diff --git a/doc/BookChapters/chapter1.dlog b/doc/BookChapters/chapter1.dlog
index ce4fec57d..f01f5e236 100644
--- a/doc/BookChapters/chapter1.dlog
+++ b/doc/BookChapters/chapter1.dlog
@@ -97,3 +97,13 @@ found info about 5 exercises
*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
output in chapter1.ipynb
+Translating doconce text in chapter1.do.txt to ipynb
+*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
+found info about 5 exercises
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
+
+*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
+output in chapter1.ipynb
diff --git a/doc/BookChapters/chapter1.do.txt b/doc/BookChapters/chapter1.do.txt
index 31f5fc14c..09aac09ea 100644
--- a/doc/BookChapters/chapter1.do.txt
+++ b/doc/BookChapters/chapter1.do.txt
@@ -262,10 +262,10 @@ y = 2x+N(0,1),
where $N(0,1)$ represents random numbers generated by the normal
distribution. From _Scikit-Learn_ we import then the
_LinearRegression_ functionality and make a prediction $\tilde{y} =
-\alpha + \beta x$ using the function _fit(x,y)_. We call the set of
+\alpha + \theta x$ using the function _fit(x,y)_. We call the set of
data $(\bm{x},\bm{y})$ for our training data. The Python package
_scikit-learn_ has also a functionality which extracts the above
-fitting parameters $\alpha$ and $\beta$ (see below). Later we will
+fitting parameters $\alpha$ and $\theta$ (see below). Later we will
distinguish between training data and test data.
For plotting we use the Python package
@@ -351,7 +351,7 @@ dimensionless.
Minimizing the cost function is a central aspect of
our discussions to come. Finding its minima as function of the model
-parameters ($\alpha$ and $\beta$ in our case) will be a recurring
+parameters ($\alpha$ and $\theta$ in our case) will be a recurring
theme in these series of lectures. Essentially all machine learning
algorithms we will discuss center around the minimization of the
chosen cost function. This depends in turn on our specific
@@ -407,7 +407,7 @@ different training data sets and study (graphically) the value of the
relative error.
As mentioned above, _Scikit-Learn_ has an impressive functionality.
-We can for example extract the values of $\alpha$ and $\beta$ and
+We can for example extract the values of $\alpha$ and $\theta$ and
their error estimates, or the variance and standard deviation and many
other properties from the statistical data analysis.
@@ -425,7 +425,7 @@ linreg = LinearRegression()
linreg.fit(x,y)
ypredict = linreg.predict(x)
print('The intercept alpha: \n', linreg.intercept_)
-print('Coefficient beta : \n', linreg.coef_)
+print('Coefficient theta : \n', linreg.coef_)
# The mean squared error
print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
# Explained variance score: 1 is perfect prediction
@@ -443,8 +443,8 @@ plt.title(r'Linear Regression fit ')
plt.show()
!ec
-The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields
-$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
+The function _coef_ gives us the parameter $\theta$ of our fit while _intercept_ yields
+$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\theta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
!bt
\[ MSE(\bm{y},\bm{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
@@ -1815,160 +1815,6 @@ print(MSE(y_test,ypredict))
!ec
-===== The Boston housing data example =====
-
-The Boston housing
-data set was originally a part of UCI Machine Learning Repository
-and has been removed now. The data set is now included in _Scikit-Learn_'s
-library. There are 506 samples and 13 feature (predictor) variables
-in this data set. The objective is to predict the value of prices of
-the house using the features (predictors) listed here.
-
-The features/predictors are
- o CRIM: Per capita crime rate by town
- o ZN: Proportion of residential land zoned for lots over 25000 square feet
- o INDUS: Proportion of non-retail business acres per town
- o CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
- o NOX: Nitric oxide concentration (parts per 10 million)
- o RM: Average number of rooms per dwelling
- o AGE: Proportion of owner-occupied units built prior to 1940
- o DIS: Weighted distances to five Boston employment centers
- o RAD: Index of accessibility to radial highways
- o TAX: Full-value property tax rate per USD10000
- o B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town
- o LSTAT: Percentage of lower status of the population
- o MEDV: Median value of owner-occupied homes in USD 1000s
-
-!split
-===== Housing data, the code =====
-We start by importing the libraries
-!bc pycod
-import numpy as np
-import matplotlib.pyplot as plt
-
-import pandas as pd
-import seaborn as sns
-!ec
-and load the Boston Housing DataSet from _Scikit-Learn_
-
-
-!bc pycod
-from sklearn.datasets import load_boston
-
-boston_dataset = load_boston()
-
-# boston_dataset is a dictionary
-# let's check what it contains
-boston_dataset.keys()
-!ec
-Then we invoke Pandas
-!bc pycod
-boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)
-boston.head()
-boston['MEDV'] = boston_dataset.target
-!ec
-and preprocess the data
-!bc pycod
-# check for missing values in all the columns
-boston.isnull().sum()
-!ec
-We can then visualize the data
-!bc pycod
-# set the size of the figure
-sns.set(rc={'figure.figsize':(11.7,8.27)})
-
-# plot a histogram showing the distribution of the target values
-sns.distplot(boston['MEDV'], bins=30)
-plt.show()
-!ec
-
-It is now useful to look at the correlation matrix
-!bc pycod
-# compute the pair wise correlation for all columns
-correlation_matrix = boston.corr().round(2)
-# use the heatmap function from seaborn to plot the correlation matrix
-# annot = True to print the values inside the square
-sns.heatmap(data=correlation_matrix, annot=True)
-!ec
-From the above coorelation plot we can see that _MEDV_ is strongly correlated to _LSTAT_ and _RM_. We see also that _RAD_ and _TAX_ are stronly correlated, but we don't include this in our features together to avoid multi-colinearity
-
-!bc pycod
-plt.figure(figsize=(20, 5))
-
-features = ['LSTAT', 'RM']
-target = boston['MEDV']
-
-for i, col in enumerate(features):
- plt.subplot(1, len(features) , i+1)
- x = boston[col]
- y = target
- plt.scatter(x, y, marker='o')
- plt.title(col)
- plt.xlabel(col)
- plt.ylabel('MEDV')
-!ec
-Now we start training our model
-!bc pycod
-X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])
-Y = boston['MEDV']
-!ec
-We split the data into training and test sets
-
-!bc pycod
-from sklearn.model_selection import train_test_split
-
-# splits the training and test data set in 80% : 20%
-# assign random_state to any value.This ensures consistency.
-X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)
-print(X_train.shape)
-print(X_test.shape)
-print(Y_train.shape)
-print(Y_test.shape)
-!ec
-Then we use the linear regression functionality from _Scikit-Learn_
-!bc pycod
-from sklearn.linear_model import LinearRegression
-from sklearn.metrics import mean_squared_error, r2_score
-
-lin_model = LinearRegression()
-lin_model.fit(X_train, Y_train)
-
-# model evaluation for training set
-
-y_train_predict = lin_model.predict(X_train)
-rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))
-r2 = r2_score(Y_train, y_train_predict)
-
-print("The model performance for training set")
-print("--------------------------------------")
-print('RMSE is {}'.format(rmse))
-print('R2 score is {}'.format(r2))
-print("\n")
-
-# model evaluation for testing set
-
-y_test_predict = lin_model.predict(X_test)
-# root mean square error of the model
-rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))
-
-# r-squared score of the model
-r2 = r2_score(Y_test, y_test_predict)
-
-print("The model performance for testing set")
-print("--------------------------------------")
-print('RMSE is {}'.format(rmse))
-print('R2 score is {}'.format(r2))
-!ec
-
-!bc pycod
-# plotting the y_test vs y_pred
-# ideally should have been a straight line
-plt.scatter(Y_test, y_test_predict)
-plt.show()
-!ec
-
-
-
===== Reducing the number of degrees of freedom, overarching view =====
diff --git a/doc/LectureNotes/_build/.doctrees/chapter1.doctree b/doc/LectureNotes/_build/.doctrees/chapter1.doctree
index 1fb9c6c33..f139420c4 100644
Binary files a/doc/LectureNotes/_build/.doctrees/chapter1.doctree and b/doc/LectureNotes/_build/.doctrees/chapter1.doctree differ
diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index 3d0ad2793..7f75321f5 100644
Binary files a/doc/LectureNotes/_build/.doctrees/environment.pickle and b/doc/LectureNotes/_build/.doctrees/environment.pickle differ
diff --git a/doc/LectureNotes/_build/html/_sources/chapter1.ipynb b/doc/LectureNotes/_build/html/_sources/chapter1.ipynb
index 231068020..3a8e25dcf 100644
--- a/doc/LectureNotes/_build/html/_sources/chapter1.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/chapter1.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
- "id": "a453b968",
+ "id": "7d0d70d8",
"metadata": {
"editable": true
},
@@ -13,7 +13,7 @@
},
{
"cell_type": "markdown",
- "id": "499b2ddb",
+ "id": "3399a55c",
"metadata": {
"editable": true
},
@@ -23,7 +23,7 @@
},
{
"cell_type": "markdown",
- "id": "c84cce7e",
+ "id": "e7513472",
"metadata": {
"editable": true
},
@@ -65,7 +65,7 @@
},
{
"cell_type": "markdown",
- "id": "8419208e",
+ "id": "17b7ba24",
"metadata": {
"editable": true
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@@ -167,7 +167,7 @@
},
{
"cell_type": "markdown",
- "id": "ceb7a805",
+ "id": "23adffca",
"metadata": {
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@@ -202,7 +202,7 @@
},
{
"cell_type": "markdown",
- "id": "d6c1062f",
+ "id": "1263a25e",
"metadata": {
"editable": true
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@@ -253,7 +253,7 @@
},
{
"cell_type": "markdown",
- "id": "50c1b706",
+ "id": "e2ab4f0a",
"metadata": {
"editable": true
},
@@ -286,7 +286,7 @@
},
{
"cell_type": "markdown",
- "id": "12e3ac84",
+ "id": "23173385",
"metadata": {
"editable": true
},
@@ -298,7 +298,7 @@
},
{
"cell_type": "markdown",
- "id": "10b4c333",
+ "id": "57b14c43",
"metadata": {
"editable": true
},
@@ -306,10 +306,10 @@
"where $N(0,1)$ represents random numbers generated by the normal\n",
"distribution. From **Scikit-Learn** we import then the\n",
"**LinearRegression** functionality and make a prediction $\\tilde{y} =\n",
- "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n",
+ "\\alpha + \\theta x$ using the function **fit(x,y)**. We call the set of\n",
"data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n",
"**scikit-learn** has also a functionality which extracts the above\n",
- "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n",
+ "fitting parameters $\\alpha$ and $\\theta$ (see below). Later we will\n",
"distinguish between training data and test data.\n",
"\n",
"For plotting we use the Python package\n",
@@ -335,7 +335,7 @@
{
"cell_type": "code",
"execution_count": 1,
- "id": "3e216ef0",
+ "id": "f7d66caa",
"metadata": {
"collapsed": false,
"editable": true
@@ -368,7 +368,7 @@
},
{
"cell_type": "markdown",
- "id": "af765a12",
+ "id": "504848dc",
"metadata": {
"editable": true
},
@@ -385,7 +385,7 @@
},
{
"cell_type": "markdown",
- "id": "2f6fd730",
+ "id": "bff268fb",
"metadata": {
"editable": true
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@@ -397,7 +397,7 @@
},
{
"cell_type": "markdown",
- "id": "1770d85d",
+ "id": "53357604",
"metadata": {
"editable": true
},
@@ -418,7 +418,7 @@
},
{
"cell_type": "markdown",
- "id": "684b72a0",
+ "id": "4e5b1dad",
"metadata": {
"editable": true
},
@@ -431,7 +431,7 @@
},
{
"cell_type": "markdown",
- "id": "e9fa8fd5",
+ "id": "b8e0f4c7",
"metadata": {
"editable": true
},
@@ -443,7 +443,7 @@
"\n",
"Minimizing the cost function is a central aspect of\n",
"our discussions to come. Finding its minima as function of the model\n",
- "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n",
+ "parameters ($\\alpha$ and $\\theta$ in our case) will be a recurring\n",
"theme in these series of lectures. Essentially all machine learning\n",
"algorithms we will discuss center around the minimization of the\n",
"chosen cost function. This depends in turn on our specific\n",
@@ -462,7 +462,7 @@
},
{
"cell_type": "markdown",
- "id": "15a4642f",
+ "id": "c4e4b512",
"metadata": {
"editable": true
},
@@ -474,7 +474,7 @@
},
{
"cell_type": "markdown",
- "id": "589ec9cb",
+ "id": "71393ea9",
"metadata": {
"editable": true
},
@@ -492,7 +492,7 @@
{
"cell_type": "code",
"execution_count": 2,
- "id": "cf21ae2b",
+ "id": "a6c77c37",
"metadata": {
"collapsed": false,
"editable": true
@@ -520,7 +520,7 @@
},
{
"cell_type": "markdown",
- "id": "41c55cbb",
+ "id": "bc21adc2",
"metadata": {
"editable": true
},
@@ -531,7 +531,7 @@
"relative error.\n",
"\n",
"As mentioned above, **Scikit-Learn** has an impressive functionality.\n",
- "We can for example extract the values of $\\alpha$ and $\\beta$ and\n",
+ "We can for example extract the values of $\\alpha$ and $\\theta$ and\n",
"their error estimates, or the variance and standard deviation and many\n",
"other properties from the statistical data analysis. \n",
"\n",
@@ -542,7 +542,7 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "d62c9412",
+ "id": "febc49fc",
"metadata": {
"collapsed": false,
"editable": true
@@ -560,7 +560,7 @@
"linreg.fit(x,y)\n",
"ypredict = linreg.predict(x)\n",
"print('The intercept alpha: \\n', linreg.intercept_)\n",
- "print('Coefficient beta : \\n', linreg.coef_)\n",
+ "print('Coefficient theta : \\n', linreg.coef_)\n",
"# The mean squared error \n",
"print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n",
"# Explained variance score: 1 is perfect prediction \n",
@@ -580,18 +580,18 @@
},
{
"cell_type": "markdown",
- "id": "ae0c6c2a",
+ "id": "3f1c6408",
"metadata": {
"editable": true
},
"source": [
- "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n",
- "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
+ "The function **coef** gives us the parameter $\\theta$ of our fit while **intercept** yields \n",
+ "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\theta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
]
},
{
"cell_type": "markdown",
- "id": "64cf13de",
+ "id": "b4e26d20",
"metadata": {
"editable": true
},
@@ -604,7 +604,7 @@
},
{
"cell_type": "markdown",
- "id": "18439dae",
+ "id": "8137fd70",
"metadata": {
"editable": true
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@@ -625,7 +625,7 @@
},
{
"cell_type": "markdown",
- "id": "3e9fb291",
+ "id": "3f40e79c",
"metadata": {
"editable": true
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@@ -637,7 +637,7 @@
},
{
"cell_type": "markdown",
- "id": "46d8744e",
+ "id": "6a8eb368",
"metadata": {
"editable": true
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@@ -647,7 +647,7 @@
},
{
"cell_type": "markdown",
- "id": "38a29b65",
+ "id": "71d8af23",
"metadata": {
"editable": true
},
@@ -659,7 +659,7 @@
},
{
"cell_type": "markdown",
- "id": "438e73b8",
+ "id": "aee25556",
"metadata": {
"editable": true
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@@ -671,7 +671,7 @@
},
{
"cell_type": "markdown",
- "id": "1403cc6a",
+ "id": "1bc1d5e2",
"metadata": {
"editable": true
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@@ -683,7 +683,7 @@
},
{
"cell_type": "markdown",
- "id": "9fbd1c1b",
+ "id": "dfb80b0d",
"metadata": {
"editable": true
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@@ -694,7 +694,7 @@
},
{
"cell_type": "markdown",
- "id": "d25da1d3",
+ "id": "1ee8b15b",
"metadata": {
"editable": true
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@@ -706,7 +706,7 @@
},
{
"cell_type": "markdown",
- "id": "c43c7452",
+ "id": "9ec15fef",
"metadata": {
"editable": true
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@@ -728,7 +728,7 @@
},
{
"cell_type": "markdown",
- "id": "ad0d5004",
+ "id": "8e386295",
"metadata": {
"editable": true
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@@ -740,7 +740,7 @@
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{
"cell_type": "markdown",
- "id": "eb9e2efb",
+ "id": "d8d8ad23",
"metadata": {
"editable": true
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@@ -755,7 +755,7 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "963f4f84",
+ "id": "0c9a0500",
"metadata": {
"collapsed": false,
"editable": true
@@ -796,7 +796,7 @@
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{
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+ "id": "c7786c3b",
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@@ -811,7 +811,7 @@
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@@ -823,7 +823,7 @@
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@@ -833,7 +833,7 @@
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@@ -845,7 +845,7 @@
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@@ -928,7 +928,7 @@
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@@ -991,7 +991,7 @@
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@@ -1008,7 +1008,7 @@
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@@ -1052,7 +1052,7 @@
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@@ -1069,7 +1069,7 @@
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+ "id": "8ac3d80a",
"metadata": {
"editable": true
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@@ -2736,7 +2736,7 @@
},
{
"cell_type": "markdown",
- "id": "4e631fe5",
+ "id": "1f0efa4f",
"metadata": {
"editable": true
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@@ -2748,7 +2748,7 @@
},
{
"cell_type": "markdown",
- "id": "a1c651fa",
+ "id": "37a13bcf",
"metadata": {
"editable": true
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@@ -2758,7 +2758,7 @@
},
{
"cell_type": "markdown",
- "id": "0ba91c05",
+ "id": "c2c1621c",
"metadata": {
"editable": true
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@@ -2770,7 +2770,7 @@
},
{
"cell_type": "markdown",
- "id": "9c7d45d8",
+ "id": "0c4b3684",
"metadata": {
"editable": true
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@@ -2780,7 +2780,7 @@
},
{
"cell_type": "markdown",
- "id": "1148dda1",
+ "id": "91e11f8d",
"metadata": {
"editable": true
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@@ -2792,7 +2792,7 @@
},
{
"cell_type": "markdown",
- "id": "899ad725",
+ "id": "3d1bf994",
"metadata": {
"editable": true
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@@ -2802,7 +2802,7 @@
},
{
"cell_type": "markdown",
- "id": "442cefde",
+ "id": "f98818a7",
"metadata": {
"editable": true
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@@ -2814,7 +2814,7 @@
},
{
"cell_type": "markdown",
- "id": "303d558b",
+ "id": "2d2b2f06",
"metadata": {
"editable": true
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@@ -2824,7 +2824,7 @@
},
{
"cell_type": "markdown",
- "id": "c8361a08",
+ "id": "f452db13",
"metadata": {
"editable": true
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@@ -2836,7 +2836,7 @@
},
{
"cell_type": "markdown",
- "id": "fad61a6c",
+ "id": "10f930c3",
"metadata": {
"editable": true
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@@ -2846,7 +2846,7 @@
},
{
"cell_type": "markdown",
- "id": "29731faf",
+ "id": "9e5bd8a6",
"metadata": {
"editable": true
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@@ -2858,7 +2858,7 @@
},
{
"cell_type": "markdown",
- "id": "ecb0789a",
+ "id": "fe7f99e3",
"metadata": {
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@@ -2868,7 +2868,7 @@
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{
"cell_type": "markdown",
- "id": "bf04c0eb",
+ "id": "9cb21aae",
"metadata": {
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@@ -2880,7 +2880,7 @@
},
{
"cell_type": "markdown",
- "id": "7fba1a68",
+ "id": "e4953c1b",
"metadata": {
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@@ -2890,7 +2890,7 @@
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{
"cell_type": "markdown",
- "id": "cb8528d5",
+ "id": "d3f4ca47",
"metadata": {
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@@ -2902,7 +2902,7 @@
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{
"cell_type": "markdown",
- "id": "2771f01a",
+ "id": "2e91b201",
"metadata": {
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@@ -2913,7 +2913,7 @@
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{
"cell_type": "markdown",
- "id": "dcf74fcf",
+ "id": "17412942",
"metadata": {
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@@ -2925,7 +2925,7 @@
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{
"cell_type": "markdown",
- "id": "a7e95721",
+ "id": "cbdf2f6f",
"metadata": {
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@@ -2937,7 +2937,7 @@
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{
"cell_type": "markdown",
- "id": "a636424d",
+ "id": "89e592b9",
"metadata": {
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@@ -2949,7 +2949,7 @@
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{
"cell_type": "markdown",
- "id": "294e5692",
+ "id": "0cd113ed",
"metadata": {
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@@ -2961,7 +2961,7 @@
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{
"cell_type": "markdown",
- "id": "923ebdc5",
+ "id": "8767c120",
"metadata": {
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@@ -2973,7 +2973,7 @@
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{
"cell_type": "markdown",
- "id": "d605b9e0",
+ "id": "489b6e55",
"metadata": {
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@@ -2983,7 +2983,7 @@
},
{
"cell_type": "markdown",
- "id": "d9e8c477",
+ "id": "2d983c3c",
"metadata": {
"editable": true
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@@ -2995,7 +2995,7 @@
},
{
"cell_type": "markdown",
- "id": "2831c92f",
+ "id": "bea2c0ec",
"metadata": {
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@@ -3007,7 +3007,7 @@
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{
"cell_type": "markdown",
- "id": "36edba71",
+ "id": "db78d47f",
"metadata": {
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@@ -3021,7 +3021,7 @@
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{
"cell_type": "markdown",
- "id": "f209ca1f",
+ "id": "c75b9d59",
"metadata": {
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@@ -3047,7 +3047,7 @@
{
"cell_type": "code",
"execution_count": 22,
- "id": "5d230f93",
+ "id": "d06bf2cc",
