Morten Hjorth-Jensen
387 lines
26 KiB
HTML
387 lines
26 KiB
HTML
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<!-- navigation toc: --> <li><a href="._week45-bs001.html#plan-for-week-45" style="font-size: 80%;"><b>Plan for week 45</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs002.html#material-for-the-lab-sessions-additional-ways-to-present-classification-results-and-other-practicalities" style="font-size: 80%;"><b>Material for the lab sessions, additional ways to present classification results and other practicalities</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs003.html#searching-for-optimal-regularization-parameters-lambda" style="font-size: 80%;"><b>Searching for Optimal Regularization Parameters \( \lambda \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs004.html#grid-search" style="font-size: 80%;"><b>Grid Search</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs005.html#randomized-grid-search" style="font-size: 80%;"><b>Randomized Grid Search</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs006.html#wisconsin-cancer-data" style="font-size: 80%;"><b>Wisconsin Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs007.html#using-the-correlation-matrix" style="font-size: 80%;"><b>Using the correlation matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs008.html#discussing-the-correlation-data" style="font-size: 80%;"><b>Discussing the correlation data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs009.html#other-ways-of-presenting-a-classification-problem" style="font-size: 80%;"><b>Other ways of presenting a classification problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs010.html#combinations-of-classification-results" style="font-size: 80%;"><b>Combinations of classification results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs011.html#positive-and-negative-prediction-values" style="font-size: 80%;"><b>Positive and negative prediction values</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs012.html#other-quantities" style="font-size: 80%;"><b>Other quantities</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs013.html#f-1-score" style="font-size: 80%;"><b>\( F_1 \) score</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs014.html#roc-curve" style="font-size: 80%;"><b>ROC curve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs015.html#cumulative-gain-curve" style="font-size: 80%;"><b>Cumulative gain curve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs016.html#other-measures-in-classification-studies-cancer-data-again" style="font-size: 80%;"><b>Other measures in classification studies: Cancer Data again</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs017.html#material-for-lecture-thursday-november-9" style="font-size: 80%;"><b>Material for Lecture Thursday November 9</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs018.html#recurrent-neural-networks-rnns-overarching-view" style="font-size: 80%;"><b>Recurrent neural networks (RNNs): Overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs019.html#a-simple-example" style="font-size: 80%;"><b>A simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs019.html#rnns" style="font-size: 80%;"> RNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#basic-layout" style="font-size: 80%;"><b>Basic layout</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#we-need-to-specify-the-initial-activity-state-of-all-the-hidden-and-output-units" style="font-size: 80%;"> We need to specify the initial activity state of all the hidden and output units</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#we-can-specify-inputs-in-several-ways" style="font-size: 80%;"> We can specify inputs in several ways</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#we-can-specify-targets-in-several-ways" style="font-size: 80%;"> We can specify targets in several ways</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#backpropagation-through-time" style="font-size: 80%;"> Backpropagation through time</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#the-backward-pass-is-linear" style="font-size: 80%;"> The backward pass is linear</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs021.html#the-problem-of-exploding-or-vanishing-gradients" style="font-size: 80%;"><b>The problem of exploding or vanishing gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#four-effective-ways-to-learn-an-rnn" style="font-size: 80%;"><b>Four effective ways to learn an RNN</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#long-short-term-memory-lstm" style="font-size: 80%;"> Long Short Term Memory (LSTM)</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#implementing-a-memory-cell-in-a-neural-network" style="font-size: 80%;"> Implementing a memory cell in a neural network</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs023.html#an-extrapolation-example" style="font-size: 80%;"><b>An extrapolation example</b></a></li>
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<!-- navigation toc: --> <li><a href="#formatting-the-data" style="font-size: 80%;"><b>Formatting the Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs025.html#predicting-new-points-with-a-trained-recurrent-neural-network" 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="._week45-bs026.html#other-things-to-try" style="font-size: 80%;"><b>Other Things to Try</b></a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs027.html#other-types-of-recurrent-neural-networks" style="font-size: 80%;"><b>Other Types of Recurrent Neural Networks</b></a></li>
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</ul>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0024"></a>
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<!-- !split -->
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<h2 id="formatting-the-data" class="anchor">Formatting the Data </h2>
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<p>The way the recurrent neural networks are trained in this program
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differs from how machine learning algorithms are usually trained.
