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Update on neural networks

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mhjensen
2017-11-22 14:45:03 +01:00
parent d52541b08e
commit 9d2bcb57ab
52 changed files with 617 additions and 792 deletions
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doc/pub/NeuralNet/html/._NeuralNet-bs000.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Nov 2, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
@@ -168,8 +168,8 @@ MathJax.Hub.Config({
</div> <!-- end container -->
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doc/pub/NeuralNet/html/._NeuralNet-bs001.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -156,8 +156,8 @@ found applications in a wide variety of other areas, including bioinformatics, e
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs002.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -163,8 +163,8 @@ Some of the most common tasks are:
</div> <!-- end container -->
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<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs003.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -167,8 +167,8 @@ a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neuron
</div> <!-- end container -->
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<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs004.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -154,8 +154,8 @@ is associated with a weight variable.
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<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs005.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
<script type="text/javascript" async
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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@@ -169,8 +169,8 @@ which gathers all the local data and produces the outputs. They have wide applic
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs006.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -153,8 +153,8 @@ sentences, making recurrent NNs especially well-suited for handwriting and speec
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs007.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -157,8 +157,8 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data
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<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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doc/pub/NeuralNet/html/._NeuralNet-bs008.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -156,8 +156,8 @@ which means that each neuron receives a weighted sum of the outputs of <em>all</
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<!-- include javascript, jQuery *first* -->
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs009.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<footer>
doc/pub/NeuralNet/html/._NeuralNet-bs010.html
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -67,8 +67,8 @@ MathJax.Hub.Config({
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@@ -169,8 +169,8 @@ $$
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<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
<!-- Bootstrap style: bootstrap -->
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@@ -159,8 +159,8 @@ which illustrates a basic property of MLPs: The only independent variables are t
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
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@@ -172,8 +172,8 @@ $$
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<title>Data Analysis and Machine Learning: Elements of machine learning</title>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs001.html#___sec0" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs002.html#___sec1" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs013.html#___sec12" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;"><b>Other types of networks</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
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<h2 id="___sec0" class="anchor">What is Machine Learning? </h2>
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Machine learning is the science of giving computers the ability to learn without being explicitly programmed.
The idea is that there exist generic algorithms which can be used to find patterns in a broad class of data sets without
having to write code specifically for each problem. The algorithm will build its own logic based on the data.
<p>
Machine learning is a subfield of computer science, and is closely related to computational statistics.
It evolved from the study of pattern recognition in artificial intelligence (AI) research, and has made contributions to
AI tasks like computer vision, natural language processing
and speech recognition. It has also, especially in later years,
found applications in a wide variety of other areas, including bioinformatics, economy, physics, finance and marketing.
<p>
<!-- !split -->
<h2 id="___sec1" class="anchor">Types of Machine Learning </h2>
<p>
The approaches to machine learning are many, but are often split into two main categories.
In <em>supervised learning</em> we know the answer to a problem,
and let the computer deduce the logic behind it. On the other hand, <em>unsupervised learning</em>
is a method for finding patterns and relationship in data sets without any prior knowledge of the system.
Some authours also operate with a third category, namely <em>reinforcement learning</em>. This is a paradigm
of learning inspired by behavioural psychology, where learning is achieved by trial-and-error,
solely from rewards and punishment.
<p>
Another way to categorize machine learning tasks is to consider the desired output of a system.
Some of the most common tasks are:
<ul>
<li> Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</li>
<li> Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</li>
<li> Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</li>
</ul>
<!-- !split -->
<h2 id="___sec2" class="anchor">Artificial neurons </h2>
The field of artificial neural networks has a long history of development, and is closely connected with
the advancement of computer science and computers in general. A model of artificial neurons
was first developed by McCulloch and Pitts in 1943 to study signal processing in the brain and
has later been refined by others. The general idea is to mimic neural networks in the human brain, which
is composed of billions of neurons that communicate with each other by sending electrical signals.
Each neuron accumulates its incoming signals,
which must exceed an activation threshold to yield an output. If the threshold is not overcome, the neuron
remains inactive, i.e. has zero output.
<p>
This behaviour has inspired a simple mathematical model for an artificial neuron.
$$
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron}
\end{equation}
$$
Here, the output \( y \) of the neuron is the value of its activation function, which have as input
a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
<p>
<!-- !split -->
<h2 id="___sec3" class="anchor">Neural network types </h2>
<p>
An artificial neural network (NN), is a computational model that consists of layers of connected neurons, or <em>nodes</em>.
It is supposed to mimic a biological nervous system by letting each neuron interact with other neurons
by sending signals in the form of mathematical functions between layers.
