adding files
This commit is contained in:
mhjensen
@@ -0,0 +1,907 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 34: Introduction to the course, Logistics and Practicalities">
|
||||
|
||||
<title>Week 34: Introduction to the course, Logistics and Practicalities</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Overview of first week', 2, None, '___sec0'),
|
||||
('Thursday August 20', 2, None, '___sec1'),
|
||||
('Lectures and ComputerLab', 2, None, '___sec2'),
|
||||
('Course Format', 2, None, '___sec3'),
|
||||
('Teachers', 2, None, '___sec4'),
|
||||
('Deadlines for projects (tentative)', 2, None, '___sec5'),
|
||||
('Recommended textbooks', 2, None, '___sec6'),
|
||||
('Prerequisites', 2, None, '___sec7'),
|
||||
('Learning outcomes', 2, None, '___sec8'),
|
||||
('Topics covered in this course: Statistical analysis and '
|
||||
'optimization of data',
|
||||
2,
|
||||
None,
|
||||
'___sec9'),
|
||||
('Topics covered in this course: Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Extremely useful tools, strongly recommended',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Other courses on Data science and Machine Learning at UiO',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Introduction', 2, None, '___sec13'),
|
||||
('What is Machine Learning?', 2, None, '___sec14'),
|
||||
('Types of Machine Learning', 2, None, '___sec15'),
|
||||
('Software and needed installations', 2, None, '___sec16'),
|
||||
('Python installers', 2, None, '___sec17'),
|
||||
('Useful Python libraries', 2, None, '___sec18'),
|
||||
('Installing R, C++, cython or Julia', 2, None, '___sec19'),
|
||||
('Installing R, C++, cython, Numba etc', 2, None, '___sec20'),
|
||||
('Numpy examples and Important Matrix and vector handling '
|
||||
'packages',
|
||||
2,
|
||||
None,
|
||||
'___sec21'),
|
||||
('Basic Matrix Features', 2, None, '___sec22'),
|
||||
('Some famous Matrices', 3, None, '___sec23'),
|
||||
('More Basic Matrix Features', 3, None, '___sec24'),
|
||||
('Numpy and arrays', 2, None, '___sec25'),
|
||||
('Matrices in Python', 2, None, '___sec26'),
|
||||
('Meet the Pandas', 2, None, '___sec27'),
|
||||
('Friday August 21', 2, None, '___sec28'),
|
||||
('Reading Data and fitting', 2, None, '___sec29'),
|
||||
('Friday August 21', 2, None, '___sec30'),
|
||||
('Simple linear regression model using _scikit-learn_',
|
||||
3,
|
||||
None,
|
||||
'___sec31'),
|
||||
('To our real data: nuclear binding energies. Brief reminder on '
|
||||
'masses and binding energies',
|
||||
3,
|
||||
None,
|
||||
'___sec32'),
|
||||
('Organizing our data', 3, None, '___sec33'),
|
||||
('Seeing the wood for the trees', 3, None, '___sec34'),
|
||||
('And what about using neural networks?', 3, None, '___sec35'),
|
||||
('A first summary', 2, None, '___sec36')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week34-bs.html">Week 34: Introduction to the course, Logistics and Practicalities</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week34-bs001.html#___sec0" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#___sec1" style="font-size: 80%;"><b>Thursday August 20</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#___sec2" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#___sec3" style="font-size: 80%;"><b>Course Format</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#___sec4" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#___sec5" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#___sec6" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#___sec7" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#___sec8" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#___sec9" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#___sec10" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#___sec11" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#___sec12" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#___sec13" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#___sec14" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#___sec15" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#___sec16" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#___sec17" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#___sec18" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#___sec19" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#___sec20" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#___sec21" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#___sec22" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#___sec23" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#___sec24" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#___sec25" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#___sec26" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#___sec27" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#___sec28" style="font-size: 80%;"><b>Friday August 21</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#___sec29" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#___sec30" style="font-size: 80%;"><b>Friday August 21</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec32" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0032"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="___sec31" class="anchor">Simple linear regression model using <b>scikit-learn</b> </h3>
|
||||
|
||||
<p>
|
||||
We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> to perform linear regression analysis on a data set produced by us.
|
||||
|
||||
<p>
|
||||
What follows is a simple Python code where we have defined a function
|
||||
\( y \) in terms of the variable \( x \). Both are defined as vectors with \( 100 \) entries.
|
||||
The numbers in the vector \( \hat{x} \) are given
|
||||
by random numbers generated with a uniform distribution with entries
|
||||
\( x_i \in [0,1] \) (more about probability distribution functions
|
||||
later). These values are then used to define a function \( y(x) \)
|
||||
(tabulated again as a vector) with a linear dependence on \( x \) plus a
|
||||
random noise added via the normal distribution.
|
||||
|
||||
<p>
|
||||
The Numpy functions are imported used the <b>import numpy as np</b>
|
||||
statement and the random number generator for the uniform distribution
|
||||
is called using the function <b>np.random.rand()</b>, where we specificy
|
||||
that we want \( 100 \) random variables. Using Numpy we define
|
||||
automatically an array with the specified number of elements, \( 100 \) in
|
||||
our case. With the Numpy function <b>randn()</b> we can compute random
|
||||
numbers with the normal distribution (mean value \( \mu \) equal to zero and
|
||||
variance \( \sigma^2 \) set to one) and produce the values of \( y \) assuming a linear
|
||||
dependence as function of \( x \)
|
||||
|
||||
$$
|
||||
y = 2x+N(0,1),
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( N(0,1) \) represents random numbers generated by the normal
|
||||
distribution. From <b>Scikit-Learn</b> we import then the
|
||||
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
|
||||
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
|
||||
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
|
||||
<b>scikit-learn</b> has also a functionality which extracts the above
|
||||
fitting parameters \( \alpha \) and \( \beta \) (see below). Later we will
|
||||
distinguish between training data and test data.
|
||||
|
||||
<p>
|
||||
For plotting we use the Python package
|
||||
<a href="https://matplotlib.org/" target="_self">matplotlib</a> which produces publication
|
||||
quality figures. Feel free to explore the extensive
|
||||
<a href="https://matplotlib.org/gallery/index.html" target="_self">gallery</a> of examples. In
|
||||
this example we plot our original values of \( x \) and \( y \) as well as the
|
||||
prediction <b>ypredict</b> (\( \tilde{y} \)), which attempts at fitting our
|
||||
data with a straight line.
|
||||
|
||||
<p>
|
||||
The Python code follows here.
