cs代写|机器学习代写machine learning代考|Support vector machines

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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
cs代写|机器学习代写machine learning代考|Support vector machines

cs代写|机器学习代写machine learning代考|Soft margin classifier

Thus far we have only discussed the linear separable case, but how about the case when there are overlapping classes? It is possible to extend the optimization problem by allowing some data points to be in the margin while penalizing these points somewhat. We therefore include some slag variables $\xi_{i}$ that reduce the effective margin for each data point, but we add a penalty term to the optimization that penalizes if the sum of these slag variables are large,
\min {\mathbf{w}, b} \frac{1}{2}|\mathbf{w}|^{2}+C \sum{i} \xi_{i}
subject to the constraints
y^{(i)}\left(\mathbf{w}^{T} \mathbf{x}+b\right) & \geq 1-\xi_{i} \
\xi_{i} & \geq 0
The constant $C$ is a free parameter in this algorithm. Making this constant large means allowing fewer points to be in the margin. This parameter must be tuned and it is advisable at least to try to vary this parameter in order to verify that the results do not dramatically depend on an initial choice.

cs代写|机器学习代写machine learning代考|Non-linear support vector machines

We have treated the case of overlapping classes while assuming that the best we can do is a linear separation. However, what if the underlying problem is separable with a function that might be more complex? An example is shown in Fig. 3.10. Nonlinear separation and regression models are of course much more common in machine learning, and we will now look into the non-linear generalization of the SVM.

Let us illustrate the basic idea with an example in two-dimensions. A linear function with two attributes that span the 2-dimensional feature space is given by
y=w_{0}+w_{1} x_{1}+w_{2} x_{2}=\mathbf{w}^{T} \mathbf{x},
1 \
x_{1} \
and weight vector
\mathbf{w}^{T}=\left(w_{0}, w_{1}, w_{2}\right) .
Let us say that we cannot separate the data with this linear function but that we could separate it with a polynomial that include second-order terms like
y=\tilde{w}{0}+\tilde{w}{1} x_{1}+\tilde{w}{2} x{2}+\tilde{w}{3} x{1} x_{2}+\tilde{w}{4} x{1}^{2}+\tilde{w}{5} x{2}^{2}=\tilde{\mathbf{w}} \phi(\mathbf{x}) .
We can view the second equation as a linear separation on a feature vector
\mathbf{x} \rightarrow \phi(\mathbf{x})=\left(\begin{array}{c}
1 \
x_{1} \
x_{2} \
x_{1} x_{2} \
x_{1}^{2} \
\end{array}\right) .
This can be seen as mapping the attribute space $\left(1, x_{1}, x_{2}\right)$ to a higher-dimensional space with the mapping function $\phi(\mathbf{x})$. We call this mapping a feature map. The separating hyperplane is then linear in this higher-dimensional space. Thus, we can use the above linear maximum margin classification method in non-linear cases if we replace all occurrences of the attribute vector $x$ with the mapped feature vector $\phi(\mathbf{x})$.
There are only three problems remaining. One is that we don’t know what the mapping function should be. The somewhat ad-hoc solution to this problem will be that we try out some functions and see which one works best. We will discuss this further later in this chapter. The second problem is that we have the problem of overfitting

as we might use too many feature dimensions and corresponding free parameters $w_{i}$. In the next section, we provide a glimpse of an argument why SVMs might address this problem. The third problem is that with an increased number of dimensions the evaluation of the equations becomes more computational intensive. However, there is a useful trick to alleviate the last problem in the case when the calculations always contain only dot products between feature vectors. An example of this is the solution of the minimization problem of the dual problem in the earlier discussions of the linear SVM. The function to be minimized in this formulation, Egn $3.26$ with the feature maps, only depends on the dot products between a vector $\mathbf{x}^{(i)}$ of one example and another example $\mathbf{x}^{(j)}$. Also, when predicting the class for a new input vector $\mathbf{x}$ from Egn $3.24$ when adding the feature maps, we only need the resulting values for the dot products $\phi\left(\mathbf{x}^{(i)}\right)^{T} \phi(\mathbf{x})$. We now discuss that such dot products can sometimes be represented with functions called kernel functions,
K(\mathbf{x}, \mathbf{z})=\phi(\mathbf{x})^{T} \phi(\mathbf{z})
Instead of actually specifying a feature map, which is often a guess to start with, we could actually specify a kernel function. For example, let us consider a quadratic kernel function between two vectors $\mathbf{x}$ and $\mathbf{z}$,
K(\mathbf{x}, \mathbf{z})=\left(\mathbf{x}^{T} \mathbf{z}+1\right)^{2}