A wide variety of different NNs have
been developed, but most of them consist of an input layer, an output layer and eventual layers in-between, called
<em>hidden layers</em>. All layers can contain an arbitrary number of nodes, and each connection between two nodes
is associated with a weight variable.
<p>
<!-- !split -->
<h2 id="___sec4" class="anchor">Feed-forward neural networks </h2>
The feed-forward neural network (FFNN) was the first and simplest type of NN devised. In this network,
the information moves in only one direction: forward through the layers.
<p>
Nodes are represented by circles, while the arrows display the connections between the nodes, including the
direction of information flow. Additionally, each arrow corresponds to a weight variable, not displayed here.
We observe that each node in a layer is connected to <em>all</em> nodes in the subsequent layer,
making this a so-called <em>fully-connected</em> FFNN.
<p>
A different variant of FFNNs are <em>convolutional neural networks</em> (CNNs), which have a connectivity pattern
inspired by the animal visual cortex. Individual neurons in the visual cortex only respond to stimuli from
small sub-regions of the visual field, called a receptive field. This makes the neurons well-suited to exploit the strong
spatially local correlation present in natural images. The response of each neuron can be approximated mathematically
as a convolution operation.
<p>
CNNs emulate the behaviour of neurons in the visual cortex by enforcing a <em>local</em> connectivity pattern
between nodes of adjacent layers: Each node
in a convolutional layer is connected only to a subset of the nodes in the previous layer,
in contrast to the fully-connected FFNN.
Often, CNNs
consist of several convolutional layers that learn local features of the input, with a fully-connected layer at the end,
which gathers all the local data and produces the outputs. They have wide applications in image and video recognition
<p>
<!-- !split -->
<h2 id="___sec5" class="anchor">Recurrent neural networks </h2>
<p>
So far we have only mentioned NNs where information flows in one direction: forward. <em>Recurrent neural networks</em> on
the other hand, have connections between nodes that form directed <em>cycles</em>. This creates a form of
internal memory which are able to capture information on what has been calculated before; the output is dependent
on the previous computations. Recurrent NNs make use of sequential information by performing the same task for
every element in a sequence, where each element depends on previous elements. An example of such information is
sentences, making recurrent NNs especially well-suited for handwriting and speech recognition.
<p>
<!-- !split -->
<h2 id="___sec6" class="anchor">Other types of networks </h2>
<p>
There are many other kinds of NNs that have been developed. One type that is specifically designed for interpolation
in multidimensional space is the radial basis function (RBF) network. RBFs are typically made up of three layers:
an input layer, a hidden layer with non-linear radial symmetric activation functions and a linear output layer (''linear'' here
means that each node in the output layer has a linear activation function). The layers are normally fully-connected and
there are no cycles, thus RBFs can be viewed as a type of fully-connected FFNN. They are however usually treated as
a separate type of NN due the unusual activation functions.
<p>
Other types of NNs could also be mentioned, but are outside the scope of this work. We will now move on to a detailed description
of how a fully-connected FFNN works, and how it can be used to interpolate data sets.
<p>
<!-- !split -->
<h2 id="___sec7" class="anchor">Mathematical model </h2>
$$
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(u)
\label{artificialNeuron2}
\end{equation}
$$
In an FFNN of such neurons, the <em>inputs</em> \( x_i \)
are the <em>outputs</em> of the neurons in the preceding layer. Furthermore, a MLP is fully-connected,
which means that each neuron receives a weighted sum of the outputs of <em>all</em> neurons in the previous layer.
<p>
<!-- !split -->
<h2 id="___sec8" class="anchor">Mathematical model </h2>
<p>
First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
$$
\begin{equation}
u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
\label{_auto1}
\end{equation}
$$
This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
producing the output \( y_i^1 \) of all neurons in layer 1,
$$
\begin{equation}
y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1}
\end{equation}
$$
where we assume that all nodes in the same layer have identical activation functions, hence the notation \( f_l \)
$$
\begin{equation}
y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer}
\end{equation}
$$
where \( N_l \) is the number of nodes in layer \( l \). When the output of all the nodes in the first hidden layer are computed,
the values of the subsequent layer can be calculated and so forth until the output is obtained.