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"editable": true
@@ -3129,7 +3129,7 @@
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{
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- "id": "b3d6f5a2",
+ "id": "5de4c8d1",
"metadata": {
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@@ -3140,7 +3140,7 @@
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{
"cell_type": "markdown",
- "id": "4b61d1b0",
+ "id": "6e491525",
"metadata": {
"editable": true
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@@ -3168,7 +3168,7 @@
{
"cell_type": "code",
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- "id": "8c25e039",
+ "id": "ff20a1d4",
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@@ -3217,7 +3217,7 @@
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{
"cell_type": "markdown",
- "id": "3c09d70c",
+ "id": "ccadceb8",
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@@ -3228,7 +3228,7 @@
{
"cell_type": "code",
"execution_count": 24,
- "id": "3e9b7ad8",
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@@ -3253,7 +3253,7 @@
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{
"cell_type": "markdown",
- "id": "18804e9f",
+ "id": "90d9e62f",
"metadata": {
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@@ -3270,7 +3270,7 @@
{
"cell_type": "code",
"execution_count": 25,
- "id": "90f453cd",
+ "id": "c8f97789",
"metadata": {
"collapsed": false,
"editable": true
@@ -3345,371 +3345,7 @@
},
{
"cell_type": "markdown",
- "id": "3d672d7a",
- "metadata": {
- "editable": true
- },
- "source": [
- "## The Boston housing data example\n",
- "\n",
- "The Boston housing \n",
- "data set was originally a part of UCI Machine Learning Repository\n",
- "and has been removed now. The data set is now included in **Scikit-Learn**'s \n",
- "library. There are 506 samples and 13 feature (predictor) variables\n",
- "in this data set. The objective is to predict the value of prices of\n",
- "the house using the features (predictors) listed here.\n",
- "\n",
- "The features/predictors are\n",
- "1. CRIM: Per capita crime rate by town\n",
- "\n",
- "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n",
- "\n",
- "3. INDUS: Proportion of non-retail business acres per town\n",
- "\n",
- "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
- "\n",
- "5. NOX: Nitric oxide concentration (parts per 10 million)\n",
- "\n",
- "6. RM: Average number of rooms per dwelling\n",
- "\n",
- "7. AGE: Proportion of owner-occupied units built prior to 1940\n",
- "\n",
- "8. DIS: Weighted distances to five Boston employment centers\n",
- "\n",
- "9. RAD: Index of accessibility to radial highways\n",
- "\n",
- "10. TAX: Full-value property tax rate per USD10000\n",
- "\n",
- "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n",
- "\n",
- "12. LSTAT: Percentage of lower status of the population\n",
- "\n",
- "13. MEDV: Median value of owner-occupied homes in USD 1000s"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "e2426f64",
- "metadata": {
- "editable": true
- },
- "source": [
- "## Housing data, the code\n",
- "We start by importing the libraries"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "id": "519e0c09",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "import numpy as np\n",
- "import matplotlib.pyplot as plt \n",
- "\n",
- "import pandas as pd \n",
- "import seaborn as sns"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "7b6fd188",
- "metadata": {
- "editable": true
- },
- "source": [
- "and load the Boston Housing DataSet from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 27,
- "id": "3665e1b0",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.datasets import load_boston\n",
- "\n",
- "boston_dataset = load_boston()\n",
- "\n",
- "# boston_dataset is a dictionary\n",
- "# let's check what it contains\n",
- "boston_dataset.keys()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "96f2fe7b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we invoke Pandas"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "id": "df2c84ca",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n",
- "boston.head()\n",
- "boston['MEDV'] = boston_dataset.target"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0c4588d5",
- "metadata": {
- "editable": true
- },
- "source": [
- "and preprocess the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 29,
- "id": "209438db",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# check for missing values in all the columns\n",
- "boston.isnull().sum()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "1b67c46a",
- "metadata": {
- "editable": true
- },
- "source": [
- "We can then visualize the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 30,
- "id": "236343bf",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# set the size of the figure\n",
- "sns.set(rc={'figure.figsize':(11.7,8.27)})\n",
- "\n",
- "# plot a histogram showing the distribution of the target values\n",
- "sns.distplot(boston['MEDV'], bins=30)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c8726877",
- "metadata": {
- "editable": true
- },
- "source": [
- "It is now useful to look at the correlation matrix"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 31,
- "id": "aaa47b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# compute the pair wise correlation for all columns \n",
- "correlation_matrix = boston.corr().round(2)\n",
- "# use the heatmap function from seaborn to plot the correlation matrix\n",
- "# annot = True to print the values inside the square\n",
- "sns.heatmap(data=correlation_matrix, annot=True)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0283f3f1",
- "metadata": {
- "editable": true
- },
- "source": [
- "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 32,
- "id": "c0823ed1",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "plt.figure(figsize=(20, 5))\n",
- "\n",
- "features = ['LSTAT', 'RM']\n",
- "target = boston['MEDV']\n",
- "\n",
- "for i, col in enumerate(features):\n",
- " plt.subplot(1, len(features) , i+1)\n",
- " x = boston[col]\n",
- " y = target\n",
- " plt.scatter(x, y, marker='o')\n",
- " plt.title(col)\n",
- " plt.xlabel(col)\n",
- " plt.ylabel('MEDV')"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "dc2cf448",
- "metadata": {
- "editable": true
- },
- "source": [
- "Now we start training our model"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 33,
- "id": "ee945b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n",
- "Y = boston['MEDV']"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "288a1417",
- "metadata": {
- "editable": true
- },
- "source": [
- "We split the data into training and test sets"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 34,
- "id": "0b640ff9",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.model_selection import train_test_split\n",
- "\n",
- "# splits the training and test data set in 80% : 20%\n",
- "# assign random_state to any value.This ensures consistency.\n",
- "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n",
- "print(X_train.shape)\n",
- "print(X_test.shape)\n",
- "print(Y_train.shape)\n",
- "print(Y_test.shape)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "32cfad5b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we use the linear regression functionality from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "id": "792df674",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.linear_model import LinearRegression\n",
- "from sklearn.metrics import mean_squared_error, r2_score\n",
- "\n",
- "lin_model = LinearRegression()\n",
- "lin_model.fit(X_train, Y_train)\n",
- "\n",
- "# model evaluation for training set\n",
- "\n",
- "y_train_predict = lin_model.predict(X_train)\n",
- "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n",
- "r2 = r2_score(Y_train, y_train_predict)\n",
- "\n",
- "print(\"The model performance for training set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))\n",
- "print(\"\\n\")\n",
- "\n",
- "# model evaluation for testing set\n",
- "\n",
- "y_test_predict = lin_model.predict(X_test)\n",
- "# root mean square error of the model\n",
- "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n",
- "\n",
- "# r-squared score of the model\n",
- "r2 = r2_score(Y_test, y_test_predict)\n",
- "\n",
- "print(\"The model performance for testing set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "id": "6f9196e5",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# plotting the y_test vs y_pred\n",
- "# ideally should have been a straight line\n",
- "plt.scatter(Y_test, y_test_predict)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "b72d5080",
+ "id": "44caf83f",
"metadata": {
"editable": true
},
@@ -3786,7 +3422,7 @@
},
{
"cell_type": "markdown",
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+ "id": "fea67f0c",
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@@ -3798,7 +3434,7 @@
},
{
"cell_type": "markdown",
- "id": "4b02a26a",
+ "id": "1af1f20c",
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"editable": true
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@@ -3817,8 +3453,8 @@
},
{
"cell_type": "code",
- "execution_count": 37,
- "id": "efd083fc",
+ "execution_count": 26,
+ "id": "e9bcb7b1",
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@@ -3852,7 +3488,7 @@
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{
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- "id": "b3590aea",
+ "id": "f3961700",
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@@ -3867,7 +3503,7 @@
},
{
"cell_type": "markdown",
- "id": "9592b7bf",
+ "id": "a33d2ba3",
"metadata": {
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@@ -3879,7 +3515,7 @@
},
{
"cell_type": "markdown",
- "id": "0941f045",
+ "id": "3bc979c2",
"metadata": {
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@@ -3889,7 +3525,7 @@
},
{
"cell_type": "markdown",
- "id": "d47fc8d3",
+ "id": "c234f16e",
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@@ -3912,8 +3548,8 @@
},
{
"cell_type": "code",
- "execution_count": 38,
- "id": "1ddb9cb5",
+ "execution_count": 27,
+ "id": "662bea3f",
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"editable": true
@@ -3957,7 +3593,7 @@
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{
"cell_type": "markdown",
- "id": "725b78e9",
+ "id": "05d214df",
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@@ -3967,7 +3603,7 @@
},
{
"cell_type": "markdown",
- "id": "b08f94e5",
+ "id": "aa18093a",
"metadata": {
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@@ -4036,7 +3672,7 @@
},
{
"cell_type": "markdown",
- "id": "95e72a9e",
+ "id": "a8f251d6",
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@@ -4049,8 +3685,8 @@
},
{
"cell_type": "code",
- "execution_count": 39,
- "id": "fb3ad8e5",
+ "execution_count": 28,
+ "id": "48099cce",
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"editable": true
@@ -4063,7 +3699,7 @@
},
{
"cell_type": "markdown",
- "id": "02fe1db6",
+ "id": "1cf31ef1",
"metadata": {
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@@ -4077,7 +3713,7 @@
},
{
"cell_type": "markdown",
- "id": "33a4aed5",
+ "id": "dde166b5",
"metadata": {
"editable": true
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@@ -4090,7 +3726,7 @@
},
{
"cell_type": "markdown",
- "id": "78a3bc86",
+ "id": "273b1e1d",
"metadata": {
"editable": true
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@@ -4101,7 +3737,7 @@
},
{
"cell_type": "markdown",
- "id": "38c3a27d",
+ "id": "f7a4a3f1",
"metadata": {
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@@ -4113,7 +3749,7 @@
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{
"cell_type": "markdown",
- "id": "7eb5c51b",
+ "id": "0b3e8071",
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@@ -4123,7 +3759,7 @@
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{
"cell_type": "markdown",
- "id": "3597b20a",
+ "id": "9f2d1bf3",
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@@ -4135,7 +3771,7 @@
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{
"cell_type": "markdown",
- "id": "82922e13",
+ "id": "73c33ded",
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@@ -4150,8 +3786,8 @@
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{
"cell_type": "code",
- "execution_count": 40,
- "id": "61cd693e",
+ "execution_count": 29,
+ "id": "c2fc4af7",
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@@ -4202,7 +3838,7 @@
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{
"cell_type": "markdown",
- "id": "2f8d2e6e",
+ "id": "a13aba09",
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@@ -4212,7 +3848,7 @@
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{
"cell_type": "markdown",
- "id": "c81f5caf",
+ "id": "dcf58060",
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@@ -4257,8 +3893,8 @@
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{
"cell_type": "code",
- "execution_count": 41,
- "id": "37ca3335",
+ "execution_count": 30,
+ "id": "f1bd9543",
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@@ -4271,7 +3907,7 @@
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{
"cell_type": "markdown",
- "id": "e6925e8c",
+ "id": "38030ca1",
"metadata": {
"editable": true
},
@@ -4281,8 +3917,8 @@
},
{
"cell_type": "code",
- "execution_count": 42,
- "id": "caecb70a",
+ "execution_count": 31,
+ "id": "e1d90b7a",
"metadata": {
"collapsed": false,
"editable": true
@@ -4297,7 +3933,7 @@
},
{
"cell_type": "markdown",
- "id": "deabdf0c",
+ "id": "4de86c60",
"metadata": {
"editable": true
},
@@ -4314,8 +3950,8 @@
},
{
"cell_type": "code",
- "execution_count": 43,
- "id": "b083bb84",
+ "execution_count": 32,
+ "id": "4aae540f",
"metadata": {
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"editable": true
@@ -4332,7 +3968,7 @@
},
{
"cell_type": "markdown",
- "id": "21102c44",
+ "id": "966d110b",
"metadata": {
"editable": true
},
@@ -4347,8 +3983,8 @@
},
{
"cell_type": "code",
- "execution_count": 44,
- "id": "b9108dab",
+ "execution_count": 33,
+ "id": "011363cd",
"metadata": {
"collapsed": false,
"editable": true
@@ -4392,7 +4028,7 @@
},
{
"cell_type": "markdown",
- "id": "cd19b575",
+ "id": "cf8dd05c",
"metadata": {
"editable": true
},
@@ -4402,7 +4038,7 @@
},
{
"cell_type": "markdown",
- "id": "eb2f6352",
+ "id": "d3c9b66f",
"metadata": {
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@@ -4413,7 +4049,7 @@
},
{
"cell_type": "markdown",
- "id": "6e09ea94",
+ "id": "3363b9e6",
"metadata": {
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@@ -4424,7 +4060,7 @@
},
{
"cell_type": "markdown",
- "id": "38857156",
+ "id": "33ac5d0b",
"metadata": {
"editable": true
},
@@ -4435,7 +4071,7 @@
},
{
"cell_type": "markdown",
- "id": "6b9115a6",
+ "id": "e80b75d3",
"metadata": {
"editable": true
},
@@ -4457,8 +4093,8 @@
},
{
"cell_type": "code",
- "execution_count": 45,
- "id": "ae7a71c1",
+ "execution_count": 34,
+ "id": "ba7cc092",
"metadata": {
"collapsed": false,
"editable": true
@@ -4471,7 +4107,7 @@
},
{
"cell_type": "markdown",
- "id": "ca615d39",
+ "id": "36b73ce3",
"metadata": {
"editable": true
},
@@ -4488,7 +4124,7 @@
},
{
"cell_type": "markdown",
- "id": "d83c8354",
+ "id": "9e7ac684",
"metadata": {
"editable": true
},
@@ -4501,7 +4137,7 @@
},
{
"cell_type": "markdown",
- "id": "74f6e912",
+ "id": "b94b5ddd",
"metadata": {
"editable": true
},
@@ -4512,7 +4148,7 @@
},
{
"cell_type": "markdown",
- "id": "93761664",
+ "id": "f8776c45",
"metadata": {
"editable": true
},
@@ -4524,7 +4160,7 @@
},
{
"cell_type": "markdown",
- "id": "be729d32",
+ "id": "40a07c84",
"metadata": {
"editable": true
},
@@ -4534,7 +4170,7 @@
},
{
"cell_type": "markdown",
- "id": "08549523",
+ "id": "55a234b6",
"metadata": {
"editable": true
},
@@ -4546,7 +4182,7 @@
},
{
"cell_type": "markdown",
- "id": "03cc0ca0",
+ "id": "59dc00fb",
"metadata": {
"editable": true
},
@@ -4562,8 +4198,8 @@
},
{
"cell_type": "code",
- "execution_count": 46,
- "id": "9b1b9378",
+ "execution_count": 35,
+ "id": "44a8d4ef",
"metadata": {
"collapsed": false,
"editable": true
@@ -4655,7 +4291,7 @@
},
{
"cell_type": "markdown",
- "id": "341e9820",
+ "id": "703ec310",
"metadata": {
"editable": true
},
@@ -4665,7 +4301,7 @@
},
{
"cell_type": "markdown",
- "id": "6feb372f",
+ "id": "980aceb3",
"metadata": {
"editable": true
},
@@ -4687,7 +4323,7 @@
},
{
"cell_type": "markdown",
- "id": "3c12062d",
+ "id": "c2c4fafc",
"metadata": {
"editable": true
},
@@ -4699,7 +4335,7 @@
},
{
"cell_type": "markdown",
- "id": "edfcd8ff",
+ "id": "84a561a2",
"metadata": {
"editable": true
},
@@ -4709,7 +4345,7 @@
},
{
"cell_type": "markdown",
- "id": "e8689eb2",
+ "id": "bb9a9639",
"metadata": {
"editable": true
},
@@ -4721,7 +4357,7 @@
},
{
"cell_type": "markdown",
- "id": "62145d09",
+ "id": "8f1dd1a6",
"metadata": {
"editable": true
},
@@ -4731,7 +4367,7 @@
},
{
"cell_type": "markdown",
- "id": "ec992c25",
+ "id": "74dcf2c0",
"metadata": {
"editable": true
},
@@ -4743,7 +4379,7 @@
},
{
"cell_type": "markdown",
- "id": "4a922090",
+ "id": "c530cce6",
"metadata": {
"editable": true
},
@@ -4758,7 +4394,7 @@
},
{
"cell_type": "markdown",
- "id": "10f294a3",
+ "id": "f337bf71",
"metadata": {
"editable": true
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@@ -4770,7 +4406,7 @@
},
{
"cell_type": "markdown",
- "id": "1dfa508a",
+ "id": "cfa053a1",
"metadata": {
"editable": true
},
@@ -4780,7 +4416,7 @@
},
{
"cell_type": "markdown",
- "id": "b5474b9f",
+ "id": "e9035e0c",
"metadata": {
"editable": true
},
@@ -4792,7 +4428,7 @@
},
{
"cell_type": "markdown",
- "id": "4dee1baf",
+ "id": "1b198445",
"metadata": {
"editable": true
},
@@ -4802,7 +4438,7 @@
},
{
"cell_type": "markdown",
- "id": "9949ee71",
+ "id": "e21f0cf0",
"metadata": {
"editable": true
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@@ -4814,7 +4450,7 @@
},
{
"cell_type": "markdown",
- "id": "73ce2a98",
+ "id": "8918cc8c",
"metadata": {
"editable": true
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@@ -4824,7 +4460,7 @@
},
{
"cell_type": "markdown",
- "id": "41e1008d",
+ "id": "bed599b9",
"metadata": {
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@@ -4836,7 +4472,7 @@
},
{
"cell_type": "markdown",
- "id": "597c4ebb",
+ "id": "4faf8bfa",
"metadata": {
"editable": true
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@@ -4846,7 +4482,7 @@
},
{
"cell_type": "markdown",
- "id": "e9163f17",
+ "id": "ebde5b6c",
"metadata": {
"editable": true
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@@ -4858,7 +4494,7 @@
},
{
"cell_type": "markdown",
- "id": "fc3d2fdb",
+ "id": "5ef54737",
"metadata": {
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@@ -4868,7 +4504,7 @@
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{
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- "id": "e7cec690",
+ "id": "6ef07a21",
"metadata": {
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@@ -4880,7 +4516,7 @@
},
{
"cell_type": "markdown",
- "id": "1a74bac9",
+ "id": "7d5bcb91",
"metadata": {
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@@ -4890,7 +4526,7 @@
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{
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- "id": "7087bc76",
+ "id": "1d35a7b9",
"metadata": {
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@@ -4902,7 +4538,7 @@
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{
"cell_type": "markdown",
- "id": "1e5a21fd",
+ "id": "90fcecec",
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@@ -4912,7 +4548,7 @@
},
{
"cell_type": "markdown",
- "id": "624d40e0",
+ "id": "4c2ab747",
"metadata": {
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@@ -4924,7 +4560,7 @@
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{
"cell_type": "markdown",
- "id": "d871f62f",
+ "id": "ae1d9316",
"metadata": {
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@@ -4934,7 +4570,7 @@
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{
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+ "id": "aa76ae3d",
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@@ -4946,7 +4582,7 @@
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@@ -4956,7 +4592,7 @@
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+ "id": "8476e871",
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@@ -4968,7 +4604,7 @@
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{
"cell_type": "markdown",
- "id": "4331c20e",
+ "id": "70fa4295",
"metadata": {
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diff --git a/doc/LectureNotes/_build/html/chapter1.html b/doc/LectureNotes/_build/html/chapter1.html
index a4f572502..5fced352b 100644
--- a/doc/LectureNotes/_build/html/chapter1.html
+++ b/doc/LectureNotes/_build/html/chapter1.html
@@ -391,16 +391,14 @@ document.write(`
3.5. Splitting our Data in Training and Test data
-3.6. The Boston housing data example
-3.7. Housing data, the code
-3.8. Reducing the number of degrees of freedom, overarching view
-3.9. Testing the Means Squared Error as function of Complexity
-3.10. Exercises
-3.11. Exercise 1: Setting up various Python environments
-3.12. Exercise 2: making your own data and exploring scikit-learn
-3.13. Exercise 3: Normalizing our data
-3.14. Exercise 4: Adding Ridge Regression
-3.15. Exercise 5: Analytical exercises
+3.6. Reducing the number of degrees of freedom, overarching view
+3.7. Testing the Means Squared Error as function of Complexity
+3.8. Exercises
+3.9. Exercise 1: Setting up various Python environments
+3.10. Exercise 2: making your own data and exploring scikit-learn
+3.11. Exercise 3: Normalizing our data
+3.12. Exercise 4: Adding Ridge Regression
+3.13. Exercise 5: Analytical exercises
@@ -621,10 +619,10 @@ y = 2x+N(0,1),
where \(N(0,1)\) represents random numbers generated by the normal
distribution. From Scikit-Learn we import then the
LinearRegression functionality and make a prediction \(\tilde{y} =
-\alpha + \beta x\) using the function fit(x,y). We call the set of
+\alpha + \theta x\) using the function fit(x,y). We call the set of
data \((\boldsymbol{x},\boldsymbol{y})\) for our training data. The Python package
scikit-learn has also a functionality which extracts the above
-fitting parameters \(\alpha\) and \(\beta\) (see below). Later we will
+fitting parameters \(\alpha\) and \(\theta\) (see below). Later we will
distinguish between training data and test data.
