From 5371b6a4ce363d3604db2c70e7bf20538ae429d9 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 17 Sep 2021 08:29:24 +0200 Subject: [PATCH] update chapter 3 --- doc/BookChapters/chapter3.do.txt | 58 +++++ .../_build/.doctrees/chapter3.doctree | Bin 465467 -> 489694 bytes .../_build/.doctrees/environment.pickle | Bin 177211 -> 177323 bytes .../_build/html/_images/chapter3_47_0.png | Bin 14342 -> 11796 bytes .../_build/html/_images/chapter3_61_1.png | Bin 0 -> 23409 bytes .../_build/html/_images/chapter3_62_6.png | Bin 0 -> 16322 bytes .../_build/html/_images/chapter3_64_1.png | Bin 0 -> 38099 bytes .../_build/html/_images/chapter3_65_11.png | Bin 0 -> 16300 bytes .../_build/html/_images/chapter3_71_0.png | Bin 0 -> 14342 bytes .../_build/html/_images/chapter3_73_1.png | Bin 0 -> 11986 bytes .../_build/html/_images/chapter3_74_0.png | Bin 32458 -> 10546 bytes .../_build/html/_images/chapter3_78_0.png | Bin 0 -> 32458 bytes .../_build/html/_sources/chapter3.ipynb | 85 ++++++- doc/LectureNotes/_build/html/chapter3.html | 143 ++++++++--- doc/LectureNotes/_build/html/searchindex.js | 2 +- .../_build/jupyter_execute/chapter3.ipynb | 224 +++++++++++++----- .../_build/jupyter_execute/chapter3.py | 53 ++++- .../_build/jupyter_execute/chapter3_47_0.png | Bin 14342 -> 11796 bytes .../_build/jupyter_execute/chapter3_61_1.png | Bin 0 -> 23409 bytes .../_build/jupyter_execute/chapter3_62_6.png | Bin 0 -> 16322 bytes .../_build/jupyter_execute/chapter3_64_1.png | Bin 0 -> 38099 bytes .../_build/jupyter_execute/chapter3_65_11.png | Bin 0 -> 16300 bytes .../_build/jupyter_execute/chapter3_71_0.png | Bin 0 -> 14342 bytes .../_build/jupyter_execute/chapter3_73_1.png | Bin 0 -> 11986 bytes .../_build/jupyter_execute/chapter3_74_0.png | Bin 32458 -> 10546 bytes .../_build/jupyter_execute/chapter3_78_0.png | Bin 0 -> 32458 bytes doc/LectureNotes/chapter3.ipynb | 85 ++++++- 27 files changed, 552 insertions(+), 98 deletions(-) create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_61_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_62_6.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_64_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_65_11.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_71_0.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_73_1.png create mode 100644 doc/LectureNotes/_build/html/_images/chapter3_78_0.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_61_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_62_6.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_64_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_65_11.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_71_0.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_73_1.png create mode 100644 doc/LectureNotes/_build/jupyter_execute/chapter3_78_0.png diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index d60f2c858..f83d7feb4 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -504,6 +504,63 @@ finite $m$, it is not always possible to find a closed form /analytic expression $\tilde{p}(x)$. +The following code starts with a Gaussian distribution with mean value +$\mu =100$ and variance $\sigma=15$. We use this to generate the data +used in the bootstrap analysis. The bootstrap analysis returns a data +set after a given number of bootstrap operations (as many as we have +data points). This data set consists of estimated mean values for each +bootstrap operation. The histogram generated by the bootstrap method +shows that the distribution for these mean values is also a Gaussian, +centered around the mean value $\mu=100$ but with standard deviation +$\sigma/\sqrt{n}$, where $n$ is the number of bootstrap samples (in +this case the same as the number of original data points). The value +of the standard deviation is what we expect from the central limit +theorem. + + +!bc pycod +import numpy as np +from time import time +from scipy.stats import norm +import matplotlib.pyplot as plt + +# Returns mean of bootstrap samples +# Bootstrap algorithm +def bootstrap(data, datapoints): + t = np.zeros(datapoints) + n = len(data) + # non-parametric bootstrap + for i in range(datapoints): + t[i] = np.mean(data[np.random.randint(0,n,n)]) + # analysis + print("Bootstrap Statistics :") + print("original bias std. error") + print("%8g %8g %14g %15g" % (np.mean(data), np.std(data),np.mean(t),np.std(t))) + return