feat: Add lua script for matrix product representation
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main.pdf filter=lfs diff=lfs merge=lfs -text
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@ -1,34 +1,22 @@
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\chapter{Linear Model}
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\section{Generalized Linear Model}
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\section{Simple Linear Regression}
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\[
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g(\EE(Y)) = X \beta
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Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i
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\]
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with $g$ being
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\begin{itemize}
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\item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
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\item Poission regression: $g(v) = \log(v)$, for instance for discrete variables.
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\end{itemize}
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\subsection{Penalized Regression}
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When the number of variables is large, e.g, when the number of explanatory variable is above the number of observations, if $p >> n$ ($p$: the number of explanatory variable, $n$ is the number of observations), we cannot estimate the parameters.
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In order to estimate the parameters, we can use penalties (additional terms).
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Lasso regression, Elastic Net, etc.
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\subsection{Simple Linear Model}
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\begin{align*}
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\Y &= \X \beta + \varepsilon \\
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\[
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\Y = \X \beta + \varepsilon.
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\]
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\[
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\begin{pmatrix}
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Y_1 \\
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Y_2 \\
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\vdots \\
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Y_n
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\end{pmatrix}
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&= \begin{pmatrix}
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=
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\begin{pmatrix}
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1 & X_1 \\
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1 & X_2 \\
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\vdots & \vdots \\
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@ -42,17 +30,35 @@ Lasso regression, Elastic Net, etc.
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\begin{pmatrix}
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\varepsilon_1 \\
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\varepsilon_2 \\
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\vdots \\
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\vdots
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\varepsilon_n
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\end{pmatrix}
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\end{align*}
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\]
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\subsection{Assumptions}
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\paragraph*{Assumptions}
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\begin{enumerate}[label={\color{primary}{($A_\arabic*$)}}]
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\item $\varepsilon_i$ are independent;
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\item $\varepsilon_i$ are identically distributed;
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\item $\varepsilon_i$ are i.i.d $\sim \Norm(0, \sigma^2)$ (homoscedasticity).
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\end{enumerate}
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\section{Generalized Linear Model}
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\[
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g(\EE(Y)) = X \beta
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\]
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with $g$ being
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\begin{itemize}
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\item
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\item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
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\item Poisson regression: $g(v) = \log(v)$, for instance for discrete variables.
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\end{itemize}
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\subsection{Penalized Regression}
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When the number of variables is large, e.g, when the number of explanatory variable is above the number of observations, if $p >> n$ ($p$: the number of explanatory variable, $n$ is the number of observations), we cannot estimate the parameters.
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In order to estimate the parameters, we can use penalties (additional terms).
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Lasso regression, Elastic Net, etc.
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\subsection{Statistical Analysis Workflow}
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@ -95,9 +101,9 @@ is noted equivalently as
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\subsection{General Case}
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If $\X^\T\X$ is invertible, the OLS estimator is:
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If $\X^T\X$ is invertible, the OLS estimator is:
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\begin{equation}
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\hat{\beta} = (\X^\T\X)^{-1} \X^\T \Y
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\hat{\beta} = (\X^T\X)^{-1} \X^T \Y
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\end{equation}
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\subsection{Ordinary Least Square Algorithm}
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@ -112,12 +118,12 @@ We want to minimize the distance between $\X\beta$ and $\Y$:
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\Rightarrow& \forall v \in w,\, vy = v proj^w(y)\\
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\Rightarrow& \forall i: \\
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& \X_i \Y = \X_i X\hat{\beta} \qquad \text{where $\hat{\beta}$ is the estimator of $\beta$} \\
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\Rightarrow& \X^\T \Y = \X^\T \X \hat{\beta} \\
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\Rightarrow& {\color{gray}(\X^\T \X)^{-1}} \X^\T \Y = {\color{gray}(\X^\T \X)^{-1}} (\X^\T\X) \hat{\beta} \\
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\Rightarrow& \hat{\beta} = (X^\T\X)^{-1} \X^\T \Y
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\Rightarrow& \X^T \Y = \X^T \X \hat{\beta} \\
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\Rightarrow& {\color{gray}(\X^T \X)^{-1}} \X^T \Y = {\color{gray}(\X^T \X)^{-1}} (\X^T\X) \hat{\beta} \\
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\Rightarrow& \hat{\beta} = (\X^T\X)^{-1} \X^T \Y
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\end{align*}
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This formula comes from the orthogonal projection of $\Y$ on the subspace define by the explanatory variables $\X$
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This formula comes from the orthogonal projection of $\Y$ on the vector subspace defined by the explanatory variables $\X$
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$\X \hat{\beta}$ is the closest point to $\Y$ in the subspace generated by $\X$.