For plotting we use the Python package
matplotlib which produces publication
@@ -704,7 +702,7 @@ however the aim of scaling the equations and make the cost function
dimensionless.
Minimizing the cost function is a central aspect of
our discussions to come. Finding its minima as function of the model
-parameters (\(\alpha\) and \(\beta\) in our case) will be a recurring
+parameters (\(\alpha\) and \(\theta\) in our case) will be a recurring
theme in these series of lectures. Essentially all machine learning
algorithms we will discuss center around the minimization of the
chosen cost function. This depends in turn on our specific
@@ -757,7 +755,7 @@ have a small or larger relative error. Try to play around with
different training data sets and study (graphically) the value of the
relative error.
As mentioned above, Scikit-Learn has an impressive functionality.
-We can for example extract the values of \(\alpha\) and \(\beta\) and
+We can for example extract the values of \(\alpha\) and \(\theta\) and
their error estimates, or the variance and standard deviation and many
other properties from the statistical data analysis.
Here we show an
@@ -775,7 +773,7 @@ linreg = LinearRegression()
linreg.fit(x,y)
ypredict = linreg.predict(x)
print('The intercept alpha: \n', linreg.intercept_)
-print('Coefficient beta : \n', linreg.coef_)
+print('Coefficient theta : \n', linreg.coef_)
# The mean squared error
print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
# Explained variance score: 1 is perfect prediction
@@ -795,8 +793,8 @@ plt.show()
-
The function coef gives us the parameter \(\beta\) of our fit while intercept yields
-\(\alpha\). Depending on the constant in front of the normal distribution, we get values near or far from \(alpha =2\) and \(\beta =5\). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
+The function coef gives us the parameter \(\theta\) of our fit while intercept yields
+\(\alpha\). Depending on the constant in front of the normal distribution, we get values near or far from \(alpha =2\) and \(\theta =5\). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
\[
MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
@@ -2038,199 +2036,8 @@ print(MSE(y_test,ypredict))
-
-3.6. The Boston housing data example
-The Boston housing
-data set was originally a part of UCI Machine Learning Repository
-and has been removed now. The data set is now included in Scikit-Learn’s
-library. There are 506 samples and 13 feature (predictor) variables
-in this data set. The objective is to predict the value of prices of
-the house using the features (predictors) listed here.
-The features/predictors are
-
-CRIM: Per capita crime rate by town
ZN: Proportion of residential land zoned for lots over 25000 square feet
INDUS: Proportion of non-retail business acres per town
CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
NOX: Nitric oxide concentration (parts per 10 million)
RM: Average number of rooms per dwelling
AGE: Proportion of owner-occupied units built prior to 1940
DIS: Weighted distances to five Boston employment centers
RAD: Index of accessibility to radial highways
TAX: Full-value property tax rate per USD10000
B: \(1000(Bk - 0.63)^2\), where \(Bk\) is the proportion of [people of African American descent] by town
LSTAT: Percentage of lower status of the population
MEDV: Median value of owner-occupied homes in USD 1000s
-
-
-3.7. Housing data, the code
-We start by importing the libraries
-
-and load the Boston Housing DataSet from Scikit-Learn
-
-Then we invoke Pandas
-
-and preprocess the data
-
-We can then visualize the data
-
-It is now useful to look at the correlation matrix
-
-From the above coorelation plot we can see that MEDV is strongly correlated to LSTAT and RM. We see also that RAD and TAX are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity
-
-Now we start training our model
-
-We split the data into training and test sets
-
-Then we use the linear regression functionality from Scikit-Learn
-
-
-
-3.8. Reducing the number of degrees of freedom, overarching view
+3.6. Reducing the number of degrees of freedom, overarching view
Many Machine Learning problems involve thousands or even millions of
features for each training instance. Not only does this make training
extremely slow, it can also make it much harder to find a good
@@ -2343,7 +2150,7 @@ x_j^{(i)} \rightarrow (b-a)\frac{x_j^{(i)} - \min(x_j)}{\max(x_j) - \min(x_j)} -
where \(\min(x_j)\) and \(\max(x_j)\) return the minimum and maximum value of \(x_j\) over the data set, respectively.
-3.9. Testing the Means Squared Error as function of Complexity
+3.7. Testing the Means Squared Error as function of Complexity
Before we proceed with a more detailed analysis of the so-called
Bias-Variance tradeoff, we present here an example of the relation
between model complexity and the mean squared error for the triaining
@@ -2394,10 +2201,10 @@ plt.show()
-3.10. Exercises
+3.8. Exercises
-3.11. Exercise 1: Setting up various Python environments
+3.9. Exercise 1: Setting up various Python environments
The first exercise here is of a mere technical art. We want you to have
git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo GitHub facilities.
We recommend using Anaconda if you are not too familiar with setting paths in a terminal environment.
-3.12. Exercise 2: making your own data and exploring scikit-learn
+3.10. Exercise 2: making your own data and exploring scikit-learn
We will generate our own dataset for a function \(y(x)\) where \(x \in [0,1]\) and defined by random numbers computed with the uniform distribution. The function \(y\) is a quadratic polynomial in \(x\) with added stochastic noise according to the normal distribution \(\cal {N}(0,1)\).
The following simple Python instructions define our \(x\) and \(y\) values (with 100 data points).
@@ -2538,7 +2345,7 @@ print(MSE(y_test,ypredict))
-3.13. Exercise 3: Normalizing our data
+3.11. Exercise 3: Normalizing our data
A much used approach before starting to train the data is to preprocess our
data. Normally the data may need a rescaling and/or may be sensitive
to extreme values. Scaling the data renders our inputs much more
@@ -2658,7 +2465,7 @@ Perform an ordinary least squares and compute the means squared error and the \(15\). Perform a standard OLS fitting of the training data and compute the MSE and \(R2\) for the training and test data and plot both test and training data MSE and \(R2\) as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
-3.14. Exercise 4: Adding Ridge Regression
+3.12. Exercise 4: Adding Ridge Regression
This exercise is a continuation of exercise 2. We will use the same function to
generate our data set, still staying with a simple function \(y(x)\)
which we want to fit using linear regression, but now extending the
@@ -2793,7 +2600,7 @@ plt.show()
-3.15. Exercise 5: Analytical exercises
+3.13. Exercise 5: Analytical exercises
In this exercise we derive the expressions for various derivatives of
products of vectors and matrices. Such derivatives are central to the
optimization of various cost functions. Although we will often use
@@ -2956,16 +2763,14 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
3.5. Splitting our Data in Training and Test data
-3.6. The Boston housing data example
-3.7. Housing data, the code
-3.8. Reducing the number of degrees of freedom, overarching view
-3.9. Testing the Means Squared Error as function of Complexity
-3.10. Exercises
-3.11. Exercise 1: Setting up various Python environments
-3.12. Exercise 2: making your own data and exploring scikit-learn
-3.13. Exercise 3: Normalizing our data
-3.14. Exercise 4: Adding Ridge Regression
-3.15. Exercise 5: Analytical exercises
+3.6. Reducing the number of degrees of freedom, overarching view
+3.7. Testing the Means Squared Error as function of Complexity
+3.8. Exercises
+3.9. Exercise 1: Setting up various Python environments
+3.10. Exercise 2: making your own data and exploring scikit-learn
+3.11. Exercise 3: Normalizing our data
+3.12. Exercise 4: Adding Ridge Regression
+3.13. Exercise 5: Analytical exercises
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 27046ee33..ee128239d 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"A Classification Tree": [[11, "a-classification-tree"]], "A Frequentist approach to data analysis": [[2, "a-frequentist-approach-to-data-analysis"], [23, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[10, "a-better-approach"]], "A first summary": [[23, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[10, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[6, "a-simple-example"]], "A soft classifier": [[10, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[3, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[15, "adam-optimizer"]], "Activation functions": [[14, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[12, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[23, "adding-error-analysis-and-training-set-up"], [24, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[3, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[11, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[12, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[6, "an-extrapolation-example"]], "An optimization/minimization problem": [[23, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[24, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[23, "and-what-about-using-neural-networks"]], "Another example, the moons again": [[11, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[17, null]], "Autocorrelation function": [[20, "autocorrelation-function"]], "Automatic differentiation": [[15, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[24, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[13, "back-to-the-cancer-data"]], "Bagging": [[12, "bagging"]], "Bagging Examples": [[12, "bagging-examples"]], "Basic Matrix Features": [[18, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[13, null]], "Basic math of the SVD": [[7, "basic-math-of-the-svd"], [24, "basic-math-of-the-svd"]], "Basics": [[9, "basics"]], "Basics of a tree": [[11, "basics-of-a-tree"]], "Batch Normalization": [[3, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[7, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[12, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[8, "bootstrap"]], "Bringing it together, first back propagation equation": [[14, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[3, null]], "Building a tree, regression": [[11, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[3, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[5, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[11, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[3, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[13, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[16, null]], "Code for SVD and Inversion of Matrices": [[7, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[16, "codes-and-approaches"]], "Codes for the SVD": [[7, "codes-for-the-svd"], [24, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[0, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[3, "collect-and-pre-process-data"]], "Communication channels": [[23, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[12, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[4, "comparing-with-a-numerical-scheme"]], "Computing the Gini index": [[11, "computing-the-gini-index"]], "Conjugate gradient method": [[15, "conjugate-gradient-method"]], "Convex functions": [[15, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[5, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[5, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[14, "convolutional-neural-network"]], "Convolutional Neural Networks": [[5, null]], "Correlation Function and Design/Feature Matrix": [[24, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[13, "correlation-matrix"], [24, "correlation-matrix"]], "Correlation Matrix with Pandas": [[24, "correlation-matrix-with-pandas"]], "Course Format": [[23, "course-format"]], "Course setting": [[19, null]], "Covariance Matrix Examples": [[24, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[24, "covariance-and-correlation-matrix"]], "Cross-validation": [[8, "cross-validation"]], "Deadlines for projects (tentative)": [[23, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[11, null]], "Deep learning methods": [[23, "deep-learning-methods"]], "Define model and architecture": [[3, "define-model-and-architecture"]], "Defining the cost function": [[3, "defining-the-cost-function"]], "Deliverables": [[0, "deliverables"], [1, "deliverables"]], "Derivatives and the chain rule": [[14, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[24, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[7, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[1, "deriving-and-implementing-ordinary-least-squares"]], "Deriving the Lasso Regression Equations": [[24, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[24, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[14, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[3, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[13, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[10, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[11, "disadvantages"]], "Discriminative Modeling": [[23, "discriminative-modeling"]], "Domains and probabilities": [[20, "domains-and-probabilities"]], "Dropout": [[3, "dropout"]], "Economy-size SVD": [[24, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[20, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[12, null]], "Entropy and the ID3 algorithm": [[11, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[23, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[3, "evaluate-model-performance-on-test-data"]], "Example 2": [[24, "example-2"]], "Example 3": [[24, "example-3"]], "Example 4": [[24, "example-4"]], "Example Matrix": [[24, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[23, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[23, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[24, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[24, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[4, "example-exponential-decay"]], "Example: Population growth": [[4, "example-population-growth"]], "Example: The diffusion equation": [[4, "example-the-diffusion-equation"]], "Example: binary classification problem": [[3, "example-binary-classification-problem"]], "Examples": [[23, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[9, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[1, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[0, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[2, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[1, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Setting up a Github repository": [[0, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[2, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[1, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[0, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Setting up a Python virtual environment": [[0, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[2, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[1, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - The train-test split": [[0, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[2, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[1, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[2, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[8, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[8, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[8, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[8, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[8, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[8, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[2, "exercises"]], "Exercises and Projects": [[8, "exercises-and-projects"]], "Exercises week 34": [[0, null]], "Exercises week 35": [[1, null]], "Expectation values": [[20, "expectation-values"]], "Extremely useful tools, strongly recommended": [[23, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[14, "feed-forward-neural-networks"]], "Feed-forward pass": [[3, "feed-forward-pass"]], "Final back propagating equation": [[14, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[3, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[2, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[24, "fixing-the-singularity"]], "Frequently used scaling functions": [[24, "frequently-used-scaling-functions"]], "From one to many layers, the universal approximation theorem": [[14, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[24, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[5, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[7, "further-properties-important-for-our-analyses-later"], [24, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[18, "gaussian-elimination"]], "General Features": [[11, "general-features"]], "General linear models and linear algebra": [[23, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[23, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [23, "id1"]], "Generative Adversarial Networks": [[6, "generative-adversarial-networks"]], "Generative Models": [[6, "generative-models"]], "Generative Versus Discriminative Modeling": [[23, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[13, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[12, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[12, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[3, "gradient-clipping"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[12, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[4, "gradient-descent"]], "Grading": [[21, "grading"], [21, "id2"], [23, "grading"]], "Housing data, the code": [[2, "housing-data-the-code"]], "How to take derivatives of Matrix-Vector expressions": [[1, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[10, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[18, "important-matrix-and-vector-handling-packages"]], "Improving performance": [[3, "improving-performance"]], "In summary": [[21, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[15, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[13, "incremental-pca"]], "Installing R, C++, cython or Julia": [[23, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[23, "installing-r-c-cython-numba-etc"]], "Instructor information": [[21, "instructor-information"]], "Interpretations and optimizing our parameters": [[23, "interpretations-and-optimizing-our-parameters"], [23, "id2"], [23, "id3"], [24, "interpretations-and-optimizing-our-parameters"], [24, "id1"], [24, "id2"]], "Interpreting the Ridge results": [[24, "interpreting-the-ridge-results"]], "Introducing JAX": [[15, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[13, "introducing-the-covariance-and-correlation-functions"], [24, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[2, "introduction"], [8, "introduction"], [17, "introduction"], [18, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[12, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[12, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[13, "kernel-pca"]], "Kernels and non-linearity": [[10, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[18, "lu-decomposition-the-inverse-of-a-matrix"]], "Layers": [[3, "layers"]], "Layers used to build CNNs": [[5, "layers-used-to-build-cnns"]], "Learning