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># Importing various packages</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">1</span>]])
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(xnew)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">"r-"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">5.0</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Simple Linear Regression'</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
This example serves several aims. It allows us to demonstrate several
|
||||
aspects of data analysis and later machine learning algorithms. The
|
||||
immediate visualization shows that our linear fit is not
|
||||
impressive. It goes through the data points, but there are many
|
||||
outliers which are not reproduced by our linear regression. We could
|
||||
now play around with this small program and change for example the
|
||||
factor in front of \( x \) and the normal distribution. Try to change the
|
||||
function \( y \) to
|
||||
|
||||
$$
|
||||
y = 10x+0.01 \times N(0,1),
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( x \) is defined as before. Does the fit look better? Indeed, by
|
||||
reducing the role of the noise given by the normal distribution we see immediately that
|
||||
our linear prediction seemingly reproduces better the training
|
||||
set. However, this testing 'by the eye' is obviouly not satisfactory in the
|
||||
long run. Here we have only defined the training data and our model, and
|
||||
have not discussed a more rigorous approach to the <b>cost</b> function.
|
||||
|
||||
<p>
|
||||
We need more rigorous criteria in defining whether we have succeeded or
|
||||
not in modeling our training data. You will be surprised to see that
|
||||
many scientists seldomly venture beyond this 'by the eye' approach. A
|
||||
standard approach for the <em>cost</em> function is the so-called \( \chi^2 \)
|
||||
function (a variant of the mean-squared error (MSE))
|
||||
|
||||
$$ \chi^2 = \frac{1}{n}
|
||||
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( \sigma_i^2 \) is the variance (to be defined later) of the entry
|
||||
\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves
|
||||
however the aim of scaling the equations and make the cost function
|
||||
dimensionless.
|
||||
|
||||
<p>
|
||||
Minimizing the cost function is a central aspect of
|
||||
our discussions to come. Finding its minima as function of the model
|
||||
parameters (\( \alpha \) and \( \beta \) in our case) will be a recurring
|
||||
theme in these series of lectures. Essentially all machine learning
|
||||
algorithms we will discuss center around the minimization of the
|
||||
chosen cost function. This depends in turn on our specific
|
||||
model for describing the data, a typical situation in supervised
|
||||
learning. Automatizing the search for the minima of the cost function is a
|
||||
central ingredient in all algorithms. Typical methods which are
|
||||
employed are various variants of <b>gradient</b> methods. These will be
|
||||
discussed in more detail later. Again, you'll be surprised to hear that
|
||||
many practitioners minimize the above function ''by the eye', popularly dubbed as
|
||||
'chi by the eye'. That is, change a parameter and see (visually and numerically) that
|
||||
the \( \chi^2 \) function becomes smaller.
|
||||
|
||||
<p>
|
||||
There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define
|
||||
the relative error (why would we prefer the MSE instead of the relative error?) as
|
||||
|
||||
$$
|
||||
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
The squared cost function results in an arithmetic mean-unbiased
|
||||
estimator, and the absolute-value cost function results in a
|
||||
median-unbiased estimator (in the one-dimensional case, and a
|
||||
geometric median-unbiased estimator for the multi-dimensional
|
||||
case). The squared cost function has the disadvantage that it has the tendency
|
||||
to be dominated by outliers.
|
||||
|
||||
<p>
|
||||
We can modify easily the above Python code and plot the relative error instead
|
||||
<p>
|
||||
|
||||
<!-- 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">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.01*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(x, np<span style="color: #666666">.</span>abs(ypredict<span style="color: #666666">-</span>y)<span style="color: #666666">/</span><span style="color: #008000">abs</span>(y), <span style="color: #BA2121">"ro"</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0.0</span>, <span style="color: #666666">0.5</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$\epsilon_{\mathrm{relative}}$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Relative error'</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
Depending on the parameter in front of the normal distribution, we may
|
||||
have a small or larger relative error. Try to play around with
|
||||
different training data sets and study (graphically) the value of the
|
||||
relative error.
|
||||
|
||||
<p>
|
||||
As mentioned above, <b>Scikit-Learn</b> has an impressive functionality.
|
||||
We can for example extract the values of \( \alpha \) and \( \beta \) and
|
||||
their error estimates, or the variance and standard deviation and many
|
||||
other properties from the statistical data analysis.
|
||||
|
||||
<p>
|
||||
Here we show an
|
||||
example of the functionality of <b>Scikit-Learn</b>.
|
||||
<p>
|
||||
|
||||
<!-- 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">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.5*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'The intercept alpha: </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>intercept_)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Coefficient beta : </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #408080; font-style: italic"># The mean squared error </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(y, ypredict))
|
||||
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(y, ypredict))
|
||||
<span style="color: #408080; font-style: italic"># Mean squared log error </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Mean squared log error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_squared_log_error(y, ypredict) )
|
||||
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(y, ypredict))
|
||||
plt<span style="color: #666666">.</span>plot(x, ypredict, <span style="color: #BA2121">"r-"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0.0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">1.5</span>, <span style="color: #666666">7.0</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Linear Regression fit '</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> yields
|
||||
\( \alpha \). Depending on the constant in front of the normal distribution, we get values near or far from \( alpha =2 \) and \( \beta =5 \). Try to play around with different parameters in front of the normal distribution. The function <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
|
||||
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
|
||||
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
The smaller the value, the better the fit. Ideally we would like to
|
||||
have an MSE equal zero. The attentive reader has probably recognized
|
||||
this function as being similar to the \( \chi^2 \) function defined above.
|
||||
|
||||
<p>
|
||||
The <b>r2score</b> function computes \( R^2 \), the coefficient of
|
||||
determination. It provides a measure of how well future samples are
|
||||
likely to be predicted by the model. Best possible score is 1.0 and it
|
||||
can be negative (because the model can be arbitrarily worse). A
|
||||
constant model that always predicts the expected value of \( \hat{y} \),
|
||||
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
|
||||
|
||||
<p>
|
||||
If \( \tilde{\hat{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as
|
||||
$$
|
||||
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
||||
$$
|
||||
|
||||
where we have defined the mean value of \( \hat{y} \) as
|
||||
$$
|
||||
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
||||
$$
|
||||
|
||||
Another quantity taht we will meet again in our discussions of regression analysis is
|
||||
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error.
|
||||
The MAE is defined as follows
|
||||
$$
|
||||
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
|
||||
$$
|
||||
|
||||
We present the
|
||||
squared logarithmic (quadratic) error
|
||||
$$
|
||||
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( \log_e (x) \) stands for the natural logarithm of \( x \). This error
|
||||
estimate is best to use when targets having exponential growth, such
|
||||
as population counts, average sales of a commodity over a span of
|
||||
years etc.