t + +# We set the mean value to 100 and the standard deviation to 15 +mu, sigma = 100, 15 +datapoints = 10000 +# We generate random numbers according to the normal distribution +x = mu + sigma*np.random.randn(datapoints) +# bootstrap returns the data sample +t = bootstrap(x, datapoints) +!ec +We see that our new variance and from that the standard deviation, agrees with the central limit theorem. + +We plot then the histogram together with a best fit for the data set. +!bc pycod +# the histogram of the bootstrapped data (normalized data if density = True) +n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75) +# add a 'best fit' line +y = norm.pdf(binsboot, np.mean(t), np.std(t)) +lt = plt.plot(binsboot, y, 'b', linewidth=1) +plt.xlabel('x') +plt.ylabel('Probability') +plt.grid(True) +plt.show() +!ec + @@ -1479,3 +1536,4 @@ and discuss the applicability of these regression methods to the type of data presented here (either the terrain data we propose or other data sets). + diff --git a/doc/LectureNotes/_build/.doctrees/chapter3.doctree b/doc/LectureNotes/_build/.doctrees/chapter3.doctree index 8ba13486cb0d08054e30314125562c65d777e116..b39c4cbe3689f7959148b4c453a34d63dbfc3a1d 100644 GIT binary patch delta 33114 zcmeFZd9d`@bsskG%|dcWMvf?v!x>6?qG)qQ7!DfUSY_UL;#v=v5LOyKFh-RM}2k@j}^jqHIpeB_*kHmWp!w 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--git a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb index 4a7aa4470..413c54051 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb @@ -735,11 +735,92 @@ "$\\tilde{p}(x)$.\n", "\n", "\n", + "The following code starts with a Gaussian distribution with mean value\n", + "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n", + "used in the bootstrap analysis. The bootstrap analysis returns a data\n", + "set after a given number of bootstrap operations (as many as we have\n", + "data points). This data set consists of estimated mean values for each\n", + "bootstrap operation. The histogram generated by the bootstrap method\n", + "shows that the distribution for these mean values is also a Gaussian,\n", + "centered around the mean value $\\mu=100$ but with standard deviation\n", + "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n", + "this case the same as the number of original data points). The value\n", + "of the standard deviation is what we expect from the central limit\n", + "theorem." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", "\n", + "import numpy as np\n", + "from time import time\n", + "from scipy.stats import norm\n", + "import matplotlib.pyplot as plt\n", "\n", + "# Returns mean of bootstrap samples \n", + "# Bootstrap algorithm\n", + "def bootstrap(data, datapoints):\n", + " t = np.zeros(datapoints)\n", + " n = len(data)\n", + " # non-parametric bootstrap \n", + " for i in range(datapoints):\n", + " t[i] = np.mean(data[np.random.randint(0,n,n)])\n", + " # analysis \n", + " print(\"Bootstrap Statistics :\")\n", + " print(\"original bias std. error\")\n", + " print(\"%8g %8g %14g %15g\" % (np.mean(data), np.std(data),np.mean(t),np.std(t)))\n", + " return t\n", "\n", + "# We set the mean value to 100 and the standard deviation to 15\n", + "mu, sigma = 100, 15\n", + "datapoints = 10000\n", + "# We generate random numbers according to the normal distribution\n", + "x = mu + sigma*np.random.randn(datapoints)\n", + "# bootstrap returns the data sample \n", + "t = bootstrap(x, datapoints)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n", "\n", - "\n", + "We plot then the histogram together with a best fit for the data set." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "# the histogram of the bootstrapped data (normalized data if density = True)\n", + "n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75)\n", + "# add a 'best fit' line \n", + "y = norm.pdf(binsboot, np.mean(t), np.std(t))\n", + "lt = plt.plot(binsboot, y, 'b', linewidth=1)\n", + "plt.xlabel('x')\n", + "plt.ylabel('Probability')\n", + "plt.grid(True)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## The bias-variance tradeoff\n", "\n", "\n", @@ -871,8 +952,6 @@ }, "outputs": [], "source": [ - "%matplotlib inline\n", - "\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 0724cc393..71542c6b8 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -630,10 +630,10 @@ number \(i\) is left out. Usin

