added random walk example

2018-05-29 11:08:41 -04:00
parent 3a2abcd62b
commit d34d39836b
8 changed files with 608 additions and 101 deletions
@@ -57,7 +57,8 @@ Automatically generated HTML file from DocOnce source
              ('Particle in one dimension and velocity distribution',
               3,
               None,
-               '___sec11')]}
+               '___sec11'),
+              ('Random walk model', 3, None, '___sec12')]}
 end of tocinfo -->

 <body>
@@ -107,6 +108,7 @@ MathJax.Hub.Config({
     <!-- navigation toc: --> <li><a href="#___sec9" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Predator-Prey model from ecology</a></li>
     <!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Simulating financial transactions</a></li>
     <!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Particle in one dimension and velocity distribution</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Random walk model</a></li>

        </ul>
      </li>
@@ -143,11 +145,8 @@ MathJax.Hub.Config({
 <center><h4>May 29, 2018</h4></center> <!-- date -->
 <br>
 <p>
-<!-- potential-jumbotron-button -->
 </div> <!-- end jumbotron -->

-<!-- !split -->
-
 <h2 id="___sec0" class="anchor">Introduction </h2>

 <p>
@@ -188,9 +187,6 @@ introduce will serve as inputs to many of our discussions later, as
 well as allowing you to set up models and produce your own data and
 get started with programming.

-<p>
-<!-- !split -->
-
 <h2 id="___sec1" class="anchor">Software and needed installations </h2>

 <p>
@@ -233,9 +229,6 @@ you can use <b>pip</b> as well and simply install Python as

 etc etc.

-<p>
-<!-- !split -->
-
 <h2 id="___sec2" class="anchor">Python installers </h2>

 <p>
@@ -263,9 +256,6 @@ distribution for scientific and analytic computing distribution and
 analysis environment, available for free and under a commercial
 license.

-<p>
-<!-- !split -->
-
 <h2 id="___sec3" class="anchor">Installing R, C++, cython or Julia </h2>

 <p>
@@ -284,9 +274,6 @@ texts.
 To install <b>R</b> with Jupyter notebook 
 <a href="https://mpacer.org/maths/r-kernel-for-ipython-notebook" target="_self">follow the link here</a>

-<p>
-<!-- !split -->
-
 <h2 id="___sec4" class="anchor">Installing R, C++, cython, Numba etc </h2>

 <p>
@@ -321,9 +308,6 @@ Finally, if you wish to use the light mark-up language
 <a href="https://github.com/hplgit/doconce" target="_self">doconce</a> you can convert a standard ascii text file into various HTML 
 formats, ipython notebooks, latex files, pdf files etc with minimal edits.

-<p>
-<!-- !split -->
-
 <h2 id="___sec5" class="anchor">Simple linear regression model using <b>scikit-learn</b> </h2>

 <p>
@@ -610,7 +594,6 @@ plt<span style="color: #666666">.</span>show()
 </pre></div>
 <p>
 Similarly, using <b>R</b>, we can perform similar studies. The following <b>R</b> code illustrates this.
-<!-- !split -->

 <h2 id="___sec6" class="anchor">Non-Linear Least squares in R  </h2>
 <div class="panel panel-default">
@@ -662,8 +645,6 @@ data <span style="color: #666666">=</span> {<span style="color: #BA2121">&#39;Na
 data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data)
 display(data_pandas)
 </pre></div>
-<p>
-<!-- !split -->

