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483 lines
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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="._week45-bs024.html#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="#other-types-of-recurrent-neural-networks" style="font-size: 80%;"><b>Other Types of Recurrent Neural Networks</b></a></li>
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<a name="part0027"></a>
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<!-- !split -->
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<h2 id="other-types-of-recurrent-neural-networks" class="anchor">Other Types of Recurrent Neural Networks </h2>
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<p>Besides a simple recurrent neural network layer, there are two other
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commonly used types of recurrent neural network layers: Long Short
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Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short
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introduction to these layers see <a href="https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b" target="_self"><tt>https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b</tt></a>
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and <a href="https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b" target="_self"><tt>https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b</tt></a>.
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</p>
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<p>The first network created below is similar to the previous network,
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but it replaces the SimpleRNN layers with LSTM layers. The second
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network below has two hidden layers made up of GRUs, which are
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preceeded by two dense (feeddorward) neural network layers. These
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dense layers "preprocess" the data before it reaches the recurrent
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layers. This architecture has been shown to improve the performance
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of recurrent neural networks (see the link above and also
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<a href="https://arxiv.org/pdf/1807.02857.pdf" target="_self"><tt>https://arxiv.org/pdf/1807.02857.pdf</tt></a>.
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</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">lstm_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 LSTM 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 on the input/output layer and the number of neurons in the hidden layer</span>
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in_out_neurons <span style="color: #666666">=</span> <span style="color: #666666">1</span>
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hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">250</span>
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<span style="color: #408080; font-style: italic"># 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"># Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)</span>
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rnn<span style="color: #666666">=</span> LSTM(hidden_neurons,
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return_sequences<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</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>, use_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, activation<span style="color: #666666">=</span><span style="color: #BA2121">'tanh'</span>)(inp)
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rnn1 <span style="color: #666666">=</span> LSTM(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">"RNN1"</span>, use_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, activation<span style="color: #666666">=</span><span style="color: #BA2121">'tanh'</span>)(rnn)
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<span style="color: #408080; font-style: italic"># Output 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>)(rnn1)
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<span style="color: #408080; font-style: italic"># Define the midel</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 model</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: #408080; font-style: italic"># Return the model</span>
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<span style="color: #008000; font-weight: bold">return</span> model
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">dnn2_gru2</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 four hidden layers (two dense followed by</span>
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<span style="color: #BA2121; font-style: italic"> two GRU 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 on the input/output layers and hidden layers</span>
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in_out_neurons <span style="color: #666666">=</span> <span style="color: #666666">1</span>
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hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">250</span>
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<span style="color: #408080; font-style: italic"># 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"># Hidden Dense (feedforward) layers</span>
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dnn <span style="color: #666666">=</span> Dense(hidden_neurons<span style="color: #666666">/2</span>, activation<span style="color: #666666">=</span><span style="color: #BA2121">'relu'</span>, name<span style="color: #666666">=</span><span style="color: #BA2121">'dnn'</span>)(inp)
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dnn1 <span style="color: #666666">=</span> Dense(hidden_neurons<span style="color: #666666">/2</span>, activation<span style="color: #666666">=</span><span style="color: #BA2121">'relu'</span>, name<span style="color: #666666">=</span><span style="color: #BA2121">'dnn1'</span>)(dnn)
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<span style="color: #408080; font-style: italic"># Hidden GRU layers</span>
