机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Regression by Support Vector Machines

statistics-lab™ 为您的留学生涯保驾护航 在代写监督学习Supervised and Unsupervised learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写监督学习Supervised and Unsupervised learning代写方面经验极为丰富，各种代写监督学习Supervised and Unsupervised learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Regression by Support Vector Machines

In the regression, we estimate the functional dependence of the dependent (output) variable $y \in \Re$ on an $m$-dimensional input variable $\mathbf{x}$. Thus, unlike in pattern recognition problems (where the desired outputs $y_{i}$ are discrete values e.g., Boolean) we deal with real valued functions and we model an $\Re^{m}$ to $\Re^{1}$ mapping here. Same as in the case of classification, this will be achieved by training the SVM model on a training data set first. Interestingly and importantly, a learning stage will end in the same shape of a dual Lagrangian as in classification, only difference being in a dimensionalities of the Hessian matrix and corresponding vectors which are of a double size now e.g., $\mathbf{H}$ is a $(2 n, 2 n)$ matrix. Initially developed for solving classification problems, SV techniques can be successfully applied in regression, i.e., for a functional approximation problems $[45,142]$. The general regression learning problem is set as follows – the learning machine is given $n$ training data from which it attempts to learn the input-output relationship (dependency, mapping or function) $f(\mathbf{x})$. A training data set $\mathcal{X}=[\mathbf{x}(i), y(i)] \in \Re^{m} \times \Re, i=1, \ldots, n$ consists of $n$ pairs $\left(\mathbf{x}{1}, y{1}\right),\left(\mathbf{x}{2}, y{2}\right), \ldots,\left(\mathbf{x}{n}, y{n}\right)$, where the inputs $\mathbf{x}$ are $m$-dimensional vectors $\mathbf{x} \in \Re^{m}$ and system responses $y \in \Re$, are continuous values. We introduce all the relevant and necessary concepts of SVM’s regression in a gentle way starting again with a linear regression hyperplane $f(\mathbf{x}, \mathbf{w})$ given as
$$f(\mathbf{x}, \mathbf{w})=\mathbf{w}^{T} \mathbf{x}+b$$
In the case of SVM’s regression, we measure the error of approximation instead of the margin used in classification. The most important difference in respect to classic regression is that we use a novel loss (error) functions here. This is the Vapnik’s linear loss function with e-insensitivity zone defined as
$$E(\mathbf{x}, y, f)=|y-f(\mathbf{x}, \mathbf{w})|_{e}= \begin{cases}0 & \text { if }|y-f(\mathbf{x}, \mathbf{w})| \leq \varepsilon \ |y-f(\mathbf{x}, \mathbf{w})|-\varepsilon & \text { otherwise }\end{cases}$$

or as,
$$E(\mathbf{x}, y, f)=\max (0,|y-f(\mathbf{x}, \mathbf{w})|-\varepsilon)$$
Thus, the loss is equal to zero if the difference between the predicted $f\left(\mathbf{x}{i}, \mathbf{w}\right)$ and the measured value $y{i}$ is less than $\varepsilon$. In contrast, if the difference is larger than $\varepsilon$, this difference is used as the error. Vapnik’s $\varepsilon$-insensitivity loss function (2.40) defines an $\varepsilon$ tube as shown in Fig. 2.18. If the predicted value is within the tube, the loss (error or cost) is zero. For all other predicted points outside the tube, the loss equals the magnitude of the difference between the predicted value and the radius $\varepsilon$ of the tube.

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Implementation Issues

In both the classification and the regression the learning problem boils down to solving the QP problem subject to the so-called ‘box-constraints’ and to the equality constraint in the case that a model with a bias term $b$ is used. The SV training works almost perfectly for not too large data basis. However, when the number of data points is large (say $n>2,000$ ) the QP problem becomes extremely difficult to solve with standard QP solvers and methods. For example, a classification training set of 50,000 examples amounts to a Hessian matrix $\mathbf{H}$ with $2.5 * 10^{9}$ (2.5 billion) elements. Using an 8 -byte floating-point representation we need 20,000 Megabytes $=20$ Gigabytes of memory [109]. This cannot be easily fit into memory of present standard computers, and this is the single basic disadvantage of the SVM method. There are three approaches that resolve the QP for large data sets. Vapnik in [144] proposed the chunking method that is the decomposition approach. Another decomposition approach is suggested in [109]. The sequential minimal optimization [115] algorithm is of different character and it seems to be an ‘error back propagation’ for an SVM learning. A systematic exposition of these various techniques is not given here, as all three would require a lot of space. However, the interested reader can find a description and discussion about the algorithms mentioned above in next chapter and $[84,150]$. The Vogt and Kecman’s chapter $[150]$ discusses the application of an active set algorithm in solving small to medium sized QP problems. For such data sets and when the high precision is required the active set approach in solving QP problems seems to be superior to other approaches (notably to the interior point methods and to the sequential minimal optimization (SMO) algorithm). Next chapter introduces the efficient iterative single data algorithm (ISDA) for solving huge data sets (say more than 100,000 or 500,000 or over 1 million training data pairs).

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Iterative Single Data Algorithm

One of the mainstream research fields in learning from empirical data by support vector machines (SVMs), and solving both the classification and the regression problems is an implementation of the iterative (incremental) learning schemes when the training data set is huge. The challenge of applying SVMs on huge data sets comes from the fact that the amount of computer memory required for solving the quadratic programming (QP) problem presented in the previous chapter increases drastically with the size of the training data set $n$. Depending on the memory requirement, all the solvers of SVMs can be classified into one of the three basic types as shown in Fig. 3.1 [150]. Direct methods (such as interior point methods) can efficiently obtain solution in machine precision, but they require at least $\mathcal{O}\left(n^{2}\right)$ of memory to store the Hessian matrix of the QP problem. As a result, they are often used to solve small-sized problems which require high precision. At the other end of the spectrum are the working-set (decomposition) algorithms whose memory requirements are only $\mathcal{O}\left(n+q^{2}\right)$ where $q$ is the size of the working-set (for the ISDAs developed in this book, $q$ is equal to 1). The reason for the low memory footprint is due to the fact that the solution is obtained iteratively instead of directly as in most of the QP solvers. They are the only possible algorithms for solving large-scale learning problems, but they are not suitable for obtaining high precision solutions because of the iterative nature of the algorithm. The relative size of the learning problem depends on the computer being used. As a result, a learning problem will be regarded as a “large” or “huge” problem in this book if the Hessian matrix of its unbounded SVs $\left(\mathbf{H}{S{f}} S_{f}\right.$ where $S_{f}$ denotes the set of free SVs) cannot be stored in the computer memory. Between the two ends of the spectrum are the active-set algorithms $[150]$ and their memory requirements are $\mathcal{O}\left(N_{F S V}^{2}\right)$, i.e. they depend on the number of unbounded support vectors of the problem. The main focus of this book is to develop efficient algorithms that can solve large-scale QP problems for SVMs in practice. Although many applications in engineering also require the solving of large-scale QP problems (and there are many solvers available), the QP problems induced by SVMs are different from these applications. In the case of SVMs, the Hessian matrix of (2.38a) is extremely dense, whereas in most of the engineering applications, the optimization problems have relatively sparse Hessian matrices. This is why many of the existing QP solvers are not suitable for SVMs and new approaches need to be invented and developed. Among several candidates that avoid the use of standard QP solvers, the two learning approaches which recently have drawn the attention are the Iterative Single Data Algorithm (ISDA), and the Sequential Minimal Optimization (SMO) $[69,78,115,148]$.

F(X,在)=在吨X+b

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。