 <h2 id="___sec7" class="anchor">Examples </h2>

@@ -1332,6 +1313,101 @@ plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-5</s
 plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
 plt<span style="color: #666666">.</span>show()
 </pre></div>
+
+<h3 id="___sec12" class="anchor">Random walk model </h3>
+
+<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.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.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
+
+steps<span style="color: #666666">=250</span>
+
+distance<span style="color: #666666">=0</span>
+x<span style="color: #666666">=0</span>
+distance_list<span style="color: #666666">=</span>[]
+steps_list<span style="color: #666666">=</span>[]
+<span style="color: #008000; font-weight: bold">while</span> x<span style="color: #666666">&lt;</span>steps:
+    distance<span style="color: #666666">+=</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(<span style="color: #666666">-1</span>,<span style="color: #666666">2</span>)
+    distance_list<span style="color: #666666">.</span>append(distance)
+    x<span style="color: #666666">+=1</span>
+    steps_list<span style="color: #666666">.</span>append(x)
+plt<span style="color: #666666">.</span>plot(steps_list,distance_list, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;green&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Random Walk Data&quot;</span>)
+
+steps_list<span style="color: #666666">=</span>np<span style="color: #666666">.</span>asarray(steps_list)
+distance_list<span style="color: #666666">=</span>np<span style="color: #666666">.</span>asarray(distance_list)
+
+X<span style="color: #666666">=</span>steps_list[:,np<span style="color: #666666">.</span>newaxis]
+
+<span style="color: #408080; font-style: italic">#Polynomial fits</span>
+
+<span style="color: #408080; font-style: italic">#Degree 2</span>
+poly_features<span style="color: #666666">=</span>PolynomialFeatures(degree<span style="color: #666666">=2</span>, include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
+X_poly<span style="color: #666666">=</span>poly_features<span style="color: #666666">.</span>fit_transform(X)
+
+lin_reg<span style="color: #666666">=</span>LinearRegression()
+poly_fit<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>fit(X_poly,distance_list)
+b<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>coef_
+c<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>intercept_
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;2nd degree coefficients:&quot;</span>)
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;zero power: &quot;</span>,c)
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;first power: &quot;</span>, b[<span style="color: #666666">0</span>])
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;second power: &quot;</span>,b[<span style="color: #666666">1</span>])
+
+z <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">0</span>, steps, <span style="color: #666666">.01</span>)
+z_mod<span style="color: #666666">=</span>b[<span style="color: #666666">1</span>]<span style="color: #666666">*</span>z<span style="color: #666666">**2+</span>b[<span style="color: #666666">0</span>]<span style="color: #666666">*</span>z<span style="color: #666666">+</span>c
+
+fit_mod<span style="color: #666666">=</span>b[<span style="color: #666666">1</span>]<span style="color: #666666">*</span>X<span style="color: #666666">**2+</span>b[<span style="color: #666666">0</span>]<span style="color: #666666">*</span>X<span style="color: #666666">+</span>c
+plt<span style="color: #666666">.</span>plot(z, z_mod, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;r&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;2nd Degree Fit&quot;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Polynomial Regression&quot;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;Steps&quot;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;Distance&quot;</span>)
+
+<span style="color: #408080; font-style: italic">#Degree 10</span>
+poly_features10<span style="color: #666666">=</span>PolynomialFeatures(degree<span style="color: #666666">=10</span>, include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
+X_poly10<span style="color: #666666">=</span>poly_features10<span style="color: #666666">.</span>fit_transform(X)
+
+poly_fit10<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>fit(X_poly10,distance_list)
+
+y_plot<span style="color: #666666">=</span>poly_fit10<span style="color: #666666">.</span>predict(X_poly10)
+plt<span style="color: #666666">.</span>plot(X, y_plot, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;black&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;10th Degree Fit&quot;</span>)
+
+plt<span style="color: #666666">.</span>legend()
+plt<span style="color: #666666">.</span>show()
+
+
+<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">=2</span>)
+regr_2<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=5</span>)
+regr_3<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=7</span>)
+regr_1<span style="color: #666666">.</span>fit(X, distance_list)
+regr_2<span style="color: #666666">.</span>fit(X, distance_list)
+regr_3<span style="color: #666666">.</span>fit(X, distance_list)
+
+X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">0.0</span>, steps, <span style="color: #666666">0.01</span>)[:, np<span style="color: #666666">.</span>newaxis]
+y_1 <span style="color: #666666">=</span> regr_1<span style="color: #666666">.</span>predict(X_test)
+y_2 <span style="color: #666666">=</span> regr_2<span style="color: #666666">.</span>predict(X_test)
+y_3<span style="color: #666666">=</span>regr_3<span style="color: #666666">.</span>predict(X_test)
+
+<span style="color: #408080; font-style: italic"># Plot the results</span>
+plt<span style="color: #666666">.</span>figure()
+plt<span style="color: #666666">.</span>scatter(X, distance_list, s<span style="color: #666666">=2.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;black&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;data&quot;</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_1, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;red&quot;</span>,
+         label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=2&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_2, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;green&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=5&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_3, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;m&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=7&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;Data&quot;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;Darget&quot;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Decision Tree Regression&quot;</span>)
+plt<span style="color: #666666">.</span>legend()
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>

 <!-- ------------------- end of main content --------------- -->
@@ -150,15 +150,7 @@ MathJax.Hub.Config({
 <p>&nbsp;<br>
 <center><h4>May 29, 2018</h4></center> <!-- date -->
 <br>
-<p>

-<center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
-</center>
-</section>
-
-
-<section>
 <h2 id="___sec0">Introduction </h2>

 <p>
@@ -198,10 +190,7 @@ tensorflow (see below for links etc).  Moreover, the examples we
 introduce will serve as inputs to many of our discussions later, as
 well as allowing you to set up models and produce your own data and
 get started with programming.
-</section>

-
-<section>
 <h2 id="___sec1">Software and needed installations </h2>

 <p>
@@ -246,10 +235,7 @@ you can use <b>pip</b> as well and simply install Python as
 <p>

 etc etc.
-</section>

-
-<section>
 <h2 id="___sec2">Python installers </h2>

 <p>
@@ -278,10 +264,7 @@ is a Python
 distribution for scientific and analytic computing distribution and
 analysis environment, available for free and under a commercial
 license.
-</section>

-
-<section>
 <h2 id="___sec3">Installing R, C++, cython or Julia </h2>

 <p>
@@ -299,10 +282,7 @@ texts.
 <p>
 To install <b>R</b> with Jupyter notebook 
 <a href="https://mpacer.org/maths/r-kernel-for-ipython-notebook" target="_blank">follow the link here</a>
-</section>

-
-<section>
 <h2 id="___sec4">Installing R, C++, cython, Numba etc </h2>

 <p>
@@ -336,10 +316,7 @@ And to add more versatility, the Python package <a href="http://www.sympy.org/en
 Finally, if you wish to use the light mark-up language 
 <a href="https://github.com/hplgit/doconce" target="_blank">doconce</a> you can convert a standard ascii text file into various HTML 
 formats, ipython notebooks, latex files, pdf files etc with minimal edits.
-</section>

-
-<section>
 <h2 id="___sec5">Simple linear regression model using <b>scikit-learn</b> </h2>