-
Runtime: 0.141725 sec
+
Runtime: 0.13859 sec
 Jackknife Statistics :
 original           bias      std. error
- 100.148        100.138          0.1492
+ 99.9054        99.8954        0.149328
@@ -804,6 +804,79 @@ correlations.

The theorem is satisfied by a large class of PDFs. Note however that for a finite \(m\), it is not always possible to find a closed form /analytic expression for \(\tilde{p}(x)\).

+

The following code starts with a Gaussian distribution with mean value +\(\mu =100\) and variance \(\sigma=15\). We use this to generate the data +used in the bootstrap analysis. The bootstrap analysis returns a data +set after a given number of bootstrap operations (as many as we have +data points). This data set consists of estimated mean values for each +bootstrap operation. The histogram generated by the bootstrap method +shows that the distribution for these mean values is also a Gaussian, +centered around the mean value \(\mu=100\) but with standard deviation +\(\sigma/\sqrt{n}\), where \(n\) is the number of bootstrap samples (in +this case the same as the number of original data points). The value +of the standard deviation is what we expect from the central limit +theorem.

+
+
+
%matplotlib inline
+
+import numpy as np
+from time import time
+from scipy.stats import norm
+import matplotlib.pyplot as plt
+
+# Returns mean of bootstrap samples 
+# Bootstrap algorithm
+def bootstrap(data, datapoints):
+    t = np.zeros(datapoints)
+    n = len(data)
+    # non-parametric bootstrap         
+    for i in range(datapoints):
+        t[i] = np.mean(data[np.random.randint(0,n,n)])
+    # analysis    
+    print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (np.mean(data), np.std(data),np.mean(t),np.std(t)))
+    return t
+
+# We set the mean value to 100 and the standard deviation to 15
+mu, sigma = 100, 15
+datapoints = 10000
+# We generate random numbers according to the normal distribution
+x = mu + sigma*np.random.randn(datapoints)
+# bootstrap returns the data sample                                    
+t = bootstrap(x, datapoints)
+
+
+
+
+
Bootstrap Statistics :
+original           bias      std. error
+ 99.7442  15.0015        99.7465        0.148127
+
+
+
+
+

We see that our new variance and from that the standard deviation, agrees with the central limit theorem.

+

We plot then the histogram together with a best fit for the data set.

+
+
+
# the histogram of the bootstrapped data (normalized data if density = True)
+n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75)
+# add a 'best fit' line  
+y = norm.pdf(binsboot, np.mean(t), np.std(t))
+lt = plt.plot(binsboot, y, 'b', linewidth=1)
+plt.xlabel('x')
+plt.ylabel('Probability')
+plt.grid(True)
+plt.show()
+
+
+
+
+_images/chapter3_47_0.png +
+
@@ -857,9 +930,7 @@ We use a more compact notation in terms of the expectation value

that is the rewriting in terms of the so-called bias, the variance of the model \(\boldsymbol{\tilde{y}}\) and the variance of \(\boldsymbol{\epsilon}\).

-
%matplotlib inline
-
-import matplotlib.pyplot as plt
+
import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -923,7 +994,7 @@ Var: 0.0010479246398391328
 0.01312157412031145 >= 0.012073649480472317 + 0.0010479246398391328 = 0.01312157412031145
 

 