|
||||
|
||||
<p>
|
||||
Finally, another cost function is the Huber cost function used in robust regression.
|
||||
|
||||
<p>
|
||||
The rationale behind this possible cost function is its reduced
|
||||
sensitivity to outliers in the data set. In our discussions on
|
||||
dimensionality reduction and normalization of data we will meet other
|
||||
ways of dealing with outliers.
|
||||
|
||||
<p>
|
||||
The Huber cost function is defined as
|
||||
$$
|
||||
H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}.
|
||||
$$
|
||||
|
||||
Here \( a=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).
|
||||
We will discuss in more
|
||||
detail these and other functions in the various lectures. We conclude this part with another example. Instead of
|
||||
a linear \( x \)-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- 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">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x<span style="color: #666666">=</span>np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0.02</span>,<span style="color: #666666">0.98</span>,<span style="color: #666666">200</span>)
|
||||
noise <span style="color: #666666">=</span> np<span style="color: #666666">.</span>asarray(random<span style="color: #666666">.</span>sample((<span style="color: #008000">range</span>(<span style="color: #666666">200</span>)),<span style="color: #666666">200</span>))
|
||||
y<span style="color: #666666">=</span>x<span style="color: #666666">**3*</span>noise
|
||||
yn<span style="color: #666666">=</span>x<span style="color: #666666">**3*100</span>
|
||||
poly3 <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=3</span>)
|
||||
X <span style="color: #666666">=</span> poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
||||
clf3 <span style="color: #666666">=</span> LinearRegression()
|
||||
clf3<span style="color: #666666">.</span>fit(X,y)
|
||||
|
||||
Xplot<span style="color: #666666">=</span>poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
||||
poly3_plot<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>plot(x, clf3<span style="color: #666666">.</span>predict(Xplot), label<span style="color: #666666">=</span><span style="color: #BA2121">'Cubic Fit'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x,yn, color<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"True Cubic"</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x, y, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data'</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'orange'</span>, s<span style="color: #666666">=15</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">error</span>(a):
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> y:
|
||||
err<span style="color: #666666">=</span>(y<span style="color: #666666">-</span>yn)<span style="color: #666666">/</span>yn
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #008000">abs</span>(np<span style="color: #666666">.</span>sum(err))<span style="color: #666666">/</span><span style="color: #008000">len</span>(err)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span> (error(y))
|
||||
</pre></div>
|
||||
|
||||
<h3 id="___sec32" class="anchor">To our real data: nuclear binding energies. Brief reminder on masses and binding energies </h3>
|
||||
|
||||
<p>
|
||||
Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding
|
||||
energies. A basic quantity which can be measured for the ground
|
||||
states of nuclei is the atomic mass \( M(N, Z) \) of the neutral atom with
|
||||
atomic mass number \( A \) and charge \( Z \). The number of neutrons is \( N \). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).
|
||||
|
||||
<p>
|
||||
Atomic masses are usually tabulated in terms of the mass excess defined by
|
||||
$$
|
||||
\Delta M(N, Z) = M(N, Z) - uA,
|
||||
$$
|
||||
|
||||
where \( u \) is the Atomic Mass Unit
|
||||
$$
|
||||
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
|
||||
$$
|
||||
|
||||
The nucleon masses are
|
||||
$$
|
||||
m_p = 1.00727646693(9)u,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
|
||||
$$
|
||||
|
||||
<p>
|
||||
In the <a href="http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf" target="_self">2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu</a>
|
||||
there are data on masses and decays of 3437 nuclei.
|
||||
|
||||
<p>
|
||||
The nuclear binding energy is defined as the energy required to break
|
||||
up a given nucleus into its constituent parts of \( N \) neutrons and \( Z \)
|
||||
protons. In terms of the atomic masses \( M(N, Z) \) the binding energy is
|
||||
defined by
|
||||
|
||||
$$
|
||||
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
|
||||
$$
|
||||
|
||||
where \( M_H \) is the mass of the hydrogen atom and \( m_n \) is the mass of the neutron.
|
||||
In terms of the mass excess the binding energy is given by
|
||||
$$
|
||||
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
|
||||
$$
|
||||
|
||||
where \( \Delta_H c^2 = 7.2890 \) MeV and \( \Delta_n c^2 = 8.0713 \) MeV.
|
||||
|
||||
<p>
|
||||
A popular and physically intuitive model which can be used to parametrize
|
||||
the experimental binding energies as function of \( A \), is the so-called
|
||||
<b>liquid drop model</b>. The ansatz is based on the following expression
|
||||
|
||||
$$
|
||||
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( A \) stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit
|
||||
to the experimental data.
|
||||
|
||||
<p>
|
||||
To arrive at the above expression we have assumed that we can make the following assumptions:
|
||||
|
||||
<ul>
|
||||
<li> There is a volume term \( a_1A \) proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.</li>
|
||||
<li> There is a surface energy term \( a_2A^{2/3} \). The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.</li>
|
||||
<li> There is a Coulomb energy term \( a_3\frac{Z^2}{A^{1/3}} \). The electric repulsion between each pair of protons in a nucleus yields less binding.</li>
|
||||
<li> There is an asymmetry term \( a_4\frac{(N-Z)^2}{A} \). This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.</li>
|
||||
</ul>
|
||||
|
||||
We could also add a so-called pairing term, which is a correction term that
|
||||
arises from the tendency of proton pairs and neutron pairs to
|
||||
occur. An even number of particles is more stable than an odd number.
|
||||
|
||||
<h3 id="___sec33" class="anchor">Organizing our data </h3>
|
||||
|
||||
<p>
|
||||
Let us start with reading and organizing our data.
|
||||
We start with the compilation of masses and binding energies from 2016.
|
||||
After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.
|
||||
|
||||
<p>
|
||||
We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of <b>scikit-learn</b>.