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rnn1 <span style="color: #666666">=</span> GRU(hidden_neurons,
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return_sequences<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</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">"RNN1"</span>, use_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)(dnn1)
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rnn <span style="color: #666666">=</span> GRU(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>, use_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)(rnn1)
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<span style="color: #408080; font-style: italic"># Output 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"># Define the model</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 mdoel</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: #408080; font-style: italic"># Return the model</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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|
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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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|
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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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<span style="color: #408080; font-style: italic"># Change the method name to reflect which network you want to use</span>
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model <span style="color: #666666">=</span> dnn2_gru2(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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<span style="color: #408080; font-style: italic"># Start the timer. Want to time training+testing</span>
|
|
start <span style="color: #666666">=</span> timer()
|
|
<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>
|
|
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>,
|
|
verbose<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>,validation_split<span style="color: #666666">=0.05</span>)
|
|
|
|
|
|
<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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<span style="color: #408080; font-style: italic"># ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)</span>
|
|
<span style="color: #408080; font-style: italic"># </span>
|
|
<span style="color: #408080; font-style: italic"># 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.</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Check to make sure the data set is complete</span>
|
|
<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)
|
|
|
|
<span style="color: #408080; font-style: italic"># This is the number of points that will be used in as the training data</span>
|
|
dim<span style="color: #666666">=12</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Separate the training data from the whole data set</span>
|
|
X_train <span style="color: #666666">=</span> X_tot[:dim]
|
|
y_train <span style="color: #666666">=</span> y_tot[:dim]
|
|
|
|
<span style="color: #408080; font-style: italic"># Reshape the data for Keras specifications</span>
|
|
X_train <span style="color: #666666">=</span> X_train<span style="color: #666666">.</span>reshape((dim, <span style="color: #666666">1</span>))
|
|
y_train <span style="color: #666666">=</span> y_train<span style="color: #666666">.</span>reshape((dim, <span style="color: #666666">1</span>))
|
|
|
|
|
|
<span style="color: #408080; font-style: italic"># Create a recurrent neural network in Keras and produce a summary of the </span>
|
|
<span style="color: #408080; font-style: italic"># machine learning model</span>
|
|
<span style="color: #408080; font-style: italic"># Set the sequence length to 1 for regular data formatting </span>
|
|
model <span style="color: #666666">=</span> rnn(length_of_sequences <span style="color: #666666">=</span> <span style="color: #666666">1</span>)
|
|
model<span style="color: #666666">.</span>summary()
|
|
|
|
<span style="color: #408080; font-style: italic"># Start the timer. Want to time training+testing</span>
|
|
start <span style="color: #666666">=</span> timer()
|
|
<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>
|
|
hist <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(X_train, y_train, batch_size<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>, epochs<span style="color: #666666">=150</span>,
|
|
verbose<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>,validation_split<span style="color: #666666">=0.05</span>)
|
|
|
|
|
|
<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 the remaining data points</span>
|
|
X_pred <span style="color: #666666">=</span> X_tot[dim:]
|
|
X_pred <span style="color: #666666">=</span> X_pred<span style="color: #666666">.</span>reshape((<span style="color: #008000">len</span>(X_pred), <span style="color: #666666">1</span>))
|
|
y_model <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(X_pred)
|
|
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate((y_tot[:dim], y_model<span style="color: #666666">.</span>flatten()))
|
|
|
|
<span style="color: #408080; font-style: italic"># Plot the known data set and the predicted data set. The red box represents the region that was used</span>
|
|
<span style="color: #408080; font-style: italic"># for the training data.</span>
|
|
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
|
ax<span style="color: #666666">.</span>plot(X_tot, y_tot, label<span style="color: #666666">=</span><span style="color: #BA2121">"true"</span>, linewidth<span style="color: #666666">=3</span>)
|
|
ax<span style="color: #666666">.</span>plot(X_tot, y_pred, <span style="color: #BA2121">'g-.'</span>,label<span style="color: #666666">=</span><span style="color: #BA2121">"predicted"</span>, linewidth<span style="color: #666666">=4</span>)
|
|
ax<span style="color: #666666">.</span>legend()
|
|
<span style="color: #408080; font-style: italic"># Created a red region to represent the points used in the training data.</span>
|
|
ax<span style="color: #666666">.</span>axvspan(X_tot[<span style="color: #666666">0</span>], X_tot[dim], alpha<span style="color: #666666">=0.25</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
|
|
<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)
|
|
</pre>
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