-_images/chapter3_57_1.png
+_images/chapter3_61_1.png
@@ -983,14 +1054,15 @@ Error: 0.32149601703519126 Bias^2: 0.3123314713548606 Var: 0.009164545680330616 0.32149601703519126 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912 -Polynomial degree: 1 +Polynomial degree: +
+
+
 1
 Error: 0.08426840630693411
 Bias^2: 0.07968918676726028
 Var: 0.004579219539673833
 0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411
-
-
-
Polynomial degree: 2
+Polynomial degree: 2
 Error: 0.10398646080125035
 Bias^2: 0.10077114273548986
 Var: 0.0032153180657605086
@@ -1010,26 +1082,26 @@ Error: 0.05227921801205707
 Bias^2: 0.048187277304303125
 Var: 0.004091940707753964
 0.05227921801205707 >= 0.048187277304303125 + 0.004091940707753964 = 0.05227921801205709
-Polynomial degree: 6
+
+
+
Polynomial degree: 6
 Error: 0.03781367141738898
 Bias^2: 0.03365768507152761
 Var: 0.004155986345861379
 0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899
-
-
-
Polynomial degree: 7
+Polynomial degree: 7
 Error: 0.027609773491022498
 Bias^2: 0.02299949826036597
 Var: 0.004610275230656537
 0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505
-Polynomial degree: 8
+
+
+
Polynomial degree: 8
 Error: 0.017355848195591973
 Bias^2: 0.010331721306655588
 Var: 0.007024126888936384
 0.017355848195591973 >= 0.010331721306655588 + 0.007024126888936384 = 0.017355848195591973
-
-
-
Polynomial degree: 9
+Polynomial degree: 9
 Error: 0.026605727637189085
 Bias^2: 0.010018312644140933
 Var: 0.016587414993048166
@@ -1044,21 +1116,24 @@ Error: 0.07160048164228314
 Bias^2: 0.01443680008897583
 Var: 0.0571636815533073
 0.07160048164228314 >= 0.01443680008897583 + 0.0571636815533073 = 0.07160048164228312
-Polynomial degree: 12
+
+
+
Polynomial degree: 12
 Error: 0.1154777721897675
 Bias^2: 0.01628578269590588
 Var: 0.09919198949386163
 0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751
+Polynomial degree:
-
Polynomial degree: 13
+
 13
 Error: 0.22842468702166951
 Bias^2: 0.01975416527163567
 Var: 0.20867052175003387
 0.22842468702166951 >= 0.01975416527163567 + 0.20867052175003387 = 0.22842468702166954
 

 

-_images/chapter3_58_5.png
+_images/chapter3_62_6.png

The bias-variance tradeoff summarizes the fundamental tension in @@ -1183,7 +1258,7 @@ set, the higher, the less likely the model generalizes correctly from the training data.

-_images/chapter3_60_1.png +_images/chapter3_64_1.png
@@ -1290,11 +1365,11 @@ Mean squared error on test data: 5.98822371 Degree of polynomial: 6 Mean squared error on training data: 3.66204648 Mean squared error on test data: 8.14812206 +Degree of polynomial: 7 +Mean squared error on training data: 0.47075725
-
Degree of polynomial:   7
-Mean squared error on training data: 0.47075725
-Mean squared error on test data: 2.00607783
+
Mean squared error on test data: 2.00607783
 Degree of polynomial:   8
 Mean squared error on training data: 0.04912436
 Mean squared error on test data: 0.21596432
@@ -1377,13 +1452,13 @@ Mean squared error on training data: 0.00060728
 Mean squared error on test data: 2988.64001211
 

 

-
<ipython-input-5-8dc29df57a8c>:73: RuntimeWarning: divide by zero encountered in log10
+
<ipython-input-7-8dc29df57a8c>:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-<ipython-input-5-8dc29df57a8c>:74: RuntimeWarning: divide by zero encountered in log10
+<ipython-input-7-8dc29df57a8c>:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 

 

-_images/chapter3_61_11.png
+_images/chapter3_65_11.png
@@ -1536,7 +1611,7 @@ cross-validation (LOOCV).

-_images/chapter3_67_0.png +_images/chapter3_71_0.png

More examples of the application of cross-validation follow here.

@@ -1613,11 +1688,11 @@ cross-validation (LOOCV).

-
<ipython-input-7-49b0ef2e51e2>:63: RuntimeWarning: divide by zero encountered in log10
+
<ipython-input-9-49b0ef2e51e2>:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 

 

-_images/chapter3_69_1.png
+_images/chapter3_73_1.png
@@ -1661,7 +1736,7 @@ cross-validation (LOOCV).

