-
@@ -156,7 +160,7 @@ MathJax.Hub.Config({
-Logistic regression will also serve as our stepping towards neural +Logistic regression will also serve as our stepping stone towards neural network algorithms and supervised deep learning. For logistic learning, the minimization of the cost function leads to a non-linear equation in the parameters \( \hat{\beta} \). The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms. @@ -151,7 +155,7 @@ models, as we will see later.
We consider the case where the dependent variables, also called the -responses or the outcomes) \( y_i \) are discrete and only take values +responses or the outcomes, \( y_i \) are discrete and only take values from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
The goal is to predict the -output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which carries \( p \) features. The +output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \) +made of \( n \) samples, each of which carries \( p \) features or predictors. The primary goal is to identify the classes to which new unseen samples belong.
-Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is +Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is $$ -y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}. +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. $$
@@ -159,7 +163,7 @@ $$
@@ -158,7 +162,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
+The code for plotting the perceptron can be seen here. This si nothing but the standard Heaviside step function. +
+ + +
+
@@ -158,7 +169,7 @@ the probability of a given category. This leads us to the logistic function.
-A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \), +The perceptron is an example of a ``hard classification" model. We +will encounter this model when we discuss neural networks as +well. Each datapoint is deterministically assigned to a category (i.e +\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" +classifier that outputs the probability of a given category rather +than a single value. For example, given \( x_i \), the classifier +outputs the probability of being in a category \( k \). Logistic regression +is the most common example of a so-called soft classifier. In logistic +regression, the probability that a data point \( x_i \) +belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, $$ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$ Note that \( 1-p(t)= p(-t) \). +The following code plots the logistic function. +
+ +
+
@@ -151,6 +169,8 @@ Note that \( 1-p(t)= p(-t) \).
-The perceptron is an example of a ``hard classification" model. We -will encounter this model when we discuss neural networks as -well. Each datapoint is deterministically assigned to a category (i.e -\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" -classifier that outputs the probability of a given category rather -than a single value. For example, given \( x_i \), the classifier -outputs the probability of being in a category \( k \). Logistic regression -is the most common example of a so-called soft classifier. In logistic -regression, the probability that a data point \( x_i \) -belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) +We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ + +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). + +
+Note that we used +$$ +p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). +$$
@@ -155,6 +164,7 @@ belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function
- + -
-We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities +In order to define the total likelihood for all possible outcomes from a +dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels +\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. +We aim thus at maximizing +the probability of seeing the observed data. We can then approximate the +likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is $$ \begin{align*} -p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ \end{align*} $$ -where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). - -
-Note that we used +from which we obtain the log-likelihood and our cost/loss function $$ -p(y_i=1) = 1-p(y_i=0). +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). $$
@@ -160,6 +165,7 @@ $$
- + -
-In order to define the total likelihood for all possible outcomes from a -dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels -\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. -We aim thus at maximizing -the probability of seeing the observed data. We can then approximate the -likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is +Reordering the logarithms, we can rewrite the cost/loss function as $$ -\begin{align*} -P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ -\end{align*} +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). $$ -from which we obtain the log-likelihood and our cost/loss function +
+The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \). +Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that $$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]. +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). $$ +This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, +in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +
@@ -161,6 +163,7 @@ $$
-Reordering the logarithms, we can rewrite the cost/loss function as +The cross entropy is a convex function of the weights \( \hat{\beta} \) and, +therefore, any local minimizer is a global minimizer. + +