goals": [[0, "learning-goals"], [1, "learning-goals"]], "Learning outcomes": [[17, "learning-outcomes"], [23, "learning-outcomes"]], "Lectures and ComputerLab": [[23, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[3, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[18, null]], "Linear Regression": [[2, null]], "Linear Regression Problems": [[24, "linear-regression-problems"]], "Linear Regression, basic elements": [[2, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[7, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[7, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[7, "linking-with-the-svd"], [24, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[22, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[9, null], [9, "id1"]], "MNIST and GANs": [[6, "mnist-and-gans"]], "Machine Learning": [[23, "machine-learning"]], "Machine learning": [[17, "machine-learning"]], "Main textbooks": [[23, "main-textbooks"]], "Making a tree": [[11, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[12, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[24, "making-your-own-test-train-splitting"]], "Mathematical Interpretation of Ordinary Least Squares": [[7, "mathematical-interpretation-of-ordinary-least-squares"], [24, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[10, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[5, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[7, "mathematics-of-the-svd-and-implications"], [24, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[23, "matrices-in-python"]], "Matrix multiplication": [[3, "matrix-multiplication"]], "Matrix-vector notation and activation": [[14, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[20, "meet-the-covariance"]], "Meet the Covariance Matrix": [[7, "meet-the-covariance-matrix"], [24, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[24, "meet-the-hessian-matrix"]], "Meet the Pandas": [[23, "meet-the-pandas"]], "Min-Max Scaling": [[24, "min-max-scaling"]], "Momentum based GD": [[15, "momentum-based-gd"]], "More complicated Example: The Ising model": [[8, "more-complicated-example-the-ising-model"]], "More interpretations": [[24, "more-interpretations"]], "More on Dimensionalities": [[5, "more-on-dimensionalities"]], "More on Rescaling data": [[8, "more-on-rescaling-data"]], "More preprocessing": [[24, "more-preprocessing"]], "Multilayer perceptrons": [[14, "multilayer-perceptrons"]], "Network requirements": [[4, "network-requirements"]], "Neural Networks vs CNNs": [[5, "neural-networks-vs-cnns"]], "Neural networks": [[14, null]], "Note about SVD Calculations": [[24, "note-about-svd-calculations"]], "Numerical experiments and the covariance, central limit theorem": [[20, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[18, "numpy-and-arrays"], [23, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[23, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization, the central part of any Machine Learning algortithm": [[15, null]], "Optimizing our parameters": [[23, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[23, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[3, "optimizing-the-cost-function"]], "Organizing our data": [[2, "organizing-our-data"], [23, "organizing-our-data"]], "Other Matrix and Vector Operations": [[18, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[6, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[23, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[23, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[23, "other-popular-texts"]], "Other techniques": [[13, "other-techniques"]], "Other types of networks": [[14, "other-types-of-networks"]], "Other ways of visualizing the trees": [[11, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[23, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[23, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[23, "own-code-for-ordinary-least-squares"], [24, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[13, "pca-and-scikit-learn"]], "Pandas AI": [[23, "pandas-ai"]], "Partial Differential Equations": [[4, "partial-differential-equations"]], "Plans for week 35": [[24, "plans-for-week-35"]], "Practical tips": [[15, "practical-tips"]], "Practicalities": [[21, "practicalities"], [21, "id1"]], "Predicting New Points With A Trained Recurrent Neural Network": [[6, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[24, "preprocessing-our-data"]], "Prerequisites": [[23, "prerequisites"]], "Prerequisites and background": [[17, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[5, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[20, "probability-distribution-functions"]], "Program for stochastic gradient": [[15, "program-for-stochastic-gradient"]], "Properties of PDFs": [[20, "properties-of-pdfs"]], "Pros and cons of trees, pros": [[11, "pros-and-cons-of-trees-pros"]], "Python installers": [[17, "python-installers"], [23, "python-installers"]], "RMS prop": [[15, "rms-prop"]], "Random Numbers": [[20, "random-numbers"]], "Random forests": [[12, "random-forests"]], "Randomized PCA": [[13, "randomized-pca"]], "Reading material": [[23, "reading-material"]], "Reading recommendations:": [[24, "reading-recommendations"]], "Reading suggestions week 34": [[23, "reading-suggestions-week-34"]], "Recurrent neural networks": [[14, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[6, null]], "Reducing the number of degrees of freedom, overarching view": [[2, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [24, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[4, "reformulating-the-problem"]], "Regression Case": [[12, "regression-case"]], "Regression analysis, overarching aims": [[23, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[23, "regression-analysis-overarching-aims-ii"]], "Regularization": [[3, "regularization"]], "Reminder from last week": [[24, "reminder-from-last-week"]], "Reminder on Statistics": [[8, "reminder-on-statistics"]], "Replace or not": [[15, "replace-or-not"]], "Required Technologies": [[17, "required-technologies"]], "Resampling Methods": [[8, null]], "Resampling methods": [[8, "id1"]], "Residual Error": [[24, "residual-error"]], "Resources on differential equations and deep learning": [[4, "resources-on-differential-equations-and-deep-learning"]], "Revisiting our Linear Regression Solvers": [[15, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[24, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the fitting procedure as a linear algebra problem": [[23, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[23, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge and LASSO Regression": [[24, "ridge-and-lasso-regression"]], "Ridge and Lasso Regression": [[7, null], [7, "id1"]], "Same code but now with momentum gradient descent": [[15, "same-code-but-now-with-momentum-gradient-descent"]], "Schedule first week": [[23, "schedule-first-week"]], "Schematic Regression Procedure": [[11, "schematic-regression-procedure"]], "Setting up the Back propagation algorithm": [[14, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[24, "setting-up-the-matrix-to-be-inverted"]], "Setting up the network using Autograd; The full program": [[4, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[15, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[11, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[24, "simple-case"]], "Simple linear regression model using scikit-learn": [[2, "simple-linear-regression-model-using-scikit-learn"], [23, "simple-linear-regression-model-using-scikit-learn"]], "Software and needed installations": [[23, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[4, null]], "Solving the one dimensional Poisson equation": [[4, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation with Neural Networks": [[4, "solving-the-wave-equation-with-neural-networks"]], "Some famous Matrices": [[18, "some-famous-matrices"]], "Some simple problems": [[15, "some-simple-problems"]], "Some useful matrix and vector expressions": [[24, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[2, "splitting-our-data-in-training-and-test-data"], [24, "splitting-our-data-in-training-and-test-data"]], "Standard steepest descent": [[15, "standard-steepest-descent"]], "Statistical analysis and optimization of data": [[17, "statistical-analysis-and-optimization-of-data"], [23, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[15, "steepest-descent"]], "Stochastic 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+Search.setIndex({"alltitles": {"A Classification Tree": [[11, "a-classification-tree"]], "A Frequentist approach to data analysis": [[2, "a-frequentist-approach-to-data-analysis"], [23, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[10, "a-better-approach"]], "A first summary": [[23, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[10, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[6, "a-simple-example"]], "A soft classifier": [[10, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[3, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[15, "adam-optimizer"]], "Activation functions": [[14, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[12, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[23, "adding-error-analysis-and-training-set-up"], [24, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[3, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[11, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[12, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[6, "an-extrapolation-example"]], "An optimization/minimization problem": [[23, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[24, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[23, "and-what-about-using-neural-networks"]], "Another example, the moons again": [[11, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[17, null]], "Autocorrelation function": [[20, "autocorrelation-function"]], "Automatic differentiation": [[15, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[24, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[13, "back-to-the-cancer-data"]], "Bagging": [[12, "bagging"]], "Bagging Examples": [[12, "bagging-examples"]], "Basic Matrix Features": [[18, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[13, null]], "Basic math of the SVD": [[7, "basic-math-of-the-svd"], [24, "basic-math-of-the-svd"]], "Basics": [[9, "basics"]], "Basics of a tree": [[11, "basics-of-a-tree"]], "Batch Normalization": [[3, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[7, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[12, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[8, "bootstrap"]], "Bringing it together, first back propagation equation": [[14, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[3, null]], "Building a tree, regression": [[11, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[3, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[5, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[11, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[3, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[13, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[16, null]], "Code for SVD and Inversion of Matrices": [[7, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[16, "codes-and-approaches"]], "Codes for the SVD": [[7, "codes-for-the-svd"], [24, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[0, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[3, "collect-and-pre-process-data"]], "Communication channels": [[23, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[12, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[4, "comparing-with-a-numerical-scheme"]], "Computing the Gini index": [[11, "computing-the-gini-index"]], "Conjugate gradient method": [[15, "conjugate-gradient-method"]], "Convex functions": [[15, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[5, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[5, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[14, "convolutional-neural-network"]], "Convolutional Neural Networks": [[5, null]], "Correlation Function and Design/Feature Matrix": [[24, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[13, "correlation-matrix"], [24, "correlation-matrix"]], "Correlation Matrix with Pandas": [[24, "correlation-matrix-with-pandas"]], "Course Format": [[23, "course-format"]], "Course setting": [[19, null]], "Covariance Matrix Examples": [[24, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[24, "covariance-and-correlation-matrix"]], "Cross-validation": [[8, "cross-validation"]], "Deadlines for projects (tentative)": [[23, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[11, null]], "Deep learning methods": [[23, "deep-learning-methods"]], "Define model and architecture": [[3, "define-model-and-architecture"]], "Defining the cost function": [[3, "defining-the-cost-function"]], "Deliverables": [[0, "deliverables"], [1, "deliverables"]], "Derivatives and the chain rule": [[14, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[24, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[7, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[1, "deriving-and-implementing-ordinary-least-squares"]], "Deriving the Lasso Regression Equations": [[24, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[24, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[14, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[3, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[13, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[10, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[11, "disadvantages"]], "Discriminative Modeling": [[23, "discriminative-modeling"]], "Domains and probabilities": [[20, "domains-and-probabilities"]], "Dropout": [[3, "dropout"]], "Economy-size SVD": [[24, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[20, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[12, null]], "Entropy and the ID3 algorithm": [[11, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[23, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[3, "evaluate-model-performance-on-test-data"]], "Example 2": [[24, "example-2"]], "Example 3": [[24, "example-3"]], "Example 4": [[24, "example-4"]], "Example Matrix": [[24, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[23, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[23, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[24, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[24, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[4, "example-exponential-decay"]], "Example: Population growth": [[4, "example-population-growth"]], "Example: The diffusion equation": [[4, "example-the-diffusion-equation"]], "Example: binary classification problem": [[3, "example-binary-classification-problem"]], "Examples": [[23, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[9, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[1, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[0, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[2, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[1, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Setting up a Github repository": [[0, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[2, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[1, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[0, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Setting up a Python virtual environment": [[0, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[2, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[1, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - The train-test split": [[0, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[2, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[1, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[2, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[8, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[8, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[8, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[8, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[8, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[8, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[2, "exercises"]], "Exercises and Projects": [[8, "exercises-and-projects"]], "Exercises week 34": [[0, null]], "Exercises week 35": [[1, null]], "Expectation values": [[20, "expectation-values"]], "Extremely useful tools, strongly recommended": [[23, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[14, "feed-forward-neural-networks"]], "Feed-forward pass": [[3, "feed-forward-pass"]], "Final back propagating equation": [[14, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[3, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[2, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[24, "fixing-the-singularity"]], "Frequently used scaling functions": [[24, "frequently-used-scaling-functions"]], "From one to many layers, the universal approximation theorem": [[14, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[24, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[5, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[7, "further-properties-important-for-our-analyses-later"], [24, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[18, "gaussian-elimination"]], "General Features": [[11, "general-features"]], "General linear models and linear algebra": [[23, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[23, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [23, "id1"]], "Generative Adversarial Networks": [[6, "generative-adversarial-networks"]], "Generative Models": [[6, "generative-models"]], "Generative Versus Discriminative Modeling": [[23, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[13, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[12, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[12, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[3, "gradient-clipping"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[12, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[4, "gradient-descent"]], "Grading": [[21, "grading"], [21, "id2"], [23, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[1, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[10, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[18, "important-matrix-and-vector-handling-packages"]], "Improving performance": [[3, "improving-performance"]], "In summary": [[21, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[15, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[13, "incremental-pca"]], "Installing R, C++, cython or Julia": [[23, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[23, "installing-r-c-cython-numba-etc"]], "Instructor information": [[21, "instructor-information"]], "Interpretations and optimizing our parameters": [[23, "interpretations-and-optimizing-our-parameters"], [23, "id2"], [23, "id3"], [24, "interpretations-and-optimizing-our-parameters"], [24, "id1"], [24, "id2"]], "Interpreting the Ridge results": [[24, "interpreting-the-ridge-results"]], "Introducing JAX": [[15, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[13, "introducing-the-covariance-and-correlation-functions"], [24, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[2, "introduction"], [8, "introduction"], [17, "introduction"], [18, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[12, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[12, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[13, "kernel-pca"]], "Kernels and