-_images/chapter3_70_0.png +_images/chapter3_74_0.png
@@ -1750,7 +1825,7 @@ which polynomial fits the data best.

-_images/chapter3_74_0.png +_images/chapter3_78_0.png
@@ -1907,7 +1982,7 @@ Python program using

---------------------------------------------------------------------------
 NameError                                 Traceback (most recent call last)
-<ipython-input-10-d985fb40c43d> in <module>
+<ipython-input-12-d985fb40c43d> in <module>
 ----> 1 scipy.misc.imread
 
 NameError: name 'scipy' is not defined
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 7f03a111a..8e0b42c94 100644
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+++ b/doc/LectureNotes/_build/html/searchindex.js
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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
index 41bef246b..f83b4276c 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
@@ -397,10 +397,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Runtime: 0.141725 sec\n",
+      "Runtime: 0.13859 sec\n",
       "Jackknife Statistics :\n",
       "original           bias      std. error\n",
-      " 100.148        100.138          0.1492\n"
+      " 99.9054        99.8954        0.149328\n"
      ]
     }
    ],
@@ -746,11 +746,118 @@
     "$\\tilde{p}(x)$.\n",
     "\n",
     "\n",
+    "The following code starts with a Gaussian distribution with mean value\n",
+    "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n",
+    "used in the bootstrap analysis. The bootstrap analysis returns a data\n",
+    "set after a given number of bootstrap operations (as many as we have\n",
+    "data points). This data set consists of estimated mean values for each\n",
+    "bootstrap operation. The histogram generated by the bootstrap method\n",
+    "shows that the distribution for these mean values is also a Gaussian,\n",
+    "centered around the mean value $\\mu=100$ but with standard deviation\n",
+    "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n",
+    "this case the same as the number of original data points). The value\n",
+    "of the standard deviation is what we expect from the central limit\n",
+    "theorem."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Bootstrap Statistics :\n",
+      "original           bias      std. error\n",
+      " 99.7442  15.0015        99.7465        0.148127\n"
+     ]
+    }
+   ],
+   "source": [
+    "%matplotlib inline\n",
     "\n",
+    "import numpy as np\n",
+    "from time import time\n",
+    "from scipy.stats import norm\n",
+    "import matplotlib.pyplot as plt\n",
     "\n",
+    "# Returns mean of bootstrap samples \n",
+    "# Bootstrap algorithm\n",
+    "def bootstrap(data, datapoints):\n",
+    "    t = np.zeros(datapoints)\n",
+    "    n = len(data)\n",
+    "    # non-parametric bootstrap         \n",
+    "    for i in range(datapoints):\n",
+    "        t[i] = np.mean(data[np.random.randint(0,n,n)])\n",
+    "    # analysis    \n",
+    "    print(\"Bootstrap Statistics :\")\n",
+    "    print(\"original           bias      std. error\")\n",
+    "    print(\"%8g %8g %14g %15g\" % (np.mean(data), np.std(data),np.mean(t),np.std(t)))\n",
+    "    return t\n",
     "\n",
+    "# We set the mean value to 100 and the standard deviation to 15\n",
+    "mu, sigma = 100, 15\n",
+    "datapoints = 10000\n",
+    "# We generate random numbers according to the normal distribution\n",
+    "x = mu + sigma*np.random.randn(datapoints)\n",
+    "# bootstrap returns the data sample                                    \n",
+    "t = bootstrap(x, datapoints)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n",
     "\n",
-    "\n",
+    "We plot then the histogram together with a best fit for the data set."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "
"
+      ]
+     },
+     "metadata": {
+      "filenames": {
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_47_0.png"
+      },
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# the histogram of the bootstrapped data (normalized data if density = True)\n",
+    "n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75)\n",
+    "# add a 'best fit' line  \n",
+    "y = norm.pdf(binsboot, np.mean(t), np.std(t))\n",
+    "lt = plt.plot(binsboot, y, 'b', linewidth=1)\n",
+    "plt.xlabel('x')\n",
