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# 计算机代写|机器学习代写Machine Learning代考|QBUS3820

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## 计算机代写|机器学习代写Machine Learning代考|Error-Correcting Output Codes

Given a set of $N$ classes to be learned in an ECOC framework, $n$ different bipartitions (groups of classes) are formed, and $n$ binary problems (dichotomizers) over the partitions are trained. As a result, a codeword of length $n$ is obtained for each class, where each position (bit) of the code corresponds to a response of a given dichotomizer (coded by +1 or -1 according to their class set membership). Arranging the codewords as rows of a matrix, we define a coding matrix $M$, where $M \in{-1,+1}^{N \times n}$ in the binary case. In Fig. 2.1 we show an example of a binary coding matrix $M$. The matrix is coded using five dichotomizers $\left{h_1, \ldots, h_5\right}$ for a 4-class problem $\left{c_1, \ldots, c_4\right}$ of respective codewords $\left{y_1, \ldots, y_4\right}$. The hypotheses are trained by considering the labelled training data samples $\left{\left(\rho_1, l\left(\rho_1\right)\right), \ldots,\left(\rho_m, l\left(\rho_m\right)\right)\right}$ for a set of $m$ data samples. The white and black regions of the coding matrix $M$ are coded by +1 and -1 , respectively. For example, the first classifier is trained to discriminate $c_3$ against $c_1, c_2$, and $c_4$; the second one classifies $c_2$ and $c_3$ against $c_1$ and $c_4$, and so on, as follows:
$$h_1(x)=\left{\begin{array}{ll} 1 & \text { if } x \in\left{c_3\right} \ -1 & \text { if } x \in\left{c_1, c_2, c_4\right} \end{array}, \ldots, \quad h_5(x)=\left{\begin{array}{ll} 1 & \text { if } x \in\left{c_2, c_4\right} \ -1 & \text { if } x \in\left{c_1, c_3\right} \end{array}\right}\right.$$

The standard binary coding designs are the one-versus-all [19] strategy with $N$ dichotomizers and the dense random strategy [2], with $10 \log _2 N$ classifiers. In the case of the ternary symbol-based ECOC, the coding matrix becomes $M \in$ ${-1,0,+1}^{N \times n}$. In this case, the symbol zero means that a particular class is not considered for a given classifier. In this ternary framework, the standard designs are the one-versus-one strategy [13] and the sparse random strategy [2], with $\frac{N(N-1)}{2}$ and $15 \log _2 N$ binary problems, respectively.

During the decoding process, applying $n$ binary classifiers, a code $x$ is obtained for each data sample $\rho$ in the test set. This code is compared to the base codewords $\left(y_i, i \in[1, . ., N]\right)$ of each class defined in the matrix $M$, and the data sample is assigned to the class with the closest codeword. In Fig. 2.1, the new code $x$ is compared to the class codewords $\left{y_1, \ldots, y_4\right}$ using Hamming [19] and Euclidean Decoding [2]. The test sample is classified by class $c_2$ in both cases, correcting one bit error.

In the literature, there roughly exists three different lines for decoding [9]: those based on similarity measurements, including the Hamming and Euclidean decoding [19], probabilistic approaches [21], and loss-functions strategies [2].

## 计算机代写|机器学习代写Machine Learning代考|Compact ECOC Coding

Although the use of large codewords was initially suggested in order to correct as many errors as possible at the decoding step, high effort has been put into improving the robustness of each individual dichotomizer so that compact codewords can be defined in order to save time. In this way, the one-versus-all ECOC has been widely applied for several years in the binary ECOC framework (see Fig. 2.2). Although the use of a reduced number of binary problems often implies dealing with more data per classifier (i.e. compared to the one-versus-one coding), this approach has been defended by some authors in the literature demonstrating that the one-versusall technique can reach comparable results to the rest of combining strategies if the base classifier is properly tuned [23]. Recently, this codeword length has been reduced to $N-1$ in the DECOC approach of [22], where the authors codify $N-1$ nodes of a binary tree structure as dichotomizers of a ternary problem-dependent ECOC design. In the same line, several problem-dependent designs have been recently proposed $[5,10,22,24]$. The new techniques are based on exploiting the problem domain by selecting the representative binary problems that increase the generalization performance while keeping the code length “relatively” small. Figure 2.2 shows the number ofdichotomizers required for the ECOC configurations of the state-of-the-art for different number of classes. The considered codings are: one-versus-all, one-versus-one, Dense random, Sparse random, DECOC and Compact ECOC $[2,10,13,19,22]$.

Although one-versus-all, DECOC, dense, and sparse random approaches have a relatively small codeword length, we can take advantage of the information theory principles to obtain a more compact definition of the codewords. Having a $N$-class problem, the minimum number of bits necessary to codify and univocally distinguish $N$ codes is:
$$B=\left\lceil\log _2 N\right\rceil,$$
where $\lceil$.$] rounds to the upper integer.$

## 计算机代写|机器学习代写Machine Learning代考|Error-Correcting Output Codes

$$h_1(x)=\left{\begin{array}{ll} 1 & \text { if } x \in\left{c_3\right} \ -1 & \text { if } x \in\left{c_1, c_2, c_4\right} \end{array}, \ldots, \quad h_5(x)=\left{\begin{array}{ll} 1 & \text { if } x \in\left{c_2, c_4\right} \ -1 & \text { if } x \in\left{c_1, c_3\right} \end{array}\right}\right.$$

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。