<p>
<!-- !split -->
<h2 id="___sec9" class="anchor">Mathematical model </h2>
<p>
The output of neuron \( i \) in layer 2 is thus,
$$
\begin{align}
y_i^2 &= f_2\left(\sum_{j=1}^3 w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2}\\
&= f_2\left[\sum_{j=1}^3 w_{ij}^2f_1\left(\sum_{k=1}^2 w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2}
\end{align}
$$
where we have substituted \( y_m^1 \) with. Finally, the NN output yields,
$$
\begin{align}
y_1^3 &= f_3\left(\sum_{j=1}^3 w_{1m}^3 y_j^2 + b_1^3\right)
\label{_auto3}\\
&= f_3\left[\sum_{j=1}^3 w_{1j}^3 f_2\left(\sum_{k=1}^3 w_{jk}^2 f_1\left(\sum_{m=1}^2 w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4}
\end{align}
$$
<p>
<!-- !split -->
<h2 id="___sec10" class="anchor">Mathematical model </h2>
<p>
We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form
is,
$$
\begin{align}
&y^{l+1}_1\! = \!f_{l+1}\!\left[\!\sum_{j=1}^{N_l}\! w_{1j}^3 f_l\!\left(\!\sum_{k=1}^{N_{l-1}}\! w_{jk}^2 f_{l-1}\!\left(\!
\dots \!f_1\!\left(\!\sum_{n=1}^{N_0} \!w_{mn}^1 x_n\! + \!b_m^1\!\right)
\!\dots \!\right) \!+ \!b_k^2\!\right)
\!+ \!b_1^3\!\right] &&
\label{completeNN}
\end{align}
$$
which illustrates a basic property of MLPs: The only independent variables are the input values \( x_n \).
<p>
<!-- !split -->
<h2 id="___sec11" class="anchor">Mathematical model </h2>
<p>
This confirms that an MLP,
despite its quite convoluted mathematical form, is nothing more than an analytic function, specifically a
mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow \vec{y} \in \mathbb{R}^m \).
In our example, \( n=2 \) and \( m=1 \). Consequentially,
the number of input and output values of the function we want to fit must be equal to the number of inputs and outputs of our MLP.
<p>
Furthermore, the flexibility and universality of a MLP can be illustrated by realizing that
the expression is essentially a nested sum of scaled activation functions of the form
$$
\begin{equation}
h(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5}
\end{equation}
$$
where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions
can be shifted up and down or left and right, change slope or be rescaled
which is the key to the flexibility of a NN.
$$
\begin{equation}
f_o = f(u_o) = u_o
\label{outputActivation}
\end{equation}
$$
<p>
<!-- !split -->
<h3 id="___sec12" class="anchor">Matrix-vector notation </h3>
We can introduce a more convenient notation for the activations in a NN.
<p>
Additionally, we can represent the biases and activations
as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
<p>
We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
With this notation, the sum in becomes a matrix-vector multiplication, and we can write
the equation for the activations of hidden layer 2 in
$$
\begin{equation}
\vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
w^2_{31} &w^2_{32} &w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right)
\label{_auto6}
\end{equation}
$$
and we see that the activation of node \( i \) in layer 2 is
$$
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right)
\label{_auto7}
\end{equation}
$$
which is in accordance with. Note that
This is not just a convenient and compact notation, but also
a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications
and vector additions that are used as input to the activation functions. For each operation
\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
<p>
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<br>
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<center><h4>Nov 2, 2017</h4></center> <!-- date -->
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@@ -7,9 +7,10 @@
"<!-- dom:TITLE: Data Analysis and Machine Learning: Elements of machine learning -->\n",
"# Data Analysis and Machine Learning: Elements of machine learning\n",
"<!-- dom:AUTHOR: Morten Hjorth-Jensen at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University -->\n",
"<!-- Author: --> **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"<!-- Author: --> \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
"Date: **Nov 2, 2017**\n",
"Date: **Nov 22, 2017**\n",
"\n",
"Copyright 1999-2017, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -485,5 +486,5 @@
],
"metadata": {},
"nbformat": 4,
"nbformat_minor": 1
"nbformat_minor": 2
}
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@@ -168,7 +168,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Nov 2, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
@@ -199,8 +199,8 @@ MathJax.Hub.Config({
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<!-- Bootstrap footer
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doc/pub/Regression/html/._Regression-bs001.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<script type="text/javascript"
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@@ -196,8 +196,8 @@ A regression model aims at finding a likelihood function \( p(y\vert \hat{x}) \)
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<footer>
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<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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<script type="text/javascript" async
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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@@ -193,8 +193,8 @@ where \( \epsilon_i \) is the error in our approximation.