non-linearity": [[10, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[18, "lu-decomposition-the-inverse-of-a-matrix"]], "Layers": [[3, "layers"]], "Layers used to build CNNs": [[5, "layers-used-to-build-cnns"]], "Learning goals": [[0, "learning-goals"], [1, "learning-goals"]], "Learning outcomes": [[17, "learning-outcomes"], [23, "learning-outcomes"]], "Lectures and ComputerLab": [[23, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[3, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[18, null]], "Linear Regression": [[2, null]], "Linear Regression Problems": [[24, "linear-regression-problems"]], "Linear Regression, basic elements": [[2, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[7, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[7, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[7, "linking-with-the-svd"], [24, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[22, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[9, null], [9, "id1"]], "MNIST and GANs": [[6, "mnist-and-gans"]], "Machine Learning": [[23, "machine-learning"]], "Machine learning": [[17, "machine-learning"]], "Main textbooks": [[23, "main-textbooks"]], "Making a tree": [[11, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[12, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[24, "making-your-own-test-train-splitting"]], "Mathematical Interpretation of Ordinary Least Squares": [[7, "mathematical-interpretation-of-ordinary-least-squares"], [24, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[10, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[5, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[7, "mathematics-of-the-svd-and-implications"], [24, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[23, "matrices-in-python"]], "Matrix multiplication": [[3, "matrix-multiplication"]], "Matrix-vector notation and activation": [[14, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[20, "meet-the-covariance"]], "Meet the Covariance Matrix": [[7, "meet-the-covariance-matrix"], [24, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[24, "meet-the-hessian-matrix"]], "Meet the Pandas": [[23, "meet-the-pandas"]], "Min-Max Scaling": [[24, "min-max-scaling"]], "Momentum based GD": [[15, "momentum-based-gd"]], "More complicated Example: The Ising model": [[8, "more-complicated-example-the-ising-model"]], "More interpretations": [[24, "more-interpretations"]], "More on Dimensionalities": [[5, "more-on-dimensionalities"]], "More on Rescaling data": [[8, "more-on-rescaling-data"]], "More preprocessing": [[24, "more-preprocessing"]], "Multilayer perceptrons": [[14, "multilayer-perceptrons"]], "Network requirements": [[4, "network-requirements"]], "Neural Networks vs CNNs": [[5, "neural-networks-vs-cnns"]], "Neural networks": [[14, null]], "Note about SVD Calculations": [[24, "note-about-svd-calculations"]], "Numerical experiments and the covariance, central limit theorem": [[20, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[18, "numpy-and-arrays"], [23, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[23, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization, the central part of any Machine Learning algortithm": [[15, null]], "Optimizing our parameters": [[23, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[23, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[3, "optimizing-the-cost-function"]], "Organizing our data": [[2, "organizing-our-data"], [23, "organizing-our-data"]], "Other Matrix and Vector Operations": [[18, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[6, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[23, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[23, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[23, "other-popular-texts"]], "Other techniques": [[13, "other-techniques"]], "Other types of networks": [[14, "other-types-of-networks"]], "Other ways of visualizing the trees": [[11, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[23, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[23, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[23, "own-code-for-ordinary-least-squares"], [24, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[13, "pca-and-scikit-learn"]], "Pandas AI": [[23, "pandas-ai"]], "Partial Differential Equations": [[4, "partial-differential-equations"]], "Plans for week 35": [[24, "plans-for-week-35"]], "Practical tips": [[15, "practical-tips"]], "Practicalities": [[21, "practicalities"], [21, "id1"]], "Predicting New Points With A Trained Recurrent Neural Network": [[6, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[24, "preprocessing-our-data"]], "Prerequisites": [[23, "prerequisites"]], "Prerequisites and background": [[17, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[5, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[20, "probability-distribution-functions"]], "Program for stochastic gradient": [[15, "program-for-stochastic-gradient"]], "Properties of PDFs": [[20, "properties-of-pdfs"]], "Pros and cons of trees, pros": [[11, "pros-and-cons-of-trees-pros"]], "Python installers": [[17, "python-installers"], [23, "python-installers"]], "RMS prop": [[15, "rms-prop"]], "Random Numbers": [[20, "random-numbers"]], "Random forests": [[12, "random-forests"]], "Randomized PCA": [[13, "randomized-pca"]], "Reading material": [[23, "reading-material"]], "Reading recommendations:": [[24, "reading-recommendations"]], "Reading suggestions week 34": [[23, "reading-suggestions-week-34"]], "Recurrent neural networks": [[14, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[6, null]], "Reducing the number of degrees of freedom, overarching view": [[2, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [24, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[4, "reformulating-the-problem"]], "Regression Case": [[12, "regression-case"]], "Regression analysis, overarching aims": [[23, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[23, "regression-analysis-overarching-aims-ii"]], "Regularization": [[3, "regularization"]], "Reminder from last week": [[24, "reminder-from-last-week"]], "Reminder on Statistics": [[8, "reminder-on-statistics"]], "Replace or not": [[15, "replace-or-not"]], "Required Technologies": [[17, "required-technologies"]], "Resampling Methods": [[8, null]], "Resampling methods": [[8, "id1"]], "Residual Error": [[24, "residual-error"]], "Resources on differential equations and deep learning": [[4, "resources-on-differential-equations-and-deep-learning"]], "Revisiting our Linear Regression Solvers": [[15, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[24, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the fitting procedure as a linear algebra problem": [[23, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[23, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge and LASSO Regression": [[24, "ridge-and-lasso-regression"]], "Ridge and Lasso Regression": [[7, null], [7, "id1"]], "Same code but now with momentum gradient descent": [[15, "same-code-but-now-with-momentum-gradient-descent"]], "Schedule first week": [[23, "schedule-first-week"]], "Schematic Regression Procedure": [[11, "schematic-regression-procedure"]], "Setting up the Back propagation algorithm": [[14, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[24, "setting-up-the-matrix-to-be-inverted"]], "Setting up the network using Autograd; The full program": [[4, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[15, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[11, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[24, "simple-case"]], "Simple linear regression model using scikit-learn": [[2, "simple-linear-regression-model-using-scikit-learn"], [23, "simple-linear-regression-model-using-scikit-learn"]], "Software and needed installations": [[23, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[4, 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diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
index 4b22b16cd..56d1bc4e8 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
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+ "id": "17b7ba24",
"metadata": {
"editable": true
},
@@ -167,7 +167,7 @@
},
{
"cell_type": "markdown",
- "id": "ceb7a805",
+ "id": "23adffca",
"metadata": {
"editable": true
},
@@ -202,7 +202,7 @@
},
{
"cell_type": "markdown",
- "id": "d6c1062f",
+ "id": "1263a25e",
"metadata": {
"editable": true
},
@@ -253,7 +253,7 @@
},
{
"cell_type": "markdown",
- "id": "50c1b706",
+ "id": "e2ab4f0a",
"metadata": {
"editable": true
},
@@ -286,7 +286,7 @@
},
{
"cell_type": "markdown",
- "id": "12e3ac84",
+ "id": "23173385",
"metadata": {
"editable": true
},
@@ -298,7 +298,7 @@
},
{
"cell_type": "markdown",
- "id": "10b4c333",
+ "id": "57b14c43",
"metadata": {
"editable": true
},
@@ -306,10 +306,10 @@
"where $N(0,1)$ represents random numbers generated by the normal\n",
"distribution. From **Scikit-Learn** we import then the\n",
"**LinearRegression** functionality and make a prediction $\\tilde{y} =\n",
- "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n",
+ "\\alpha + \\theta x$ using the function **fit(x,y)**. We call the set of\n",
"data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n",
"**scikit-learn** has also a functionality which extracts the above\n",
- "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n",
+ "fitting parameters $\\alpha$ and $\\theta$ (see below). Later we will\n",
"distinguish between training data and test data.\n",
"\n",
"For plotting we use the Python package\n",
@@ -335,7 +335,7 @@
{
"cell_type": "code",
"execution_count": 1,
- "id": "3e216ef0",
+ "id": "f7d66caa",
"metadata": {
"collapsed": false,
"editable": true
@@ -368,7 +368,7 @@
},
{
"cell_type": "markdown",
- "id": "af765a12",
+ "id": "504848dc",
"metadata": {
"editable": true
},
@@ -385,7 +385,7 @@
},
{
"cell_type": "markdown",
- "id": "2f6fd730",
+ "id": "bff268fb",
"metadata": {
"editable": true
},
@@ -397,7 +397,7 @@
},
{
"cell_type": "markdown",
- "id": "1770d85d",
+ "id": "53357604",
"metadata": {
"editable": true
},
@@ -418,7 +418,7 @@
},
{
"cell_type": "markdown",
- "id": "684b72a0",
+ "id": "4e5b1dad",
"metadata": {
"editable": true
},
@@ -431,7 +431,7 @@
},
{
"cell_type": "markdown",
- "id": "e9fa8fd5",
+ "id": "b8e0f4c7",
"metadata": {
"editable": true
},
@@ -443,7 +443,7 @@
"\n",
"Minimizing the cost function is a central aspect of\n",
"our discussions to come. Finding its minima as function of the model\n",
- "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n",
+ "parameters ($\\alpha$ and $\\theta$ in our case) will be a recurring\n",
"theme in these series of lectures. Essentially all machine learning\n",
"algorithms we will discuss center around the minimization of the\n",
"chosen cost function. This depends in turn on our specific\n",
@@ -462,7 +462,7 @@
},
{
"cell_type": "markdown",
- "id": "15a4642f",
+ "id": "c4e4b512",
"metadata": {
"editable": true
},
@@ -474,7 +474,7 @@
},
{
"cell_type": "markdown",
- "id": "589ec9cb",
+ "id": "71393ea9",
"metadata": {
"editable": true
},
@@ -492,7 +492,7 @@
{
"cell_type": "code",
"execution_count": 2,
- "id": "cf21ae2b",
+ "id": "a6c77c37",
"metadata": {
"collapsed": false,
"editable": true
@@ -520,7 +520,7 @@
},
{
"cell_type": "markdown",
- "id": "41c55cbb",
+ "id": "bc21adc2",
"metadata": {
"editable": true
},
@@ -531,7 +531,7 @@
"relative error.\n",
"\n",
"As mentioned above, **Scikit-Learn** has an impressive functionality.\n",
- "We can for example extract the values of $\\alpha$ and $\\beta$ and\n",
+ "We can for example extract the values of $\\alpha$ and $\\theta$ and\n",
"their error estimates, or the variance and standard deviation and many\n",
"other properties from the statistical data analysis. \n",
"\n",
@@ -542,7 +542,7 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "d62c9412",
+ "id": "febc49fc",
"metadata": {
"collapsed": false,
"editable": true
@@ -560,7 +560,7 @@
"linreg.fit(x,y)\n",
"ypredict = linreg.predict(x)\n",
"print('The intercept alpha: \\n', linreg.intercept_)\n",
- "print('Coefficient beta : \\n', linreg.coef_)\n",
+ "print('Coefficient theta : \\n', linreg.coef_)\n",
"# The mean squared error \n",
"print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n",
"# Explained variance score: 1 is perfect prediction \n",
@@ -580,18 +580,18 @@
},
{
"cell_type": "markdown",
- "id": "ae0c6c2a",
+ "id": "3f1c6408",
"metadata": {
"editable": true
},
"source": [
- "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n",
- "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
+ "The function **coef** gives us the parameter $\\theta$ of our fit while **intercept** yields \n",
+ "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\theta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
]
},
{
"cell_type": "markdown",
- "id": "64cf13de",
+ "id": "b4e26d20",
"metadata": {
"editable": true
},
@@ -604,7 +604,7 @@
},
{
"cell_type": "markdown",
- "id": "18439dae",
+ "id": "8137fd70",
"metadata": {
"editable": true
},
@@ -625,7 +625,7 @@
},
{
"cell_type": "markdown",
- "id": "3e9fb291",
+ "id": "3f40e79c",
"metadata": {
"editable": true
},
@@ -637,7 +637,7 @@
},
{
"cell_type": "markdown",
- "id": "46d8744e",
+ "id": "6a8eb368",
"metadata": {
"editable": true
},
@@ -647,7 +647,7 @@
},
{
"cell_type": "markdown",
- "id": "38a29b65",
+ "id": "71d8af23",
"metadata": {
"editable": true
},
@@ -659,7 +659,7 @@
},
{
"cell_type": "markdown",
- "id": "438e73b8",
+ "id": "aee25556",
"metadata": {
"editable": true
},
@@ -671,7 +671,7 @@
},
{
"cell_type": "markdown",
- "id": "1403cc6a",
+ "id": "1bc1d5e2",
"metadata": {
"editable": true
},
@@ -683,7 +683,7 @@
},
{
"cell_type": "markdown",
- "id": "9fbd1c1b",
+ "id": "dfb80b0d",
"metadata": {
"editable": true
},
@@ -694,7 +694,7 @@
},
{
"cell_type": "markdown",
- "id": "d25da1d3",
+ "id": "1ee8b15b",
"metadata": {
"editable": true
},
@@ -706,7 +706,7 @@
},
{
"cell_type": "markdown",
- "id": "c43c7452",
+ "id": "9ec15fef",
"metadata": {
"editable": true
},
@@ -728,7 +728,7 @@
},
{
"cell_type": "markdown",
- "id": "ad0d5004",
+ "id": "8e386295",
"metadata": {
"editable": true
},
@@ -740,7 +740,7 @@
},
{
"cell_type": "markdown",
- "id": "eb9e2efb",
+ "id": "d8d8ad23",
"metadata": {
"editable": true
},
@@ -755,7 +755,7 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "963f4f84",
+ "id": "0c9a0500",
"metadata": {
"collapsed": false,
"editable": true
@@ -796,7 +796,7 @@
},
{
"cell_type": "markdown",
- "id": "d530f60a",
+ "id": "c7786c3b",
"metadata": {
"editable": true
},
@@ -811,7 +811,7 @@
},
{
"cell_type": "markdown",
- "id": "f8951307",
+ "id": "83f5d1f7",
"metadata": {
"editable": true
},
@@ -823,7 +823,7 @@
},
{
"cell_type": "markdown",
- "id": "913bb6a6",
+ "id": "36b41230",
"metadata": {
"editable": true
},
@@ -833,7 +833,7 @@
},
{
"cell_type": "markdown",
- "id": "25cb82ce",
+ "id": "5476ceac",
"metadata": {
"editable": true
},
@@ -845,7 +845,7 @@
},
{
"cell_type": "markdown",
- "id": "d04f788d",
+ "id": "a0252aa4",
"metadata": {
"editable": true
},
@@ -855,7 +855,7 @@
},
{
"cell_type": "markdown",
- "id": "866bf69f",
+ "id": "993c3fe8",
"metadata": {
"editable": true
},
@@ -867,7 +867,7 @@
},
{
"cell_type": "markdown",
- "id": "a0d3421f",
+ "id": "699624c4",
"metadata": {
"editable": true
},
@@ -877,7 +877,7 @@
},
{
"cell_type": "markdown",
- "id": "680eefd0",
+ "id": "09e37178",
"metadata": {
"editable": true
},
@@ -889,7 +889,7 @@
},
{
"cell_type": "markdown",
- "id": "de97e875",
+ "id": "5dca3cfd",
"metadata": {
"editable": true
},
@@ -905,7 +905,7 @@
},
{
"cell_type": "markdown",
- "id": "5d7dbc49",
+ "id": "6e758f39",
"metadata": {
"editable": true
},
@@ -917,7 +917,7 @@
},
{
"cell_type": "markdown",
- "id": "5a9a6d4d",
+ "id": "cd2a7727",
"metadata": {
"editable": true
},
@@ -928,7 +928,7 @@
},
{
"cell_type": "markdown",
- "id": "fef247ce",
+ "id": "e8b8a9d8",
"metadata": {
"editable": true
},
@@ -940,7 +940,7 @@
},
{
"cell_type": "markdown",
- "id": "3ed61266",
+ "id": "b390292b",
"metadata": {
"editable": true
},
@@ -954,7 +954,7 @@
},
{
"cell_type": "markdown",
- "id": "9882fb4e",
+ "id": "cfdb1cf9",
"metadata": {
"editable": true
},
@@ -966,7 +966,7 @@
},
{
"cell_type": "markdown",
- "id": "992b3ae7",
+ "id": "10b030a0",
"metadata": {
"editable": true
},
@@ -991,7 +991,7 @@
},
{
"cell_type": "markdown",
- "id": "6d772376",
+ "id": "cb289c64",
"metadata": {
"editable": true
},
@@ -1008,7 +1008,7 @@
{
"cell_type": "code",
"execution_count": 5,
- "id": "f861cd3f",
+ "id": "cca06315",
"metadata": {
"collapsed": false,
"editable": true
@@ -1052,7 +1052,7 @@
},
{
"cell_type": "markdown",
- "id": "da27a328",
+ "id": "518d9d53",
"metadata": {
"editable": true
},
@@ -1069,7 +1069,7 @@
{
"cell_type": "code",
"execution_count": 6,
- "id": "b4ef8b5a",
+ "id": "38f65b25",
"metadata": {
"collapsed": false,
"editable": true
@@ -1090,7 +1090,7 @@
},
{
"cell_type": "markdown",
- "id": "1352f7d2",
+ "id": "7df95f40",
"metadata": {
"editable": true
},
@@ -1104,7 +1104,7 @@
{
"cell_type": "code",
"execution_count": 7,
- "id": "c5b3333d",
+ "id": "bdf76b5e",
"metadata": {
"collapsed": false,
"editable": true
@@ -1133,7 +1133,7 @@
},
{
"cell_type": "markdown",
- "id": "6652db3a",
+ "id": "becf6238",
"metadata": {
"editable": true
},
@@ -1153,7 +1153,7 @@
{
"cell_type": "code",
"execution_count": 8,
- "id": "715c03ae",
+ "id": "280c0700",
"metadata": {
"collapsed": false,
"editable": true
@@ -1170,7 +1170,7 @@
},
{
"cell_type": "markdown",
- "id": "eaff6065",
+ "id": "46814807",
"metadata": {
"editable": true
},
@@ -1182,7 +1182,7 @@
{
"cell_type": "code",
"execution_count": 9,
- "id": "4053e205",
+ "id": "0f758df8",
"metadata": {
"collapsed": false,
"editable": true
@@ -1200,7 +1200,7 @@
},
{
"cell_type": "markdown",
- "id": "587289d6",
+ "id": "fda4c0ab",
"metadata": {
"editable": true
},
@@ -1216,7 +1216,7 @@
{
"cell_type": "code",
"execution_count": 10,
- "id": "52f1c9dc",
+ "id": "8b295d45",
"metadata": {
"collapsed": false,
"editable": true
@@ -1229,7 +1229,7 @@
},
{
"cell_type": "markdown",
- "id": "25241a4e",
+ "id": "d26a94a9",
"metadata": {
"editable": true
},
@@ -1241,7 +1241,7 @@
{
"cell_type": "code",
"execution_count": 11,
- "id": "06020cef",
+ "id": "c346d782",
"metadata": {
"collapsed": false,
"editable": true
@@ -1271,7 +1271,7 @@
},
{
"cell_type": "markdown",
- "id": "8d07d616",
+ "id": "947211f8",
"metadata": {
"editable": true
},
@@ -1282,7 +1282,7 @@
{
"cell_type": "code",
"execution_count": 12,
- "id": "1c4a2fd6",
+ "id": "3be55471",
"metadata": {
"collapsed": false,
"editable": true
@@ -1323,7 +1323,7 @@
},
{
"cell_type": "markdown",
- "id": "b9728189",
+ "id": "91b65208",
"metadata": {
"editable": true
},
@@ -1345,7 +1345,7 @@
{
"cell_type": "code",
"execution_count": 13,
- "id": "8b5b6d13",
+ "id": "c80885e3",
"metadata": {
"collapsed": false,
"editable": true
@@ -1385,7 +1385,7 @@
},
{
"cell_type": "markdown",
- "id": "ec654f7a",
+ "id": "df0689a5",
"metadata": {
"editable": true
},
@@ -1455,7 +1455,7 @@
},
{
"cell_type": "markdown",
- "id": "08d4a54f",
+ "id": "49bf1460",
"metadata": {
"editable": true
},
@@ -1467,7 +1467,7 @@
},
{
"cell_type": "markdown",
- "id": "df21cf23",
+ "id": "1881006d",
"metadata": {
"editable": true
},
@@ -1486,7 +1486,7 @@
},
{
"cell_type": "markdown",
- "id": "6359332f",
+ "id": "e34d3a6a",
"metadata": {
"editable": true
},
@@ -1498,7 +1498,7 @@
},
{
"cell_type": "markdown",
- "id": "6c8e31ed",
+ "id": "a45cb8e0",
"metadata": {
"editable": true
},
@@ -1510,7 +1510,7 @@
},
{
"cell_type": "markdown",
- "id": "0cc24030",
