lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

small update on neural networks

Browse Source
This commit is contained in:
mhjensen
2018-10-16 13:54:54 +02:00
parent ce0c90eccd
commit 9bdb28895e
75 changed files with 6318 additions and 5703 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/pub/NeuralNet/html/._NeuralNet-bs037.html
+100 -95
View File
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
'___sec30'),
('Defining the cost function', 2, None, '___sec31'),
('Example: binary classification problem', 2, None, '___sec32'),
('The Softmax function', 2, None, '___sec33'),
('Developing a code for doing neural networks with back '
'propagation',
2,
None,
'___sec33'),
('Collect and pre-process data', 2, None, '___sec34'),
('Train and test datasets', 2, None, '___sec35'),
('Define model and architecture', 2, None, '___sec36'),
('Layers', 2, None, '___sec37'),
('Weights and biases', 2, None, '___sec38'),
('Feed-forward pass', 2, None, '___sec39'),
('Matrix multiplications', 2, None, '___sec40'),
('Choose cost function and optimizer', 2, None, '___sec41'),
('Optimizing the cost function', 2, None, '___sec42'),
('Regularization', 2, None, '___sec43'),
('Matrix multiplication', 2, None, '___sec44'),
('Improving performance', 2, None, '___sec45'),
('Full object-oriented implementation', 2, None, '___sec46'),
('Evaluate model performance on test data', 2, None, '___sec47'),
('Adjust hyperparameters', 2, None, '___sec48'),
('Visualization', 2, None, '___sec49'),
('scikit-learn implementation', 2, None, '___sec50'),
('Visualization', 2, None, '___sec51'),
'___sec34'),
('Collect and pre-process data', 2, None, '___sec35'),
('Train and test datasets', 2, None, '___sec36'),
('Define model and architecture', 2, None, '___sec37'),
('Layers', 2, None, '___sec38'),
('Weights and biases', 2, None, '___sec39'),
('Feed-forward pass', 2, None, '___sec40'),
('Matrix multiplications', 2, None, '___sec41'),
('Choose cost function and optimizer', 2, None, '___sec42'),
('Optimizing the cost function', 2, None, '___sec43'),
('Regularization', 2, None, '___sec44'),
('Matrix multiplication', 2, None, '___sec45'),
('Improving performance', 2, None, '___sec46'),
('Full object-oriented implementation', 2, None, '___sec47'),
('Evaluate model performance on test data', 2, None, '___sec48'),
('Adjust hyperparameters', 2, None, '___sec49'),
('Visualization', 2, None, '___sec50'),
('scikit-learn implementation', 2, None, '___sec51'),
('Visualization', 2, None, '___sec52'),
('Building neural networks in Tensorflow and Keras',
2,
None,
'___sec52'),
('Tensorflow', 2, None, '___sec53'),
('Collect and pre-process data', 2, None, '___sec54'),
('Using TensorFlow backend', 2, None, '___sec55'),
('Optimizing and using gradient descent', 2, None, '___sec56'),
('Using Keras', 2, None, '___sec57'),
('Which activation function should I use?', 2, None, '___sec58'),
'___sec53'),
('Tensorflow', 2, None, '___sec54'),
('Collect and pre-process data', 2, None, '___sec55'),
('Using TensorFlow backend', 2, None, '___sec56'),
('Optimizing and using gradient descent', 2, None, '___sec57'),
('Using Keras', 2, None, '___sec58'),
('Which activation function should I use?', 2, None, '___sec59'),
('Is the Logistic activation function (Sigmoid) our choice?',
2,
None,
'___sec59'),
('The derivative of the Logistic funtion', 2, None, '___sec60'),
('The RELU function family', 2, None, '___sec61'),
('Which activation function should we use?', 2, None, '___sec62'),
'___sec60'),
('The derivative of the Logistic funtion', 2, None, '___sec61'),
('The RELU function family', 2, None, '___sec62'),
('Which activation function should we use?', 2, None, '___sec63'),
('A top-down perspective on Neural networks',
2,
None,
'___sec63'),
'___sec64'),
('Limitations of supervised learning with deep networks',
2,
None,
'___sec64')]}
'___sec65')]}
end of tocinfo -->
<body>
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._NeuralNet-bs031.html#___sec30" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs032.html#___sec31" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs033.html#___sec32" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs034.html#___sec33" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs035.html#___sec34" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs036.html#___sec35" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs038.html#___sec37" style="font-size: 80%;"><b>Layers</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs039.html#___sec38" style="font-size: 80%;"><b>Weights and biases</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs040.html#___sec39" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs041.html#___sec40" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs042.html#___sec41" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs043.html#___sec42" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs044.html#___sec43" style="font-size: 80%;"><b>Regularization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs045.html#___sec44" style="font-size: 80%;"><b>Matrix multiplication</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs046.html#___sec45" style="font-size: 80%;"><b>Improving performance</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs047.html#___sec46" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs048.html#___sec47" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs049.html#___sec48" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs050.html#___sec49" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs051.html#___sec50" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs052.html#___sec51" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs053.html#___sec52" style="font-size: 80%;"><b>Building neural networks in Tensorflow and Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs054.html#___sec53" style="font-size: 80%;"><b>Tensorflow</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs055.html#___sec54" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs056.html#___sec55" style="font-size: 80%;"><b>Using TensorFlow backend</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs057.html#___sec56" style="font-size: 80%;"><b>Optimizing and using gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs058.html#___sec57" style="font-size: 80%;"><b>Using Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs059.html#___sec58" style="font-size: 80%;"><b>Which activation function should I use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs060.html#___sec59" style="font-size: 80%;"><b>Is the Logistic activation function (Sigmoid) our choice?