 <p>
@@ -644,10 +621,7 @@ plt.show()
 </pre></div>
 <p>
 Similarly, using <b>R</b>, we can perform similar studies. The following <b>R</b> code illustrates this.
-</section>

-
-<section>
 <h2 id="___sec6">Non-Linear Least squares in R  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
@@ -695,10 +669,7 @@ data = {<span style="color: #CD5555">&#39;Name&#39;</span>: [<span style="color:
 data_pandas = pd.DataFrame(data)
 display(data_pandas)
 </pre></div>
-</section>

-
-<section>
 <h2 id="___sec7">Examples </h2>

 <p>
@@ -1396,6 +1367,107 @@ plt.axis([-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5
 plt.grid(<span style="color: #658b00">True</span>)
 plt.show()
 </pre></div>
+
+<h3 id="___sec12">Random walk model </h3>
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> PolynomialFeatures
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearRegression
+
+steps=<span style="color: #B452CD">250</span>
+
+distance=<span style="color: #B452CD">0</span>
+x=<span style="color: #B452CD">0</span>
+distance_list=[]
+steps_list=[]
+<span style="color: #8B008B; font-weight: bold">while</span> x&lt;steps:
+    distance+=np.random.randint(-<span style="color: #B452CD">1</span>,<span style="color: #B452CD">2</span>)
+    distance_list.append(distance)
+    x+=<span style="color: #B452CD">1</span>
+    steps_list.append(x)
+plt.plot(steps_list,distance_list, color=<span style="color: #CD5555">&#39;green&#39;</span>, label=<span style="color: #CD5555">&quot;Random Walk Data&quot;</span>)
+
+steps_list=np.asarray(steps_list)
+distance_list=np.asarray(distance_list)
+
+X=steps_list[:,np.newaxis]
+
+<span style="color: #228B22">#Polynomial fits</span>
+
+<span style="color: #228B22">#Degree 2</span>
+poly_features=PolynomialFeatures(degree=<span style="color: #B452CD">2</span>, include_bias=<span style="color: #658b00">False</span>)
+X_poly=poly_features.fit_transform(X)
+
+lin_reg=LinearRegression()
+poly_fit=lin_reg.fit(X_poly,distance_list)
+b=lin_reg.coef_
+c=lin_reg.intercept_
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;2nd degree coefficients:&quot;</span>)
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;zero power: &quot;</span>,c)
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;first power: &quot;</span>, b[<span style="color: #B452CD">0</span>])
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;second power: &quot;</span>,b[<span style="color: #B452CD">1</span>])
+
+z = np.arange(<span style="color: #B452CD">0</span>, steps, .<span style="color: #B452CD">01</span>)
+z_mod=b[<span style="color: #B452CD">1</span>]*z**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*z+c
+
+fit_mod=b[<span style="color: #B452CD">1</span>]*X**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*X+c
+plt.plot(z, z_mod, color=<span style="color: #CD5555">&#39;r&#39;</span>, label=<span style="color: #CD5555">&quot;2nd Degree Fit&quot;</span>)
+plt.title(<span style="color: #CD5555">&quot;Polynomial Regression&quot;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&quot;Steps&quot;</span>)
+plt.ylabel(<span style="color: #CD5555">&quot;Distance&quot;</span>)
+
+<span style="color: #228B22">#Degree 10</span>
+poly_features10=PolynomialFeatures(degree=<span style="color: #B452CD">10</span>, include_bias=<span style="color: #658b00">False</span>)
+X_poly10=poly_features10.fit_transform(X)
+
+poly_fit10=lin_reg.fit(X_poly10,distance_list)
+
+y_plot=poly_fit10.predict(X_poly10)
+plt.plot(X, y_plot, color=<span style="color: #CD5555">&#39;black&#39;</span>, label=<span style="color: #CD5555">&quot;10th Degree Fit&quot;</span>)
+
+plt.legend()
+plt.show()
+
+
+<span style="color: #228B22">#Decision Tree Regression</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.tree</span> <span style="color: #8B008B; font-weight: bold">import</span> DecisionTreeRegressor
+regr_1=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">2</span>)
+regr_2=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">5</span>)
+regr_3=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">7</span>)
+regr_1.fit(X, distance_list)
+regr_2.fit(X, distance_list)
+regr_3.fit(X, distance_list)
+
+X_test = np.arange(<span style="color: #B452CD">0.0</span>, steps, <span style="color: #B452CD">0.01</span>)[:, np.newaxis]
+y_1 = regr_1.predict(X_test)
+y_2 = regr_2.predict(X_test)
+y_3=regr_3.predict(X_test)
+
+<span style="color: #228B22"># Plot the results</span>
+plt.figure()
+plt.scatter(X, distance_list, s=<span style="color: #B452CD">2.5</span>, c=<span style="color: #CD5555">&quot;black&quot;</span>, label=<span style="color: #CD5555">&quot;data&quot;</span>)
+plt.plot(X_test, y_1, color=<span style="color: #CD5555">&quot;red&quot;</span>,
+         label=<span style="color: #CD5555">&quot;max_depth=2&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+plt.plot(X_test, y_2, color=<span style="color: #CD5555">&quot;green&quot;</span>, label=<span style="color: #CD5555">&quot;max_depth=5&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+plt.plot(X_test, y_3, color=<span style="color: #CD5555">&quot;m&quot;</span>, label=<span style="color: #CD5555">&quot;max_depth=7&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+
+plt.xlabel(<span style="color: #CD5555">&quot;Data&quot;</span>)
+plt.ylabel(<span style="color: #CD5555">&quot;Darget&quot;</span>)
+plt.title(<span style="color: #CD5555">&quot;Decision Tree Regression&quot;</span>)
+plt.legend()
+plt.show()
+</pre></div>
+<p>
+
+
+<center style="font-size:80%">
+<!-- copyright --> &copy; 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+</center>
 </section>