+ "id": "905abab9",
"metadata": {
"editable": true
},
@@ -1528,7 +1528,7 @@
},
{
"cell_type": "markdown",
- "id": "146809cb",
+ "id": "6a1f7209",
"metadata": {
"editable": true
},
@@ -1538,7 +1538,7 @@
},
{
"cell_type": "markdown",
- "id": "4796a596",
+ "id": "a90bc55b",
"metadata": {
"editable": true
},
@@ -1550,7 +1550,7 @@
},
{
"cell_type": "markdown",
- "id": "edfabcdf",
+ "id": "e91b18ae",
"metadata": {
"editable": true
},
@@ -1560,7 +1560,7 @@
},
{
"cell_type": "markdown",
- "id": "0b4a6062",
+ "id": "8afcf3ab",
"metadata": {
"editable": true
},
@@ -1572,7 +1572,7 @@
},
{
"cell_type": "markdown",
- "id": "e419ff68",
+ "id": "7bf8b155",
"metadata": {
"editable": true
},
@@ -1582,7 +1582,7 @@
},
{
"cell_type": "markdown",
- "id": "dd55bff1",
+ "id": "09e4774d",
"metadata": {
"editable": true
},
@@ -1594,7 +1594,7 @@
},
{
"cell_type": "markdown",
- "id": "df571b73",
+ "id": "adcffacb",
"metadata": {
"editable": true
},
@@ -1604,7 +1604,7 @@
},
{
"cell_type": "markdown",
- "id": "c1c6ffaa",
+ "id": "31ef2d69",
"metadata": {
"editable": true
},
@@ -1623,7 +1623,7 @@
},
{
"cell_type": "markdown",
- "id": "592bff5d",
+ "id": "3f74f04e",
"metadata": {
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},
@@ -1633,7 +1633,7 @@
},
{
"cell_type": "markdown",
- "id": "9b776152",
+ "id": "641f1a69",
"metadata": {
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@@ -1645,7 +1645,7 @@
},
{
"cell_type": "markdown",
- "id": "29c95a7a",
+ "id": "d3c3656a",
"metadata": {
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@@ -1661,7 +1661,7 @@
},
{
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- "id": "b53b0001",
+ "id": "5ff9ce6b",
"metadata": {
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@@ -1681,7 +1681,7 @@
},
{
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- "id": "f2fea940",
+ "id": "489081af",
"metadata": {
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@@ -1693,7 +1693,7 @@
},
{
"cell_type": "markdown",
- "id": "26411342",
+ "id": "09adf537",
"metadata": {
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@@ -1712,7 +1712,7 @@
},
{
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- "id": "d432e651",
+ "id": "ad443b1c",
"metadata": {
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},
@@ -1722,7 +1722,7 @@
},
{
"cell_type": "markdown",
- "id": "e536d0b3",
+ "id": "1d14f721",
"metadata": {
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@@ -1734,7 +1734,7 @@
},
{
"cell_type": "markdown",
- "id": "bbe0dd39",
+ "id": "9c202983",
"metadata": {
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@@ -1746,7 +1746,7 @@
},
{
"cell_type": "markdown",
- "id": "51d53970",
+ "id": "7b83a961",
"metadata": {
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@@ -1766,7 +1766,7 @@
},
{
"cell_type": "markdown",
- "id": "4778eaaf",
+ "id": "6ba4ceac",
"metadata": {
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@@ -1783,7 +1783,7 @@
{
"cell_type": "code",
"execution_count": 14,
- "id": "87ed061d",
+ "id": "c6247298",
"metadata": {
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"editable": true
@@ -1863,7 +1863,7 @@
},
{
"cell_type": "markdown",
- "id": "42b8f7c4",
+ "id": "80fa67c7",
"metadata": {
"editable": true
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@@ -1873,7 +1873,7 @@
},
{
"cell_type": "markdown",
- "id": "73ea9a01",
+ "id": "f0235c00",
"metadata": {
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@@ -1885,7 +1885,7 @@
},
{
"cell_type": "markdown",
- "id": "253891dd",
+ "id": "27f3ed42",
"metadata": {
"editable": true
},
@@ -1897,7 +1897,7 @@
},
{
"cell_type": "markdown",
- "id": "d2d20886",
+ "id": "9283df88",
"metadata": {
"editable": true
},
@@ -1909,7 +1909,7 @@
},
{
"cell_type": "markdown",
- "id": "480c4c58",
+ "id": "e3c659c0",
"metadata": {
"editable": true
},
@@ -1919,7 +1919,7 @@
},
{
"cell_type": "markdown",
- "id": "7a782da9",
+ "id": "a5701a60",
"metadata": {
"editable": true
},
@@ -1931,7 +1931,7 @@
},
{
"cell_type": "markdown",
- "id": "c1c60d77",
+ "id": "7b04480d",
"metadata": {
"editable": true
},
@@ -1941,7 +1941,7 @@
},
{
"cell_type": "markdown",
- "id": "60d26064",
+ "id": "13f43e44",
"metadata": {
"editable": true
},
@@ -1953,7 +1953,7 @@
},
{
"cell_type": "markdown",
- "id": "8a19c134",
+ "id": "953031c1",
"metadata": {
"editable": true
},
@@ -1966,7 +1966,7 @@
},
{
"cell_type": "markdown",
- "id": "f21d525f",
+ "id": "aafd8557",
"metadata": {
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@@ -1978,7 +1978,7 @@
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+ "id": "b45abbf0",
"metadata": {
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@@ -1990,7 +1990,7 @@
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@@ -2002,7 +2002,7 @@
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@@ -2013,7 +2013,7 @@
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@@ -2025,7 +2025,7 @@
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@@ -2044,7 +2044,7 @@
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@@ -2057,7 +2057,7 @@
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@@ -2067,7 +2067,7 @@
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@@ -2079,7 +2079,7 @@
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@@ -2089,7 +2089,7 @@
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@@ -2101,7 +2101,7 @@
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@@ -2111,7 +2111,7 @@
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@@ -2123,7 +2123,7 @@
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@@ -2133,7 +2133,7 @@
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@@ -2145,7 +2145,7 @@
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@@ -2155,7 +2155,7 @@
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@@ -2167,7 +2167,7 @@
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@@ -2177,7 +2177,7 @@
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@@ -2189,7 +2189,7 @@
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@@ -2367,7 +2367,7 @@
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@@ -2377,7 +2377,7 @@
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@@ -2405,7 +2405,7 @@
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@@ -2420,7 +2420,7 @@
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@@ -2431,7 +2431,7 @@
{
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@@ -2444,7 +2444,7 @@
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@@ -2455,7 +2455,7 @@
{
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@@ -2478,7 +2478,7 @@
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@@ -2490,7 +2490,7 @@
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@@ -2636,7 +2636,7 @@
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@@ -2646,7 +2646,7 @@
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@@ -2658,7 +2658,7 @@
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@@ -2668,7 +2668,7 @@
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@@ -2680,7 +2680,7 @@
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@@ -2692,7 +2692,7 @@
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@@ -2704,7 +2704,7 @@
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@@ -2714,7 +2714,7 @@
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@@ -2726,7 +2726,7 @@
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@@ -2736,7 +2736,7 @@
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@@ -2748,7 +2748,7 @@
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@@ -2758,7 +2758,7 @@
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@@ -2770,7 +2770,7 @@
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@@ -2780,7 +2780,7 @@
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@@ -2792,7 +2792,7 @@
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@@ -2802,7 +2802,7 @@
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@@ -2880,7 +2880,7 @@
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@@ -2961,7 +2961,7 @@
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@@ -2973,7 +2973,7 @@
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@@ -2983,7 +2983,7 @@
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@@ -2995,7 +2995,7 @@
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{
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},
{
"cell_type": "markdown",
- "id": "3d672d7a",
- "metadata": {
- "editable": true
- },
- "source": [
- "## The Boston housing data example\n",
- "\n",
- "The Boston housing \n",
- "data set was originally a part of UCI Machine Learning Repository\n",
- "and has been removed now. The data set is now included in **Scikit-Learn**'s \n",
- "library. There are 506 samples and 13 feature (predictor) variables\n",
- "in this data set. The objective is to predict the value of prices of\n",
- "the house using the features (predictors) listed here.\n",
- "\n",
- "The features/predictors are\n",
- "1. CRIM: Per capita crime rate by town\n",
- "\n",
- "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n",
- "\n",
- "3. INDUS: Proportion of non-retail business acres per town\n",
- "\n",
- "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
- "\n",
- "5. NOX: Nitric oxide concentration (parts per 10 million)\n",
- "\n",
- "6. RM: Average number of rooms per dwelling\n",
- "\n",
- "7. AGE: Proportion of owner-occupied units built prior to 1940\n",
- "\n",
- "8. DIS: Weighted distances to five Boston employment centers\n",
- "\n",
- "9. RAD: Index of accessibility to radial highways\n",
- "\n",
- "10. TAX: Full-value property tax rate per USD10000\n",
- "\n",
- "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n",
- "\n",
- "12. LSTAT: Percentage of lower status of the population\n",
- "\n",
- "13. MEDV: Median value of owner-occupied homes in USD 1000s"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "e2426f64",
- "metadata": {
- "editable": true
- },
- "source": [
- "## Housing data, the code\n",
- "We start by importing the libraries"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "id": "519e0c09",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "import numpy as np\n",
- "import matplotlib.pyplot as plt \n",
- "\n",
- "import pandas as pd \n",
- "import seaborn as sns"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "7b6fd188",
- "metadata": {
- "editable": true
- },
- "source": [
- "and load the Boston Housing DataSet from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 27,
- "id": "3665e1b0",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.datasets import load_boston\n",
- "\n",
- "boston_dataset = load_boston()\n",
- "\n",
- "# boston_dataset is a dictionary\n",
- "# let's check what it contains\n",
- "boston_dataset.keys()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "96f2fe7b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we invoke Pandas"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "id": "df2c84ca",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n",
- "boston.head()\n",
- "boston['MEDV'] = boston_dataset.target"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0c4588d5",
- "metadata": {
- "editable": true
- },
- "source": [
- "and preprocess the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 29,
- "id": "209438db",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# check for missing values in all the columns\n",
- "boston.isnull().sum()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "1b67c46a",
- "metadata": {
- "editable": true
- },
- "source": [
- "We can then visualize the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 30,
- "id": "236343bf",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# set the size of the figure\n",
- "sns.set(rc={'figure.figsize':(11.7,8.27)})\n",
- "\n",
- "# plot a histogram showing the distribution of the target values\n",
- "sns.distplot(boston['MEDV'], bins=30)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c8726877",
- "metadata": {
- "editable": true
- },
- "source": [
- "It is now useful to look at the correlation matrix"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 31,
- "id": "aaa47b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# compute the pair wise correlation for all columns \n",
- "correlation_matrix = boston.corr().round(2)\n",
- "# use the heatmap function from seaborn to plot the correlation matrix\n",
- "# annot = True to print the values inside the square\n",
- "sns.heatmap(data=correlation_matrix, annot=True)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0283f3f1",
- "metadata": {
- "editable": true
- },
- "source": [
- "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 32,
- "id": "c0823ed1",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "plt.figure(figsize=(20, 5))\n",
- "\n",
- "features = ['LSTAT', 'RM']\n",
- "target = boston['MEDV']\n",
- "\n",
- "for i, col in enumerate(features):\n",
- " plt.subplot(1, len(features) , i+1)\n",
- " x = boston[col]\n",
- " y = target\n",
- " plt.scatter(x, y, marker='o')\n",
- " plt.title(col)\n",
- " plt.xlabel(col)\n",
- " plt.ylabel('MEDV')"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "dc2cf448",
- "metadata": {
- "editable": true
- },
- "source": [
- "Now we start training our model"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 33,
- "id": "ee945b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n",
- "Y = boston['MEDV']"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "288a1417",
- "metadata": {
- "editable": true
- },
- "source": [
- "We split the data into training and test sets"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 34,
- "id": "0b640ff9",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.model_selection import train_test_split\n",
- "\n",
- "# splits the training and test data set in 80% : 20%\n",
- "# assign random_state to any value.This ensures consistency.\n",
- "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n",
- "print(X_train.shape)\n",
- "print(X_test.shape)\n",
- "print(Y_train.shape)\n",
- "print(Y_test.shape)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "32cfad5b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we use the linear regression functionality from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "id": "792df674",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.linear_model import LinearRegression\n",
- "from sklearn.metrics import mean_squared_error, r2_score\n",
- "\n",
- "lin_model = LinearRegression()\n",
- "lin_model.fit(X_train, Y_train)\n",
- "\n",
- "# model evaluation for training set\n",
- "\n",
- "y_train_predict = lin_model.predict(X_train)\n",
- "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n",
- "r2 = r2_score(Y_train, y_train_predict)\n",
- "\n",
- "print(\"The model performance for training set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))\n",
- "print(\"\\n\")\n",
- "\n",
- "# model evaluation for testing set\n",
- "\n",
- "y_test_predict = lin_model.predict(X_test)\n",
- "# root mean square error of the model\n",
- "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n",
- "\n",
- "# r-squared score of the model\n",
- "r2 = r2_score(Y_test, y_test_predict)\n",
- "\n",
- "print(\"The model performance for testing set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "id": "6f9196e5",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# plotting the y_test vs y_pred\n",
- "# ideally should have been a straight line\n",
- "plt.scatter(Y_test, y_test_predict)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "b72d5080",
+ "id": "44caf83f",
"metadata": {
"editable": true
},
@@ -3786,7 +3422,7 @@
},
{
"cell_type": "markdown",
- "id": "009b948b",
+ "id": "fea67f0c",
"metadata": {
"editable": true
},
@@ -3798,7 +3434,7 @@
},
{
"cell_type": "markdown",
- "id": "4b02a26a",
+ "id": "1af1f20c",
"metadata": {
"editable": true
},
@@ -3817,8 +3453,8 @@
},
{
"cell_type": "code",
- "execution_count": 37,
- "id": "efd083fc",
+ "execution_count": 26,
+ "id": "e9bcb7b1",
"metadata": {
"collapsed": false,
"editable": true
@@ -3852,7 +3488,7 @@
},
{
"cell_type": "markdown",
- "id": "b3590aea",
+ "id": "f3961700",
"metadata": {
"editable": true
},
@@ -3867,7 +3503,7 @@
},
{
"cell_type": "markdown",
- "id": "9592b7bf",
+ "id": "a33d2ba3",
"metadata": {
"editable": true
},
@@ -3879,7 +3515,7 @@
},
{
"cell_type": "markdown",
- "id": "0941f045",
+ "id": "3bc979c2",
"metadata": {
"editable": true
},
@@ -3889,7 +3525,7 @@
},
{
"cell_type": "markdown",
- "id": "d47fc8d3",
+ "id": "c234f16e",
"metadata": {
"editable": true
},
@@ -3912,8 +3548,8 @@
},
{
"cell_type": "code",
- "execution_count": 38,
- "id": "1ddb9cb5",
+ "execution_count": 27,
+ "id": "662bea3f",
"metadata": {
"collapsed": false,
"editable": true
@@ -3957,7 +3593,7 @@
},
{
"cell_type": "markdown",
- "id": "725b78e9",
+ "id": "05d214df",
"metadata": {
"editable": true
},
@@ -3967,7 +3603,7 @@
},
{
"cell_type": "markdown",
- "id": "b08f94e5",
+ "id": "aa18093a",
"metadata": {
"editable": true
},
@@ -4036,7 +3672,7 @@
},
{
"cell_type": "markdown",
- "id": "95e72a9e",
+ "id": "a8f251d6",
"metadata": {
"editable": true
},
@@ -4049,8 +3685,8 @@
},
{
"cell_type": "code",
- "execution_count": 39,
- "id": "fb3ad8e5",
+ "execution_count": 28,
+ "id": "48099cce",
"metadata": {
"collapsed": false,
"editable": true
@@ -4063,7 +3699,7 @@
},
{
"cell_type": "markdown",
- "id": "02fe1db6",
+ "id": "1cf31ef1",
"metadata": {
"editable": true
},
@@ -4077,7 +3713,7 @@
},
{
"cell_type": "markdown",
- "id": "33a4aed5",
+ "id": "dde166b5",
"metadata": {
"editable": true
},
@@ -4090,7 +3726,7 @@
},
{
"cell_type": "markdown",
- "id": "78a3bc86",
+ "id": "273b1e1d",
"metadata": {
"editable": true
},
@@ -4101,7 +3737,7 @@
},
{
"cell_type": "markdown",
- "id": "38c3a27d",
+ "id": "f7a4a3f1",
"metadata": {
"editable": true
},
@@ -4113,7 +3749,7 @@
},
{
"cell_type": "markdown",
- "id": "7eb5c51b",
+ "id": "0b3e8071",
"metadata": {
"editable": true
},
@@ -4123,7 +3759,7 @@
},
{
"cell_type": "markdown",
- "id": "3597b20a",
+ "id": "9f2d1bf3",
"metadata": {
"editable": true
},
@@ -4135,7 +3771,7 @@
},
{
"cell_type": "markdown",
- "id": "82922e13",
+ "id": "73c33ded",
"metadata": {
"editable": true
},
@@ -4150,8 +3786,8 @@
},
{
"cell_type": "code",
- "execution_count": 40,
- "id": "61cd693e",
+ "execution_count": 29,
+ "id": "c2fc4af7",
"metadata": {
"collapsed": false,
"editable": true
@@ -4202,7 +3838,7 @@
},
{
"cell_type": "markdown",
- "id": "2f8d2e6e",
+ "id": "a13aba09",
"metadata": {
"editable": true
},
@@ -4212,7 +3848,7 @@
},
{
"cell_type": "markdown",
- "id": "c81f5caf",
+ "id": "dcf58060",
"metadata": {
"editable": true
},
@@ -4257,8 +3893,8 @@
},
{
"cell_type": "code",
- "execution_count": 41,
- "id": "37ca3335",
+ "execution_count": 30,
+ "id": "f1bd9543",
"metadata": {
"collapsed": false,
"editable": true
@@ -4271,7 +3907,7 @@
},
{
"cell_type": "markdown",
- "id": "e6925e8c",
+ "id": "38030ca1",
"metadata": {
"editable": true
},
@@ -4281,8 +3917,8 @@
},
{
"cell_type": "code",
- "execution_count": 42,
- "id": "caecb70a",
+ "execution_count": 31,
+ "id": "e1d90b7a",
"metadata": {
"collapsed": false,
"editable": true
@@ -4297,7 +3933,7 @@
},
{
"cell_type": "markdown",
- "id": "deabdf0c",
+ "id": "4de86c60",
"metadata": {
"editable": true
},
@@ -4314,8 +3950,8 @@
},
{
"cell_type": "code",
- "execution_count": 43,
- "id": "b083bb84",
+ "execution_count": 32,
+ "id": "4aae540f",
"metadata": {
"collapsed": false,
"editable": true
@@ -4332,7 +3968,7 @@
},
{
"cell_type": "markdown",
- "id": "21102c44",
+ "id": "966d110b",
"metadata": {
"editable": true