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@ -31,12 +31,12 @@ Let $u = \begin{pmatrix}
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We may use $\scalar{u, v}$ or $u \cdot v$ notations.
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\end{definition}
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\paragraph{Dot product properties}
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\begin{itemize}
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\item $\scalar{u, v} = \scalar{v, u}$
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\item $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
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\item $\scalar{u, v}$
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\item $\scalar{\vec{u}, \vec{v}} = \norm{\vec{u}} \times \norm{\vec{v}} \times \cos(\widehat{\vec{u}, \vec{v}})$
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\end{itemize}
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\begin{description}
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\item[Commutative] $\scalar{u, v} = \scalar{v, u}$
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\item[Distributive] $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
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\item $\scalar{u, v} = \norm{u} \times \norm{v} \times \cos(\widehat{u, v})$
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\item $\scalar{a, a} = \norm{a}^2$
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\end{description}
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\begin{definition}[Norm]
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Length of the vector.
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@ -99,7 +99,7 @@ Let $u = \begin{pmatrix}
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\end{align*}
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\end{proof}
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\begin{theorem}{Pythagorean theorem}
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\begin{theorem}[Pythagorean theorem]
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If $u \perp v$, then $\norm{u+v}^2 = \norm{u}^2 + \norm{v}^2$ .
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\end{theorem}
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@ -110,7 +110,7 @@ Let $y = \begin{pmatrix}
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y_1 \\
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. \\
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y_n
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\end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$
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\end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$.
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$\mathcal{Y}$ can be written as the orthogonal projection of $y$ on $w$:
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\[
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\mathcal{Y} = proj^w(y) + z,
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@ -178,9 +178,26 @@ The scalar product between $z$ and (?) is zero.
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x_3 \\
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x_4
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\end{pmatrix}
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& = \begin{pmatrix}
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a x_1 + b x_2 + c x_3 \ldots
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\end{pmatrix}
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=
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\begin{pmatrix}
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\luadirect{
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local matrix_product = require("scripts.matrix_product")
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local m1 = {
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{"a", "b", "c", "d"},
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{"e", "f", "g", "h"},
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{"i", "j", "k", "l"}
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}
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local m2 = {
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{"x_1"},
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{"x_2"},
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{"x_3"},
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{"x_4"}
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}
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local product_matrix = matrix_product.matrix_product_repr(m1,m2)
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local matrix_dump = matrix_product.dump_matrix(product_matrix)
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tex.print(matrix_dump)
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}
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\end{pmatrix}
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\end{align*}
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\end{example}
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@ -190,7 +207,7 @@ The number of columns has to be the same as the dimension of the vector to which
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Let $A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}$, then $A^\T = \begin{pmatrix}
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\end{pmatrix}$, then $A^T = \begin{pmatrix}
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a & c \\
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b & d
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\end{pmatrix}$
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fill-opacity="1"
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<use
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xlink:href="#glyph-0-1"
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x="103.093"
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g61">
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<use
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xlink:href="#glyph-0-1"
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x="96.693"
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y="96.162"
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id="use61" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g62">
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<use
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xlink:href="#glyph-0-1"
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x="131.542"
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id="use62" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g63">
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<use
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xlink:href="#glyph-0-1"
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x="150.79"
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y="86.629"
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id="use63" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g64">
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<use
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xlink:href="#glyph-0-1"
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x="30.961"
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id="use64" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g65">
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<use
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xlink:href="#glyph-0-1"
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x="139.911"
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y="47.335"
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id="use65" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g66">
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<use
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xlink:href="#glyph-0-1"
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x="87.263"
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y="61.833"
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id="use66" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g67">
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<use
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xlink:href="#glyph-0-1"
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x="63.363"
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id="use67" />
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g68">
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<use
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xlink:href="#glyph-0-1"
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x="46.762"
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y="94.27"
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id="use68" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g69">