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs003.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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@@ -193,8 +193,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs004.html
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<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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@@ -213,8 +213,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs005.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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@@ -198,8 +198,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs006.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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@@ -204,8 +204,8 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs007.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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@@ -199,8 +199,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs008.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<link href="http://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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@@ -202,8 +202,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
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<!-- Bootstrap footer
<footer>
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<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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@@ -216,8 +216,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<!-- Bootstrap footer
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doc/pub/Regression/html/._Regression-bs010.html
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<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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@@ -203,8 +203,8 @@ $$
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs011.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<!-- not necessary
<link href="http://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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<script type="text/javascript"
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<script type="text/javascript" async
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@@ -204,8 +204,8 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="http://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
<!-- Bootstrap footer
<footer>
doc/pub/Regression/html/._Regression-bs012.html
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@@ -11,9 +11,9 @@ Automatically generated HTML file from DocOnce source
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
<!-- Bootstrap style: bootstrap -->
<link href="http://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<link href="http://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
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@@ -93,8 +93,8 @@ MathJax.Hub.Config({
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@@ -198,8 +198,8 @@ where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as
</div> <!-- end container -->
<!-- include javascript, jQuery *first* -->
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<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">General linear models</a></li>
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, follows</a></li>
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Optimizing our parameters</a></li>
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
<!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs001.html#___sec0" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">General linear models</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, follows</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Nov 21, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
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<!-- !split -->
<h2 id="___sec0" class="anchor">Regression analysis, overarching aims </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) varies as function of another variable or a set of such variables \( \hat{x} =[x_0, x_1,\dots, x_p]^T \).
The first variable is called the <b>dependent</b>, the <b>outcome</b> or the <b>response</b> variable while the set of variables \( \hat{x} \) is called the independent variable, or the predictor variable or the explanatory variable.
<p>
A regression model aims at finding a likelihood function \( p(y\vert \hat{x}) \), that is the conditional distribution for \( y \) with a given \( \hat{x} \). The estimation of \( p(y\vert \hat{x}) \) is made using a data set with
<ul>
<li> \( n \) cases \( i = 0, 1, 2, \dots, n-1 \)</li>
<li> Response (dependent or outcome) variable \( y_i \) with \( i = 0, 1, 2, \dots, n-1 \)</li>
<li> \( p \) Explanatory (independent or predictor) variables \( \hat{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip}] \) with \( i = 0, 1, 2, \dots, n-1 \)</li>
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The goal of the regression analysis is to extract/exploit relationship between \( y_i \) and \( \hat{x}_i \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions .
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec1" class="anchor">General linear models </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \hat{y}=[y_0,y_1,\dots,y_{n-1}] \). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \( \hat{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.
<p>
Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \( y \) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \( n-1 \) with \( n \) points, that is
$$
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_i x_i^j+\epsilon_i,
$$
where \( \epsilon_i \) is the error in our approximation.
<p>
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec2" class="anchor">Rewriting the fitting procedure as a linear algebra problem </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
For every set of values \( y_i,x_i \) we have thus the corresponding set of equations
$$
\begin{align*}
y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
\dots & \dots \\
y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec3" class="anchor">Rewriting the fitting procedure as a linear algebra problem, follows </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
Defining the vectors
$$
\hat{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
$$
$$
\hat{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
$$
$$
\hat{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
$$
and the matrix
$$
\hat{X}=
\begin{bmatrix}
1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\
1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\
1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\
\dots& \dots &\dots& \dots & \dots &\dots\\
1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\
\end{bmatrix}
$$
we can rewrite our equations as
$$
\hat{y} = \hat{X}\hat{\beta}+\hat{\epsilon}.
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec4" class="anchor">Generalizing the fitting procedure as a linear algebra problem </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
We are obviously not limited to the above polynomial. We could replace the various powers of \( x \) with elements of Fourier series, that is, instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j x_i)} \), or time series or other orthogonal functions.
For every set of values \( y_i,x_i \) we can then generalize the equations to
$$
\begin{align*}
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
\dots & \dots \\
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
\dots & \dots \\
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec5" class="anchor">Generalizing the fitting procedure as a linear algebra problem </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
We redefine in turn the matrix \( \hat{X} \) as
$$
\hat{X}=
\begin{bmatrix}
x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\
x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\
x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\
\dots& \dots &\dots& \dots & \dots &\dots\\
x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\
\end{bmatrix}
$$
and without loss of generality we rewrite again our equations as
$$
\hat{y} = \hat{X}\hat{\beta}+\hat{\epsilon}.
$$
The left-hand side of this equation forms know. Our error vector \( \hat{\epsilon} \) and the parameter vector \( \hat{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec6" class="anchor">Optimizing our parameters </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
We have defined the matrix \( \hat{X} \)
$$
\begin{align*}
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
\dots & \dots \\
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
\dots & \dots \\
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec7" class="anchor">Optimizing our parameters, more details </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
We well use this matrix to define the approximation \( \hat{\tilde{y}} \) via the unknown quantity \( \hat{\beta} \) as
$$
\hat{\tilde{y}}= \hat{X}\hat{\beta},
$$
and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parametrized values \( \tilde{y}_i \), namely
$$
Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right),
$$
or using the matrix \( \hat{X} \) as
$$
Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right).
