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<h2 id="___sec19" class="anchor">Simple linear regression model using <b>scikit-learn</b> </h2>
<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.
What follows is a simple Python code where we have defined function \( y \) in terms of the variable \( x \). Both are defined as vectors of dimension \( 1\times 100 \). The entries to 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">&quot;r-&quot;</span>)
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</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&#39;$x$&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Simple Linear Regression&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<p>
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