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 bears \( p \) features. Of cours e the +made of \( n \) samples, each of which bears \( p \) features. The primary goal is to identify the classes to which new unseen samples belong. diff --git a/doc/pub/Regression/html/._Regression-bs104.html b/doc/pub/Regression/html/._Regression-bs104.html index b210240fd..f7dcffe23 100644 --- a/doc/pub/Regression/html/._Regression-bs104.html +++ b/doc/pub/Regression/html/._Regression-bs104.html @@ -396,7 +396,18 @@ MathJax.Hub.Config({
-This function takes values on the entire real axis. In the case of logistic regression, however, the labels \( y_i \) are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to \( \{0,1\} \), \( f(s_i)= \) sign$(s_i) = 1$ if \( s_i\ge 0 \) and 0 if otherwise. Indeed, this is commonly known as the "perceptron" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft" classifier that outputs the probability of a given category. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). One such function is the logistic (or sigmoid) function: +This function takes values on the entire real axis. In the case of +logistic regression, however, the labels \( y_i \) are discrete +variables. One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to \( \{0,1\} \), +\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, +this is commonly known as the "perceptron" in the machine learning +literature. This model is extremely simple, and it is favorable in +many cases (e.g. noisy data) to have a ``soft" classifier that outputs +the probability of a given category. For example, given +\( \mathbf{x}_i \), the classifier outputs the probability of being in +category \( m \). One such function is the logistic (or sigmoid) function: + $$ \begin{equation} f(s) = \frac{1}{1+\mathrm e^{-s}}. diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html index ef7046053..49c3a7cb8 100644 --- a/doc/pub/Regression/html/Regression-reveal.html +++ b/doc/pub/Regression/html/Regression-reveal.html @@ -3488,7 +3488,7 @@ 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 bears \( p \) features. Of cours e the +made of \( n \) samples, each of which bears \( p \) features. The primary goal is to identify the classes to which new unseen samples belong. @@ -3517,7 +3517,18 @@ where we use the short-hand notation
-This function takes values on the entire real axis. In the case of logistic regression, however, the labels \( y_i \) are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to \( \{0,1\} \), \( f(s_i)= \) sign$(s_i) = 1$ if \( s_i\ge 0 \) and 0 if otherwise. Indeed, this is commonly known as the "perceptron" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft" classifier that outputs the probability of a given category. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). One such function is the logistic (or sigmoid) function: +This function takes values on the entire real axis. In the case of +logistic regression, however, the labels \( y_i \) are discrete +variables. One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to \( \{0,1\} \), +\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, +this is commonly known as the "perceptron" in the machine learning +literature. This model is extremely simple, and it is favorable in +many cases (e.g. noisy data) to have a ``soft" classifier that outputs +the probability of a given category. For example, given +\( \mathbf{x}_i \), the classifier outputs the probability of being in +category \( m \). One such function is the logistic (or sigmoid) function: +
$$
\begin{equation}
diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html
index 42ef73561..c11e09b23 100644
--- a/doc/pub/Regression/html/Regression-solarized.html
+++ b/doc/pub/Regression/html/Regression-solarized.html
@@ -3431,7 +3431,7 @@ 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 bears \( p \) features. Of cours e the +made of \( n \) samples, each of which bears \( p \) features. The primary goal is to identify the classes to which new unseen samples belong. @@ -3458,7 +3458,18 @@ where we use the short-hand notation
-This function takes values on the entire real axis. In the case of logistic regression, however, the labels \( y_i \) are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to \( \{0,1\} \), \( f(s_i)= \) sign$(s_i) = 1$ if \( s_i\ge 0 \) and 0 if otherwise. Indeed, this is commonly known as the "perceptron" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft" classifier that outputs the probability of a given category. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). One such function is the logistic (or sigmoid) function: +This function takes values on the entire real axis. In the case of +logistic regression, however, the labels \( y_i \) are discrete +variables. One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to \( \{0,1\} \), +\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, +this is commonly known as the "perceptron" in the machine learning +literature. This model is extremely simple, and it is favorable in +many cases (e.g. noisy data) to have a ``soft" classifier that outputs +the probability of a given category. For example, given +\( \mathbf{x}_i \), the classifier outputs the probability of being in +category \( m \). One such function is the logistic (or sigmoid) function: + $$ \begin{equation} f(s) = \frac{1}{1+\mathrm e^{-s}}. diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index aae956a02..58412808d 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -3436,7 +3436,7 @@ 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 bears \( p \) features. Of cours e the +made of \( n \) samples, each of which bears \( p \) features. The primary goal is to identify the classes to which new unseen samples belong. @@ -3463,7 +3463,18 @@ where we use the short-hand notation