},
@@ -4347,8 +3983,8 @@
},
{
"cell_type": "code",
- "execution_count": 44,
- "id": "b9108dab",
+ "execution_count": 33,
+ "id": "011363cd",
"metadata": {
"collapsed": false,
"editable": true
@@ -4392,7 +4028,7 @@
},
{
"cell_type": "markdown",
- "id": "cd19b575",
+ "id": "cf8dd05c",
"metadata": {
"editable": true
},
@@ -4402,7 +4038,7 @@
},
{
"cell_type": "markdown",
- "id": "eb2f6352",
+ "id": "d3c9b66f",
"metadata": {
"editable": true
},
@@ -4413,7 +4049,7 @@
},
{
"cell_type": "markdown",
- "id": "6e09ea94",
+ "id": "3363b9e6",
"metadata": {
"editable": true
},
@@ -4424,7 +4060,7 @@
},
{
"cell_type": "markdown",
- "id": "38857156",
+ "id": "33ac5d0b",
"metadata": {
"editable": true
},
@@ -4435,7 +4071,7 @@
},
{
"cell_type": "markdown",
- "id": "6b9115a6",
+ "id": "e80b75d3",
"metadata": {
"editable": true
},
@@ -4457,8 +4093,8 @@
},
{
"cell_type": "code",
- "execution_count": 45,
- "id": "ae7a71c1",
+ "execution_count": 34,
+ "id": "ba7cc092",
"metadata": {
"collapsed": false,
"editable": true
@@ -4471,7 +4107,7 @@
},
{
"cell_type": "markdown",
- "id": "ca615d39",
+ "id": "36b73ce3",
"metadata": {
"editable": true
},
@@ -4488,7 +4124,7 @@
},
{
"cell_type": "markdown",
- "id": "d83c8354",
+ "id": "9e7ac684",
"metadata": {
"editable": true
},
@@ -4501,7 +4137,7 @@
},
{
"cell_type": "markdown",
- "id": "74f6e912",
+ "id": "b94b5ddd",
"metadata": {
"editable": true
},
@@ -4512,7 +4148,7 @@
},
{
"cell_type": "markdown",
- "id": "93761664",
+ "id": "f8776c45",
"metadata": {
"editable": true
},
@@ -4524,7 +4160,7 @@
},
{
"cell_type": "markdown",
- "id": "be729d32",
+ "id": "40a07c84",
"metadata": {
"editable": true
},
@@ -4534,7 +4170,7 @@
},
{
"cell_type": "markdown",
- "id": "08549523",
+ "id": "55a234b6",
"metadata": {
"editable": true
},
@@ -4546,7 +4182,7 @@
},
{
"cell_type": "markdown",
- "id": "03cc0ca0",
+ "id": "59dc00fb",
"metadata": {
"editable": true
},
@@ -4562,8 +4198,8 @@
},
{
"cell_type": "code",
- "execution_count": 46,
- "id": "9b1b9378",
+ "execution_count": 35,
+ "id": "44a8d4ef",
"metadata": {
"collapsed": false,
"editable": true
@@ -4655,7 +4291,7 @@
},
{
"cell_type": "markdown",
- "id": "341e9820",
+ "id": "703ec310",
"metadata": {
"editable": true
},
@@ -4665,7 +4301,7 @@
},
{
"cell_type": "markdown",
- "id": "6feb372f",
+ "id": "980aceb3",
"metadata": {
"editable": true
},
@@ -4687,7 +4323,7 @@
},
{
"cell_type": "markdown",
- "id": "3c12062d",
+ "id": "c2c4fafc",
"metadata": {
"editable": true
},
@@ -4699,7 +4335,7 @@
},
{
"cell_type": "markdown",
- "id": "edfcd8ff",
+ "id": "84a561a2",
"metadata": {
"editable": true
},
@@ -4709,7 +4345,7 @@
},
{
"cell_type": "markdown",
- "id": "e8689eb2",
+ "id": "bb9a9639",
"metadata": {
"editable": true
},
@@ -4721,7 +4357,7 @@
},
{
"cell_type": "markdown",
- "id": "62145d09",
+ "id": "8f1dd1a6",
"metadata": {
"editable": true
},
@@ -4731,7 +4367,7 @@
},
{
"cell_type": "markdown",
- "id": "ec992c25",
+ "id": "74dcf2c0",
"metadata": {
"editable": true
},
@@ -4743,7 +4379,7 @@
},
{
"cell_type": "markdown",
- "id": "4a922090",
+ "id": "c530cce6",
"metadata": {
"editable": true
},
@@ -4758,7 +4394,7 @@
},
{
"cell_type": "markdown",
- "id": "10f294a3",
+ "id": "f337bf71",
"metadata": {
"editable": true
},
@@ -4770,7 +4406,7 @@
},
{
"cell_type": "markdown",
- "id": "1dfa508a",
+ "id": "cfa053a1",
"metadata": {
"editable": true
},
@@ -4780,7 +4416,7 @@
},
{
"cell_type": "markdown",
- "id": "b5474b9f",
+ "id": "e9035e0c",
"metadata": {
"editable": true
},
@@ -4792,7 +4428,7 @@
},
{
"cell_type": "markdown",
- "id": "4dee1baf",
+ "id": "1b198445",
"metadata": {
"editable": true
},
@@ -4802,7 +4438,7 @@
},
{
"cell_type": "markdown",
- "id": "9949ee71",
+ "id": "e21f0cf0",
"metadata": {
"editable": true
},
@@ -4814,7 +4450,7 @@
},
{
"cell_type": "markdown",
- "id": "73ce2a98",
+ "id": "8918cc8c",
"metadata": {
"editable": true
},
@@ -4824,7 +4460,7 @@
},
{
"cell_type": "markdown",
- "id": "41e1008d",
+ "id": "bed599b9",
"metadata": {
"editable": true
},
@@ -4836,7 +4472,7 @@
},
{
"cell_type": "markdown",
- "id": "597c4ebb",
+ "id": "4faf8bfa",
"metadata": {
"editable": true
},
@@ -4846,7 +4482,7 @@
},
{
"cell_type": "markdown",
- "id": "e9163f17",
+ "id": "ebde5b6c",
"metadata": {
"editable": true
},
@@ -4858,7 +4494,7 @@
},
{
"cell_type": "markdown",
- "id": "fc3d2fdb",
+ "id": "5ef54737",
"metadata": {
"editable": true
},
@@ -4868,7 +4504,7 @@
},
{
"cell_type": "markdown",
- "id": "e7cec690",
+ "id": "6ef07a21",
"metadata": {
"editable": true
},
@@ -4880,7 +4516,7 @@
},
{
"cell_type": "markdown",
- "id": "1a74bac9",
+ "id": "7d5bcb91",
"metadata": {
"editable": true
},
@@ -4890,7 +4526,7 @@
},
{
"cell_type": "markdown",
- "id": "7087bc76",
+ "id": "1d35a7b9",
"metadata": {
"editable": true
},
@@ -4902,7 +4538,7 @@
},
{
"cell_type": "markdown",
- "id": "1e5a21fd",
+ "id": "90fcecec",
"metadata": {
"editable": true
},
@@ -4912,7 +4548,7 @@
},
{
"cell_type": "markdown",
- "id": "624d40e0",
+ "id": "4c2ab747",
"metadata": {
"editable": true
},
@@ -4924,7 +4560,7 @@
},
{
"cell_type": "markdown",
- "id": "d871f62f",
+ "id": "ae1d9316",
"metadata": {
"editable": true
},
@@ -4934,7 +4570,7 @@
},
{
"cell_type": "markdown",
- "id": "675aea37",
+ "id": "aa76ae3d",
"metadata": {
"editable": true
},
@@ -4946,7 +4582,7 @@
},
{
"cell_type": "markdown",
- "id": "746b74e2",
+ "id": "473eca6a",
"metadata": {
"editable": true
},
@@ -4956,7 +4592,7 @@
},
{
"cell_type": "markdown",
- "id": "6381d6ee",
+ "id": "8476e871",
"metadata": {
"editable": true
},
@@ -4968,7 +4604,7 @@
},
{
"cell_type": "markdown",
- "id": "4331c20e",
+ "id": "70fa4295",
"metadata": {
"editable": true
},
diff --git a/doc/LectureNotes/chapter1.ipynb b/doc/LectureNotes/chapter1.ipynb
index 231068020..3a8e25dcf 100644
--- a/doc/LectureNotes/chapter1.ipynb
+++ b/doc/LectureNotes/chapter1.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
- "id": "a453b968",
+ "id": "7d0d70d8",
"metadata": {
"editable": true
},
@@ -13,7 +13,7 @@
},
{
"cell_type": "markdown",
- "id": "499b2ddb",
+ "id": "3399a55c",
"metadata": {
"editable": true
},
@@ -23,7 +23,7 @@
},
{
"cell_type": "markdown",
- "id": "c84cce7e",
+ "id": "e7513472",
"metadata": {
"editable": true
},
@@ -65,7 +65,7 @@
},
{
"cell_type": "markdown",
- "id": "8419208e",
+ "id": "17b7ba24",
"metadata": {
"editable": true
},
@@ -167,7 +167,7 @@
},
{
"cell_type": "markdown",
- "id": "ceb7a805",
+ "id": "23adffca",
"metadata": {
"editable": true
},
@@ -202,7 +202,7 @@
},
{
"cell_type": "markdown",
- "id": "d6c1062f",
+ "id": "1263a25e",
"metadata": {
"editable": true
},
@@ -253,7 +253,7 @@
},
{
"cell_type": "markdown",
- "id": "50c1b706",
+ "id": "e2ab4f0a",
"metadata": {
"editable": true
},
@@ -286,7 +286,7 @@
},
{
"cell_type": "markdown",
- "id": "12e3ac84",
+ "id": "23173385",
"metadata": {
"editable": true
},
@@ -298,7 +298,7 @@
},
{
"cell_type": "markdown",
- "id": "10b4c333",
+ "id": "57b14c43",
"metadata": {
"editable": true
},
@@ -306,10 +306,10 @@
"where $N(0,1)$ represents random numbers generated by the normal\n",
"distribution. From **Scikit-Learn** we import then the\n",
"**LinearRegression** functionality and make a prediction $\\tilde{y} =\n",
- "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n",
+ "\\alpha + \\theta x$ using the function **fit(x,y)**. We call the set of\n",
"data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n",
"**scikit-learn** has also a functionality which extracts the above\n",
- "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n",
+ "fitting parameters $\\alpha$ and $\\theta$ (see below). Later we will\n",
"distinguish between training data and test data.\n",
"\n",
"For plotting we use the Python package\n",
@@ -335,7 +335,7 @@
{
"cell_type": "code",
"execution_count": 1,
- "id": "3e216ef0",
+ "id": "f7d66caa",
"metadata": {
"collapsed": false,
"editable": true
@@ -368,7 +368,7 @@
},
{
"cell_type": "markdown",
- "id": "af765a12",
+ "id": "504848dc",
"metadata": {
"editable": true
},
@@ -385,7 +385,7 @@
},
{
"cell_type": "markdown",
- "id": "2f6fd730",
+ "id": "bff268fb",
"metadata": {
"editable": true
},
@@ -397,7 +397,7 @@
},
{
"cell_type": "markdown",
- "id": "1770d85d",
+ "id": "53357604",
"metadata": {
"editable": true
},
@@ -418,7 +418,7 @@
},
{
"cell_type": "markdown",
- "id": "684b72a0",
+ "id": "4e5b1dad",
"metadata": {
"editable": true
},
@@ -431,7 +431,7 @@
},
{
"cell_type": "markdown",
- "id": "e9fa8fd5",
+ "id": "b8e0f4c7",
"metadata": {
"editable": true
},
@@ -443,7 +443,7 @@
"\n",
"Minimizing the cost function is a central aspect of\n",
"our discussions to come. Finding its minima as function of the model\n",
- "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n",
+ "parameters ($\\alpha$ and $\\theta$ in our case) will be a recurring\n",
"theme in these series of lectures. Essentially all machine learning\n",
"algorithms we will discuss center around the minimization of the\n",
"chosen cost function. This depends in turn on our specific\n",
@@ -462,7 +462,7 @@
},
{
"cell_type": "markdown",
- "id": "15a4642f",
+ "id": "c4e4b512",
"metadata": {
"editable": true
},
@@ -474,7 +474,7 @@
},
{
"cell_type": "markdown",
- "id": "589ec9cb",
+ "id": "71393ea9",
"metadata": {
"editable": true
},
@@ -492,7 +492,7 @@
{
"cell_type": "code",
"execution_count": 2,
- "id": "cf21ae2b",
+ "id": "a6c77c37",
"metadata": {
"collapsed": false,
"editable": true
@@ -520,7 +520,7 @@
},
{
"cell_type": "markdown",
- "id": "41c55cbb",
+ "id": "bc21adc2",
"metadata": {
"editable": true
},
@@ -531,7 +531,7 @@
"relative error.\n",
"\n",
"As mentioned above, **Scikit-Learn** has an impressive functionality.\n",
- "We can for example extract the values of $\\alpha$ and $\\beta$ and\n",
+ "We can for example extract the values of $\\alpha$ and $\\theta$ and\n",
"their error estimates, or the variance and standard deviation and many\n",
"other properties from the statistical data analysis. \n",
"\n",
@@ -542,7 +542,7 @@
{
"cell_type": "code",
"execution_count": 3,
- "id": "d62c9412",
+ "id": "febc49fc",
"metadata": {
"collapsed": false,
"editable": true
@@ -560,7 +560,7 @@
"linreg.fit(x,y)\n",
"ypredict = linreg.predict(x)\n",
"print('The intercept alpha: \\n', linreg.intercept_)\n",
- "print('Coefficient beta : \\n', linreg.coef_)\n",
+ "print('Coefficient theta : \\n', linreg.coef_)\n",
"# The mean squared error \n",
"print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n",
"# Explained variance score: 1 is perfect prediction \n",
@@ -580,18 +580,18 @@
},
{
"cell_type": "markdown",
- "id": "ae0c6c2a",
+ "id": "3f1c6408",
"metadata": {
"editable": true
},
"source": [
- "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n",
- "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
+ "The function **coef** gives us the parameter $\\theta$ of our fit while **intercept** yields \n",
+ "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\theta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as"
]
},
{
"cell_type": "markdown",
- "id": "64cf13de",
+ "id": "b4e26d20",
"metadata": {
"editable": true
},
@@ -604,7 +604,7 @@
},
{
"cell_type": "markdown",
- "id": "18439dae",
+ "id": "8137fd70",
"metadata": {
"editable": true
},
@@ -625,7 +625,7 @@
},
{
"cell_type": "markdown",
- "id": "3e9fb291",
+ "id": "3f40e79c",
"metadata": {
"editable": true
},
@@ -637,7 +637,7 @@
},
{
"cell_type": "markdown",
- "id": "46d8744e",
+ "id": "6a8eb368",
"metadata": {
"editable": true
},
@@ -647,7 +647,7 @@
},
{
"cell_type": "markdown",
- "id": "38a29b65",
+ "id": "71d8af23",
"metadata": {
"editable": true
},
@@ -659,7 +659,7 @@
},
{
"cell_type": "markdown",
- "id": "438e73b8",
+ "id": "aee25556",
"metadata": {
"editable": true
},
@@ -671,7 +671,7 @@
},
{
"cell_type": "markdown",
- "id": "1403cc6a",
+ "id": "1bc1d5e2",
"metadata": {
"editable": true
},
@@ -683,7 +683,7 @@
},
{
"cell_type": "markdown",
- "id": "9fbd1c1b",
+ "id": "dfb80b0d",
"metadata": {
"editable": true
},
@@ -694,7 +694,7 @@
},
{
"cell_type": "markdown",
- "id": "d25da1d3",
+ "id": "1ee8b15b",
"metadata": {
"editable": true
},
@@ -706,7 +706,7 @@
},
{
"cell_type": "markdown",
- "id": "c43c7452",
+ "id": "9ec15fef",
"metadata": {
"editable": true
},
@@ -728,7 +728,7 @@
},
{
"cell_type": "markdown",
- "id": "ad0d5004",
+ "id": "8e386295",
"metadata": {
"editable": true
},
@@ -740,7 +740,7 @@
},
{
"cell_type": "markdown",
- "id": "eb9e2efb",
+ "id": "d8d8ad23",
"metadata": {
"editable": true
},
@@ -755,7 +755,7 @@
{
"cell_type": "code",
"execution_count": 4,
- "id": "963f4f84",
+ "id": "0c9a0500",
"metadata": {
"collapsed": false,
"editable": true
@@ -796,7 +796,7 @@
},
{
"cell_type": "markdown",
- "id": "d530f60a",
+ "id": "c7786c3b",
"metadata": {
"editable": true
},
@@ -811,7 +811,7 @@
},
{
"cell_type": "markdown",
- "id": "f8951307",
+ "id": "83f5d1f7",
"metadata": {
"editable": true
},
@@ -823,7 +823,7 @@
},
{
"cell_type": "markdown",
- "id": "913bb6a6",
+ "id": "36b41230",
"metadata": {
"editable": true
},
@@ -833,7 +833,7 @@
},
{
"cell_type": "markdown",
- "id": "25cb82ce",
+ "id": "5476ceac",
"metadata": {
"editable": true
},
@@ -845,7 +845,7 @@
},
{
"cell_type": "markdown",
- "id": "d04f788d",
+ "id": "a0252aa4",
"metadata": {
"editable": true
},
@@ -855,7 +855,7 @@
},
{
"cell_type": "markdown",
- "id": "866bf69f",
+ "id": "993c3fe8",
"metadata": {
"editable": true
},
@@ -867,7 +867,7 @@
},
{
"cell_type": "markdown",
- "id": "a0d3421f",
+ "id": "699624c4",
"metadata": {
"editable": true
},
@@ -877,7 +877,7 @@
},
{
"cell_type": "markdown",
- "id": "680eefd0",
+ "id": "09e37178",
"metadata": {
"editable": true
},
@@ -889,7 +889,7 @@
},
{
"cell_type": "markdown",
- "id": "de97e875",
+ "id": "5dca3cfd",
"metadata": {
"editable": true
},
@@ -905,7 +905,7 @@
},
{
"cell_type": "markdown",
- "id": "5d7dbc49",
+ "id": "6e758f39",
"metadata": {
"editable": true
},
@@ -917,7 +917,7 @@
},
{
"cell_type": "markdown",
- "id": "5a9a6d4d",
+ "id": "cd2a7727",
"metadata": {
"editable": true
},
@@ -928,7 +928,7 @@
},
{
"cell_type": "markdown",
- "id": "fef247ce",
+ "id": "e8b8a9d8",
"metadata": {
"editable": true
},
@@ -940,7 +940,7 @@
},
{
"cell_type": "markdown",
- "id": "3ed61266",
+ "id": "b390292b",
"metadata": {
"editable": true
},
@@ -954,7 +954,7 @@
},
{
"cell_type": "markdown",
- "id": "9882fb4e",
+ "id": "cfdb1cf9",
"metadata": {
"editable": true
},
@@ -966,7 +966,7 @@
},
{
"cell_type": "markdown",
- "id": "992b3ae7",
+ "id": "10b030a0",
"metadata": {
"editable": true
},
@@ -991,7 +991,7 @@
},
{
"cell_type": "markdown",
- "id": "6d772376",
+ "id": "cb289c64",
"metadata": {
"editable": true
},
@@ -1008,7 +1008,7 @@
{
"cell_type": "code",
"execution_count": 5,
- "id": "f861cd3f",
+ "id": "cca06315",
"metadata": {
"collapsed": false,
"editable": true
@@ -1052,7 +1052,7 @@
},
{
"cell_type": "markdown",
- "id": "da27a328",
+ "id": "518d9d53",
"metadata": {
"editable": true
},
@@ -1069,7 +1069,7 @@
{
"cell_type": "code",
"execution_count": 6,
- "id": "b4ef8b5a",
+ "id": "38f65b25",
"metadata": {
"collapsed": false,
"editable": true
@@ -1090,7 +1090,7 @@
},
{
"cell_type": "markdown",
- "id": "1352f7d2",
+ "id": "7df95f40",
"metadata": {
"editable": true
},
@@ -1104,7 +1104,7 @@
{
"cell_type": "code",
"execution_count": 7,
- "id": "c5b3333d",
+ "id": "bdf76b5e",
"metadata": {
"collapsed": false,
"editable": true
@@ -1133,7 +1133,7 @@
},
{
"cell_type": "markdown",
- "id": "6652db3a",
+ "id": "becf6238",
"metadata": {
"editable": true
},
@@ -1153,7 +1153,7 @@
{
"cell_type": "code",
"execution_count": 8,
- "id": "715c03ae",
+ "id": "280c0700",
"metadata": {
"collapsed": false,
"editable": true
@@ -1170,7 +1170,7 @@
},
{
"cell_type": "markdown",
- "id": "eaff6065",
+ "id": "46814807",
"metadata": {
"editable": true
},
@@ -1182,7 +1182,7 @@
{
"cell_type": "code",
"execution_count": 9,
- "id": "4053e205",
+ "id": "0f758df8",
"metadata": {
"collapsed": false,
"editable": true
@@ -1200,7 +1200,7 @@
},
{
"cell_type": "markdown",
- "id": "587289d6",
+ "id": "fda4c0ab",
"metadata": {
"editable": true
},
@@ -1216,7 +1216,7 @@
{
"cell_type": "code",
"execution_count": 10,
- "id": "52f1c9dc",
+ "id": "8b295d45",
"metadata": {
"collapsed": false,
"editable": true
@@ -1229,7 +1229,7 @@
},
{
"cell_type": "markdown",
- "id": "25241a4e",
+ "id": "d26a94a9",
"metadata": {
"editable": true
},
@@ -1241,7 +1241,7 @@
{
"cell_type": "code",
"execution_count": 11,
- "id": "06020cef",
+ "id": "c346d782",
"metadata": {
"collapsed": false,
"editable": true
@@ -1271,7 +1271,7 @@
},
{
"cell_type": "markdown",
- "id": "8d07d616",
+ "id": "947211f8",
"metadata": {
"editable": true
},
@@ -1282,7 +1282,7 @@
{
"cell_type": "code",
"execution_count": 12,
- "id": "1c4a2fd6",
+ "id": "3be55471",
"metadata": {
"collapsed": false,
"editable": true
@@ -1323,7 +1323,7 @@
},
{
"cell_type": "markdown",
- "id": "b9728189",
+ "id": "91b65208",
"metadata": {
"editable": true
},
@@ -1345,7 +1345,7 @@
{
"cell_type": "code",
"execution_count": 13,
- "id": "8b5b6d13",
+ "id": "c80885e3",
"metadata": {
"collapsed": false,
"editable": true
@@ -1385,7 +1385,7 @@
},
{
"cell_type": "markdown",
- "id": "ec654f7a",
+ "id": "df0689a5",
"metadata": {
"editable": true
},
@@ -1455,7 +1455,7 @@
},
{
"cell_type": "markdown",
- "id": "08d4a54f",
+ "id": "49bf1460",
"metadata": {
"editable": true
},
@@ -1467,7 +1467,7 @@
},
{
"cell_type": "markdown",
- "id": "df21cf23",
+ "id": "1881006d",
"metadata": {
"editable": true
},
@@ -1486,7 +1486,7 @@
},
{
"cell_type": "markdown",
- "id": "6359332f",
+ "id": "e34d3a6a",
"metadata": {
"editable": true
},
@@ -1498,7 +1498,7 @@
},
{
"cell_type": "markdown",
- "id": "6c8e31ed",
+ "id": "a45cb8e0",
"metadata": {
"editable": true
},
@@ -1510,7 +1510,7 @@
},
{
"cell_type": "markdown",
- "id": "0cc24030",
+ "id": "905abab9",
"metadata": {
"editable": true
},
@@ -1528,7 +1528,7 @@
},
{
"cell_type": "markdown",
- "id": "146809cb",
+ "id": "6a1f7209",
"metadata": {
"editable": true
},
@@ -1538,7 +1538,7 @@
},
{
"cell_type": "markdown",
- "id": "4796a596",
+ "id": "a90bc55b",
"metadata": {
"editable": true
},
@@ -1550,7 +1550,7 @@
},
{
"cell_type": "markdown",
- "id": "edfabcdf",
+ "id": "e91b18ae",
"metadata": {
"editable": true
},
@@ -1560,7 +1560,7 @@
},
{
"cell_type": "markdown",
- "id": "0b4a6062",
+ "id": "8afcf3ab",
"metadata": {
"editable": true
},
@@ -1572,7 +1572,7 @@
},
{
"cell_type": "markdown",
- "id": "e419ff68",
+ "id": "7bf8b155",
"metadata": {
"editable": true
},
@@ -1582,7 +1582,7 @@
},
{
"cell_type": "markdown",
- "id": "dd55bff1",
+ "id": "09e4774d",
"metadata": {
"editable": true
},
@@ -1594,7 +1594,7 @@
},
{
"cell_type": "markdown",
- "id": "df571b73",
+ "id": "adcffacb",
"metadata": {
"editable": true
},
@@ -1604,7 +1604,7 @@
},
{
"cell_type": "markdown",
- "id": "c1c6ffaa",
+ "id": "31ef2d69",
"metadata": {
"editable": true
},
@@ -1623,7 +1623,7 @@
},
{
"cell_type": "markdown",
- "id": "592bff5d",
+ "id": "3f74f04e",
"metadata": {
"editable": true
},
@@ -1633,7 +1633,7 @@
},
{
"cell_type": "markdown",
- "id": "9b776152",
+ "id": "641f1a69",
"metadata": {
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},
@@ -1645,7 +1645,7 @@
},
{
"cell_type": "markdown",
- "id": "29c95a7a",
+ "id": "d3c3656a",
"metadata": {
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@@ -1661,7 +1661,7 @@
},
{
"cell_type": "markdown",
- "id": "b53b0001",
+ "id": "5ff9ce6b",
"metadata": {
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@@ -1681,7 +1681,7 @@
},
{
"cell_type": "markdown",
- "id": "f2fea940",
+ "id": "489081af",
"metadata": {
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},
@@ -1693,7 +1693,7 @@