+Minimizing this +cost function with respect to the two parameters \( \beta_0 \) and \( \beta_1 \) we obtain + $$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right). +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), $$ +and +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ + +
diff --git a/doc/pub/LogReg/html/._LogReg-bs011.html b/doc/pub/LogReg/html/._LogReg-bs011.html index 0a94cd42d..26ff89a72 100644 --- a/doc/pub/LogReg/html/._LogReg-bs011.html +++ b/doc/pub/LogReg/html/._LogReg-bs011.html @@ -46,14 +46,16 @@ Automatically generated HTML file from DocOnce source ('Linear classifier', 2, None, '___sec3'), ('Some selected properties', 2, None, '___sec4'), ('The logistic function', 2, None, '___sec5'), - ('Probabilistic classifiers', 2, None, '___sec6'), - ('Two parameters', 2, None, '___sec7'), - ('Maximum likelihood', 2, None, '___sec8'), - ('The cost function rewritten', 2, None, '___sec9'), - ('Minimizing the cross entropy', 2, None, '___sec10'), - ('A more compact expression', 2, None, '___sec11'), - ('A _scikit-learn_ example', 2, None, '___sec12'), - ('A simple classification problem', 2, None, '___sec13')]} + ('Two parameters', 2, None, '___sec6'), + ('Maximum likelihood', 2, None, '___sec7'), + ('The cost function rewritten', 2, None, '___sec8'), + ('Minimizing the cross entropy', 2, None, '___sec9'), + ('A more compact expression', 2, None, '___sec10'), + ('Extending to more predictors', 2, None, '___sec11'), + ('Including more classes', 2, None, '___sec12'), + ('Optimizing the cost function', 2, None, '___sec13'), + ('A _scikit-learn_ example', 2, None, '___sec14'), + ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -97,14 +99,16 @@ MathJax.Hub.Config({
+Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an +\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a +vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first +derivative of cost function as + $$ -\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right). +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). $$ -This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, -in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +
+If we in addition define a diagonal matrix \( \hat{W} \) with elements +\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$
@@ -148,6 +164,7 @@ in practice we often supplement the cross-entropy with additional regularization
-The cross entropy is a convex function of the weights \( \hat{\beta} \) and, -therefore, any local minimizer is a global minimizer. - -
-Minimizing this -cost function with respect to the two parameters \( \beta_0 \) and \( \beta_1 \) we obtain - -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), -$$ - -and -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). -$$ +
@@ -158,6 +145,7 @@ $$
-Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an -\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a -vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first -derivative of cost function as - -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). -$$ - -
-If we in addition define a diagonal matrix \( \hat{W} \) with elements -\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as - -$$ -\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. -$$ +
@@ -158,6 +144,7 @@ $$
+Newton's method and gradient descent methods - -
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-iris = datasets.load_iris()
-list(iris.keys())
-['data', 'target_names', 'feature_names', 'target', 'DESCR']
-X = iris["data"][:, 3:] # petal width
-y = (iris["target"] == 2).astype(np.int) # 1 if Iris-Virginica, else 0
-
-from sklearn.linear_model import LogisticRegression
-log_reg = LogisticRegression()
-log_reg.fit(X, y)
-
-X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
-y_proba = log_reg.predict_proba(X_new)
-plt.plot(X_new, y_proba[:, 1], "g-", label="Iris-Virginica")
-plt.plot(X_new, y_proba[:, 0], "b--", label="Not Iris-Virginica")
-plt.show()
-
@@ -161,6 +146,7 @@ plt.show()
import numpy as np
-from sklearn import datasets, linear_model
import matplotlib.pyplot as plt
+from sklearn import datasets
+iris = datasets.load_iris()
+list(iris.keys())
+['data', 'target_names', 'feature_names', 'target', 'DESCR']
+X = iris["data"][:, 3:] # petal width
+y = (iris["target"] == 2).astype(np.int) # 1 if Iris-Virginica, else 0
+from sklearn.linear_model import LogisticRegression
+log_reg = LogisticRegression()
+log_reg.fit(X, y)
-def generate_data():
- np.random.seed(0)
- X, y = datasets.make_moons(200, noise=0.20)
- return X, y
-
-
-def visualize(X, y, clf):
- # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
- # plt.show()
- plot_decision_boundary(lambda x: clf.predict(x), X, y)
- plt.title("Logistic Regression")
-
-
-def plot_decision_boundary(pred_func, X, y):
- # Set min and max values and give it some padding
- x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
- y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
- h = 0.01
- # Generate a grid of points with distance h between them
- xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
- # Predict the function value for the whole gid
- Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
- Z = Z.reshape(xx.shape)
- # Plot the contour and training examples
- plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
- plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
- plt.show()
-
-
-def classify(X, y):
- clf = linear_model.LogisticRegressionCV()
- clf.fit(X, y)