-This function takes values on the entire real axis. In the case of logistic regression, however, the labels \( y_i \) are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to \( \{0,1\} \), \( f(s_i)= \) sign$(s_i) = 1$ if \( s_i\ge 0 \) and 0 if otherwise. Indeed, this is commonly known as the "perceptron" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft" classifier that outputs the probability of a given category. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). One such function is the logistic (or sigmoid) function: +This function takes values on the entire real axis. In the case of +logistic regression, however, the labels \( y_i \) are discrete +variables. One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to \( \{0,1\} \), +\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, +this is commonly known as the "perceptron" in the machine learning +literature. This model is extremely simple, and it is favorable in +many cases (e.g. noisy data) to have a ``soft" classifier that outputs +the probability of a given category. For example, given +\( \mathbf{x}_i \), the classifier outputs the probability of being in +category \( m \). One such function is the logistic (or sigmoid) function: + $$ \begin{equation} f(s) = \frac{1}{1+\mathrm e^{-s}}. diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb index 1615fc1bc..bc61ef14c 100644 --- a/doc/pub/Regression/ipynb/Regression.ipynb +++ b/doc/pub/Regression/ipynb/Regression.ipynb @@ -4435,7 +4435,7 @@ "\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 bears $p$ features. Of cours e the\n", + "made of $n$ samples, each of which bears $p$ features. The\n", "primary goal is to identify the classes to which new unseen samples\n", "belong.\n", "\n", @@ -4469,7 +4469,17 @@ "\n", "## Some selected properties\n", "\n", - "This function takes values on the entire real axis. In the case of logistic regression, however, the labels $y_i$ are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to $\\{0,1\\}$, $f(s_i)=$ sign$(s_i) = 1$ if $s_i\\ge 0$ and 0 if otherwise. Indeed, this is commonly known as the \"perceptron\" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft\" classifier that outputs the probability of a given category. For example, given $\\mathbf{x}_i$, the classifier outputs the probability of being in category $m$. One such function is the logistic (or sigmoid) function:" + "This function takes values on the entire real axis. In the case of\n", + "logistic regression, however, the labels $y_i$ are discrete\n", + "variables. One simple way to get a discrete output is to have sign\n", + "functions that map the output of a linear regressor to $\\{0,1\\}$,\n", + "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. Indeed,\n", + "this is commonly known as the \"perceptron\" in the machine learning\n", + "literature. This model is extremely simple, and it is favorable in\n", + "many cases (e.g. noisy data) to have a ``soft\" classifier that outputs\n", + "the probability of a given category. For example, given\n", + "$\\mathbf{x}_i$, the classifier outputs the probability of being in\n", + "category $m$. One such function is the logistic (or sigmoid) function:" ] }, { diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz index db1df1813..883168687 100644 Binary files a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz and b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz differ diff --git a/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf b/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf index 025f01c9e..815c78910 100644 Binary files a/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf and b/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf differ diff --git a/doc/pub/Regression/pdf/Regression-beamer.pdf b/doc/pub/Regression/pdf/Regression-beamer.pdf index 1178c2081..d252562c0 100644 Binary files a/doc/pub/Regression/pdf/Regression-beamer.pdf and b/doc/pub/Regression/pdf/Regression-beamer.pdf differ diff --git a/doc/pub/Regression/pdf/Regression-minted.pdf b/doc/pub/Regression/pdf/Regression-minted.pdf index 0bd32443c..e9cfe6a97 100644 Binary files a/doc/pub/Regression/pdf/Regression-minted.pdf and b/doc/pub/Regression/pdf/Regression-minted.pdf differ diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt index ff9b7a4c7..fae509785 100644 --- a/doc/src/Regression/Regression.do.txt +++ b/doc/src/Regression/Regression.do.txt @@ -2830,7 +2830,7 @@ 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 bears $p$ features. Of cours e the +made of $n$ samples, each of which bears $p$ features. The primary goal is to identify the classes to which new unseen samples belong. @@ -2850,7 +2850,18 @@ $\mathbf{x}_i = (1,\boldsymbol{x}_i)$ and $\mathbf{w}_i = (b_0,\boldsymbol{w}_i) !split ===== Some selected properties ===== -This function takes values on the entire real axis. In the case of logistic regression, however, the labels $y_i$ are discrete variables. One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to $\{0,1\}$, $f(s_i)=$ sign$(s_i) = 1$ if $s_i\ge 0$ and 0 if otherwise. Indeed, this is commonly known as the ``perceptron" in the machine learning literature. This model is extremely simple, and it is favorable in many cases (e.g. noisy data) to have a ``soft" classifier that outputs the probability of a given category. For example, given $\mathbf{x}_i$, the classifier outputs the probability of being in category $m$. One such function is the logistic (or sigmoid) function: +This function takes values on the entire real axis. In the case of +logistic regression, however, the labels $y_i$ are discrete +variables. One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to $\{0,1\}$, +$f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise. Indeed, +this is commonly known as the ``perceptron" in the machine learning +literature. This model is extremely simple, and it is favorable in +many cases (e.g. noisy data) to have a ``soft" classifier that outputs +the probability of a given category. For example, given +$\mathbf{x}_i$, the classifier outputs the probability of being in +category $m$. One such function is the logistic (or sigmoid) function: + !bt \begin{equation} f(s) = \frac{1}{1+\mathrm e^{-s}}.