notation foret et arbre

reports/bolsonaro.tex

@@ -5,6 +5,13 @@
 \usepackage{algpseudocode}
 \usepackage{algorithm}
 \usepackage{float}
+\usepackage{dsfont}
+\usepackage{amsmath}
+
+
+\DeclareMathOperator*{\argmax}{arg\,max}
+\DeclareMathOperator*{\argmin}{arg\,min}
+
 
 \algnewcommand\algorithmicforeach{\textbf{for each}}
 \algdef{S}[FOR]{ForEach}[1]{\algorithmicforeach\ #1\ \algorithmicdo}
@@ -23,7 +30,28 @@
 \section{Introduction}
 \subsection{Notation}
 
-$S = \{(x_i, y_i)\}^n_{i=1}$ the dataset, with $x_i \in X$ and $y_i \in Y$. $T = \{t_1, t_2, \dots, t_d\}$ the random forest of $d$ trees, such that $t_j : X \rightarrow Y$.
+Let $ X \in \mathbb{R}^{n \times d}$ be the matrix data and $Y \in \mathbb{R}^{n}$ be the labels vector associated to the matrix $X$, where for each $i$, $\textbf{x}_i \in \cal{X} \subseteq \mathbb{R}^d$ and $y_i \in {\cal Y} \subseteq \mathbb{R}$. \\
+
+A random forest $F_{t_1, \dots, t_l}$ is a classifier  made of a collection of $l$ trees ${t_1, \dots, t_l}$. This forest can be seen as a function, and noted as:
+%
+$$\begin{array}{ccccc}
+F_{t_1, \dots, t_l} & : & \cal{X} & \to & \cal{Y} \\
+ & & \textbf{x} & \mapsto & F_{t_1, \dots, t_l}(\textbf{x}) = f(\{t_1, \dots, t_l\} , \textbf{x}) \\
+\end{array}$$
+%
+where $f$ is a function which depend on the task. In a regression setup, where ${\cal Y} = \mathbb{R}$, this function can be defined as: 
+%
+$$f(\{t_1, \dots, t_l \} , \textbf{x}) = \sum_{i = 1}^{l} \alpha_i t_i(x) \ \text{ where } \alpha_i \in \mathbb{R},$$
+%
+while in a classification setup, in which ${\cal Y} = \{ c_1, \dots, c_m \}$, $f$ will be a majority vote function:
+%
+$$f(\{t_1, \dots, t_l \} , \textbf{x}) = \argmax_{c \in {\cal Y}} \sum_{i = 1}^{l}  \mathds{1}(t_i(\textbf{x}) = c).$$
+%
+We will need to define the vector prediction of a forest for all the data matrix: $F_{t_1, \dots, t_l}(X)  = \begin{pmatrix}
+   F_{t_1, \dots, t_l}(x_1) \\
+   \dots \\
+   F_{t_1, \dots, t_l}(x_n) 
+\end{pmatrix}.$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Orthogonal Matching Pursuit (OMP)}