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs061.html#___sec60" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs062.html#___sec61" style="font-size: 80%;"><b>The RELU function family</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs063.html#___sec62" style="font-size: 80%;"><b>Which activation function should we use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs064.html#___sec63" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs065.html#___sec64" style="font-size: 80%;"><b>Limitations of supervised learning with deep networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs034.html#___sec33" style="font-size: 80%;"><b>The Softmax function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs035.html#___sec34" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs036.html#___sec35" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs038.html#___sec37" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs039.html#___sec38" style="font-size: 80%;"><b>Layers</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs040.html#___sec39" style="font-size: 80%;"><b>Weights and biases</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs041.html#___sec40" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs042.html#___sec41" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs043.html#___sec42" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs044.html#___sec43" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs045.html#___sec44" style="font-size: 80%;"><b>Regularization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs046.html#___sec45" style="font-size: 80%;"><b>Matrix multiplication</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs047.html#___sec46" style="font-size: 80%;"><b>Improving performance</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs048.html#___sec47" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs049.html#___sec48" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs050.html#___sec49" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs051.html#___sec50" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs052.html#___sec51" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs053.html#___sec52" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs054.html#___sec53" style="font-size: 80%;"><b>Building neural networks in Tensorflow and Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs055.html#___sec54" style="font-size: 80%;"><b>Tensorflow</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs056.html#___sec55" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs057.html#___sec56" style="font-size: 80%;"><b>Using TensorFlow backend</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs058.html#___sec57" style="font-size: 80%;"><b>Optimizing and using gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs059.html#___sec58" style="font-size: 80%;"><b>Using Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs060.html#___sec59" style="font-size: 80%;"><b>Which activation function should I use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs061.html#___sec60" style="font-size: 80%;"><b>Is the Logistic activation function (Sigmoid) our choice?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs062.html#___sec61" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs063.html#___sec62" style="font-size: 80%;"><b>The RELU function family</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs064.html#___sec63" style="font-size: 80%;"><b>Which activation function should we use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs065.html#___sec64" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs066.html#___sec65" style="font-size: 80%;"><b>Limitations of supervised learning with deep networks</b></a></li>
</ul>
</li>
@@ -257,49 +259,52 @@ MathJax.Hub.Config({
<a name="part0037"></a>
<!-- !split -->
<h2 id="___sec36" class="anchor">Define model and architecture </h2>
<h2 id="___sec36" class="anchor">Train and test datasets </h2>
<p>
Our simple feed-forward neural network will consist of an <em>input</em> layer, a single <em>hidden</em> layer and an <em>output</em> layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have
$$ z = \sum_{i=1}^n w_i a_i ,$$
$$ y = f(z) ,$$
Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
<p>
where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer
and \( w_i \) is the weight to input \( i \).
The activation of the neurons in the input layer is just the features (e.g. a pixel value).
We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
<p>
The simplest activation function for a neuron is the <em>Heaviside</em> function:
$$ f(z) =
\begin{cases}
1, & z > 0\\
0, & \text{otherwise}
\end{cases}
$$
It is important that the train and test datasets are drawn randomly from our dataset, to ensure
no bias in the sampling.
Say you are taking measurements of weather data to predict the weather in the coming 5 days.
You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
collected from 12.00 to 24.00.
<p>
A feed-forward neural network with this activation is known as a <em>perceptron</em>.
For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.
This activation can be generalized to \( k \) classes (using e.g. the <em>one-against-all</em> strategy),
and we call these architectures <em>multiclass perceptrons</em>.
<p>
However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and
Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<p>
Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).
We will be using the sigmoid function \( \sigma(x) \):
<span style="color: #408080; font-style: italic"># one-liner from scikit-learn library</span>
train_size <span style="color: #666666">=</span> <span style="color: #666666">0.8</span>
test_size <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> train_size
X_train, X_test, Y_train, Y_test <span style="color: #666666">=</span> train_test_split(inputs, labels, train_size<span style="color: #666666">=</span>train_size,
test_size<span style="color: #666666">=</span>test_size)
$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
<span style="color: #408080; font-style: italic"># equivalently in numpy</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">train_test_split_numpy</span>(inputs, labels, train_size, test_size):
n_inputs <span style="color: #666666">=</span> <span style="color: #008000">len</span>(inputs)
inputs_shuffled <span style="color: #666666">=</span> inputs<span style="color: #666666">.</span>copy()
labels_shuffled <span style="color: #666666">=</span> labels<span style="color: #666666">.</span>copy()
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>shuffle(inputs_shuffled)
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>shuffle(labels_shuffled)
train_end <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n_inputs<span style="color: #666666">*</span>train_size)
X_train, X_test <span style="color: #666666">=</span> inputs_shuffled[:train_end], inputs_shuffled[train_end:]
Y_train, Y_test <span style="color: #666666">=</span> labels_shuffled[:train_end], labels_shuffled[train_end:]
<span style="color: #008000; font-weight: bold">return</span> X_train, X_test, Y_train, Y_test
<p>
which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.
<span style="color: #408080; font-style: italic">#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)</span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Number of training images: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(<span style="color: #008000">len</span>(X_train)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Number of test images: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(<span style="color: #008000">len</span>(X_test)))
</pre></div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
@@ -326,7 +331,7 @@ which is inspired by probability theory (see logistic regression) and was most c
<li><a href="._NeuralNet-bs045.html">46</a></li>
<li><a href="._NeuralNet-bs046.html">47</a></li>
<li><a href="">...</a></li>
<li><a href="._NeuralNet-bs065.html">66</a></li>
<li><a href="._NeuralNet-bs066.html">67</a></li>
<li><a href="._NeuralNet-bs038.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->