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec8" class="anchor">Interpretations and optimizing our parameters </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
The function
$$
Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right),
$$
can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value
$$
y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i,
$$
where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
<p>
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring
$$
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
$$
which results in
$$
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
$$
or in a matrix-vector form as
$$
\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right).
$$
<p>
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec9" class="anchor">Interpretations and optimizing our parameters </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
We can rewrite
$$
\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right),
$$
as
$$
\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta},
$$
and if the matrix \( \hat{X}^T\hat{X} \) is invertible we have the solution
$$
\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}.
$$
<p>
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec10" class="anchor">Interpretations and optimizing our parameters </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
The residuals \( \hat{\epsilon} \) are in turn given by
$$
\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta},
$$
and with
$$
\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0,
$$
we have
$$
\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0,
$$
meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.
<p>
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec11" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
<p>
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function as
$$
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
$$
where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
<p>
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec12" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\hat{\beta}) \) by requiring
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
$$
which results in
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
$$
or in a matrix-vector form as
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
$$
where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \hat{b} \) with elements \( b_i = y_i/\sigma_i \).
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec13" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
We can rewrite
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
$$
as
$$
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
$$
and if the matrix \( \hat{A}^T\hat{A} \) is invertible we have the solution
$$
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec14" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
If we then introduce the matrix
$$
\hat{H} = \hat{A}^T\hat{A},
$$
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \hat{H} \) are \( h_{ij} \))
$$
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
$$
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as
$$
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
$$
resulting in
$$
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec15" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
$$
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
$$
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
$$
and
$$
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
$$
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec16" class="anchor">The \( \chi^2 \) function </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
We define then
$$
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
$$
$$
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
$$
$$
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
$$
$$
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
$$
$$
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
$$
and show that
$$
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
$$
$$
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
$$
<p>
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
</div>
</div>
<p>
<!-- !split -->
<h2 id="___sec17" class="anchor">The singular value decompostion </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
How can we use the singular value decomposition to find the parameters \( \beta_j \)? More details will come. We first note that a general \( m\times n \) matrix \( \hat{A} \) can be written in terms of a diagonal matrix \( \hat{\Sigma} \) of dimensionality \( n\times n \) and two orthognal matrices \( \hat{U} \) and \( \hat{V} \), where the first has dimensionality \( m \times n \) and the last dimensionality \( n\times n \). We have then
$$
\hat{A} = \hat{U}\hat{\Sigma}\hat{V}
$$
</div>
</div>
<p>
<!-- ------------------- end of main content --------------- -->
</div> <!-- end container -->
doc/pub/Regression/html/Regression-reveal.html
+1 -1
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@@ -148,7 +148,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>&nbsp;<br>
<center><h4>Nov 21, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
doc/pub/Regression/html/Regression-solarized.html
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@@ -66,8 +66,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'___sec2'),
('Rewriting the fitting procedure as a linear algebra problem, '
'follows',
('Rewriting the fitting procedure as a linear algebra problem, follows',
2,
None,
'___sec3'),
@@ -141,7 +140,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Nov 21, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/Regression/html/Regression.html
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@@ -71,8 +71,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'___sec2'),
('Rewriting the fitting procedure as a linear algebra problem, '
'follows',
('Rewriting the fitting procedure as a linear algebra problem, follows',
2,
None,
'___sec3'),
@@ -146,7 +145,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Nov 21, 2017</h4></center> <!-- date -->
<center><h4>Nov 22, 2017</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz
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#!/bin/sh
v#!/bin/sh
set -x
function system {
@@ -44,7 +44,7 @@ system doconce split_html $html.html --method=space10
# Bootstrap style
html=${name}-bs
system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
system doconce split_html $html.html --method=split --pagination --nav_button=bottom
#system doconce split_html $html.html --method=split --pagination --nav_button=bottom
# IPython notebook
system doconce format ipynb $name $opt
doc/src/Regression/make.sh
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@@ -44,10 +44,10 @@ system doconce split_html $html.html --method=space10
# Bootstrap style
html=${name}-bs
system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
#system doconce split_html $html.html --method=split --pagination --nav_button=bottom
system doconce split_html $html.html --method=split --pagination --nav_button=bottom
# IPython notebook
system doconce format ipynb $name $opt
#system doconce format ipynb $name $opt
# LaTeX Beamer slides
beamertheme=red_plain