- return clf
-
-
-def main():
- X, y = generate_data()
- # visualize(X, y)
- clf = classify(X, y)
- visualize(X, y, clf)
-
-
-if __name__ == "__main__":
- main()
+X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
+y_proba = log_reg.predict_proba(X_new)
+plt.plot(X_new, y_proba[:, 1], "g-", label="Iris-Virginica")
+plt.plot(X_new, y_proba[:, 0], "b--", label="Not Iris-Virginica")
+plt.show()
-
diff --git a/doc/pub/LogReg/html/LogReg-bs.html b/doc/pub/LogReg/html/LogReg-bs.html index ae97688d0..b583d2d04 100644 --- a/doc/pub/LogReg/html/LogReg-bs.html +++ b/doc/pub/LogReg/html/LogReg-bs.html @@ -46,14 +46,16 @@ Automatically generated HTML file from DocOnce source ('Linear classifier', 2, None, '___sec3'), ('Some selected properties', 2, None, '___sec4'), ('The logistic function', 2, None, '___sec5'), - ('Probabilistic classifiers', 2, None, '___sec6'), - ('Two parameters', 2, None, '___sec7'), - ('Maximum likelihood', 2, None, '___sec8'), - ('The cost function rewritten', 2, None, '___sec9'), - ('Minimizing the cross entropy', 2, None, '___sec10'), - ('A more compact expression', 2, None, '___sec11'), - ('A _scikit-learn_ example', 2, None, '___sec12'), - ('A simple classification problem', 2, None, '___sec13')]} + ('Two parameters', 2, None, '___sec6'), + ('Maximum likelihood', 2, None, '___sec7'), + ('The cost function rewritten', 2, None, '___sec8'), + ('Minimizing the cross entropy', 2, None, '___sec9'), + ('A more compact expression', 2, None, '___sec10'), + ('Extending to more predictors', 2, None, '___sec11'), + ('Including more classes', 2, None, '___sec12'), + ('Optimizing the cost function', 2, None, '___sec13'), + ('A _scikit-learn_ example', 2, None, '___sec14'), + ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -97,14 +99,16 @@ MathJax.Hub.Config({
-
@@ -156,7 +160,7 @@ MathJax.Hub.Config({
-
@@ -197,7 +197,7 @@ failure etc.
-Logistic regression will also serve as our stepping towards neural +Logistic regression will also serve as our stepping stone towards neural network algorithms and supervised deep learning. For logistic learning, the minimization of the cost function leads to a non-linear equation in the parameters \( \hat{\beta} \). The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms. @@ -214,21 +214,21 @@ models, as we will see later.
We consider the case where the dependent variables, also called the -responses or the outcomes) \( y_i \) are discrete and only take values +responses or the outcomes, \( y_i \) are discrete and only take values from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
The goal is to predict the -output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which carries \( p \) features. The +output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \) +made of \( n \) samples, each of which carries \( p \) features or predictors. The primary goal is to identify the classes to which new unseen samples belong.
-Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is +Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
$$
-y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}.
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -246,14 +246,14 @@ weighted linear combination, namely
$$
\begin{equation}
-\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon},
+\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
\tag{1}
\end{equation}
$$
where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
-\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators.
+\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
@@ -274,27 +274,20 @@ We will encounter this model in our first demonstration of neural networks. Hist
literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
+
+
+The code for plotting the perceptron can be seen here. This si nothing but the standard Heaviside step function. +
+ + +
+-A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \), -
-$$
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-$$
-
-
-Note that \( 1-p(t)= p(-t) \).
-
The perceptron is an example of a ``hard classification" model. We will encounter this model when we discuss neural networks as @@ -305,12 +298,25 @@ than a single value. For example, given \( x_i \), the classifier outputs the probability of being in a category \( k \). Logistic regression is the most common example of a so-called soft classifier. In logistic regression, the probability that a data point \( x_i \) -belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) +belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, +
+$$
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+$$
+
+
+Note that \( 1-p(t)= p(-t) \).
+The following code plots the logistic function.
+
+ + +
+We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities @@ -329,19 +335,19 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas Note that we used
$$
-p(y_i=1) = 1-p(y_i=0).
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
$$
In order to define the total likelihood for all possible outcomes from a dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels -\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. +\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. We aim thus at maximizing the probability of seeing the observed data. We can then approximate the likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is @@ -356,31 +362,29 @@ $$ from which we obtain the log-likelihood and our cost/loss function
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right].