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># Common imports</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</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
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
||||
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
||||
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
||||
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
||||
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, format<span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
|
||||
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"MassEval2016.dat"</span>),<span style="color: #BA2121">'r'</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various <b>matplotlib</b> commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.
|
||||
<p>
|
||||
|
||||
<!-- 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">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
||||
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
||||
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MakePlot</span>(x,y, styles, labels, axlabels):
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(x)):
|
||||
plt<span style="color: #666666">.</span>plot(x[i], y[i], styles[i], label <span style="color: #666666">=</span> labels[i])
|
||||
plt<span style="color: #666666">.</span>xlabel(axlabels[<span style="color: #666666">0</span>])
|
||||
plt<span style="color: #666666">.</span>ylabel(axlabels[<span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=0</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Our next step is to read the data on experimental binding energies and
|
||||
reorganize them as functions of the mass number \( A \), the number of
|
||||
protons \( Z \) and neutrons \( N \) using <b>pandas</b>. Before we do this it is
|
||||
always useful (unless you have a binary file or other types of compressed
|
||||
data) to actually open the file and simply take a look at it!
|
||||
|
||||
<p>
|
||||
In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with <b>pandas</b>. The file begins with some basic format information.
|
||||
<p>
|
||||
|
||||
<!-- 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: #BA2121; font-style: italic">""" </span>
|
||||
<span style="color: #BA2121; font-style: italic">This is taken from the data file of the mass 2016 evaluation. </span>
|
||||
<span style="color: #BA2121; font-style: italic">All files are 3436 lines long with 124 character per line. </span>
|
||||
<span style="color: #BA2121; font-style: italic"> Headers are 39 lines long. </span>
|
||||
<span style="color: #BA2121; font-style: italic"> col 1 : Fortran character control: 1 = page feed 0 = line feed </span>
|
||||
<span style="color: #BA2121; font-style: italic"> format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 </span>
|
||||
<span style="color: #BA2121; font-style: italic"> These formats are reflected in the pandas widths variable below, see the statement </span>
|
||||
<span style="color: #BA2121; font-style: italic"> widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), </span>
|
||||
<span style="color: #BA2121; font-style: italic"> Pandas has also a variable header, with length 39 in this case. </span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
The data we are interested in are in columns 2, 3, 4 and 11, giving us
|
||||
the number of neutrons, protons, mass numbers and binding energies,
|
||||
respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
|
||||
covert them into the <b>pandas</b> DataFrame structure.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># Read the experimental data with Pandas</span>
|
||||
Masses <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_fwf(infile, usecols<span style="color: #666666">=</span>(<span style="color: #666666">2</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">6</span>,<span style="color: #666666">11</span>),
|
||||
names<span style="color: #666666">=</span>(<span style="color: #BA2121">'N'</span>, <span style="color: #BA2121">'Z'</span>, <span style="color: #BA2121">'A'</span>, <span style="color: #BA2121">'Element'</span>, <span style="color: #BA2121">'Ebinding'</span>),
|
||||
widths<span style="color: #666666">=</span>(<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">1</span>,<span style="color: #666666">13</span>,<span style="color: #666666">11</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">2</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">1</span>,<span style="color: #666666">12</span>,<span style="color: #666666">11</span>,<span style="color: #666666">1</span>),
|
||||
header<span style="color: #666666">=39</span>,
|
||||
index_col<span style="color: #666666">=</span><span style="color: #008000">False</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by '#' in place of the decimal place, so</span>
|
||||
<span style="color: #408080; font-style: italic"># the Ebinding column won't be numeric. Coerce to float and drop these entries.</span>
|
||||
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">'Ebinding'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>dropna()
|
||||
<span style="color: #408080; font-style: italic"># Convert from keV to MeV.</span>
|
||||
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">/=</span> <span style="color: #666666">1000</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Group the DataFrame by nucleon number, A.</span>
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>groupby(<span style="color: #BA2121">'A'</span>)
|
||||
<span style="color: #408080; font-style: italic"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>apply(<span style="color: #008000; font-weight: bold">lambda</span> t: t[t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">==</span>t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">.</span>max()])
|
||||
</pre></div>
|
||||
<p>
|
||||
We have now read in the data, grouped them according to the variables we are interested in.
|
||||
We see how easy it is to reorganize the data using <b>pandas</b>. If we
|
||||
were to do these operations in C/C++ or Fortran, we would have had to
|
||||
write various functions/subroutines which perform the above
|
||||
reorganizations for us. Having reorganized the data, we can now start
|
||||
to make some simple fits using both the functionalities in <b>numpy</b> and
|
||||
<b>Scikit-Learn</b> afterwards.
|
||||
|
||||
<p>
|
||||
Now we define five variables which contain
|
||||
the number of nucleons \( A \), the number of protons \( Z \) and the number of neutrons \( N \), the element name and finally the energies themselves.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'A'</span>]
|
||||
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Z'</span>]
|
||||
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'N'</span>]
|
||||
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Element'</span>]
|
||||
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Ebinding'</span>]
|
||||
<span style="color: #008000; font-weight: bold">print</span>(Masses)
|
||||
</pre></div>
|
||||
<p>
|
||||
The next step, and we will define this mathematically later, is to set up the so-called <b>design matrix</b>. We will throughout call this matrix \( \boldsymbol{X} \).
|
||||
It has dimensionality \( p\times n \), where \( n \) is the number of data points and \( p \) are the so-called predictors. In our case here they are given by the number of polynomials in \( A \) we wish to include in the fit.
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># Now we set up the design matrix X</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(A),<span style="color: #666666">5</span>))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
||||
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> A
|
||||
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
||||
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0/3.0</span>)
|
||||
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
With <b>scikitlearn</b> we are now ready to use linear regression and fit our data.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
|
||||
fity <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
Pretty simple!
|
||||
Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># The mean squared error </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(Energies, fity))
|
||||
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(Energies, fity))
|
||||
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, fity))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
|
||||
|
||||
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> fity
|
||||
<span style="color: #408080; font-style: italic"># Generate a plot comparing the experimental with the fitted values values.</span>
|
||||
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r'$A = N + Z$'</span>)
|
||||
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r'$E_\mathrm{bind}\,/\mathrm{MeV}$'</span>)
|
||||
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Ebinding'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">'Ame2016'</span>)
|
||||
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Eapprox'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'m'</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">'Fit'</span>)
|
||||
ax<span style="color: #666666">.</span>legend()
|
||||
save_fig(<span style="color: #BA2121">"Masses2016"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
|
||||
<h3 id="___sec34" class="anchor">Seeing the wood for the trees </h3>
|
||||
|
||||
<p>
|
||||
As a teaser, let us now see how we can do this with decision trees using <b>scikit-learn</b>. Later we will switch to so-called <b>random forests</b>!