@@ -77,7 +77,8 @@ div { text-align: justify; text-justify: inter-word; }
              ('Particle in one dimension and velocity distribution',
               3,
               None,
-               '___sec11')]}
+               '___sec11'),
+              ('Random walk model', 3, None, '___sec12')]}
 end of tocinfo -->

 <body>
@@ -121,8 +122,6 @@ MathJax.Hub.Config({
 <p>
 <center><h4>May 29, 2018</h4></center> <!-- date -->
 <br>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec0">Introduction </h2>

@@ -164,9 +163,6 @@ introduce will serve as inputs to many of our discussions later, as
 well as allowing you to set up models and produce your own data and
 get started with programming.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec1">Software and needed installations </h2>

 <p>
@@ -209,9 +205,6 @@ you can use <b>pip</b> as well and simply install Python as

 etc etc.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec2">Python installers </h2>

 <p>
@@ -239,9 +232,6 @@ distribution for scientific and analytic computing distribution and
 analysis environment, available for free and under a commercial
 license.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec3">Installing R, C++, cython or Julia </h2>

 <p>
@@ -260,9 +250,6 @@ texts.
 To install <b>R</b> with Jupyter notebook 
 <a href="https://mpacer.org/maths/r-kernel-for-ipython-notebook" target="_blank">follow the link here</a>

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec4">Installing R, C++, cython, Numba etc </h2>

 <p>
@@ -297,9 +284,6 @@ Finally, if you wish to use the light mark-up language
 <a href="https://github.com/hplgit/doconce" target="_blank">doconce</a> you can convert a standard ascii text file into various HTML 
 formats, ipython notebooks, latex files, pdf files etc with minimal edits.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec5">Simple linear regression model using <b>scikit-learn</b> </h2>

 <p>
@@ -586,7 +570,6 @@ plt.show()
 </pre></div>
 <p>
 Similarly, using <b>R</b>, we can perform similar studies. The following <b>R</b> code illustrates this.
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec6">Non-Linear Least squares in R  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
@@ -637,8 +620,6 @@ data = {<span style="color: #CD5555">&#39;Name&#39;</span>: [<span style="color:
 data_pandas = pd.DataFrame(data)
 display(data_pandas)
 </pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec7">Examples </h2>