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
$$
Reordering the logarithms, we can rewrite the cost/loss function as
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
-
The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \). Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
$$
-\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -416,7 +420,7 @@ $$
Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -443,7 +447,25 @@ $$
+Newton's method and gradient descent methods
+
@@ -471,7 +493,7 @@ plt.show()
diff --git a/doc/pub/LogReg/html/LogReg-solarized.html b/doc/pub/LogReg/html/LogReg-solarized.html
index f37054fab..6211d4f48 100644
--- a/doc/pub/LogReg/html/LogReg-solarized.html
+++ b/doc/pub/LogReg/html/LogReg-solarized.html
@@ -40,14 +40,16 @@ div { text-align: justify; text-justify: inter-word; }
('Linear classifier', 2, None, '___sec3'),
('Some selected properties', 2, None, '___sec4'),
('The logistic function', 2, None, '___sec5'),
- ('Probabilistic classifiers', 2, None, '___sec6'),
- ('Two parameters', 2, None, '___sec7'),
- ('Maximum likelihood', 2, None, '___sec8'),
- ('The cost function rewritten', 2, None, '___sec9'),
- ('Minimizing the cross entropy', 2, None, '___sec10'),
- ('A more compact expression', 2, None, '___sec11'),
- ('A _scikit-learn_ example', 2, None, '___sec12'),
- ('A simple classification problem', 2, None, '___sec13')]}
+ ('Two parameters', 2, None, '___sec6'),
+ ('Maximum likelihood', 2, None, '___sec7'),
+ ('The cost function rewritten', 2, None, '___sec8'),
+ ('Minimizing the cross entropy', 2, None, '___sec9'),
+ ('A more compact expression', 2, None, '___sec10'),
+ ('Extending to more predictors', 2, None, '___sec11'),
+ ('Including more classes', 2, None, '___sec12'),
+ ('Optimizing the cost function', 2, None, '___sec13'),
+ ('A _scikit-learn_ example', 2, None, '___sec14'),
+ ('A simple classification problem', 2, None, '___sec15')]}
end of tocinfo -->
-
@@ -132,7 +134,7 @@ failure etc.
-Logistic regression will also serve as our stepping towards neural
+Logistic regression will also serve as our stepping stone towards neural
network algorithms and supervised deep learning. For logistic
learning, the minimization of the cost function leads to a non-linear
equation in the parameters \( \hat{\beta} \). The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms.
@@ -149,20 +151,20 @@ models, as we will see later.
We consider the case where the dependent variables, also called the
-responses or the outcomes) \( y_i \) are discrete and only take values
+responses or the outcomes, \( y_i \) are discrete and only take values
from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
The goal is to predict the
-output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \)
-made of \( n \) samples, each of which carries \( p \) features. The
+output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \)
+made of \( n \) samples, each of which carries \( p \) features or predictors. The
primary goal is to identify the classes to which new unseen samples
belong.
-Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is
+Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
$$
-y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}.
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -178,13 +180,13 @@ We would then have our
weighted linear combination, namely
$$
\begin{equation}
-\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon},
+\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
\label{_auto1}
\end{equation}
$$
where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
-\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators.
+\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
+The code for plotting the perceptron can be seen here. This si nothing but the standard Heaviside step function.
+
+
+
+
-A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \),
-$$
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-$$
-
-Note that \( 1-p(t)= p(-t) \).
-
-
-
The perceptron is an example of a ``hard classification" model. We
will encounter this model when we discuss neural networks as
@@ -234,12 +230,22 @@ than a single value. For example, given \( x_i \), the classifier
outputs the probability of being in a category \( k \). Logistic regression
is the most common example of a so-called soft classifier. In logistic
regression, the probability that a data point \( x_i \)
-belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid)
+belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
+$$
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+$$
+Note that \( 1-p(t)= p(-t) \).
+The following code plots the logistic function.
+
+
+
+
We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities
@@ -255,18 +261,18 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
Note that we used
$$
-p(y_i=1) = 1-p(y_i=0).
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
$$
-
In order to define the total likelihood for all possible outcomes from a
dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
-\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
+\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
We aim thus at maximizing
the probability of seeing the observed data. We can then approximate the
likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is
@@ -278,25 +284,25 @@ $$
from which we obtain the log-likelihood and our cost/loss function
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right].