|
||||
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic">#Decision Tree Regression</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeRegressor
|
||||
regr_1<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=5</span>)
|
||||
regr_2<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=7</span>)
|
||||
regr_3<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=9</span>)
|
||||
regr_1<span style="color: #666666">.</span>fit(X, Energies)
|
||||
regr_2<span style="color: #666666">.</span>fit(X, Energies)
|
||||
regr_3<span style="color: #666666">.</span>fit(X, Energies)
|
||||
|
||||
|
||||
y_1 <span style="color: #666666">=</span> regr_1<span style="color: #666666">.</span>predict(X)
|
||||
y_2 <span style="color: #666666">=</span> regr_2<span style="color: #666666">.</span>predict(X)
|
||||
y_3<span style="color: #666666">=</span>regr_3<span style="color: #666666">.</span>predict(X)
|
||||
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> y_3
|
||||
<span style="color: #408080; font-style: italic"># Plot the results</span>
|
||||
plt<span style="color: #666666">.</span>figure()
|
||||
plt<span style="color: #666666">.</span>plot(A, Energies, color<span style="color: #666666">=</span><span style="color: #BA2121">"blue"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Data"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_1, color<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=5"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_2, color<span style="color: #666666">=</span><span style="color: #BA2121">"green"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=7"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_3, color<span style="color: #666666">=</span><span style="color: #BA2121">"m"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=9"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"$A$"</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"$E$[MeV]"</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Decision Tree Regression"</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
save_fig(<span style="color: #BA2121">"Masses2016Trees"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
<span style="color: #008000; font-weight: bold">print</span>(Masses)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean( (Energies<span style="color: #666666">-</span>y_1)<span style="color: #666666">**2</span>))
|
||||
</pre></div>
|
||||
|
||||
<h3 id="___sec35" class="anchor">And what about using neural networks? </h3>
|
||||
|
||||
The <b>seaborn</b> package allows us to visualize data in an efficient way. Note that we use <b>scikit-learn</b>'s multi-layer perceptron (or feed forward neural network)
|
||||
functionality.
|
||||
<p>
|
||||
|
||||
<!-- 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.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPRegressor
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
|
||||
|
||||
X_train <span style="color: #666666">=</span> X
|
||||
Y_train <span style="color: #666666">=</span> Energies
|
||||
n_hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
epochs <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
<span style="color: #408080; font-style: italic"># store models for later use</span>
|
||||
eta_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
||||
lmbd_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
||||
<span style="color: #408080; font-style: italic"># store the models for later use</span>
|
||||
DNN_scikit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)), dtype<span style="color: #666666">=</span><span style="color: #008000">object</span>)
|
||||
train_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
|
||||
sns<span style="color: #666666">.</span>set()
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, eta <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(eta_vals):
|
||||
<span style="color: #008000; font-weight: bold">for</span> j, lmbd <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmbd_vals):
|
||||
dnn <span style="color: #666666">=</span> MLPRegressor(hidden_layer_sizes<span style="color: #666666">=</span>(n_hidden_neurons), activation<span style="color: #666666">=</span><span style="color: #BA2121">'logistic'</span>,
|
||||
alpha<span style="color: #666666">=</span>lmbd, learning_rate_init<span style="color: #666666">=</span>eta, max_iter<span style="color: #666666">=</span>epochs)
|
||||
dnn<span style="color: #666666">.</span>fit(X_train, Y_train)
|
||||
DNN_scikit[i][j] <span style="color: #666666">=</span> dnn
|
||||
train_accuracy[i][j] <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>score(X_train, Y_train)
|
||||
|
||||
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
|
||||
sns<span style="color: #666666">.</span>heatmap(train_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">"viridis"</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">"Training Accuracy"</span>)
|
||||
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">"$\eta$"</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">"$\lambda$"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
|
||||
<h2 id="___sec36" class="anchor">A first summary </h2>
|
||||
|
||||
<p>
|
||||
The aim behind these introductory words was to present to you various
|
||||
Python libraries and their functionalities, in particular libraries like
|
||||
<b>numpy</b>, <b>pandas</b>, <b>xarray</b> and <b>matplotlib</b> and other that make our life much easier
|
||||
in handling various data sets and visualizing data.
|
||||
|
||||
<p>
|
||||
Furthermore,
|
||||
<b>Scikit-Learn</b> allows us with few lines of code to implement popular
|
||||
Machine Learning algorithms for supervised learning. Later we will meet <b>Tensorflow</b>, a powerful library for deep learning.
|
||||
Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.
|
||||
|
||||
<p>
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week34-bs031.html">«</a></li>
|
||||
<li><a href="._week34-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week34-bs024.html">25</a></li>
|
||||
<li><a href="._week34-bs025.html">26</a></li>
|
||||
<li><a href="._week34-bs026.html">27</a></li>
|
||||
<li><a href="._week34-bs027.html">28</a></li>
|
||||
<li><a href="._week34-bs028.html">29</a></li>
|
||||
<li><a href="._week34-bs029.html">30</a></li>
|
||||
<li><a href="._week34-bs030.html">31</a></li>
|
||||
<li><a href="._week34-bs031.html">32</a></li>
|
||||
<li class="active"><a href="._week34-bs032.html">33</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,262 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 35: Linear Regression and Review of Statistical Analysis and Probability Theory">
|
||||
|
||||
<title>Week 35: Linear Regression and Review of Statistical Analysis and Probability Theory</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 35, August 24-28', 2, None, '___sec0'),
|
||||
('Thursday August 27', 2, None, '___sec1'),
|
||||
('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec2'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec3'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec4'),
|
||||
('Examples', 2, None, '___sec5'),
|
||||
('General linear models', 2, None, '___sec6'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec9'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters', 2, None, '___sec11'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec13'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec16'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec18'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec24'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec25'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The code', 2, None, '___sec27'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
('The Boston housing data example', 2, None, '___sec29'),
|
||||
('Housing data, the code', 2, None, '___sec30'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec31'),
|
||||
('Preprocessing our data', 2, None, '___sec32'),
|
||||
('More preprocessing', 2, None, '___sec33'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec34'),
|
||||
('Friday August 28', 2, None, '___sec35')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week35-bs.html">Week 35: Linear Regression and Review of Statistical Analysis and Probability Theory</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week35-bs001.html#___sec0" style="font-size: 80%;">Plans for week 35, August 24-28</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs002.html#___sec1" style="font-size: 80%;">Thursday August 27</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs003.html#___sec2" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs004.html#___sec3" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs005.html#___sec4" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs006.html#___sec5" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs007.html#___sec6" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs008.html#___sec7" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs009.html#___sec8" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs010.html#___sec9" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs011.html#___sec10" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs013.html#___sec12" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs014.html#___sec13" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs015.html#___sec14" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs017.html#___sec16" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs018.html#___sec17" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs019.html#___sec18" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs020.html#___sec19" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs025.html#___sec24" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs026.html#___sec25" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs027.html#___sec26" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs028.html#___sec27" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs029.html#___sec28" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs030.html#___sec29" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs031.html#___sec30" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs032.html#___sec31" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs033.html#___sec32" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs034.html#___sec33" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week35-bs035.html#___sec34" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;">Friday August 28</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0036"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec35" class="anchor">Friday August 28 </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureAug28.mp4?vrtx=view-as-webpage" target="_self">Video of Lecture</a> and <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesAugust28.pdf" target="_self">handwritten notes</a>
|
||||
|
||||
<p>
|
||||
More material will be added here, see handwritten notes also.