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<use
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xlink:href="#glyph-0-1"
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x="69.139"
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y="61.006"
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id="use69" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g70">
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<use
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xlink:href="#glyph-0-1"
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x="94.176"
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y="58.936"
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id="use70" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g71">
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<use
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xlink:href="#glyph-0-1"
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x="92.871"
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y="94.973"
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id="use71" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g72">
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<use
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xlink:href="#glyph-0-1"
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x="144.642"
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y="58.179"
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id="use72" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g73">
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<use
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xlink:href="#glyph-0-1"
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x="122.192"
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y="81.657"
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id="use73" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g74">
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<use
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xlink:href="#glyph-0-1"
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x="63.488"
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y="100.304"
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id="use74" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g75">
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<use
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xlink:href="#glyph-0-1"
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x="37.946"
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y="68.026"
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id="use75" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g76">
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<use
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xlink:href="#glyph-0-1"
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x="30.259"
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y="82.34"
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id="use76" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g77">
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<use
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xlink:href="#glyph-0-1"
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x="49.831"
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y="67.561"
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id="use77" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g78">
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<use
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xlink:href="#glyph-0-1"
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x="108.444"
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y="97.003"
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id="use78" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g79">
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<use
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xlink:href="#glyph-0-1"
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x="41.836"
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y="78.118"
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id="use79" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g80">
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<use
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xlink:href="#glyph-0-1"
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x="75.475"
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y="111.147"
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id="use80" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g81">
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<use
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xlink:href="#glyph-0-1"
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x="123.115"
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y="52.529"
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id="use81" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g82">
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<use
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xlink:href="#glyph-0-1"
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x="155.535"
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y="93.242"
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id="use82" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g83">
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<use
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xlink:href="#glyph-0-1"
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x="47.094"
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y="108.603"
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g84">
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<use
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xlink:href="#glyph-0-1"
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x="99.424"
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y="75.869"
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id="use84" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g85">
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<use
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xlink:href="#glyph-0-1"
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x="45.483"
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y="112.404"
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id="use85" />
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g86">
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<use
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xlink:href="#glyph-0-1"
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x="93.806"
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y="68.145"
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id="use86" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g87">
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<use
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xlink:href="#glyph-0-1"
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x="81.735"
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y="87.214"
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id="use87" />
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</g>
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<g
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fill="rgb(0%, 0%, 0%)"
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fill-opacity="1"
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id="g88">
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<use
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xlink:href="#glyph-0-1"
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x="92.588"
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y="98.852"
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id="use88" />
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</g>
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<path
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fill="none"
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stroke-width="0.79701"
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stroke-linecap="butt"
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stroke-linejoin="miter"
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stroke="rgb(0%, 0%, 100%)"
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stroke-opacity="1"
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stroke-miterlimit="10"
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d="M 0.00165625 56.694844 L 141.732125 85.038594 "
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transform="matrix(1, 0, 0, -1, 15.346, 156.105)"
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id="path88" />
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</svg>
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|
After Width: | Height: | Size: 24 KiB |
Binary file not shown.
