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计算机代写|机器学习代写Machine Learning代考|COMP30027

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计算机代写|机器学习代写Machine Learning代考|ecomposing Multi-class Classification Problems

This section considers the main elements needed for decomposing a multi-class classification problem into a set of binary problems: the decomposition scheme, the encoding stage, and the decoding stage. More precisely, Sect. 3.3.1 formalizes the concept of decomposition scheme and presents two of the most well-known decomposition schemes. Section 3.3.2 provides a detailed explanation of the encoding and decoding stages.

Decomposition Schemes
Consider a multi-class classification problem $M C P$ determined on a class set $Y$ of size $K>2$. To show how to decompose $M C P$ into $L$ binary classification problems $B C P_l$ we define the notion of a binary class partition in Definition 3.1.

Definition 3.1. (Binary Class Partition) Given a class set $Y$, the set $P(Y)$ is said to be a binary class partition of $Y$ iff $P(Y)$ consists of two non-empty sets $Y^{-}$and $Y^{+}$ such that $Y^{-} \cup Y^{+}=Y$ and $Y^{-} \cap Y^{+}=\emptyset$.

Definition 3.1 allows us to introduce the notion of a decomposition scheme. A decomposition scheme describes how to decompose a multi-class classification problem $M C P$ into $L$ binary classification problems $B C P_l$, as given in Definition 3.2.
Definition 3.2. (Decomposition Scheme) Given a multi-class classification problem $M C P$ and positive integer $L$, the decomposition scheme of $M C P$ is a set $S P(Y)$ of $L$ binary class partitions $P_l(Y)$ such that for any two classes $y_1, y_2 \in Y$ there exists a binary class partition $P_m(Y) \in S P(Y)$ so that $\neg\left(y_1, y_2 \in Y_m^{-}\right) \wedge \neg\left(y_1, y_2 \in Y_m^{+}\right)$ where $Y_m^{-}, Y_m^{+} \in P_m(Y)$.

By Definition 3.2 any decomposition scheme $S P(Y)$ consists of $L$ binary class partitions $P_l(Y)$. The partitions $P_l(Y) \in S P(Y)$ are chosen so that each class $y \in Y$ can be uniquely determined.

A natural representation for a decomposition scheme $S P(Y)$ is a decomposition matrix $M$. The matrix $M$ is defined as a binary matrix ${-1,+1}^{K \times L}$. Its encoding is realized according to the following rule:
$$M_{k, l}= \begin{cases}-1 & \text { if class } y_k \in Y \text { belongs to } Y_l^{-} \text {of } P_l(Y) \ +1 & \text { if class } y_k \in Y \text { belongs to } Y_l^{+} \text {of } P_l(Y)\end{cases}$$

计算机代写|机器学习代写Machine Learning代考|Encoding and Decoding

To solve a multi-class classification problem $M C P$ according to a decomposition scheme $S P(Y)$ we need to pass two stages: encoding and decoding. Below we describe each of these stages in detail.

During the encoding stage we first generate binary classification problems $B C P_l$ according to a given decomposition scheme $S P(Y)$. Each $B C P_l$ is uniquely determined by a particular binary class partition $P_l(Y) \in S P(Y) . B C P_l$ is defined on the instance space $X$ and a class set given by the binary class partition $P_l(Y)$. The training data $D_l$ for $B C P_l$ consists of instances $\left(x, Y_l^{ \pm}\right) \in X \times P_l(Y)$ and for any instance $\left(x, Y_I^{ \pm}\right) \in D_l$ there exists an instance $(x, y)$ from the training data $D$ of the multiclass classification problem $M C P$ such that $y \in Y_l^{ \pm}$. Thus, the decomposition scheme $S P(Y)$ reduces the multi-class classification problem $M C P$ to $L$ binary classification problems $B C P_l$.

Once the binary classification problems $B C P_l$ have been determined, we train a binary classifier $h_{P(Y)}: X \rightarrow P(Y)$ for each $B C P_l$. The binary classifiers $h_{P(Y)}$ together form an ensemble classifier $h_{S P(Y)}: X \rightarrow Y$ equal to $\left{h_{P(Y)}\right}_{P(Y) \in S P(Y)}$.

During the decoding stage, given an instance $x \in X$ to be classified and an ensemble classifier $h_{S P(Y)}$, we need to decode the predictions provided by the binary classifiers $h_{P(Y)} \in h_{S P(Y)}$ to form a class estimate $y \in Y$ for the instance $x$. The OA and eECOC decomposition schemes both use the same decoding technique. This technique first takes the class score $S\left(x, y \mid h_{P(Y)}\right)$ provided by each binary classifier $h_{P(Y)} \in h_{S P(Y)}$ (see Definition 3.3 below) and then computes the final score $S\left(x, y \mid h_{S P(Y)}\right)$ of the ensemble classifier $h_{S P(Y)}$ as the sum of scores $S\left(x, y \mid h_{P(Y)}\right)$ over all the classifiers $h_{P(Y)} \in h_{S P(Y)}$ (see Definition 3.4 below).

Definition 3.3. Given a binary class partition $P(Y) \in S P(Y)$, a binary classifier $h_{P(Y)}: X \rightarrow P(Y)$, an instance $x \in X$ to be classified and a class $y \in Y$, the class score $S\left(x, y \mid h_{P(Y)}\right)$ for $x$ and $y$ provided by $h_{P(Y)}$ is defined as follows:
$$S\left(x, y \mid h_{P(Y)}\right)= \begin{cases}1 & \text { if class } y \in h_{P(Y)}(x) ; \ 0 & \text { if class } y \notin h_{P(Y)}(x) .\end{cases}$$

计算机代写|机器学习代写Machine Learning代考|Decomposing Multi-class Classification Problems

3.1.定义(二进制类分区)给定一个类集$Y$，集$P(Y)$被称为$Y$的二进制类分区，因为$P(Y)$包含两个非空集$Y^{-}$和$Y^{+}$，使得$Y^{-} \cup Y^{+}=Y$和$Y^{-} \cap Y^{+}=\emptyset$。

3.2.定义(分解方案)给定一个多类分类问题 $M C P$ 一个正整数 $L$的分解方案 $M C P$ 是一个集合 $S P(Y)$ 的 $L$ 二进制类分区 $P_l(Y)$ 对于任意两个类 $y_1, y_2 \in Y$ 存在二进制类分区 $P_m(Y) \in S P(Y)$ 如此……以至于…… $\neg\left(y_1, y_2 \in Y_m^{-}\right) \wedge \neg\left(y_1, y_2 \in Y_m^{+}\right)$ 在哪里 $Y_m^{-}, Y_m^{+} \in P_m(Y)$．

$$M_{k, l}= \begin{cases}-1 & \text { if class } y_k \in Y \text { belongs to } Y_l^{-} \text {of } P_l(Y) \ +1 & \text { if class } y_k \in Y \text { belongs to } Y_l^{+} \text {of } P_l(Y)\end{cases}$$

计算机代写|机器学习代写Machine Learning代考|Encoding and Decoding

3.3.定义给定一个二进制类分区$P(Y) \in S P(Y)$、一个二进制分类器$h_{P(Y)}: X \rightarrow P(Y)$、一个待分类的实例$x \in X$和一个类$y \in Y$，则$h_{P(Y)}$提供的$x$和$y$的类分数$S\left(x, y \mid h_{P(Y)}\right)$定义如下:
$$S\left(x, y \mid h_{P(Y)}\right)= \begin{cases}1 & \text { if class } y \in h_{P(Y)}(x) ; \ 0 & \text { if class } y \notin h_{P(Y)}(x) .\end{cases}$$

Matlab代写

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