|
||||
|
||||
<p>
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week35-bs035.html">«</a></li>
|
||||
<li><a href="._week35-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week35-bs028.html">29</a></li>
|
||||
<li><a href="._week35-bs029.html">30</a></li>
|
||||
<li><a href="._week35-bs030.html">31</a></li>
|
||||
<li><a href="._week35-bs031.html">32</a></li>
|
||||
<li><a href="._week35-bs032.html">33</a></li>
|
||||
<li><a href="._week35-bs033.html">34</a></li>
|
||||
<li><a href="._week35-bs034.html">35</a></li>
|
||||
<li><a href="._week35-bs035.html">36</a></li>
|
||||
<li class="active"><a href="._week35-bs036.html">37</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,311 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 36: Resampling techniques and Ordinary Least Square">
|
||||
|
||||
<title>Week 36: Resampling techniques and Ordinary Least Square</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 36', 2, None, '___sec0'),
|
||||
('Thursday September 3', 2, None, '___sec1'),
|
||||
('Why resampling methods', 2, None, '___sec2'),
|
||||
('Resampling methods', 2, None, '___sec3'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec4'),
|
||||
('Why resampling methods ?', 2, None, '___sec5'),
|
||||
('Statistical analysis', 2, None, '___sec6'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Assumptions made', 2, None, '___sec8'),
|
||||
('Expectation value and variance', 2, None, '___sec9'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Resampling methods', 2, None, '___sec11'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec13'),
|
||||
('Jackknife code example', 2, None, '___sec14'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec15'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec16'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec18'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec19'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec20'),
|
||||
('Various steps in cross-validation', 2, None, '___sec21'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec22'),
|
||||
('Cross-validation in brief', 2, None, '___sec23'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec25'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec26'),
|
||||
('Understanding what happens', 2, None, '___sec27'),
|
||||
('Summing up', 2, None, '___sec28'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec30'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec31'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec32'),
|
||||
('Friday September 4', 2, None, '___sec33')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week36-bs.html">Week 36: Resampling techniques and Ordinary Least Square</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs005.html#___sec4" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs006.html#___sec5" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs008.html#___sec7" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs010.html#___sec9" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs011.html#___sec10" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs013.html#___sec12" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs014.html#___sec13" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs015.html#___sec14" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs016.html#___sec15" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs017.html#___sec16" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs018.html#___sec17" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs019.html#___sec18" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs020.html#___sec19" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs021.html#___sec20" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs022.html#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs027.html#___sec26" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs028.html#___sec27" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs031.html#___sec30" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs033.html#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs034.html#___sec33" style="font-size: 80%;">Friday September 4</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0032"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec31" class="anchor">The same example but now with cross-validation </h2>
|
||||
|
||||
<p>
|
||||
|
||||
<!-- 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: #408080; font-style: italic"># Common imports</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
|
||||
<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> KFold
|
||||
<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> cross_val_score
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
||||
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
||||
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
||||
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
||||
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, format<span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
|
||||
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"EoS.csv"</span>),<span style="color: #BA2121">'r'</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
|
||||
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">'Density'</span>, <span style="color: #BA2121">'Energy'</span>))
|
||||
EoS[<span style="color: #BA2121">'Energy'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">'Energy'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
||||
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
|
||||
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Energy'</span>]
|
||||
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Density'</span>]
|
||||
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
|
||||
|
||||
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">30</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),Maxpolydegree))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
polynomial <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
k <span style="color: #666666">=5</span>
|
||||
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> polydegree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>, Maxpolydegree):
|
||||
polynomial[polydegree] <span style="color: #666666">=</span> polydegree
|
||||
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(polydegree):
|
||||
X[:,degree] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(degree<span style="color: #666666">/3.0</span>)
|
||||
OLS <span style="color: #666666">=</span> LinearRegression()
|
||||
<span style="color: #408080; font-style: italic"># loop over trials in order to estimate the expectation value of the MSE</span>
|
||||
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(OLS, X, Energies, scoring<span style="color: #666666">=</span><span style="color: #BA2121">'neg_mean_squared_error'</span>, cv<span style="color: #666666">=</span>kfold)
|
||||
<span style="color: #408080; font-style: italic">#[:, np.newaxis]</span>
|
||||
estimated_mse_sklearn[polydegree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(estimated_mse_sklearn), label<span style="color: #666666">=</span><span style="color: #BA2121">'Test Error'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Polynomial degree'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'log10[MSE]'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week36-bs031.html">«</a></li>
|
||||
<li><a href="._week36-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week36-bs024.html">25</a></li>
|
||||
<li><a href="._week36-bs025.html">26</a></li>
|
||||
<li><a href="._week36-bs026.html">27</a></li>
|
||||
<li><a href="._week36-bs027.html">28</a></li>
|
||||
<li><a href="._week36-bs028.html">29</a></li>
|
||||
<li><a href="._week36-bs029.html">30</a></li>
|
||||
<li><a href="._week36-bs030.html">31</a></li>
|
||||
<li><a href="._week36-bs031.html">32</a></li>
|
||||
<li class="active"><a href="._week36-bs032.html">33</a></li>
|
||||
<li><a href="._week36-bs033.html">34</a></li>
|
||||
<li><a href="._week36-bs034.html">35</a></li>
|
||||
<li><a href="._week36-bs033.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,277 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 36: Resampling techniques and Ordinary Least Square">
|
||||
|
||||
<title>Week 36: Resampling techniques and Ordinary Least Square</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 36', 2, None, '___sec0'),
|
||||
('Thursday September 3', 2, None, '___sec1'),
|
||||
('Why resampling methods', 2, None, '___sec2'),
|
||||
('Resampling methods', 2, None, '___sec3'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec4'),
|
||||
('Why resampling methods ?', 2, None, '___sec5'),
|
||||
('Statistical analysis', 2, None, '___sec6'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Assumptions made', 2, None, '___sec8'),
|
||||
('Expectation value and variance', 2, None, '___sec9'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Resampling methods', 2, None, '___sec11'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec13'),
|
||||
('Jackknife code example', 2, None, '___sec14'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec15'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec16'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec18'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec19'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec20'),
|
||||
('Various steps in cross-validation', 2, None, '___sec21'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec22'),
|
||||
('Cross-validation in brief', 2, None, '___sec23'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec25'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec26'),
|
||||
('Understanding what happens', 2, None, '___sec27'),
|
||||
('Summing up', 2, None, '___sec28'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec30'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec31'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec32'),
|
||||
('Friday September 4', 2, None, '___sec33')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week36-bs.html">Week 36: Resampling techniques and Ordinary Least Square</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs005.html#___sec4" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs006.html#___sec5" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs008.html#___sec7" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs010.html#___sec9" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs011.html#___sec10" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs013.html#___sec12" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs014.html#___sec13" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs015.html#___sec14" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs016.html#___sec15" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs017.html#___sec16" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs018.html#___sec17" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs019.html#___sec18" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs020.html#___sec19" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs021.html#___sec20" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs022.html#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs027.html#___sec26" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs028.html#___sec27" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs031.html#___sec30" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs032.html#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs034.html#___sec33" style="font-size: 80%;">Friday September 4</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0033"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec32" class="anchor">Cross-validation with Ridge </h2>