+    "plt.ylabel('Probability')\n",
+    "plt.grid(True)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "## The bias-variance tradeoff\n",
     "\n",
     "\n",
@@ -875,7 +982,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -900,7 +1007,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_57_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_61_1.png"
       },
       "needs_background": "light"
      },
@@ -908,8 +1015,6 @@
     }
    ],
    "source": [
-    "%matplotlib inline\n",
-    "\n",
     "import matplotlib.pyplot as plt\n",
     "import numpy as np\n",
     "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n",
@@ -968,7 +1073,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 5,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -983,17 +1088,18 @@
       "Bias^2: 0.3123314713548606\n",
       "Var: 0.009164545680330616\n",
       "0.32149601703519126 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912\n",
-      "Polynomial degree: 1\n",
-      "Error: 0.08426840630693411\n",
-      "Bias^2: 0.07968918676726028\n",
-      "Var: 0.004579219539673833\n",
-      "0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411\n"
+      "Polynomial degree:"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      " 1\n",
+      "Error: 0.08426840630693411\n",
+      "Bias^2: 0.07968918676726028\n",
+      "Var: 0.004579219539673833\n",
+      "0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411\n",
       "Polynomial degree: 2\n",
       "Error: 0.10398646080125035\n",
       "Bias^2: 0.10077114273548986\n",
@@ -1013,34 +1119,34 @@
       "Error: 0.05227921801205707\n",
       "Bias^2: 0.048187277304303125\n",
       "Var: 0.004091940707753964\n",
-      "0.05227921801205707 >= 0.048187277304303125 + 0.004091940707753964 = 0.05227921801205709\n",
+      "0.05227921801205707 >= 0.048187277304303125 + 0.004091940707753964 = 0.05227921801205709\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "Polynomial degree: 6\n",
       "Error: 0.03781367141738898\n",
       "Bias^2: 0.03365768507152761\n",
       "Var: 0.004155986345861379\n",
-      "0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899\n",
       "Polynomial degree: 7\n",
       "Error: 0.027609773491022498\n",
       "Bias^2: 0.02299949826036597\n",
       "Var: 0.004610275230656537\n",
-      "0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505\n",
-      "Polynomial degree: 8\n",
-      "Error: 0.017355848195591973\n",
-      "Bias^2: 0.010331721306655588\n",
-      "Var: 0.007024126888936384\n",
-      "0.017355848195591973 >= 0.010331721306655588 + 0.007024126888936384 = 0.017355848195591973\n"
+      "0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Polynomial degree: 8\n",
+      "Error: 0.017355848195591973\n",
+      "Bias^2: 0.010331721306655588\n",
+      "Var: 0.007024126888936384\n",
+      "0.017355848195591973 >= 0.010331721306655588 + 0.007024126888936384 = 0.017355848195591973\n",
       "Polynomial degree: 9\n",
       "Error: 0.026605727637189085\n",
       "Bias^2: 0.010018312644140933\n",
@@ -1055,19 +1161,26 @@
       "Error: 0.07160048164228314\n",
       "Bias^2: 0.01443680008897583\n",
       "Var: 0.0571636815533073\n",
-      "0.07160048164228314 >= 0.01443680008897583 + 0.0571636815533073 = 0.07160048164228312\n",
-      "Polynomial degree: 12\n",
-      "Error: 0.1154777721897675\n",
-      "Bias^2: 0.01628578269590588\n",
-      "Var: 0.09919198949386163\n",
-      "0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751\n"
+      "0.07160048164228314 >= 0.01443680008897583 + 0.0571636815533073 = 0.07160048164228312\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Polynomial degree: 13\n",
+      "Polynomial degree: 12\n",
+      "Error: 0.1154777721897675\n",
+      "Bias^2: 0.01628578269590588\n",
+      "Var: 0.09919198949386163\n",
+      "0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751\n",
+      "Polynomial degree:"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      " 13\n",
       "Error: 0.22842468702166951\n",
       "Bias^2: 0.01975416527163567\n",
       "Var: 0.20867052175003387\n",
@@ -1083,7 +1196,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_58_5.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_62_6.png"
       },