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Typically a machine learning algorithm is trained by learning the
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relationship between the x data and the y data. In this program, the
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recurrent neural network will be trained to recognize the relationship
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in a sequence of y values. This is type of data formatting is
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typically used time series forcasting, but it can also be used in any
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extrapolation (time series forecasting is just a specific type of
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extrapolation along the time axis). This method of data formatting
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does not use the x data and assumes that the y data are evenly spaced.
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</p>
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<p>For a standard machine learning algorithm, the training data has the
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form of (x,y) so the machine learning algorithm learns to assiciate a
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y value with a given x value. This is useful when the test data has x
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values within the same range as the training data. However, for this
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application, the x values of the test data are outside of the x values
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of the training data and the traditional method of training a machine
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learning algorithm does not work as well. For this reason, the
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recurrent neural network is trained on sequences of y values of the
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form ((y1, y2), y3), so that the network is concerned with learning
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the pattern of the y data and not the relation between the x and y
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data. As long as the pattern of y data outside of the training region
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stays relatively stable compared to what was inside the training
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region, this method of training can produce accurate extrapolations to
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y values far removed from the training data set.
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</p>
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<!-- -->
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<!-- 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). -->
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<!-- -->
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<!-- 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. -->
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<!-- -->
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<!-- 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. -->
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># FORMAT_DATA</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">format_data</span>(data, length_of_sequence <span style="color: #666666">=</span> <span style="color: #666666">2</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"> data(a numpy array): the data that will be the inputs to the recurrent neural</span>
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<span style="color: #BA2121; font-style: italic"> network</span>
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<span style="color: #BA2121; font-style: italic"> length_of_sequence (an int): the number of elements in one iteration of the</span>
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<span style="color: #BA2121; font-style: italic"> sequence patter. For a function approximator use length_of_sequence = 2.</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"> rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its</span>
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<span style="color: #BA2121; font-style: italic"> dimensions are length of data - length of sequence, length of sequence, </span>
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<span style="color: #BA2121; font-style: italic"> dimnsion of data</span>
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<span style="color: #BA2121; font-style: italic"> rnn_output (a numpy array): the training data for the neural network</span>
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<span style="color: #BA2121; font-style: italic"> Formats data to be used in a recurrent neural network.</span>
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<span style="color: #BA2121; font-style: italic"> """</span>
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X, Y <span style="color: #666666">=</span> [], []
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(data)<span style="color: #666666">-</span>length_of_sequence):
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<span style="color: #408080; font-style: italic"># Get the next length_of_sequence elements</span>
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a <span style="color: #666666">=</span> data[i:i<span style="color: #666666">+</span>length_of_sequence]
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<span style="color: #408080; font-style: italic"># Get the element that immediately follows that</span>
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b <span style="color: #666666">=</span> data[i<span style="color: #666666">+</span>length_of_sequence]
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<span style="color: #408080; font-style: italic"># Reshape so that each data point is contained in its own array</span>
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a <span style="color: #666666">=</span> np<span style="color: #666666">.</span>reshape (a, (<span style="color: #008000">len</span>(a), <span style="color: #666666">1</span>))
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X<span style="color: #666666">.</span>append(a)
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Y<span style="color: #666666">.</span>append(b)
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rnn_input <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array(X)
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rnn_output <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array(Y)
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<span style="color: #008000; font-weight: bold">return</span> rnn_input, rnn_output
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<span style="color: #408080; font-style: italic"># ## Defining the Recurrent Neural Network Using Keras</span>
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<span style="color: #408080; font-style: italic"># </span>
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<span style="color: #408080; font-style: italic"># The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">rnn</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 one hidden layer 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</span>
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hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">200</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"># Define the hidden layer as a simple RNN layer with a set number of neurons and add it to </span>
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<span style="color: #408080; font-style: italic"># the network immediately after the input layer</span>
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rnn <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>,
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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">"RNN"</span>)(inp)
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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>)(rnn)
|
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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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