + +# Start the timer. Want to time training+testing +start = timer() +# Fit the model using the training data genenerated above using 150 training iterations and a 5% +# validation split. Setting verbose to True prints information about each training iteration. +hist = model.fit(X_train, y_train, batch_size=None, epochs=150, + verbose=True,validation_split=0.05) + + +# This section plots the training loss and the validation loss as a function of training iteration. +# This is not required for analyzing the couple cluster data but can help determine if the network is +# being overtrained. +for label in ["loss","val_loss"]: + plt.plot(hist.history[label],label=label) + +plt.ylabel("loss") +plt.xlabel("epoch") +plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1])) +plt.legend() +plt.show() + +# Use the trained neural network to predict the remaining data points +X_pred = X_tot[dim:] +X_pred = X_pred.reshape((len(X_pred), 1)) +y_model = model.predict(X_pred) +y_pred = np.concatenate((y_tot[:dim], y_model.flatten())) + +# Plot the known data set and the predicted data set. The red box represents the region that was used +# for the training data. +fig, ax = plt.subplots() +ax.plot(X_tot, y_tot, label="true", linewidth=3) +ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4) +ax.legend() +# Created a red region to represent the points used in the training data. +ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red') +plt.show() + +# Stop the timer and calculate the total time needed. +end = timer() +print('Time: ', end-start) + +!ec + +