       "needs_background": "light"
      },
@@ -1178,7 +1291,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 6,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1220,7 +1333,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_60_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_64_1.png"
       }
      },
      "output_type": "display_data"
@@ -1302,7 +1415,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 7,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1335,15 +1448,16 @@
       "Mean squared error on test data: 5.98822371\n",
       "Degree of polynomial:   6\n",
       "Mean squared error on training data: 3.66204648\n",
-      "Mean squared error on test data: 8.14812206\n"
+      "Mean squared error on test data: 8.14812206\n",
+      "Degree of polynomial:   7\n",
+      "Mean squared error on training data: 0.47075725"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Degree of polynomial:   7\n",
-      "Mean squared error on training data: 0.47075725\n",
+      "\n",
       "Mean squared error on test data: 2.00607783\n",
       "Degree of polynomial:   8\n",
       "Mean squared error on training data: 0.04912436\n",
@@ -1459,9 +1573,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      ":73: RuntimeWarning: divide by zero encountered in log10\n",
+      ":73: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n",
-      ":74: RuntimeWarning: divide by zero encountered in log10\n",
+      ":74: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(testerror), label='Test Error')\n"
      ]
     },
@@ -1474,7 +1588,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_61_11.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_65_11.png"
       },
       "needs_background": "light"
      },
@@ -1654,7 +1768,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 8,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1669,7 +1783,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_67_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_71_0.png"
       },
       "needs_background": "light"
      },
@@ -1777,7 +1891,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 9,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1787,7 +1901,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      ":63: RuntimeWarning: divide by zero encountered in log10\n",
+      ":63: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n"
      ]
     },
@@ -1800,7 +1914,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_69_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_73_1.png"
       },
       "needs_background": "light"
      },
@@ -1879,7 +1993,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 10,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1894,7 +2008,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_70_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_74_0.png"
       },
       "needs_background": "light"
      },
@@ -2007,7 +2121,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 11,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2022,7 +2136,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_74_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_78_0.png"
       },
       "needs_background": "light"
      },
@@ -2314,7 +2428,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 12,
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2327,7 +2441,7 @@
      "traceback": [
       "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
       "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
-      "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mscipy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimread\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mscipy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimread\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
       "\u001b[0;31mNameError\u001b[0m: name 'scipy' is not defined"
      ]
     }
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.py b/doc/LectureNotes/_build/jupyter_execute/chapter3.py
index f964d3f28..c827ef0fe 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.py
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.py
@@ -504,10 +504,61 @@ finite $m$, it is not always possible to find a closed form /analytic expression
 $\tilde{p}(x)$.
 