@@ -1300,6 +1281,101 @@ plt.axis([-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5
 plt.grid(<span style="color: #658b00">True</span>)
 plt.show()
 </pre></div>
+
+<h3 id="___sec12">Random walk model </h3>
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> PolynomialFeatures
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearRegression
+
+steps=<span style="color: #B452CD">250</span>
+
+distance=<span style="color: #B452CD">0</span>
+x=<span style="color: #B452CD">0</span>
+distance_list=[]
+steps_list=[]
+<span style="color: #8B008B; font-weight: bold">while</span> x&lt;steps:
+    distance+=np.random.randint(-<span style="color: #B452CD">1</span>,<span style="color: #B452CD">2</span>)
+    distance_list.append(distance)
+    x+=<span style="color: #B452CD">1</span>
+    steps_list.append(x)
+plt.plot(steps_list,distance_list, color=<span style="color: #CD5555">&#39;green&#39;</span>, label=<span style="color: #CD5555">&quot;Random Walk Data&quot;</span>)
+
+steps_list=np.asarray(steps_list)
+distance_list=np.asarray(distance_list)
+
+X=steps_list[:,np.newaxis]
+
+<span style="color: #228B22">#Polynomial fits</span>
+
+<span style="color: #228B22">#Degree 2</span>
+poly_features=PolynomialFeatures(degree=<span style="color: #B452CD">2</span>, include_bias=<span style="color: #658b00">False</span>)
+X_poly=poly_features.fit_transform(X)
+
+lin_reg=LinearRegression()
+poly_fit=lin_reg.fit(X_poly,distance_list)
+b=lin_reg.coef_
+c=lin_reg.intercept_
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;2nd degree coefficients:&quot;</span>)
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;zero power: &quot;</span>,c)
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;first power: &quot;</span>, b[<span style="color: #B452CD">0</span>])
+<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">&quot;second power: &quot;</span>,b[<span style="color: #B452CD">1</span>])
+
+z = np.arange(<span style="color: #B452CD">0</span>, steps, .<span style="color: #B452CD">01</span>)
+z_mod=b[<span style="color: #B452CD">1</span>]*z**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*z+c
+
+fit_mod=b[<span style="color: #B452CD">1</span>]*X**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*X+c
+plt.plot(z, z_mod, color=<span style="color: #CD5555">&#39;r&#39;</span>, label=<span style="color: #CD5555">&quot;2nd Degree Fit&quot;</span>)
+plt.title(<span style="color: #CD5555">&quot;Polynomial Regression&quot;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&quot;Steps&quot;</span>)
+plt.ylabel(<span style="color: #CD5555">&quot;Distance&quot;</span>)
+
+<span style="color: #228B22">#Degree 10</span>
+poly_features10=PolynomialFeatures(degree=<span style="color: #B452CD">10</span>, include_bias=<span style="color: #658b00">False</span>)
+X_poly10=poly_features10.fit_transform(X)
+
+poly_fit10=lin_reg.fit(X_poly10,distance_list)
+
+y_plot=poly_fit10.predict(X_poly10)
+plt.plot(X, y_plot, color=<span style="color: #CD5555">&#39;black&#39;</span>, label=<span style="color: #CD5555">&quot;10th Degree Fit&quot;</span>)
+
+plt.legend()
+plt.show()
+
+
+<span style="color: #228B22">#Decision Tree Regression</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.tree</span> <span style="color: #8B008B; font-weight: bold">import</span> DecisionTreeRegressor
+regr_1=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">2</span>)
+regr_2=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">5</span>)
+regr_3=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">7</span>)
+regr_1.fit(X, distance_list)
+regr_2.fit(X, distance_list)
+regr_3.fit(X, distance_list)
+
+X_test = np.arange(<span style="color: #B452CD">0.0</span>, steps, <span style="color: #B452CD">0.01</span>)[:, np.newaxis]
+y_1 = regr_1.predict(X_test)
+y_2 = regr_2.predict(X_test)
+y_3=regr_3.predict(X_test)
+
+<span style="color: #228B22"># Plot the results</span>
+plt.figure()
+plt.scatter(X, distance_list, s=<span style="color: #B452CD">2.5</span>, c=<span style="color: #CD5555">&quot;black&quot;</span>, label=<span style="color: #CD5555">&quot;data&quot;</span>)
+plt.plot(X_test, y_1, color=<span style="color: #CD5555">&quot;red&quot;</span>,
+         label=<span style="color: #CD5555">&quot;max_depth=2&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+plt.plot(X_test, y_2, color=<span style="color: #CD5555">&quot;green&quot;</span>, label=<span style="color: #CD5555">&quot;max_depth=5&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+plt.plot(X_test, y_3, color=<span style="color: #CD5555">&quot;m&quot;</span>, label=<span style="color: #CD5555">&quot;max_depth=7&quot;</span>, linewidth=<span style="color: #B452CD">2</span>)
+
+plt.xlabel(<span style="color: #CD5555">&quot;Data&quot;</span>)
+plt.ylabel(<span style="color: #CD5555">&quot;Darget&quot;</span>)
+plt.title(<span style="color: #CD5555">&quot;Decision Tree Regression&quot;</span>)
+plt.legend()
+plt.show()
+</pre></div>
 <p>

 <!-- ------------------- end of main content --------------- -->
@@ -82,7 +82,8 @@ div { text-align: justify; text-justify: inter-word; }
              ('Particle in one dimension and velocity distribution',
               3,
               None,
-               '___sec11')]}
+               '___sec11'),
+              ('Random walk model', 3, None, '___sec12')]}
 end of tocinfo -->

 <body>
@@ -126,8 +127,6 @@ MathJax.Hub.Config({
 <p>
 <center><h4>May 29, 2018</h4></center> <!-- date -->
 <br>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec0">Introduction </h2>

@@ -169,9 +168,6 @@ introduce will serve as inputs to many of our discussions later, as
 well as allowing you to set up models and produce your own data and
 get started with programming.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec1">Software and needed installations </h2>

 <p>
@@ -214,9 +210,6 @@ you can use <b>pip</b> as well and simply install Python as

 etc etc.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec2">Python installers </h2>

 <p>
@@ -244,9 +237,6 @@ distribution for scientific and analytic computing distribution and
 analysis environment, available for free and under a commercial
 license.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec3">Installing R, C++, cython or Julia </h2>

 <p>
@@ -265,9 +255,6 @@ texts.
 To install <b>R</b> with Jupyter notebook 
 <a href="https://mpacer.org/maths/r-kernel-for-ipython-notebook" target="_blank">follow the link here</a>

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec4">Installing R, C++, cython, Numba etc </h2>

 <p>
@@ -302,9 +289,6 @@ Finally, if you wish to use the light mark-up language
 <a href="https://github.com/hplgit/doconce" target="_blank">doconce</a> you can convert a standard ascii text file into various HTML 
 formats, ipython notebooks, latex files, pdf files etc with minimal edits.

-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
 <h2 id="___sec5">Simple linear regression model using <b>scikit-learn</b> </h2>

 <p>
@@ -591,7 +575,6 @@ plt<span style="color: #666666">.</span>show()
 </pre></div>
 <p>
 Similarly, using <b>R</b>, we can perform similar studies. The following <b>R</b> code illustrates this.
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec6">Non-Linear Least squares in R  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
@@ -642,8 +625,6 @@ data <span style="color: #666666">=</span> {<span style="color: #BA2121">&#39;Na
 data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data)
 display(data_pandas)
 </pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>