cs代写|机器学习代写machine learning代考|Statistical learning theory and VC dimension

SVMs are good and practical classification algorithms for several reasons. In particular, they are formulated as a convex optimization problem that has many good theoretical properties and that can be solved with quadratic programming. They are formulated to

take advantage of the kernel trick, they have a compact representation of the decision hyperplane with support vectors, and turn out to be fairly robust with respect to the hyper parameters. However, in order to act as a good learner, they need to moderate the overfitting problem discussed earlier. A great theoretical contributions of Vapnik and colleagues was the embedding of supervised learning into statistical learning theory and to derive some bounds that make statements on the average ability to learn form data. We briefly outline here the ideas and state some of the results without too much details, and we discuss this issue here entirely in the context of binary classification. However, similar observations can be made in the case of multiclass classification and regression. This section uses language from probability theory that we only introduce in more detail later. Therefore, this section might be best viewed at a later stage. Again, the main reason in placing this section is to outline the deeper reasoning for specific models.

As can’t be stressed enough, our objective in supervised machine learning is to find a good model which minimizes the generalization error. To state this differently by using nomenclature common in these discussions, we call the error function here the risk function $R$; in particular, the expected risk. In the case of binary classification, this is the probability of missclassification,
R(h)=P(h(x) \neq y)
Of course, we generally do not know this density function. We assume here that the samples are iid (independent and identical distributed) data, and we can then estimate what is called the empirical risk with the help of the test data,
\hat{R}(h)=\frac{1}{m} \sum_{i=1}^{m} \mathbb{1}\left(h\left(\mathbf{x}^{(i)} ; \theta\right)=y^{(i)}\right)

cs代写|机器学习代写machine learning代考|Support vector machines


cs代写|机器学习代写machine learning代考|Soft margin classifier



是(一世)(在吨X+b)≥1−X一世 X一世≥0

cs代写|机器学习代写machine learning代考|Non-linear support vector machines

我们已经处理了重叠类的情况,同时假设我们能做的最好的是线性分离。但是,如果潜在问题可以与可能更复杂的函数分开怎么办?示例如图 3.10 所示。非线性分离和回归模型当然在机器学习中更为常见,我们现在将研究 SVM 的非线性泛化。



X=(1 X1 X2)



X→φ(X)=(1 X1 X2 X1X2 X12 X22).
这可以看作是映射属性空间(1,X1,X2)到具有映射函数的高维空间φ(X). 我们称这种映射为特征图。分离的超平面在这个高维空间中是线性的。因此,如果我们替换所有出现的属性向量,我们可以在非线性情况下使用上述线性最大边距分类方法X与映射的特征向量φ(X).

因为我们可能会使用太多的特征维度和相应的自由参数在一世. 在下一节中,我们将简要介绍为什么 SVM 可以解决这个问题。第三个问题是,随着维数的增加,方程的评估变得更加计算密集。然而,当计算总是只包含特征向量之间的点积时,有一个有用的技巧可以缓解最后一个问题。这方面的一个例子是前面讨论的线性 SVM 中对偶问题的最小化问题的解决方案。在这个公式中要最小化的函数,Egn3.26使用特征图,仅取决于向量之间的点积X(一世)一个例子和另一个例子X(j). 此外,在预测新输入向量的类别时X来自 Egn3.24添加特征图时,我们只需要点积的结果值φ(X(一世))吨φ(X). 我们现在讨论这种点积有时可以用称为核函数的函数来表示,



cs代写|机器学习代写machine learning代考|Statistical learning theory and VC dimension


利用内核技巧,它们具有带有支持向量的决策超平面的紧凑表示,并且在超参数方面相当稳健。然而,为了成为一个好的学习者,他们需要缓和前面讨论的过度拟合问题。Vapnik 及其同事的一个重要理论贡献是将监督学习嵌入到统计学习理论中,并得出了一些关于学习表格数据的平均能力的陈述。我们在这里简要概述了这些想法并在没有太多细节的情况下陈述了一些结果,并且我们在这里完全在二进制分类的背景下讨论了这个问题。但是,在多类分类和回归的情况下可以进行类似的观察。本节使用概率论中的语言,稍后我们将更详细地介绍。因此,最好在稍后阶段查看此部分。同样,放置本节的主要原因是概述特定模型的更深层次的推理。

怎么强调都不为过,我们在监督机器学习中的目标是找到一个好的模型来最小化泛化误差。为了通过使用这些讨论中常见的命名法来不同地说明这一点,我们将这里的误差函数称为风险函数R; 特别是预期风险。在二分类的情况下,这是错误分类的概率,

当然,我们一般不知道这个密度函数。我们在这里假设样本是 iid(独立同分布)数据,然后我们可以借助测试数据估计所谓的经验风险,


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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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