 
+The following code starts with a Gaussian distribution with mean value
+$\mu =100$ and variance $\sigma=15$. We use this to generate the data
+used in the bootstrap analysis. The bootstrap analysis returns a data
+set after a given number of bootstrap operations (as many as we have
+data points). This data set consists of estimated mean values for each
+bootstrap operation. The histogram generated by the bootstrap method
+shows that the distribution for these mean values is also a Gaussian,
+centered around the mean value $\mu=100$ but with standard deviation
+$\sigma/\sqrt{n}$, where $n$ is the number of bootstrap samples (in
+this case the same as the number of original data points). The value
+of the standard deviation is what we expect from the central limit
+theorem.
 
+%matplotlib inline
 
+import numpy as np
+from time import time
+from scipy.stats import norm
+import matplotlib.pyplot as plt
 
+# Returns mean of bootstrap samples 
+# Bootstrap algorithm
+def bootstrap(data, datapoints):
+    t = np.zeros(datapoints)
+    n = len(data)
+    # non-parametric bootstrap         
+    for i in range(datapoints):
+        t[i] = np.mean(data[np.random.randint(0,n,n)])
+    # analysis    
+    print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (np.mean(data), np.std(data),np.mean(t),np.std(t)))
+    return t
 
+# We set the mean value to 100 and the standard deviation to 15
+mu, sigma = 100, 15
+datapoints = 10000
+# We generate random numbers according to the normal distribution
+x = mu + sigma*np.random.randn(datapoints)
+# bootstrap returns the data sample                                    
+t = bootstrap(x, datapoints)
+
+We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
+
+We plot then the histogram together with a best fit for the data set.
+
+# the histogram of the bootstrapped data (normalized data if density = True)
+n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75)
+# add a 'best fit' line  
+y = norm.pdf(binsboot, np.mean(t), np.std(t))
+lt = plt.plot(binsboot, y, 'b', linewidth=1)
+plt.xlabel('x')
+plt.ylabel('Probability')
+plt.grid(True)
+plt.show()
 
 ## The bias-variance tradeoff
 
@@ -570,8 +621,6 @@ $$
 
 that is the rewriting in terms of the so-called bias, the variance of the model $\boldsymbol{\tilde{y}}$ and the variance of $\boldsymbol{\epsilon}$.
 
-%matplotlib inline
-
 import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
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diff --git a/doc/LectureNotes/chapter3.ipynb b/doc/LectureNotes/chapter3.ipynb
index 4a7aa4470..413c54051 100644
--- a/doc/LectureNotes/chapter3.ipynb
+++ b/doc/LectureNotes/chapter3.ipynb
@@ -735,11 +735,92 @@
     "$\\tilde{p}(x)$.\n",
     "\n",
     "\n",
+    "The following code starts with a Gaussian distribution with mean value\n",
+    "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n",
+    "used in the bootstrap analysis. The bootstrap analysis returns a data\n",
+    "set after a given number of bootstrap operations (as many as we have\n",
+    "data points). This data set consists of estimated mean values for each\n",
+    "bootstrap operation. The histogram generated by the bootstrap method\n",
+    "shows that the distribution for these mean values is also a Gaussian,\n",
+    "centered around the mean value $\\mu=100$ but with standard deviation\n",
+    "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n",
+    "this case the same as the number of original data points). The value\n",
+    "of the standard deviation is what we expect from the central limit\n",
+    "theorem."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
     "\n",
+    "import numpy as np\n",
+    "from time import time\n",
+    "from scipy.stats import norm\n",
+    "import matplotlib.pyplot as plt\n",
     "\n",
+    "# Returns mean of bootstrap samples \n",
+    "# Bootstrap algorithm\n",
+    "def bootstrap(data, datapoints):\n",
+    "    t = np.zeros(datapoints)\n",
+    "    n = len(data)\n",
+    "    # non-parametric bootstrap         \n",
+    "    for i in range(datapoints):\n",
+    "        t[i] = np.mean(data[np.random.randint(0,n,n)])\n",
+    "    # analysis    \n",
+    "    print(\"Bootstrap Statistics :\")\n",
+    "    print(\"original           bias      std. error\")\n",
+    "    print(\"%8g %8g %14g %15g\" % (np.mean(data), np.std(data),np.mean(t),np.std(t)))\n",
+    "    return t\n",
     "\n",
+    "# We set the mean value to 100 and the standard deviation to 15\n",
+    "mu, sigma = 100, 15\n",
+    "datapoints = 10000\n",
+    "# We generate random numbers according to the normal distribution\n",
+    "x = mu + sigma*np.random.randn(datapoints)\n",
+    "# bootstrap returns the data sample                                    \n",
+    "t = bootstrap(x, datapoints)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n",
     "\n",
-    "\n",
+    "We plot then the histogram together with a best fit for the data set."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# the histogram of the bootstrapped data (normalized data if density = True)\n",
+    "n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75)\n",
+    "# add a 'best fit' line  \n",
+    "y = norm.pdf(binsboot, np.mean(t), np.std(t))\n",
+    "lt = plt.plot(binsboot, y, 'b', linewidth=1)\n",
+    "plt.xlabel('x')\n",
+    "plt.ylabel('Probability')\n",
+    "plt.grid(True)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "## The bias-variance tradeoff\n",
     "\n",
     "\n",
@@ -871,8 +952,6 @@
    },
    "outputs": [],
    "source": [
-    "%matplotlib inline\n",
-    "\n",
     "import matplotlib.pyplot as plt\n",
     "import numpy as np\n",
     "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n",