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
$$
Reordering the logarithms, we can rewrite the cost/loss function as
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
-
The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
$$
-\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression,
@@ -305,7 +311,7 @@ in practice we often supplement the cross-entropy with additional regularization
The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -327,7 +333,7 @@ $$
Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -350,7 +356,25 @@ $$
+
+
+Newton's method and gradient descent methods
+
+
+
@@ -377,7 +401,7 @@ plt.show()
diff --git a/doc/pub/LogReg/html/LogReg.html b/doc/pub/LogReg/html/LogReg.html
index 1dab7dca5..02994c5c2 100644
--- a/doc/pub/LogReg/html/LogReg.html
+++ b/doc/pub/LogReg/html/LogReg.html
@@ -45,14 +45,16 @@ div { text-align: justify; text-justify: inter-word; }
('Linear classifier', 2, None, '___sec3'),
('Some selected properties', 2, None, '___sec4'),
('The logistic function', 2, None, '___sec5'),
- ('Probabilistic classifiers', 2, None, '___sec6'),
- ('Two parameters', 2, None, '___sec7'),
- ('Maximum likelihood', 2, None, '___sec8'),
- ('The cost function rewritten', 2, None, '___sec9'),
- ('Minimizing the cross entropy', 2, None, '___sec10'),
- ('A more compact expression', 2, None, '___sec11'),
- ('A _scikit-learn_ example', 2, None, '___sec12'),
- ('A simple classification problem', 2, None, '___sec13')]}
+ ('Two parameters', 2, None, '___sec6'),
+ ('Maximum likelihood', 2, None, '___sec7'),
+ ('The cost function rewritten', 2, None, '___sec8'),
+ ('Minimizing the cross entropy', 2, None, '___sec9'),
+ ('A more compact expression', 2, None, '___sec10'),
+ ('Extending to more predictors', 2, None, '___sec11'),
+ ('Including more classes', 2, None, '___sec12'),
+ ('Optimizing the cost function', 2, None, '___sec13'),
+ ('A _scikit-learn_ example', 2, None, '___sec14'),
+ ('A simple classification problem', 2, None, '___sec15')]}
end of tocinfo -->
-
@@ -137,7 +139,7 @@ failure etc.
-Logistic regression will also serve as our stepping towards neural
+Logistic regression will also serve as our stepping stone towards neural
network algorithms and supervised deep learning. For logistic
learning, the minimization of the cost function leads to a non-linear
equation in the parameters \( \hat{\beta} \). The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms.
@@ -154,20 +156,20 @@ models, as we will see later.
We consider the case where the dependent variables, also called the
-responses or the outcomes) \( y_i \) are discrete and only take values
+responses or the outcomes, \( y_i \) are discrete and only take values
from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
The goal is to predict the
-output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \)
-made of \( n \) samples, each of which carries \( p \) features. The
+output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \)
+made of \( n \) samples, each of which carries \( p \) features or predictors. The
primary goal is to identify the classes to which new unseen samples
belong.
-Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is
+Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
$$
-y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}.
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -183,13 +185,13 @@ We would then have our
weighted linear combination, namely
$$
\begin{equation}
-\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon},
+\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
\label{_auto1}
\end{equation}
$$
where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
-\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators.
+\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
+The code for plotting the perceptron can be seen here. This si nothing but the standard Heaviside step function.
+
+
+
+
-A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \),
-$$
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-$$
-
-Note that \( 1-p(t)= p(-t) \).
-
-
-
The perceptron is an example of a ``hard classification" model. We
will encounter this model when we discuss neural networks as
@@ -239,12 +235,22 @@ than a single value. For example, given \( x_i \), the classifier
outputs the probability of being in a category \( k \). Logistic regression
is the most common example of a so-called soft classifier. In logistic
regression, the probability that a data point \( x_i \)
-belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid)
+belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
+$$
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+$$
+Note that \( 1-p(t)= p(-t) \).
+The following code plots the logistic function.
+
+
+
+
We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities
@@ -260,18 +266,18 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
Note that we used
$$
-p(y_i=1) = 1-p(y_i=0).
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
$$
-
In order to define the total likelihood for all possible outcomes from a
dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
-\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
+\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
We aim thus at maximizing
the probability of seeing the observed data. We can then approximate the
likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is
@@ -283,25 +289,25 @@ $$
from which we obtain the log-likelihood and our cost/loss function
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right].