|
||||
<p>
|
||||
|
||||
<!-- 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">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</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> KFold
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
||||
<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> cross_val_score
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
|
||||
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
|
||||
<span style="color: #408080; font-style: italic"># Generate the data.</span>
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
|
||||
poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">10</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
|
||||
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
|
||||
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
|
||||
<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
|
||||
k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
|
||||
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
|
||||
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
|
||||
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
|
||||
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, x, y, scoring<span style="color: #666666">=</span><span style="color: #BA2121">'neg_mean_squared_error'</span>, cv<span style="color: #666666">=</span>kfold)
|
||||
estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
|
||||
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
||||
plt<span style="color: #666666">.</span>figure()
|
||||
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">'cross_val_score'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week36-bs032.html">«</a></li>
|
||||
<li><a href="._week36-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week36-bs025.html">26</a></li>
|
||||
<li><a href="._week36-bs026.html">27</a></li>
|
||||
<li><a href="._week36-bs027.html">28</a></li>
|
||||
<li><a href="._week36-bs028.html">29</a></li>
|
||||
<li><a href="._week36-bs029.html">30</a></li>
|
||||
<li><a href="._week36-bs030.html">31</a></li>
|
||||
<li><a href="._week36-bs031.html">32</a></li>
|
||||
<li><a href="._week36-bs032.html">33</a></li>
|
||||
<li class="active"><a href="._week36-bs033.html">34</a></li>
|
||||
<li><a href="._week36-bs034.html">35</a></li>
|
||||
<li><a href="._week36-bs034.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,244 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 36: Resampling techniques and Ordinary Least Square">
|
||||
|
||||
<title>Week 36: Resampling techniques and Ordinary Least Square</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 36', 2, None, '___sec0'),
|
||||
('Thursday September 3', 2, None, '___sec1'),
|
||||
('Why resampling methods', 2, None, '___sec2'),
|
||||
('Resampling methods', 2, None, '___sec3'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec4'),
|
||||
('Why resampling methods ?', 2, None, '___sec5'),
|
||||
('Statistical analysis', 2, None, '___sec6'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Assumptions made', 2, None, '___sec8'),
|
||||
('Expectation value and variance', 2, None, '___sec9'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Resampling methods', 2, None, '___sec11'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec13'),
|
||||
('Jackknife code example', 2, None, '___sec14'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec15'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec16'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec18'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec19'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec20'),
|
||||
('Various steps in cross-validation', 2, None, '___sec21'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec22'),
|
||||
('Cross-validation in brief', 2, None, '___sec23'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec25'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec26'),
|
||||
('Understanding what happens', 2, None, '___sec27'),
|
||||
('Summing up', 2, None, '___sec28'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec30'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec31'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec32'),
|
||||
('Friday September 4', 2, None, '___sec33')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week36-bs.html">Week 36: Resampling techniques and Ordinary Least Square</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs005.html#___sec4" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs006.html#___sec5" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs008.html#___sec7" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs010.html#___sec9" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs011.html#___sec10" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs013.html#___sec12" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs014.html#___sec13" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs015.html#___sec14" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs016.html#___sec15" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs017.html#___sec16" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs018.html#___sec17" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs019.html#___sec18" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs020.html#___sec19" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs021.html#___sec20" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs022.html#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs027.html#___sec26" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs028.html#___sec27" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs031.html#___sec30" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs032.html#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs033.html#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;">Friday September 4</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0034"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec33" class="anchor">Friday September 4 </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember4.mp4?vrtx=view-as-webpage" target="_self">Video of Lecture</a> and <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember4.pdf" target="_self">handwritten notes</a>
|
||||
|
||||
<p>
|
||||
More material will be added here, see handwritten notes also.
|
||||
|
||||
<p>
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week36-bs033.html">«</a></li>
|
||||
<li><a href="._week36-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week36-bs026.html">27</a></li>
|
||||
<li><a href="._week36-bs027.html">28</a></li>
|
||||
<li><a href="._week36-bs028.html">29</a></li>
|
||||
<li><a href="._week36-bs029.html">30</a></li>
|
||||
<li><a href="._week36-bs030.html">31</a></li>
|
||||
<li><a href="._week36-bs031.html">32</a></li>
|
||||
<li><a href="._week36-bs032.html">33</a></li>
|
||||
<li><a href="._week36-bs033.html">34</a></li>
|
||||
<li class="active"><a href="._week36-bs034.html">35</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,248 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 37: Ridge and Lasso Regression">
|
||||
|
||||
<title>Week 37: Ridge and Lasso Regression</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 37', 2, None, '___sec0'),
|
||||
('Thursday September 10', 2, None, '___sec1'),
|
||||
('A Bayesian approach to develop intuition about skrinkage '
|
||||
'methods',
|
||||
2,
|
||||
None,
|
||||
'___sec2'),
|
||||
('The singular value decomposition', 2, None, '___sec3'),
|
||||
('Linear Regression Problems', 2, None, '___sec4'),
|
||||
('Fixing the singularity', 2, None, '___sec5'),
|
||||
('Basic math of the SVD', 2, None, '___sec6'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec7'),
|
||||
('Economy-size SVD', 2, None, '___sec8'),
|
||||
('Codes for the SVD', 2, None, '___sec9'),
|
||||
('Mathematical Properties', 2, None, '___sec10'),
|
||||
('Friday September 12', 2, None, '___sec11'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec12'),
|
||||
('More on Ridge Regression', 2, None, '___sec13'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec14'),
|
||||
('More interpretations', 2, None, '___sec15'),
|
||||
('A better understanding of regularization', 2, None, '___sec16'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec18'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec19'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec20'),
|
||||
('Correlation Matrix', 2, None, '___sec21'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec22'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('Linking with SVD', 2, None, '___sec25')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week37-bs.html">Week 37: Ridge and Lasso Regression</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week37-bs001.html#___sec0" style="font-size: 80%;">Plans for week 37</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs002.html#___sec1" style="font-size: 80%;">Thursday September 10</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs003.html#___sec2" style="font-size: 80%;">A Bayesian approach to develop intuition about skrinkage methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs004.html#___sec3" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs005.html#___sec4" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs006.html#___sec5" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs007.html#___sec6" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs008.html#___sec7" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs009.html#___sec8" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs010.html#___sec9" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs011.html#___sec10" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs012.html#___sec11" style="font-size: 80%;">Friday September 12</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs013.html#___sec12" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs014.html#___sec13" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs015.html#___sec14" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs016.html#___sec15" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs017.html#___sec16" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs018.html#___sec17" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs019.html#___sec18" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs020.html#___sec19" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#___sec20" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#___sec21" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#___sec22" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#___sec25" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0025"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec24" class="anchor">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
|
||||
$$
|
||||
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}].