 <h2 id="___sec7">Examples </h2>

@@ -1305,6 +1286,101 @@ plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-5</s
 plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
 plt<span style="color: #666666">.</span>show()
 </pre></div>
+
+<h3 id="___sec12">Random walk model </h3>
+
+<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.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.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
+
+steps<span style="color: #666666">=250</span>
+
+distance<span style="color: #666666">=0</span>
+x<span style="color: #666666">=0</span>
+distance_list<span style="color: #666666">=</span>[]
+steps_list<span style="color: #666666">=</span>[]
+<span style="color: #008000; font-weight: bold">while</span> x<span style="color: #666666">&lt;</span>steps:
+    distance<span style="color: #666666">+=</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(<span style="color: #666666">-1</span>,<span style="color: #666666">2</span>)
+    distance_list<span style="color: #666666">.</span>append(distance)
+    x<span style="color: #666666">+=1</span>
+    steps_list<span style="color: #666666">.</span>append(x)
+plt<span style="color: #666666">.</span>plot(steps_list,distance_list, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;green&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Random Walk Data&quot;</span>)
+
+steps_list<span style="color: #666666">=</span>np<span style="color: #666666">.</span>asarray(steps_list)
+distance_list<span style="color: #666666">=</span>np<span style="color: #666666">.</span>asarray(distance_list)
+
+X<span style="color: #666666">=</span>steps_list[:,np<span style="color: #666666">.</span>newaxis]
+
+<span style="color: #408080; font-style: italic">#Polynomial fits</span>
+
+<span style="color: #408080; font-style: italic">#Degree 2</span>
+poly_features<span style="color: #666666">=</span>PolynomialFeatures(degree<span style="color: #666666">=2</span>, include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
+X_poly<span style="color: #666666">=</span>poly_features<span style="color: #666666">.</span>fit_transform(X)
+
+lin_reg<span style="color: #666666">=</span>LinearRegression()
+poly_fit<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>fit(X_poly,distance_list)
+b<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>coef_
+c<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>intercept_
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;2nd degree coefficients:&quot;</span>)
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;zero power: &quot;</span>,c)
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;first power: &quot;</span>, b[<span style="color: #666666">0</span>])
+<span style="color: #008000; font-weight: bold">print</span> (<span style="color: #BA2121">&quot;second power: &quot;</span>,b[<span style="color: #666666">1</span>])
+
+z <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">0</span>, steps, <span style="color: #666666">.01</span>)
+z_mod<span style="color: #666666">=</span>b[<span style="color: #666666">1</span>]<span style="color: #666666">*</span>z<span style="color: #666666">**2+</span>b[<span style="color: #666666">0</span>]<span style="color: #666666">*</span>z<span style="color: #666666">+</span>c
+
+fit_mod<span style="color: #666666">=</span>b[<span style="color: #666666">1</span>]<span style="color: #666666">*</span>X<span style="color: #666666">**2+</span>b[<span style="color: #666666">0</span>]<span style="color: #666666">*</span>X<span style="color: #666666">+</span>c
+plt<span style="color: #666666">.</span>plot(z, z_mod, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;r&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;2nd Degree Fit&quot;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Polynomial Regression&quot;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;Steps&quot;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;Distance&quot;</span>)
+
+<span style="color: #408080; font-style: italic">#Degree 10</span>
+poly_features10<span style="color: #666666">=</span>PolynomialFeatures(degree<span style="color: #666666">=10</span>, include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
+X_poly10<span style="color: #666666">=</span>poly_features10<span style="color: #666666">.</span>fit_transform(X)
+
+poly_fit10<span style="color: #666666">=</span>lin_reg<span style="color: #666666">.</span>fit(X_poly10,distance_list)
+
+y_plot<span style="color: #666666">=</span>poly_fit10<span style="color: #666666">.</span>predict(X_poly10)
+plt<span style="color: #666666">.</span>plot(X, y_plot, color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;black&#39;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;10th Degree Fit&quot;</span>)
+
+plt<span style="color: #666666">.</span>legend()
+plt<span style="color: #666666">.</span>show()
+
+
+<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">=2</span>)
+regr_2<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=5</span>)
+regr_3<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=7</span>)
+regr_1<span style="color: #666666">.</span>fit(X, distance_list)
+regr_2<span style="color: #666666">.</span>fit(X, distance_list)
+regr_3<span style="color: #666666">.</span>fit(X, distance_list)
+
+X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">0.0</span>, steps, <span style="color: #666666">0.01</span>)[:, np<span style="color: #666666">.</span>newaxis]
+y_1 <span style="color: #666666">=</span> regr_1<span style="color: #666666">.</span>predict(X_test)
+y_2 <span style="color: #666666">=</span> regr_2<span style="color: #666666">.</span>predict(X_test)
+y_3<span style="color: #666666">=</span>regr_3<span style="color: #666666">.</span>predict(X_test)
+
+<span style="color: #408080; font-style: italic"># Plot the results</span>
+plt<span style="color: #666666">.</span>figure()
+plt<span style="color: #666666">.</span>scatter(X, distance_list, s<span style="color: #666666">=2.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&quot;black&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;data&quot;</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_1, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;red&quot;</span>,
+         label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=2&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_2, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;green&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=5&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+plt<span style="color: #666666">.</span>plot(X_test, y_3, color<span style="color: #666666">=</span><span style="color: #BA2121">&quot;m&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;max_depth=7&quot;</span>, linewidth<span style="color: #666666">=2</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;Data&quot;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;Darget&quot;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Decision Tree Regression&quot;</span>)
+plt<span style="color: #666666">.</span>legend()
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>