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
$$
Reordering the logarithms, we can rewrite the cost/loss function as
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
-
The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
$$
-\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression,
@@ -310,7 +316,7 @@ in practice we often supplement the cross-entropy with additional regularization
The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -332,7 +338,7 @@ $$
Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -355,7 +361,25 @@ $$
+
+
+Newton's method and gradient descent methods
+
+
+
@@ -382,7 +406,7 @@ plt.show()
diff --git a/doc/pub/LogReg/ipynb/LogReg.ipynb b/doc/pub/LogReg/ipynb/LogReg.ipynb
index b6532eb40..43615a1d1 100644
--- a/doc/pub/LogReg/ipynb/LogReg.ipynb
+++ b/doc/pub/LogReg/ipynb/LogReg.ipynb
@@ -10,7 +10,7 @@
" \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
- "Date: **Sep 25, 2018**\n",
+ "Date: **Sep 26, 2018**\n",
"\n",
"Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -50,7 +50,7 @@
"\n",
"## Optimization and Deep learning\n",
"\n",
- "Logistic regression will also serve as our stepping towards neural\n",
+ "Logistic regression will also serve as our stepping stone towards neural\n",
"network algorithms and supervised deep learning. For logistic\n",
"learning, the minimization of the cost function leads to a non-linear\n",
"equation in the parameters $\\hat{\\beta}$. The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms. \n",
@@ -64,16 +64,16 @@
"## Basics\n",
"\n",
"We consider the case where the dependent variables, also called the\n",
- "responses or the outcomes) $y_i$ are discrete and only take values\n",
+ "responses or the outcomes, $y_i$ are discrete and only take values\n",
"from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n",
"\n",
"The goal is to predict the\n",
- "output classes from the design matrix $X\\in\\mathbb{R}^{n\\times p}$\n",
- "made of $n$ samples, each of which carries $p$ features. The\n",
+ "output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n",
+ "made of $n$ samples, each of which carries $p$ features or predictors. The\n",
"primary goal is to identify the classes to which new unseen samples\n",
"belong.\n",
"\n",
- "Let us specialize to the case of two classes only, with outputs $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is"
+ "Let us specialize to the case of two classes only, with outputs $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is"
]
},
{
@@ -81,7 +81,7 @@
"metadata": {},
"source": [
"$$\n",
- "y_i = \\begin{bmatrix} 0 & \\mathrm{no default}\\\\ 1 & \\mathrm{default} \\end{bmatrix}.\n",
+ "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n",
"$$"
]
},
@@ -106,7 +106,7 @@
"\n",
"$$\n",
"\\begin{equation}\n",
- "\\hat{y} = \\hat{X}_i^T\\hat{\\beta} + \\hat{\\epsilon},\n",
+ "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n",
"\\label{_auto1} \\tag{1}\n",
"\\end{equation}\n",
"$$"
@@ -117,7 +117,7 @@
"metadata": {},
"source": [
"where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n",
- "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators.\n",
+ "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n",
"\n",
"## Some selected properties\n",
"\n",
@@ -134,9 +134,25 @@
"favorable to use a ``soft\" classifier that outputs\n",
"the probability of a given category. This leads us to the logistic function.\n",
"\n",
+ "The code for plotting the perceptron can be seen here. This si nothing but the standard [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
"## The logistic function\n",
"\n",
- "A widely used model is the so-called logistic (or sigmoid) function which ranges from $0$ to $1$,"
+ "The perceptron is an example of a ``hard classification\" model. We\n",
+ "will encounter this model when we discuss neural networks as\n",
+ "well. Each datapoint is deterministically assigned to a category (i.e\n",
+ "$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n",
+ "classifier that outputs the probability of a given category rather\n",
+ "than a single value. For example, given $x_i$, the classifier\n",
+ "outputs the probability of being in a category $k$. Logistic regression\n",
+ "is the most common example of a so-called soft classifier. In logistic\n",
+ "regression, the probability that a data point $x_i$\n",
+ "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,"
]
},
{
@@ -153,21 +169,13 @@
"metadata": {},
"source": [
"Note that $1-p(t)= p(-t)$.\n",
- "\n",
- "## Probabilistic classifiers\n",
- "\n",
- "The perceptron is an example of a ``hard classification\" model. We\n",
- "will encounter this model when we discuss neural networks as\n",
- "well. Each datapoint is deterministically assigned to a category (i.e\n",
- "$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n",
- "classifier that outputs the probability of a given category rather\n",
- "than a single value. For example, given $x_i$, the classifier\n",
- "outputs the probability of being in a category $k$. Logistic regression\n",