|
|
@ -0,0 +1,26 @@
|
|||
\documentclass[tikz,border=3.14mm]{standalone}
|
||||
\usepackage{tikz-3dplot}
|
||||
\begin{document}
|
||||
\tdplotsetmaincoords{105}{-30}
|
||||
\usetikzlibrary{patterns}
|
||||
\begin{tikzpicture}[tdplot_main_coords,font=\sffamily]
|
||||
\tdplotsetrotatedcoords{00}{30}{0}
|
||||
\begin{scope}[tdplot_rotated_coords]
|
||||
\begin{scope}[canvas is xy plane at z=0]
|
||||
\draw[fill opacity=0,pattern=north west lines,pattern color=gray] (-2,-3) rectangle (2,3);
|
||||
\draw[gray,fill=lightgray,fill opacity=0.75] (-2,-3) rectangle (2,3);
|
||||
\draw[very thick] (-2,0) -- (2,0);
|
||||
\path (-150:2) coordinate (H) (-1.5,0) coordinate(X);
|
||||
\pgflowlevelsynccm
|
||||
\draw[very thick,-stealth,gray] (0,0) -- (-30:1.5);
|
||||
\end{scope}
|
||||
\draw[stealth-] (H) -- ++ (-1,0,0.2) node[pos=1.3]{$H$};
|
||||
\draw[stealth-] (X) -- ++ (0,1,0.2) node[pos=1.3]{$X$};
|
||||
\draw[very thick,-stealth] (0,0,0) coordinate (O) -- (0,0,3) node[right]{$p$};
|
||||
\end{scope}
|
||||
\pgfmathsetmacro{\Radius}{1.5}
|
||||
\draw[-stealth] (O)-- (2.5*\Radius,0,0) node[pos=1.15] {$x$};
|
||||
\draw[-stealth] (O) -- (0,3.5*\Radius,0) node[pos=1.15] {$z$};
|
||||
\draw[-stealth] (O) -- (0,0,2.5*\Radius) node[pos=1.05] {$y$};
|
||||
\end{tikzpicture}
|
||||
\end{document}
|
||||
|
|
@ -3,4 +3,4 @@
|
|||
\usepackage{standalone}
|
||||
\usepackage{tikz-3dplot}
|
||||
\usepackage{tkz-euclide}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{nicematrix}
|
||||
|
|
@ -0,0 +1,57 @@
|
|||
local function matrix_product_repr(m1, m2)
|
||||
if #m1[1] ~= #m2 then -- inner matrix-dimensions must agree
|
||||
return nil
|
||||
end
|
||||
|
||||
local res = {}
|
||||
|
||||
for i = 1, #m1 do
|
||||
res[i] = {}
|
||||
for j = 1, #m2[1] do
|
||||
res[i][j] = " "
|
||||
for k = 1, #m2 do
|
||||
if k ~= 1 then
|
||||
res[i][j] = res[i][j] .. " + "
|
||||
end
|
||||
res[i][j] = res[i][j] .. m1[i][k] .. " " .. m2[k][j]
|
||||
end
|
||||
end
|
||||
end
|
||||
return res
|
||||
end
|
||||
|
||||
local function dump_matrix(matrix)
|
||||
local repr = ""
|
||||
for i, row in ipairs(matrix) do
|
||||
for j, cell in ipairs(row) do
|
||||
repr = repr .. " " .. cell
|
||||
if j ~= #row then
|
||||
repr = repr .. " & "
|
||||
end
|
||||
end
|
||||
if i ~= #matrix then
|
||||
repr = repr .. [[ \\ ]]
|
||||
end
|
||||
repr = repr .. "\n"
|
||||
end
|
||||
return repr
|
||||
end
|
||||
|
||||
local m1 = {
|
||||
{"a", "b", "c", "d"},
|
||||
{"e", "f", "g", "h"},
|
||||
{"i", "j", "k", "l"}
|
||||
}
|
||||
local m2 = {
|
||||
{"x_1"},
|
||||
{"x_2"},
|
||||
{"x_3"},
|
||||
{"x_4"}
|
||||
}
|
||||
|
||||
print(dump_matrix(matrix_product_repr(m1, m2)))
|
||||
|
||||
return {
|
||||
matrix_product_repr = matrix_product_repr,
|
||||
dump_matrix = dump_matrix
|
||||
}
|
||||
Loading…
Reference in New Issue