统计代写|机器学习代写machine learning代考|The ν–Trick and Other Variants

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我们提供的机器学习machine learning及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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
统计代写|机器学习代写machine learning代考|The ν–Trick and Other Variants

统计代写|机器学习代写machine learning代考|Multiclass Support Vector Machines

In order to extend the SV learning algorithm to $K=|\mathcal{Y}|>2$ classes two different strategies have been suggested.

  1. The first method is to learn $K$ SV classifiers $f_{j}$ by labeling all training points having $y_{i}=j$ with $+1$ and $y_{i} \neq j$ with $-1$ during the training of the $j$ th classifier. In the test stage, the final decision is obtained by
    $f_{\text {multiple }}(x)=\underset{y \in \mathcal{Y}}{\operatorname{argmax}} f_{y}(x) .$
    Clearly, this method learns one classifier for each of the $K$ classes against all the other classes and is hence known as the one-versus-rest (o-v-r) method. It can be

shown that it is possible to solve the $K$ optimization problems at once. Note that the computational effort is of order $\mathcal{O}\left(K m^{2}\right)$.

  1. The second method is to learn $K(K-1) / 2 \mathrm{SV}$ classifiers. If $1 \leq i<j \leq K$ the classifiers $f_{i, j}$ is learned using only the training samples from the class $i$ and $j$, labeling them $+1$ and $-1$, respectively. This method has become known as the one-versus-one (o-v-o) method. Given a new test object $x \in \mathcal{X}$, the frequency $n_{i}$ of “wins” for class $i$ is computed by applying $f_{i, j}$ for all $j$. This results in a vector $\mathbf{n}=\left(n_{1} ; \ldots ; n_{K}\right)$ of frequencies of “wins” of each class. The final decision is made for the most frequent class, i.e.,
    $f_{\text {multiple }}(x)=\underset{y \in \mathcal{Y}}{\operatorname{argmax}} n_{y}$
    Using a probabilistic model for the frequencies $\mathbf{n}$, different prior probabilities of the classes $y \in \mathcal{Y}$ can be incorporated, resulting in better generalization ability. Instead of solving $K(K-1) / 2$ separate optimization problems, it is again possible to combine them in a single optimization problem. If the prior probabilities $\mathbf{P}_{\mathbf{Y}}(j)$ for the $K$ classes are roughly $\frac{1}{K}$, the method scales as $\mathcal{O}\left(m^{2}\right)$ and is independent of the number of classes.

Recently, a different method for combining the single pairwise decisions has been suggested. By specifying a directed acyclic graph (DAG) of consecutive pairwise classifications, it is possible to introduce a class hierarchy. The leaves of such a DAG contain the final decisions which are obtained by exclusion rather than by voting. This method compares favorably with the $o-v-o$ and $o-v-r$ methods.

统计代写|机器学习代写machine learning代考|Support Vector Regression Estimation

In the regression estimation problem we are given a sample of $m$ real target values $\boldsymbol{t}=\left(t_{1}, \ldots, t_{m}\right) \in \mathbb{R}^{m}$, rather than $m$ classes $\boldsymbol{y}=\left(y_{1}, \ldots, y_{m}\right) \in \mathcal{Y}^{m} .$ In order to extend the SV learning algorithm to this task, we note that an “inversion” of the linear loss $l_{\text {lin }}$ suffices in order to use the $S V$ machinery for real-valued outputs $t_{i}$. In classification the linear loss $l_{\operatorname{lin}}(f(x), \cdot)$ adds to the total cost, if the real-valued output of $|f(x)|$ is smaller than 1. For regression estimation it is desirable to have the opposite true, i.e., incurred costs result if $|t-f(x)|$ is very large instead of small. This requirement is formally captured by the $\varepsilon-i n$ sensitive loss
l_{\varepsilon}(f(x), t)= \begin{cases}0 & \text { if }|t-f(x)| \leq \varepsilon \ |t-f(x)|-\varepsilon & \text { if }|t-f(x)|>\varepsilon\end{cases}
$$ Then, one obtains a quadratic programming problem similar to $(2.46)$, this time in $2 m$ dual variables $\alpha_{i}$ and $\tilde{\alpha}{i}$-two corresponding to each training point constraint. This is simply due to the fact that $f$ can fail to attain a deviation less than $\varepsilon$ on both sides of the given real-valued output $t{i}$, i.e., $t_{i}-\varepsilon$ and $t_{i}+\varepsilon$. An appealing feature of this loss is that it leads to sparse solutions, i.e., only a few of the $\alpha_{i}$ (or $\tilde{\alpha}_{i}$ ) are non-zero. For further references that cover the regression estimation problem the interested reader is referred to Section 2.6.