|
||||
$$
|
||||
|
||||
<p>
|
||||
To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
|
||||
$$
|
||||
\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{00} & x_{01}\\
|
||||
x_{10} & x_{11}\\
|
||||
\end{bmatrix}=\begin{bmatrix}
|
||||
\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
|
||||
\end{bmatrix}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
If we then compute the expectation value
|
||||
$$
|
||||
\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
|
||||
x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
which is just
|
||||
$$
|
||||
\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
|
||||
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
|
||||
|
||||
<p>
|
||||
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week37-bs024.html">«</a></li>
|
||||
<li><a href="._week37-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs017.html">18</a></li>
|
||||
<li><a href="._week37-bs018.html">19</a></li>
|
||||
<li><a href="._week37-bs019.html">20</a></li>
|
||||
<li><a href="._week37-bs020.html">21</a></li>
|
||||
<li><a href="._week37-bs021.html">22</a></li>
|
||||
<li><a href="._week37-bs022.html">23</a></li>
|
||||
<li><a href="._week37-bs023.html">24</a></li>
|
||||
<li><a href="._week37-bs024.html">25</a></li>
|
||||
<li class="active"><a href="._week37-bs025.html">26</a></li>
|
||||
<li><a href="._week37-bs026.html">27</a></li>
|
||||
<li><a href="._week37-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,207 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 37: Ridge and Lasso Regression">
|
||||
|
||||
<title>Week 37: Ridge and Lasso Regression</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
<!-- not necessary
|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
-->
|
||||
|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
|
||||
display:block;
|
||||
height:50px; /* fixed header height for style bootstrap */
|
||||
margin:-50px 0 0; /* negative fixed header height */
|
||||
}
|
||||
</style>
|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Plans for week 37', 2, None, '___sec0'),
|
||||
('Thursday September 10', 2, None, '___sec1'),
|
||||
('A Bayesian approach to develop intuition about skrinkage '
|
||||
'methods',
|
||||
2,
|
||||
None,
|
||||
'___sec2'),
|
||||
('The singular value decomposition', 2, None, '___sec3'),
|
||||
('Linear Regression Problems', 2, None, '___sec4'),
|
||||
('Fixing the singularity', 2, None, '___sec5'),
|
||||
('Basic math of the SVD', 2, None, '___sec6'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec7'),
|
||||
('Economy-size SVD', 2, None, '___sec8'),
|
||||
('Codes for the SVD', 2, None, '___sec9'),
|
||||
('Mathematical Properties', 2, None, '___sec10'),
|
||||
('Friday September 12', 2, None, '___sec11'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec12'),
|
||||
('More on Ridge Regression', 2, None, '___sec13'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec14'),
|
||||
('More interpretations', 2, None, '___sec15'),
|
||||
('A better understanding of regularization', 2, None, '___sec16'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec18'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec19'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec20'),
|
||||
('Correlation Matrix', 2, None, '___sec21'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec22'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('Linking with SVD', 2, None, '___sec25')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
TeX: {
|
||||
equationNumbers: { autoNumber: "none" },
|
||||
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
|
||||
|
||||
<!-- Bootstrap navigation bar -->
|
||||
<div class="navbar navbar-default navbar-fixed-top">
|
||||
<div class="navbar-header">
|
||||
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week37-bs.html">Week 37: Ridge and Lasso Regression</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
<ul class="nav navbar-nav navbar-right">
|
||||
<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="._week37-bs001.html#___sec0" style="font-size: 80%;">Plans for week 37</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs002.html#___sec1" style="font-size: 80%;">Thursday September 10</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs003.html#___sec2" style="font-size: 80%;">A Bayesian approach to develop intuition about skrinkage methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs004.html#___sec3" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs005.html#___sec4" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs006.html#___sec5" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs007.html#___sec6" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs008.html#___sec7" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs009.html#___sec8" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs010.html#___sec9" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs011.html#___sec10" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs012.html#___sec11" style="font-size: 80%;">Friday September 12</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs013.html#___sec12" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs014.html#___sec13" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs015.html#___sec14" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs016.html#___sec15" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs017.html#___sec16" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs018.html#___sec17" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs019.html#___sec18" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs020.html#___sec19" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#___sec20" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#___sec21" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#___sec22" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#___sec24" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0026"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec25" class="anchor">Linking with SVD </h2>
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._week37-bs025.html">«</a></li>
|
||||
<li><a href="._week37-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs018.html">19</a></li>
|
||||
<li><a href="._week37-bs019.html">20</a></li>
|
||||
<li><a href="._week37-bs020.html">21</a></li>
|
||||
<li><a href="._week37-bs021.html">22</a></li>
|
||||
<li><a href="._week37-bs022.html">23</a></li>
|
||||
<li><a href="._week37-bs023.html">24</a></li>
|
||||
<li><a href="._week37-bs024.html">25</a></li>
|
||||
<li><a href="._week37-bs025.html">26</a></li>
|
||||
<li class="active"><a href="._week37-bs026.html">27</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
</div> <!-- end container -->
|
||||
<!-- include javascript, jQuery *first* -->
|
||||
<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>
|
||||
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
Reference in New Issue
Block a user