 <!-- ------------------- end of main content --------------- -->
@@ -17,6 +17,7 @@
    "\n",
    "\n",
    "\n",
+    "\n",
    "## Introduction\n",
    "\n",
    "Our emphasis throughout this series of lectures  \n",
@@ -57,6 +58,7 @@
    "\n",
    "\n",
    "\n",
+    "\n",
    "## Software and needed installations\n",
    "\n",
    "We will make extensive use of Python as programming language and its\n",
@@ -91,6 +93,7 @@
    "\n",
    "etc etc. \n",
    "\n",
+    "\n",
    "## Python installers\n",
    "\n",
    "If you don't want to perform these operations separately and venture\n",
@@ -114,6 +117,7 @@
    "license.\n",
    "\n",
    "\n",
+    "\n",
    "## Installing R, C++, cython or Julia\n",
    "\n",
    "You will also find it convenient to utilize R. Although we will mainly\n",
@@ -132,6 +136,7 @@
    "\n",
    "\n",
    "\n",
+    "\n",
    "## Installing R, C++, cython, Numba etc\n",
    "\n",
    "\n",
@@ -170,6 +175,7 @@
    "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n",
    "formats, ipython notebooks, latex files, pdf files etc with minimal edits.\n",
    "\n",
+    "\n",
    "## Simple linear regression model using **scikit-learn**\n",
    "\n",
    "We start with perhaps our simplest possible example, using **scikit-learn** to perform linear regression analysis on a data set produced by us. \n",
@@ -582,6 +588,8 @@
   "metadata": {},
   "source": [
    "Similarly, using **R**, we can perform similar studies. The following **R** code illustrates this.\n",
+    "\n",
+    "\n",
    "## Non-Linear Least squares in R"
   ]
  },
@@ -1573,6 +1581,111 @@
    "plt.grid(True)\n",
    "plt.show()"
   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Random walk model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.preprocessing import PolynomialFeatures\n",
+    "from sklearn.linear_model import LinearRegression\n",
+    "\n",
+    "steps=250\n",
+    "\n",
+    "distance=0\n",
+    "x=0\n",
+    "distance_list=[]\n",
+    "steps_list=[]\n",
+    "while x<steps:\n",
+    "    distance+=np.random.randint(-1,2)\n",
+    "    distance_list.append(distance)\n",
+    "    x+=1\n",
+    "    steps_list.append(x)\n",
+    "plt.plot(steps_list,distance_list, color='green', label=\"Random Walk Data\")\n",
+    "\n",
+    "steps_list=np.asarray(steps_list)\n",
+    "distance_list=np.asarray(distance_list)\n",
+    "\n",
+    "X=steps_list[:,np.newaxis]\n",
+    "\n",
+    "#Polynomial fits\n",
+    "\n",
+    "#Degree 2\n",
+    "poly_features=PolynomialFeatures(degree=2, include_bias=False)\n",
+    "X_poly=poly_features.fit_transform(X)\n",
+    "\n",
+    "lin_reg=LinearRegression()\n",
+    "poly_fit=lin_reg.fit(X_poly,distance_list)\n",
+    "b=lin_reg.coef_\n",
+    "c=lin_reg.intercept_\n",
+    "print (\"2nd degree coefficients:\")\n",
+    "print (\"zero power: \",c)\n",
+    "print (\"first power: \", b[0])\n",
+    "print (\"second power: \",b[1])\n",
+    "\n",
+    "z = np.arange(0, steps, .01)\n",
+    "z_mod=b[1]*z**2+b[0]*z+c\n",
+    "\n",
+    "fit_mod=b[1]*X**2+b[0]*X+c\n",
+    "plt.plot(z, z_mod, color='r', label=\"2nd Degree Fit\")\n",
+    "plt.title(\"Polynomial Regression\")\n",
+    "\n",
+    "plt.xlabel(\"Steps\")\n",
+    "plt.ylabel(\"Distance\")\n",
+    "\n",
+    "#Degree 10\n",
+    "poly_features10=PolynomialFeatures(degree=10, include_bias=False)\n",
+    "X_poly10=poly_features10.fit_transform(X)\n",
+    "\n",
+    "poly_fit10=lin_reg.fit(X_poly10,distance_list)\n",
+    "\n",
+    "y_plot=poly_fit10.predict(X_poly10)\n",
+    "plt.plot(X, y_plot, color='black', label=\"10th Degree Fit\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.show()\n",
+    "\n",
+    "\n",
+    "#Decision Tree Regression\n",
+    "from sklearn.tree import DecisionTreeRegressor\n",
+    "regr_1=DecisionTreeRegressor(max_depth=2)\n",
+    "regr_2=DecisionTreeRegressor(max_depth=5)\n",
+    "regr_3=DecisionTreeRegressor(max_depth=7)\n",
+    "regr_1.fit(X, distance_list)\n",
+    "regr_2.fit(X, distance_list)\n",
+    "regr_3.fit(X, distance_list)\n",
+    "\n",
+    "X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]\n",
+    "y_1 = regr_1.predict(X_test)\n",
+    "y_2 = regr_2.predict(X_test)\n",
+    "y_3=regr_3.predict(X_test)\n",
+    "\n",
+    "# Plot the results\n",
+    "plt.figure()\n",
+    "plt.scatter(X, distance_list, s=2.5, c=\"black\", label=\"data\")\n",
+    "plt.plot(X_test, y_1, color=\"red\",\n",
+    "         label=\"max_depth=2\", linewidth=2)\n",
+    "plt.plot(X_test, y_2, color=\"green\", label=\"max_depth=5\", linewidth=2)\n",
+    "plt.plot(X_test, y_3, color=\"m\", label=\"max_depth=7\", linewidth=2)\n",
+    "\n",
+    "plt.xlabel(\"Data\")\n",
+    "plt.ylabel(\"Darget\")\n",
+    "plt.title(\"Decision Tree Regression\")\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
  }
 ],
 "metadata": {},
@@ -3,7 +3,7 @@ AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of
 DATE: today


-!split
+
 ===== Introduction =====

 Our emphasis throughout this series of lectures  
@@ -44,7 +44,7 @@ get started with programming.