统计代写|机器学习代写machine learning代考|Support Vector Machines for Classification

A major drawback of the soft margin SV learning algorithm given in the form (2.48) is the lack of control over how many training points will be considered as margin errors or “outliers”, that is, how many have $\tilde{\gamma}{i}\left(w{\mathrm{SVM}}\right)<1$. This is essentially due to the fact that we fixed the functional margin to one. By a simple reparameterization it is possible to make the functional margin itself a variable of the optimization problem. One can show that the solution of the following optimization problem has the property that the new parameter $v$ bounds the fraction of margin errors $\frac{1}{m}\left|\left{\left(x_{i}, y_{i}\right) \in z \mid \tilde{\gamma}{i}\left(\mathbf{w}{\text {SVM }}\right)<\rho\right}\right|$ from above:
\operatorname{minimize} \quad & \frac{1}{m} \sum_{i=1}^{m} \xi_{i}-v \rho+\frac{1}{2}|\mathbf{w}|^{2} \
\text { subject to } & y_{i}\left\langle\mathbf{x}{i}, \mathbf{w}\right\rangle \geq \rho-\xi{i} \quad i=1, \ldots, m, \
& \boldsymbol{\xi} \geq \mathbf{0}, \rho \geq 0 .
It can be shown that, for each value of $v \in[0,1]$, there exists a value of $\lambda \in \mathbb{R}^{+}$ such that the solution $\mathbf{w}{\nu}$ and $\mathbf{w}{\lambda}$ found by solving $(2.52)$ and (2.48) have the same geometrical margins $\gamma_{z}\left(\mathbf{w}{\nu}\right)=\gamma{z}\left(\mathbf{w}{\lambda}\right)$. Thus we could try different values of $\lambda$ in the standard linear soft margin SVM to obtain a required fraction of margin errors. The appealing property of the problem $(2.52)$ is that this adjustment is done within the one optimization problem (see Section B.5). Another property which can be proved is that, for all probability models where neither $\mathbf{P}{X}({X, 1})$ nor $\mathbf{P}{\mathbf{X}}({\mathrm{X},-1})$ contains any discrete component, $v$ asymptotically equals the fraction of margin errors. Hence, we can incorporate prior knowledge of the noise level $\mathbf{E}{\mathrm{X}}\left[1-\max {y \in \mathcal{Y}}\left(\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x}(y)\right)\right]$ via $v$. Excluding all training points for which the real-valued output is less than $\rho$ in absolute value, the geometrical margin of the solution on the remaining training points is $\rho /|\mathbf{w}|$.

统计代写|机器学习代写machine learning代考|The ν–Trick and Other Variants


统计代写|机器学习代写machine learning代考|Multiclass Support Vector Machines

为了将 SV 学习算法扩展到ķ=|是|>2类提出了两种不同的策略。

  1. 第一种方法是学习ķSV分类器Fj通过标记所有训练点是一世=j和+1和是一世≠j和−1在训练期间j分类器。在测试阶段,最终的决定是通过
    F多 (X)=最大参数是∈是F是(X).