- "is the most common example of a so-called soft classifier. In logistic\n",
- "regression, the probability that a data point $x_i$\n",
- "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid)\n",
- "\n",
- "\n",
+ "The following code plots the logistic function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
"## Two parameters\n",
"\n",
"We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities"
@@ -199,7 +207,7 @@
"metadata": {},
"source": [
"$$\n",
- "p(y_i=1) = 1-p(y_i=0).\n",
+ "p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n",
"$$"
]
},
@@ -212,7 +220,7 @@
"\n",
"In order to define the total likelihood for all possible outcomes from a \n",
"dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n",
- "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. \n",
+ "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n",
"We aim thus at maximizing \n",
"the probability of seeing the observed data. We can then approximate the \n",
"likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is"
@@ -241,7 +249,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right].\n",
+ "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n",
"$$"
]
},
@@ -259,7 +267,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n",
+ "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
"$$"
]
},
@@ -276,7 +284,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n",
+ "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
"$$"
]
},
@@ -364,6 +372,16 @@
"cell_type": "markdown",
"metadata": {},
"source": [
+ "## Extending to more predictors\n",
+ "\n",
+ "## Including more classes\n",
+ "\n",
+ "## Optimizing the cost function\n",
+ "\n",
+ "Newton's method and gradient descent methods\n",
+ "\n",
+ "\n",
+ "\n",
"## A **scikit-learn** example"
]
},
diff --git a/doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz b/doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz
index d76ecdcb2..c730247af 100644
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diff --git a/doc/pub/LogReg/pdf/LogReg-beamer-handouts2x3.pdf b/doc/pub/LogReg/pdf/LogReg-beamer-handouts2x3.pdf
index 5bdf2d6a0..dfca58001 100644
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index 0efc20dbd..4b3c6ae9e 100644
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diff --git a/doc/pub/LogReg/pdf/LogReg-minted.pdf b/doc/pub/LogReg/pdf/LogReg-minted.pdf
index 08ffea899..21ceff7ff 100644
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diff --git a/doc/src/LogisticRegression/LogReg.do.txt b/doc/src/LogisticRegression/LogReg.do.txt
index 62f125d95..fed5ac392 100644
--- a/doc/src/LogisticRegression/LogReg.do.txt
+++ b/doc/src/LogisticRegression/LogReg.do.txt
@@ -37,7 +37,7 @@ failure etc.
!split
===== Optimization and Deep learning =====
-Logistic regression will also serve as our stepping towards neural
+Logistic regression will also serve as our stepping stone towards neural
network algorithms and supervised deep learning. For logistic
learning, the minimization of the cost function leads to a non-linear
equation in the parameters $\hat{\beta}$. The optmization of the problem calls therefore for minimization algorithms. This forms the bottle neck of all machine learning algorithms, namely how to find reliable minima of a multi-variable function. This leads us to the family of gradient descent methods. The latter are the working horses of basically all modern machine learning algorithms.
@@ -51,19 +51,19 @@ models, as we will see later.
===== Basics =====
We consider the case where the dependent variables, also called the
-responses or the outcomes) $y_i$ are discrete and only take values
+responses or the outcomes, $y_i$ are discrete and only take values
from $k=0,\dots,K-1$ (i.e. $K$ classes).
The goal is to predict the
-output classes from the design matrix $X\in\mathbb{R}^{n\times p}$
-made of $n$ samples, each of which carries $p$ features. The
+output classes from the design matrix $\hat{X}\in\mathbb{R}^{n\times p}$
+made of $n$ samples, each of which carries $p$ features or predictors. The
primary goal is to identify the classes to which new unseen samples
belong.
-Let us specialize to the case of two classes only, with outputs $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a credit card use who could default or not on her/his debt. That is
+Let us specialize to the case of two classes only, with outputs $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
!bt
\[
-y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}.
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
\]
!et
@@ -78,11 +78,11 @@ We would then have our
weighted linear combination, namely
!bt
\begin{equation}
-\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon},
+\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
\end{equation}
!et
where $\hat{y}$ is a vector representing the possible outcomes, $\hat{X}$ is our
-$n\times p$ design matrix and $\hat{\beta}$ represents our estimators.
+$n\times p$ design matrix and $\hat{\beta}$ represents our estimators/predictors.
