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394 lines
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'sections': [('Plans for week 43', 2, None, '___sec0'),
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('Reading Recommendations', 2, None, '___sec1'),
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('Summary on Deep Learning Methods', 2, None, '___sec2'),
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('CNNs in brief', 2, None, '___sec3'),
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('Recurrent neural networks: Overarching view',
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2,
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None,
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'___sec4'),
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('Set up of an RNN', 2, None, '___sec5'),
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('A simple example', 2, None, '___sec6'),
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('An extrapolation example', 2, None, '___sec7'),
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('Formatting the Data', 2, None, '___sec8'),
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('Predicting New Points With A Trained Recurrent Neural Network',
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2,
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None,
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'___sec9'),
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('Other Things to Try', 2, None, '___sec10'),
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('Other Types of Recurrent Neural Networks', 2, None, '___sec11'),
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('Generative Models', 2, None, '___sec12'),
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('Generative Adversarial Networks', 2, None, '___sec13'),
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('Discriminator', 2, None, '___sec14'),
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('Learning Process', 2, None, '___sec15'),
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('More about the Learning Process', 2, None, '___sec16'),
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('Additional References', 2, None, '___sec17'),
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('Writing Our First Generative Adversarial Network',
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2,
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None,
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'___sec18'),
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('MNIST and GANs', 2, None, '___sec19'),
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('Other Models', 2, None, '___sec20'),
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('Training Step', 2, None, '___sec21'),
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('Checkpoints', 2, None, '___sec22'),
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('Exploring the Latent Space', 2, None, '___sec23'),
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('Getting Results', 2, None, '___sec24'),
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('Interpolating Between MNIST Digits', 2, None, '___sec25'),
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('Basic ideas of the Principal Component Analysis (PCA)',
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2,
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None,
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'___sec26'),
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('Introducing the Covariance and Correlation functions',
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2,
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None,
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'___sec27'),
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('More on the covariance', 2, None, '___sec28'),
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('Reminding ourselves about Linear Regression',
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2,
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None,
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'___sec29'),
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('Simple Example', 2, None, '___sec30'),
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('The Correlation Matrix', 2, None, '___sec31'),
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('Numpy Functionality', 2, None, '___sec32'),
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('Correlation Matrix again', 2, None, '___sec33'),
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('Using Pandas', 2, None, '___sec34'),
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('And then the Franke Function', 2, None, '___sec35'),
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('Lnks with the Design Matrix', 2, None, '___sec36'),
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('Computing the Expectation Values', 2, None, '___sec37'),
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('Towards the PCA theorem', 2, None, '___sec38'),
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('More on the PCA Theorem', 2, None, '___sec39'),
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('The Algorithm before the Theorem', 2, None, '___sec40'),
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('Writing our own PCA code', 2, None, '___sec41'),
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('Implementing it', 2, None, '___sec42'),
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('First Step', 2, None, '___sec43'),
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('Scaling', 2, None, '___sec44'),
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('Centered Data', 2, None, '___sec45'),
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('Exploring', 2, None, '___sec46'),
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('Diagonalize the sample covariance matrix to obtain the '
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'principal components',
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2,
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None,
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'___sec47'),
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('Collecting all Steps', 2, None, '___sec48'),
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('Classical PCA Theorem', 2, None, '___sec49'),
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('The PCA Theorem', 2, None, '___sec50'),
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('Geometric Interpretation and link with Singular Value '
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'Decomposition',
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2,
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None,
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('PCA and scikit-learn', 2, None, '___sec52'),
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('Back to the Cancer Data', 2, None, '___sec53'),
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('Incremental PCA', 2, None, '___sec54'),
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('Randomized PCA', 3, None, '___sec55'),
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('Kernel PCA', 3, None, '___sec56'),
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('Other techniques', 2, None, '___sec57')]}
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<a class="navbar-brand" href="week43-bs.html">Week 43: Deep Learning: Recurrent Neural Networks and other Deep Learning Methods. Principal Component analysis</a>