  1. 第二种方法是学习ķ(ķ−1)/2小号在分类器。如果1≤一世<j≤ķ分类器F一世,j仅使用类中的训练样本学习一世和j, 标记它们+1和−1, 分别。这种方法已被称为一对一(ovo)方法。给定一个新的测试对象X∈X, 频率n一世课堂上的“胜利”一世通过应用计算F一世,j对全部j. 这导致一个向量n=(n1;…;nķ)每个类别的“获胜”频率。最终决定是为最频繁的类做出的,即
    F多 (X)=最大参数是∈是n是
    使用频率的概率模型n, 类的不同先验概率是∈是可以合并,从而获得更好的泛化能力。而不是解决ķ(ķ−1)/2单独的优化问题,可以再次将它们组合成一个优化问题。如果先验概率磷是(j)为了ķ类大致1ķ, 该方法缩放为这(米2)并且与类的数量无关。

最近,有人提出了一种组合单个成对决策的不同方法。通过指定连续成对分类的有向无环图(DAG),可以引入类层次结构。这种 DAG 的叶子包含通过排除而不是投票获得的最终决定。该方法与这−在−这和这−在−r方法。

统计代写|机器学习代写machine learning代考|Support Vector Regression Estimation

在回归估计问题中,我们给出了一个样本米实际目标值吨=(吨1,…,吨米)∈R米, 而不是米班级是=(是1,…,是米)∈是米.为了将 SV 学习算法扩展到此任务,我们注意到线性损失的“反转”llin 足以使用小号在实际价值产出的机器吨一世. 在分类中线性损失llin(F(X),⋅)增加总成本,如果实际价值产出|F(X)|小于 1。对于回归估计,最好有相反的结果,即,如果发生成本结果,|吨−F(X)|很大而不是很小。该要求由e−一世n敏感损失
le(F(X),吨)={0 如果 |吨−F(X)|≤e |吨−F(X)|−e 如果 |吨−F(X)|>e然后,得到一个二次规划问题,类似于(2.46), 这次在2米对偶变量一种一世和一种~一世-两个对应于每个训练点约束。这仅仅是因为F可能无法获得小于以下的偏差e在给定实值输出的两边吨一世, IE,吨一世−e和吨一世+e. 这种损失的一个吸引人的特点是它会导致稀疏的解决方案,即只有少数一种一世(或者一种~一世) 非零。有关回归估计问题的更多参考资料,感兴趣的读者请参阅第 2.6 节。

统计代写|机器学习代写machine learning代考|Support Vector Machines for Classification

以 (2.48) 形式给出的软边距 SV 学习算法的一个主要缺点是无法控制有多少训练点将被视为边际误差或“异常值”,即有多少有C~一世(在小号在米)<1. 这主要是因为我们将功能边距固定为 1。通过简单的重新参数化,可以使功能裕度本身成为优化问题的变量。可以证明下列优化问题的解具有新参数的性质在限制边际误差的分数\frac{1}{m}\left|\left{\left(x_{i}, y_{i}\right) \in z \mid \tilde{\gamma}{i}\left(\mathbf{w }{\text {SVM }}\right)<\rho\right}\right|\frac{1}{m}\left|\left{\left(x_{i}, y_{i}\right) \in z \mid \tilde{\gamma}{i}\left(\mathbf{w }{\text {SVM }}\right)<\rho\right}\right|从上面:
最小化1米∑一世=1米X一世−在ρ+12|在|2  受制于 是一世⟨X一世,在⟩≥ρ−X一世一世=1,…,米, X≥0,ρ≥0.
可以证明,对于每个值在∈[0,1], 存在一个值λ∈R+这样的解决方案在ν和在λ通过求解发现(2.52)和 (2.48) 具有相同的几何边距C和(在ν)=C和(在λ). 因此我们可以尝试不同的值λ在标准线性软裕度 SVM 中获得所需的裕度误差分数。问题的吸引力(2.52)是这种调整是在一个优化问题中完成的(参见第 B.5 节)。另一个可以证明的性质是,对于所有概率模型,磷X(X,1)也不磷X(X,−1)包含任何分立组件,在渐近地等于边际误差的分数。因此,我们可以结合噪声水平的先验知识和X[1−最大限度是∈是(磷是∣X=X(是))]通过在. 排除所有实值输出小于的训练点ρ在绝对值上,剩余训练点上解的几何裕度为ρ/|在|.

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



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