-!split
+
 ===== Software and needed installations =====

 We will make extensive use of Python as programming language and its
@@ -79,7 +79,7 @@ o sudo apt-get install python3  (or python for pyhton2.7)

 etc etc. 

-!split
+
 ===== Python installers =====

 If you don't want to perform these operations separately and venture
@@ -103,7 +103,7 @@ analysis environment, available for free and under a commercial
 license.


-!split
+
 ===== Installing R, C++, cython or Julia =====

 You will also find it convenient to utilize R. Although we will mainly
@@ -122,7 +122,7 @@ To install _R_ with Jupyter notebook



-!split
+
 ===== Installing R, C++, cython, Numba etc =====


@@ -153,7 +153,7 @@ Finally, if you wish to use the light mark-up language
 "doconce":"https://github.com/hplgit/doconce" you can convert a standard ascii text file into various HTML 
 formats, ipython notebooks, latex files, pdf files etc with minimal edits.

-!split
+
 ===== Simple linear regression model using _scikit-learn_ =====

 We start with perhaps our simplest possible example, using _scikit-learn_ to perform linear regression analysis on a data set produced by us. 
@@ -430,7 +430,8 @@ print (error(y))
 !ec

 Similarly, using _R_, we can perform similar studies. The following _R_ code illustrates this.
-!split
+
+
 ===== Non-Linear Least squares in R  =====
 !bblock
 !bc r
@@ -473,7 +474,7 @@ display(data_pandas)
 !ec


-!split
+
 ===== Examples =====

 We present here several examples, with pertinent Python codes that we
@@ -1026,3 +1027,96 @@ plt.show()



+
+=== Random walk model ===
+!bc pycod
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression
+
+steps=250
+
+distance=0
+x=0
+distance_list=[]
+steps_list=[]
+while x<steps:
+    distance+=np.random.randint(-1,2)
+    distance_list.append(distance)
+    x+=1
+    steps_list.append(x)
+plt.plot(steps_list,distance_list, color='green', label="Random Walk Data")
+
+steps_list=np.asarray(steps_list)
+distance_list=np.asarray(distance_list)
+
+X=steps_list[:,np.newaxis]
+
+#Polynomial fits
+
+#Degree 2
+poly_features=PolynomialFeatures(degree=2, include_bias=False)
+X_poly=poly_features.fit_transform(X)
+
+lin_reg=LinearRegression()
+poly_fit=lin_reg.fit(X_poly,distance_list)
+b=lin_reg.coef_
+c=lin_reg.intercept_
+print ("2nd degree coefficients:")
+print ("zero power: ",c)
+print ("first power: ", b[0])
+print ("second power: ",b[1])
+
+z = np.arange(0, steps, .01)
+z_mod=b[1]*z**2+b[0]*z+c
+
+fit_mod=b[1]*X**2+b[0]*X+c
+plt.plot(z, z_mod, color='r', label="2nd Degree Fit")
+plt.title("Polynomial Regression")
+
+plt.xlabel("Steps")
+plt.ylabel("Distance")
+
+#Degree 10
+poly_features10=PolynomialFeatures(degree=10, include_bias=False)
+X_poly10=poly_features10.fit_transform(X)
+
+poly_fit10=lin_reg.fit(X_poly10,distance_list)
+
+y_plot=poly_fit10.predict(X_poly10)
+plt.plot(X, y_plot, color='black', label="10th Degree Fit")
+
+plt.legend()
+plt.show()
+
+
+#Decision Tree Regression
+from sklearn.tree import DecisionTreeRegressor
+regr_1=DecisionTreeRegressor(max_depth=2)
+regr_2=DecisionTreeRegressor(max_depth=5)
+regr_3=DecisionTreeRegressor(max_depth=7)
+regr_1.fit(X, distance_list)
+regr_2.fit(X, distance_list)
+regr_3.fit(X, distance_list)
+
+X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]
+y_1 = regr_1.predict(X_test)
+y_2 = regr_2.predict(X_test)
+y_3=regr_3.predict(X_test)
+
+# Plot the results
+plt.figure()
+plt.scatter(X, distance_list, s=2.5, c="black", label="data")
+plt.plot(X_test, y_1, color="red",
+         label="max_depth=2", linewidth=2)
+plt.plot(X_test, y_2, color="green", label="max_depth=5", linewidth=2)
+plt.plot(X_test, y_3, color="m", label="max_depth=7", linewidth=2)
+
+plt.xlabel("Data")
+plt.ylabel("Darget")
+plt.title("Decision Tree Regression")
+plt.legend()
+plt.show()
+
+!ec