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<!-- navigation toc: --> <li><a href="._week43-bs001.html#___sec0" style="font-size: 80%;"><b>Plans for week 43</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs002.html#___sec1" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec2" style="font-size: 80%;"><b>Summary on Deep Learning Methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs004.html#___sec3" style="font-size: 80%;"><b>CNNs in brief</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs005.html#___sec4" style="font-size: 80%;"><b>Recurrent neural networks: Overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs006.html#___sec5" style="font-size: 80%;"><b>Set up of an RNN</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs007.html#___sec6" style="font-size: 80%;"><b>A simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs008.html#___sec7" style="font-size: 80%;"><b>An extrapolation example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs009.html#___sec8" style="font-size: 80%;"><b>Formatting the Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs010.html#___sec9" style="font-size: 80%;"><b>Predicting New Points With A Trained Recurrent Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;"><b>Other Things to Try</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs012.html#___sec11" style="font-size: 80%;"><b>Other Types of Recurrent Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs013.html#___sec12" style="font-size: 80%;"><b>Generative Models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs014.html#___sec13" style="font-size: 80%;"><b>Generative Adversarial Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec14" style="font-size: 80%;"><b>Discriminator</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs016.html#___sec15" style="font-size: 80%;"><b>Learning Process</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs017.html#___sec16" style="font-size: 80%;"><b>More about the Learning Process</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec17" style="font-size: 80%;"><b>Additional References</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec18" style="font-size: 80%;"><b>Writing Our First Generative Adversarial Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec19" style="font-size: 80%;"><b>MNIST and GANs</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec20" style="font-size: 80%;"><b>Other Models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec21" style="font-size: 80%;"><b>Training Step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec22" style="font-size: 80%;"><b>Checkpoints</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec23" style="font-size: 80%;"><b>Exploring the Latent Space</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs025.html#___sec24" style="font-size: 80%;"><b>Getting Results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs026.html#___sec25" style="font-size: 80%;"><b>Interpolating Between MNIST Digits</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec26" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec27" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec28" style="font-size: 80%;"><b>More on the covariance</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec29" style="font-size: 80%;"><b>Reminding ourselves about Linear Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec30" style="font-size: 80%;"><b>Simple Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec31" style="font-size: 80%;"><b>The Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec32" style="font-size: 80%;"><b>Numpy Functionality</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec33" style="font-size: 80%;"><b>Correlation Matrix again</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec34" style="font-size: 80%;"><b>Using Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec35" style="font-size: 80%;"><b>And then the Franke Function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec36" style="font-size: 80%;"><b>Lnks with the Design Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec37" style="font-size: 80%;"><b>Computing the Expectation Values</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec38" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec39" style="font-size: 80%;"><b>More on the PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec40" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec41" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec42" style="font-size: 80%;"><b>Implementing it</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec43" style="font-size: 80%;"><b>First Step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec44" style="font-size: 80%;"><b>Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec45" style="font-size: 80%;"><b>Centered Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec46" style="font-size: 80%;"><b>Exploring</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec47" style="font-size: 80%;"><b>Diagonalize the sample covariance matrix to obtain the principal components</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec48" style="font-size: 80%;"><b>Collecting all Steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec49" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec50" style="font-size: 80%;"><b>The PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec51" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec52" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec53" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec54" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec55" style="font-size: 80%;"> Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec56" style="font-size: 80%;"> Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec57" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</ul>
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</li>
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</ul>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0011"></a>
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<!-- !split -->
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<h2 id="___sec10" class="anchor">Other Things to Try </h2>
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<p>
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Changing the size of the recurrent neural network and its parameters
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can drastically change the results you get from the model. The below
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code takes the simple recurrent neural network from above and adds a
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second hidden layer, changes the number of neurons in the hidden
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layer, and explicitly declares the activation function of the hidden
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layers to be a sigmoid function. The loss function and optimizer can
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also be changed but are kept the same as the above network. These
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parameters can be tuned to provide the optimal result from the
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network. For some ideas on how to improve the performance of a
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<a href="https://danijar.com/tips-for-training-recurrent-neural-networks" target="_self">recurrent neural network</a>.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">rnn_2layers</span>(length_of_sequences, batch_size <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">None</span>, stateful <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">False</span>):
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<span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #BA2121; font-style: italic"> Inputs:</span>
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<span style="color: #BA2121; font-style: italic"> length_of_sequences (an int): the number of y values in "x data". This is determined</span>
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<span style="color: #BA2121; font-style: italic"> when the data is formatted</span>
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<span style="color: #BA2121; font-style: italic"> batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.</span>
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<span style="color: #BA2121; font-style: italic"> stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.</span>
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<span style="color: #BA2121; font-style: italic"> Returns:</span>
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<span style="color: #BA2121; font-style: italic"> model (a Keras model): The recurrent neural network that is built and compiled by this</span>
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<span style="color: #BA2121; font-style: italic"> method</span>
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<span style="color: #BA2121; font-style: italic"> Builds and compiles a recurrent neural network with two hidden layers and returns the model.</span>
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<span style="color: #BA2121; font-style: italic"> """</span>
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<span style="color: #408080; font-style: italic"># Number of neurons in the input and output layers</span>
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in_out_neurons <span style="color: #666666">=</span> <span style="color: #666666">1</span>
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<span style="color: #408080; font-style: italic"># Number of neurons in the hidden layer, increased from the first network</span>
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hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">500</span>
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<span style="color: #408080; font-style: italic"># Define the input layer</span>
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inp <span style="color: #666666">=</span> Input(batch_shape<span style="color: #666666">=</span>(batch_size,