!split
===== Some selected properties =====
@@ -100,20 +100,15 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
+The code for plotting the perceptron can be seen here. This si nothing but the standard "Heaviside step function":"https://en.wikipedia.org/wiki/Heaviside_step_function".
+!bc pycod
+
+!ec
+
+
!split
===== The logistic function =====
-A widely used model is the so-called logistic (or sigmoid) function which ranges from $0$ to $1$,
-!bt
-\[
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-\]
-!et
-Note that $1-p(t)= p(-t)$.
-
-!split
-===== Probabilistic classifiers =====
-
The perceptron is an example of a ``hard classification'' model. We
will encounter this model when we discuss neural networks as
well. Each datapoint is deterministically assigned to a category (i.e
@@ -123,7 +118,18 @@ than a single value. For example, given $x_i$, the classifier
outputs the probability of being in a category $k$. Logistic regression
is the most common example of a so-called soft classifier. In logistic
regression, the probability that a data point $x_i$
-belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid)
+belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
+!bt
+\[
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+\]
+!et
+Note that $1-p(t)= p(-t)$.
+The following code plots the logistic function.
+!bc pycod
+
+!ec
+
!split
@@ -141,7 +147,7 @@ where $\hat{\beta}$ are the weights we wish to extract from data, in our case $\
Note that we used
!bt
\[
-p(y_i=1) = 1-p(y_i=0).
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
\]
!et
@@ -150,7 +156,7 @@ p(y_i=1) = 1-p(y_i=0).
In order to define the total likelihood for all possible outcomes from a
dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels
-$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
+$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called "Maximum Likelihood Estimation":"https://en.wikipedia.org/wiki/Maximum_likelihood_estimation" (MLE) principle.
We aim thus at maximizing
the probability of seeing the observed data. We can then approximate the
likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is
@@ -162,7 +168,7 @@ P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right
from which we obtain the log-likelihood and our _cost/loss_ function
!bt
\[
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right].
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
\]
!et
@@ -172,15 +178,15 @@ from which we obtain the log-likelihood and our _cost/loss_ function
Reordering the logarithms, we can rewrite the _cost/loss_ function as
!bt
\[
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
\]
!et
-!split
+
The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$.
Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
!bt
\[
-\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
\]
!et
This equation is known in statistics as the _cross entropy_. Finally, we note that just as in linear regression,
@@ -231,6 +237,18 @@ $p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})$, we can obtain a com
\]
!et
+!split
+===== Extending to more predictors =====
+
+!split
+===== Including more classes =====
+
+!split
+===== Optimizing the cost function =====
+
+Newton's method and gradient descent methods
+
+
!split
===== A _scikit-learn_ example =====
@@ -390,7 +394,7 @@ in practice we often supplement the cross-entropy with additional regularization
Minimizing the cross entropy
+Minimizing the cross entropy
A more compact expression
+A more compact expression
A scikit-learn example
+Extending to more predictors
+Including more classes
+Optimizing the cost function
+
+A scikit-learn example
A simple classification problem
+A simple classification problem
Sep 25, 2018
Sep 26, 2018
Optimization and Deep learning
@@ -206,24 +208,18 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
+
+
The logistic function
-
-
-Probabilistic classifiers
-
+
-Two parameters
+Two parameters
Maximum likelihood
+Maximum likelihood
-The cost function rewritten
+The cost function rewritten
+
-Minimizing the cross entropy
+Minimizing the cross entropy
-A more compact expression
+A more compact expression
-A scikit-learn example
+Extending to more predictors
+
+
+
+Including more classes
+
+
+
+Optimizing the cost function
+
+
+
+A scikit-learn example
-A simple classification problem
+A simple classification problem
Sep 25, 2018
Sep 26, 2018
Optimization and Deep learning
@@ -211,24 +213,18 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
+
+
The logistic function
-
-
-Probabilistic classifiers
-
+
-Two parameters
+Two parameters
Maximum likelihood
+Maximum likelihood
-The cost function rewritten
+The cost function rewritten
+
-Minimizing the cross entropy
+Minimizing the cross entropy
-A more compact expression
+A more compact expression
-A scikit-learn example
+Extending to more predictors
+
+
+
+Including more classes
+
+
+
+Optimizing the cost function
+
+
+
+A scikit-learn example
-A simple classification problem
+A simple classification problem