},
{
"cell_type": "markdown",
- "id": "26411342",
+ "id": "09adf537",
"metadata": {
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@@ -1712,7 +1712,7 @@
},
{
"cell_type": "markdown",
- "id": "d432e651",
+ "id": "ad443b1c",
"metadata": {
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},
@@ -1722,7 +1722,7 @@
},
{
"cell_type": "markdown",
- "id": "e536d0b3",
+ "id": "1d14f721",
"metadata": {
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},
@@ -1734,7 +1734,7 @@
},
{
"cell_type": "markdown",
- "id": "bbe0dd39",
+ "id": "9c202983",
"metadata": {
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@@ -1746,7 +1746,7 @@
},
{
"cell_type": "markdown",
- "id": "51d53970",
+ "id": "7b83a961",
"metadata": {
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@@ -1766,7 +1766,7 @@
},
{
"cell_type": "markdown",
- "id": "4778eaaf",
+ "id": "6ba4ceac",
"metadata": {
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@@ -1783,7 +1783,7 @@
{
"cell_type": "code",
"execution_count": 14,
- "id": "87ed061d",
+ "id": "c6247298",
"metadata": {
"collapsed": false,
"editable": true
@@ -1863,7 +1863,7 @@
},
{
"cell_type": "markdown",
- "id": "42b8f7c4",
+ "id": "80fa67c7",
"metadata": {
"editable": true
},
@@ -1873,7 +1873,7 @@
},
{
"cell_type": "markdown",
- "id": "73ea9a01",
+ "id": "f0235c00",
"metadata": {
"editable": true
},
@@ -1885,7 +1885,7 @@
},
{
"cell_type": "markdown",
- "id": "253891dd",
+ "id": "27f3ed42",
"metadata": {
"editable": true
},
@@ -1897,7 +1897,7 @@
},
{
"cell_type": "markdown",
- "id": "d2d20886",
+ "id": "9283df88",
"metadata": {
"editable": true
},
@@ -1909,7 +1909,7 @@
},
{
"cell_type": "markdown",
- "id": "480c4c58",
+ "id": "e3c659c0",
"metadata": {
"editable": true
},
@@ -1919,7 +1919,7 @@
},
{
"cell_type": "markdown",
- "id": "7a782da9",
+ "id": "a5701a60",
"metadata": {
"editable": true
},
@@ -1931,7 +1931,7 @@
},
{
"cell_type": "markdown",
- "id": "c1c60d77",
+ "id": "7b04480d",
"metadata": {
"editable": true
},
@@ -1941,7 +1941,7 @@
},
{
"cell_type": "markdown",
- "id": "60d26064",
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@@ -2057,7 +2057,7 @@
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@@ -2067,7 +2067,7 @@
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@@ -2079,7 +2079,7 @@
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@@ -2111,7 +2111,7 @@
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},
{
"cell_type": "markdown",
- "id": "3d672d7a",
- "metadata": {
- "editable": true
- },
- "source": [
- "## The Boston housing data example\n",
- "\n",
- "The Boston housing \n",
- "data set was originally a part of UCI Machine Learning Repository\n",
- "and has been removed now. The data set is now included in **Scikit-Learn**'s \n",
- "library. There are 506 samples and 13 feature (predictor) variables\n",
- "in this data set. The objective is to predict the value of prices of\n",
- "the house using the features (predictors) listed here.\n",
- "\n",
- "The features/predictors are\n",
- "1. CRIM: Per capita crime rate by town\n",
- "\n",
- "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n",
- "\n",
- "3. INDUS: Proportion of non-retail business acres per town\n",
- "\n",
- "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
- "\n",
- "5. NOX: Nitric oxide concentration (parts per 10 million)\n",
- "\n",
- "6. RM: Average number of rooms per dwelling\n",
- "\n",
- "7. AGE: Proportion of owner-occupied units built prior to 1940\n",
- "\n",
- "8. DIS: Weighted distances to five Boston employment centers\n",
- "\n",
- "9. RAD: Index of accessibility to radial highways\n",
- "\n",
- "10. TAX: Full-value property tax rate per USD10000\n",
- "\n",
- "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n",
- "\n",
- "12. LSTAT: Percentage of lower status of the population\n",
- "\n",
- "13. MEDV: Median value of owner-occupied homes in USD 1000s"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "e2426f64",
- "metadata": {
- "editable": true
- },
- "source": [
- "## Housing data, the code\n",
- "We start by importing the libraries"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "id": "519e0c09",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "import numpy as np\n",
- "import matplotlib.pyplot as plt \n",
- "\n",
- "import pandas as pd \n",
- "import seaborn as sns"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "7b6fd188",
- "metadata": {
- "editable": true
- },
- "source": [
- "and load the Boston Housing DataSet from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 27,
- "id": "3665e1b0",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.datasets import load_boston\n",
- "\n",
- "boston_dataset = load_boston()\n",
- "\n",
- "# boston_dataset is a dictionary\n",
- "# let's check what it contains\n",
- "boston_dataset.keys()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "96f2fe7b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we invoke Pandas"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "id": "df2c84ca",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n",
- "boston.head()\n",
- "boston['MEDV'] = boston_dataset.target"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0c4588d5",
- "metadata": {
- "editable": true
- },
- "source": [
- "and preprocess the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 29,
- "id": "209438db",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# check for missing values in all the columns\n",
- "boston.isnull().sum()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "1b67c46a",
- "metadata": {
- "editable": true
- },
- "source": [
- "We can then visualize the data"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 30,
- "id": "236343bf",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# set the size of the figure\n",
- "sns.set(rc={'figure.figsize':(11.7,8.27)})\n",
- "\n",
- "# plot a histogram showing the distribution of the target values\n",
- "sns.distplot(boston['MEDV'], bins=30)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "c8726877",
- "metadata": {
- "editable": true
- },
- "source": [
- "It is now useful to look at the correlation matrix"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 31,
- "id": "aaa47b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# compute the pair wise correlation for all columns \n",
- "correlation_matrix = boston.corr().round(2)\n",
- "# use the heatmap function from seaborn to plot the correlation matrix\n",
- "# annot = True to print the values inside the square\n",
- "sns.heatmap(data=correlation_matrix, annot=True)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0283f3f1",
- "metadata": {
- "editable": true
- },
- "source": [
- "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 32,
- "id": "c0823ed1",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "plt.figure(figsize=(20, 5))\n",
- "\n",
- "features = ['LSTAT', 'RM']\n",
- "target = boston['MEDV']\n",
- "\n",
- "for i, col in enumerate(features):\n",
- " plt.subplot(1, len(features) , i+1)\n",
- " x = boston[col]\n",
- " y = target\n",
- " plt.scatter(x, y, marker='o')\n",
- " plt.title(col)\n",
- " plt.xlabel(col)\n",
- " plt.ylabel('MEDV')"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "dc2cf448",
- "metadata": {
- "editable": true
- },
- "source": [
- "Now we start training our model"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 33,
- "id": "ee945b00",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n",
- "Y = boston['MEDV']"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "288a1417",
- "metadata": {
- "editable": true
- },
- "source": [
- "We split the data into training and test sets"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 34,
- "id": "0b640ff9",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.model_selection import train_test_split\n",
- "\n",
- "# splits the training and test data set in 80% : 20%\n",
- "# assign random_state to any value.This ensures consistency.\n",
- "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n",
- "print(X_train.shape)\n",
- "print(X_test.shape)\n",
- "print(Y_train.shape)\n",
- "print(Y_test.shape)"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "32cfad5b",
- "metadata": {
- "editable": true
- },
- "source": [
- "Then we use the linear regression functionality from **Scikit-Learn**"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "id": "792df674",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "from sklearn.linear_model import LinearRegression\n",
- "from sklearn.metrics import mean_squared_error, r2_score\n",
- "\n",
- "lin_model = LinearRegression()\n",
- "lin_model.fit(X_train, Y_train)\n",
- "\n",
- "# model evaluation for training set\n",
- "\n",
- "y_train_predict = lin_model.predict(X_train)\n",
- "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n",
- "r2 = r2_score(Y_train, y_train_predict)\n",
- "\n",
- "print(\"The model performance for training set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))\n",
- "print(\"\\n\")\n",
- "\n",
- "# model evaluation for testing set\n",
- "\n",
- "y_test_predict = lin_model.predict(X_test)\n",
- "# root mean square error of the model\n",
- "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n",
- "\n",
- "# r-squared score of the model\n",
- "r2 = r2_score(Y_test, y_test_predict)\n",
- "\n",
- "print(\"The model performance for testing set\")\n",
- "print(\"--------------------------------------\")\n",
- "print('RMSE is {}'.format(rmse))\n",
- "print('R2 score is {}'.format(r2))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "id": "6f9196e5",
- "metadata": {
- "collapsed": false,
- "editable": true
- },
- "outputs": [],
- "source": [
- "# plotting the y_test vs y_pred\n",
- "# ideally should have been a straight line\n",
- "plt.scatter(Y_test, y_test_predict)\n",
- "plt.show()"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "b72d5080",
+ "id": "44caf83f",
"metadata": {
"editable": true
},
@@ -3786,7 +3422,7 @@
},
{
"cell_type": "markdown",
- "id": "009b948b",
+ "id": "fea67f0c",
"metadata": {
"editable": true
},
@@ -3798,7 +3434,7 @@
},
{
"cell_type": "markdown",
- "id": "4b02a26a",
+ "id": "1af1f20c",
"metadata": {
"editable": true
},
@@ -3817,8 +3453,8 @@
},
{
"cell_type": "code",
- "execution_count": 37,
- "id": "efd083fc",
+ "execution_count": 26,
+ "id": "e9bcb7b1",
"metadata": {
"collapsed": false,
"editable": true
@@ -3852,7 +3488,7 @@
},
{
"cell_type": "markdown",
- "id": "b3590aea",
+ "id": "f3961700",
"metadata": {
"editable": true
},
@@ -3867,7 +3503,7 @@
},
{
"cell_type": "markdown",
- "id": "9592b7bf",
+ "id": "a33d2ba3",
"metadata": {
"editable": true
},
@@ -3879,7 +3515,7 @@
},
{
"cell_type": "markdown",
- "id": "0941f045",
+ "id": "3bc979c2",
"metadata": {
"editable": true
},
@@ -3889,7 +3525,7 @@
},
{
"cell_type": "markdown",
- "id": "d47fc8d3",
+ "id": "c234f16e",
"metadata": {
"editable": true
},
@@ -3912,8 +3548,8 @@
},
{
"cell_type": "code",
- "execution_count": 38,
- "id": "1ddb9cb5",
+ "execution_count": 27,
+ "id": "662bea3f",
"metadata": {
"collapsed": false,
"editable": true
@@ -3957,7 +3593,7 @@
},
{
"cell_type": "markdown",
- "id": "725b78e9",
+ "id": "05d214df",
"metadata": {
"editable": true
},
@@ -3967,7 +3603,7 @@
},
{
"cell_type": "markdown",
- "id": "b08f94e5",
+ "id": "aa18093a",
"metadata": {
"editable": true
},
@@ -4036,7 +3672,7 @@
},
{
"cell_type": "markdown",
- "id": "95e72a9e",
+ "id": "a8f251d6",
"metadata": {
"editable": true
},
@@ -4049,8 +3685,8 @@
},
{
"cell_type": "code",
- "execution_count": 39,
- "id": "fb3ad8e5",
+ "execution_count": 28,
+ "id": "48099cce",
"metadata": {
"collapsed": false,
"editable": true
@@ -4063,7 +3699,7 @@
},
{
"cell_type": "markdown",
- "id": "02fe1db6",
+ "id": "1cf31ef1",
"metadata": {
"editable": true
},
@@ -4077,7 +3713,7 @@
},
{
"cell_type": "markdown",
- "id": "33a4aed5",
+ "id": "dde166b5",
"metadata": {
"editable": true
},
@@ -4090,7 +3726,7 @@
},
{
"cell_type": "markdown",
- "id": "78a3bc86",
+ "id": "273b1e1d",
"metadata": {
"editable": true
},
@@ -4101,7 +3737,7 @@
},
{
"cell_type": "markdown",
- "id": "38c3a27d",
+ "id": "f7a4a3f1",
"metadata": {
"editable": true
},
@@ -4113,7 +3749,7 @@
},
{
"cell_type": "markdown",
- "id": "7eb5c51b",
+ "id": "0b3e8071",
"metadata": {
"editable": true
},
@@ -4123,7 +3759,7 @@
},
{
"cell_type": "markdown",
- "id": "3597b20a",
+ "id": "9f2d1bf3",
"metadata": {
"editable": true
},
@@ -4135,7 +3771,7 @@
},
{
"cell_type": "markdown",
- "id": "82922e13",
+ "id": "73c33ded",
"metadata": {
"editable": true
},
@@ -4150,8 +3786,8 @@
},
{
"cell_type": "code",
- "execution_count": 40,
- "id": "61cd693e",
+ "execution_count": 29,
+ "id": "c2fc4af7",
"metadata": {
"collapsed": false,
"editable": true
@@ -4202,7 +3838,7 @@
},
{
"cell_type": "markdown",
- "id": "2f8d2e6e",
+ "id": "a13aba09",
"metadata": {
"editable": true
},
@@ -4212,7 +3848,7 @@
},
{
"cell_type": "markdown",
- "id": "c81f5caf",
+ "id": "dcf58060",
"metadata": {
"editable": true
},
@@ -4257,8 +3893,8 @@
},
{
"cell_type": "code",
- "execution_count": 41,
- "id": "37ca3335",
+ "execution_count": 30,
+ "id": "f1bd9543",
"metadata": {
"collapsed": false,
"editable": true
@@ -4271,7 +3907,7 @@
},
{
"cell_type": "markdown",
- "id": "e6925e8c",
+ "id": "38030ca1",
"metadata": {
"editable": true
},
@@ -4281,8 +3917,8 @@
},
{
"cell_type": "code",
- "execution_count": 42,
- "id": "caecb70a",
+ "execution_count": 31,
+ "id": "e1d90b7a",
"metadata": {
"collapsed": false,
"editable": true
@@ -4297,7 +3933,7 @@
},
{
"cell_type": "markdown",
- "id": "deabdf0c",
+ "id": "4de86c60",
"metadata": {
"editable": true
},
@@ -4314,8 +3950,8 @@
},
{
"cell_type": "code",
- "execution_count": 43,
- "id": "b083bb84",
+ "execution_count": 32,
+ "id": "4aae540f",
"metadata": {
"collapsed": false,
"editable": true
@@ -4332,7 +3968,7 @@
},
{
"cell_type": "markdown",
- "id": "21102c44",
+ "id": "966d110b",
"metadata": {
"editable": true
},
@@ -4347,8 +3983,8 @@
},
{
"cell_type": "code",
- "execution_count": 44,
- "id": "b9108dab",
+ "execution_count": 33,
+ "id": "011363cd",
"metadata": {
"collapsed": false,
"editable": true
@@ -4392,7 +4028,7 @@
},
{
"cell_type": "markdown",
- "id": "cd19b575",
+ "id": "cf8dd05c",
"metadata": {
"editable": true
},
@@ -4402,7 +4038,7 @@
},
{
"cell_type": "markdown",
- "id": "eb2f6352",
+ "id": "d3c9b66f",
"metadata": {
"editable": true
},
@@ -4413,7 +4049,7 @@
},
{
"cell_type": "markdown",
- "id": "6e09ea94",
+ "id": "3363b9e6",
"metadata": {
"editable": true
},
@@ -4424,7 +4060,7 @@
},
{
"cell_type": "markdown",
- "id": "38857156",
+ "id": "33ac5d0b",
"metadata": {
"editable": true
},
@@ -4435,7 +4071,7 @@
},
{
"cell_type": "markdown",
- "id": "6b9115a6",
+ "id": "e80b75d3",
"metadata": {
"editable": true
},
@@ -4457,8 +4093,8 @@
},
{
"cell_type": "code",
- "execution_count": 45,
- "id": "ae7a71c1",
+ "execution_count": 34,
+ "id": "ba7cc092",
"metadata": {
"collapsed": false,
"editable": true
@@ -4471,7 +4107,7 @@
},
{
"cell_type": "markdown",
- "id": "ca615d39",
+ "id": "36b73ce3",
"metadata": {
"editable": true
},
@@ -4488,7 +4124,7 @@
},
{
"cell_type": "markdown",
- "id": "d83c8354",
+ "id": "9e7ac684",
"metadata": {
"editable": true
},
@@ -4501,7 +4137,7 @@
},
{
"cell_type": "markdown",
- "id": "74f6e912",
+ "id": "b94b5ddd",
"metadata": {
"editable": true
},
@@ -4512,7 +4148,7 @@
},
{
"cell_type": "markdown",
- "id": "93761664",
+ "id": "f8776c45",
"metadata": {
"editable": true
},
@@ -4524,7 +4160,7 @@
},
{
"cell_type": "markdown",
- "id": "be729d32",
+ "id": "40a07c84",
"metadata": {
"editable": true
},
@@ -4534,7 +4170,7 @@
},
{
"cell_type": "markdown",
- "id": "08549523",
+ "id": "55a234b6",
"metadata": {
"editable": true
},
@@ -4546,7 +4182,7 @@
},
{
"cell_type": "markdown",
- "id": "03cc0ca0",
+ "id": "59dc00fb",
"metadata": {
"editable": true
},
@@ -4562,8 +4198,8 @@
},
{
"cell_type": "code",
- "execution_count": 46,
- "id": "9b1b9378",
+ "execution_count": 35,
+ "id": "44a8d4ef",
"metadata": {
"collapsed": false,
"editable": true
@@ -4655,7 +4291,7 @@
},
{
"cell_type": "markdown",
- "id": "341e9820",
+ "id": "703ec310",
"metadata": {
"editable": true
},
@@ -4665,7 +4301,7 @@
},
{
"cell_type": "markdown",
- "id": "6feb372f",
+ "id": "980aceb3",
"metadata": {
"editable": true
},
@@ -4687,7 +4323,7 @@
},
{
"cell_type": "markdown",
- "id": "3c12062d",
+ "id": "c2c4fafc",
"metadata": {
"editable": true
},
@@ -4699,7 +4335,7 @@
},
{
"cell_type": "markdown",
- "id": "edfcd8ff",
+ "id": "84a561a2",
"metadata": {
"editable": true
},
@@ -4709,7 +4345,7 @@
},
{
"cell_type": "markdown",
- "id": "e8689eb2",
+ "id": "bb9a9639",
"metadata": {
"editable": true
},
@@ -4721,7 +4357,7 @@
},
{
"cell_type": "markdown",
- "id": "62145d09",
+ "id": "8f1dd1a6",
"metadata": {
"editable": true
},
@@ -4731,7 +4367,7 @@
},
{
"cell_type": "markdown",
- "id": "ec992c25",
+ "id": "74dcf2c0",
"metadata": {
"editable": true
},
@@ -4743,7 +4379,7 @@
},
{
"cell_type": "markdown",
- "id": "4a922090",
+ "id": "c530cce6",
"metadata": {
"editable": true
},
@@ -4758,7 +4394,7 @@
},
{
"cell_type": "markdown",
- "id": "10f294a3",
+ "id": "f337bf71",
"metadata": {
"editable": true
},
@@ -4770,7 +4406,7 @@
},
{
"cell_type": "markdown",
- "id": "1dfa508a",
+ "id": "cfa053a1",
"metadata": {
"editable": true
},
@@ -4780,7 +4416,7 @@
},
{
"cell_type": "markdown",
- "id": "b5474b9f",
+ "id": "e9035e0c",
"metadata": {
"editable": true
},
@@ -4792,7 +4428,7 @@
},
{
"cell_type": "markdown",
- "id": "4dee1baf",
+ "id": "1b198445",
"metadata": {
"editable": true
},
@@ -4802,7 +4438,7 @@
},
{
"cell_type": "markdown",
- "id": "9949ee71",
+ "id": "e21f0cf0",
"metadata": {
"editable": true
},
@@ -4814,7 +4450,7 @@
},
{
"cell_type": "markdown",
- "id": "73ce2a98",
+ "id": "8918cc8c",
"metadata": {
"editable": true
},
@@ -4824,7 +4460,7 @@
},
{
"cell_type": "markdown",
- "id": "41e1008d",
+ "id": "bed599b9",
"metadata": {
"editable": true
},
@@ -4836,7 +4472,7 @@
},
{
"cell_type": "markdown",
- "id": "597c4ebb",
+ "id": "4faf8bfa",
"metadata": {
"editable": true
},
@@ -4846,7 +4482,7 @@
},
{
"cell_type": "markdown",
- "id": "e9163f17",
+ "id": "ebde5b6c",
"metadata": {
"editable": true
},
@@ -4858,7 +4494,7 @@
},
{
"cell_type": "markdown",
- "id": "fc3d2fdb",
+ "id": "5ef54737",
"metadata": {
"editable": true
},
@@ -4868,7 +4504,7 @@
},
{
"cell_type": "markdown",
- "id": "e7cec690",
+ "id": "6ef07a21",
"metadata": {
"editable": true
},
@@ -4880,7 +4516,7 @@
},
{
"cell_type": "markdown",
- "id": "1a74bac9",
+ "id": "7d5bcb91",
"metadata": {
"editable": true
},
@@ -4890,7 +4526,7 @@
},
{
"cell_type": "markdown",
- "id": "7087bc76",
+ "id": "1d35a7b9",
"metadata": {
"editable": true
},
@@ -4902,7 +4538,7 @@
},
{
"cell_type": "markdown",
- "id": "1e5a21fd",
+ "id": "90fcecec",
"metadata": {
"editable": true
},
@@ -4912,7 +4548,7 @@
},
{
"cell_type": "markdown",
- "id": "624d40e0",
+ "id": "4c2ab747",
"metadata": {
"editable": true
},
@@ -4924,7 +4560,7 @@
},
{
"cell_type": "markdown",
- "id": "d871f62f",
+ "id": "ae1d9316",
"metadata": {
"editable": true
},
@@ -4934,7 +4570,7 @@
},
{
"cell_type": "markdown",
- "id": "675aea37",
+ "id": "aa76ae3d",
"metadata": {
"editable": true
},
@@ -4946,7 +4582,7 @@
},
{
"cell_type": "markdown",
- "id": "746b74e2",
+ "id": "473eca6a",
"metadata": {
"editable": true
},
@@ -4956,7 +4592,7 @@
},
{
"cell_type": "markdown",
- "id": "6381d6ee",
+ "id": "8476e871",
"metadata": {
"editable": true
},
@@ -4968,7 +4604,7 @@
},
{
"cell_type": "markdown",
- "id": "4331c20e",
+ "id": "70fa4295",
"metadata": {
"editable": true
},