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length_of_sequences,
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in_out_neurons))
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<span style="color: #408080; font-style: italic"># Create two hidden layers instead of one hidden layer. Explicitly set the activation</span>
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<span style="color: #408080; font-style: italic"># function to be the sigmoid function (the default value is hyperbolic tangent)</span>
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rnn1 <span style="color: #666666">=</span> SimpleRNN(hidden_neurons,
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return_sequences<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, <span style="color: #408080; font-style: italic"># This needs to be True if another hidden layer is to follow</span>
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stateful <span style="color: #666666">=</span> stateful, activation <span style="color: #666666">=</span> <span style="color: #BA2121">'sigmoid'</span>,
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name<span style="color: #666666">=</span><span style="color: #BA2121">"RNN1"</span>)(inp)
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rnn2 <span style="color: #666666">=</span> SimpleRNN(hidden_neurons,
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return_sequences<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>, activation <span style="color: #666666">=</span> <span style="color: #BA2121">'sigmoid'</span>,
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stateful <span style="color: #666666">=</span> stateful,
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name<span style="color: #666666">=</span><span style="color: #BA2121">"RNN2"</span>)(rnn1)
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<span style="color: #408080; font-style: italic"># Define the output layer as a dense neural network layer (standard neural network layer)</span>
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<span style="color: #408080; font-style: italic">#and add it to the network immediately after the hidden layer.</span>
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dens <span style="color: #666666">=</span> Dense(in_out_neurons,name<span style="color: #666666">=</span><span style="color: #BA2121">"dense"</span>)(rnn2)
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<span style="color: #408080; font-style: italic"># Create the machine learning model starting with the input layer and ending with the </span>
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<span style="color: #408080; font-style: italic"># output layer</span>
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model <span style="color: #666666">=</span> Model(inputs<span style="color: #666666">=</span>[inp],outputs<span style="color: #666666">=</span>[dens])
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<span style="color: #408080; font-style: italic"># Compile the machine learning model using the mean squared error function as the loss </span>
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<span style="color: #408080; font-style: italic"># function and an Adams optimizer.</span>
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model<span style="color: #666666">.</span>compile(loss<span style="color: #666666">=</span><span style="color: #BA2121">"mean_squared_error"</span>, optimizer<span style="color: #666666">=</span><span style="color: #BA2121">"adam"</span>)
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<span style="color: #008000; font-weight: bold">return</span> model
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<span style="color: #408080; font-style: italic"># Check to make sure the data set is complete</span>
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<span style="color: #008000; font-weight: bold">assert</span> <span style="color: #008000">len</span>(X_tot) <span style="color: #666666">==</span> <span style="color: #008000">len</span>(y_tot)
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<span style="color: #408080; font-style: italic"># This is the number of points that will be used in as the training data</span>
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dim<span style="color: #666666">=12</span>
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<span style="color: #408080; font-style: italic"># Separate the training data from the whole data set</span>
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X_train <span style="color: #666666">=</span> X_tot[:dim]
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y_train <span style="color: #666666">=</span> y_tot[:dim]
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<span style="color: #408080; font-style: italic"># Generate the training data for the RNN, using a sequence of 2</span>
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rnn_input, rnn_training <span style="color: #666666">=</span> format_data(y_train, <span style="color: #666666">2</span>)
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<span style="color: #408080; font-style: italic"># Create a recurrent neural network in Keras and produce a summary of the </span>
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<span style="color: #408080; font-style: italic"># machine learning model</span>
|
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model <span style="color: #666666">=</span> rnn_2layers(length_of_sequences <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
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model<span style="color: #666666">.</span>summary()
|
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|
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<span style="color: #408080; font-style: italic"># Start the timer. Want to time training+testing</span>
|
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start <span style="color: #666666">=</span> timer()
|
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<span style="color: #408080; font-style: italic"># Fit the model using the training data genenerated above using 150 training iterations and a 5%</span>
|
|
<span style="color: #408080; font-style: italic"># validation split. Setting verbose to True prints information about each training iteration.</span>
|
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hist <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(rnn_input, rnn_training, batch_size<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>, epochs<span style="color: #666666">=150</span>,
|
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verbose<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>,validation_split<span style="color: #666666">=0.05</span>)
|
|
|
|
|
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<span style="color: #408080; font-style: italic"># This section plots the training loss and the validation loss as a function of training iteration.</span>
|
|
<span style="color: #408080; font-style: italic"># This is not required for analyzing the couple cluster data but can help determine if the network is</span>
|
|
<span style="color: #408080; font-style: italic"># being overtrained.</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> label <span style="color: #AA22FF; font-weight: bold">in</span> [<span style="color: #BA2121">"loss"</span>,<span style="color: #BA2121">"val_loss"</span>]:
|
|
plt<span style="color: #666666">.</span>plot(hist<span style="color: #666666">.</span>history[label],label<span style="color: #666666">=</span>label)
|
|
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"loss"</span>)
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"epoch"</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"The final validation loss: </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(hist<span style="color: #666666">.</span>history[<span style="color: #BA2121">"val_loss"</span>][<span style="color: #666666">-1</span>]))
|
|
plt<span style="color: #666666">.</span>legend()
|
|
plt<span style="color: #666666">.</span>show()
|
|
|
|
<span style="color: #408080; font-style: italic"># Use the trained neural network to predict more points of the data set</span>
|
|
test_rnn(X_tot, y_tot, X_tot[<span style="color: #666666">0</span>], X_tot[dim<span style="color: #666666">-1</span>])
|
|
<span style="color: #408080; font-style: italic"># Stop the timer and calculate the total time needed.</span>
|
|
end <span style="color: #666666">=</span> timer()
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Time: '</span>, end<span style="color: #666666">-</span>start)
|
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</pre></div>
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<p>
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