统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Parallel Coordinates Plots

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• 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

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|KernelParallel Coordinates Plots

PCP is a method for representing high-dimensional data, see Inselberg (1985). Instead of plotting observations in an orthogonal coordinate system, PCP draws coordinates in parallel axes and connects them with straight lines. This method helps in representing data with more than four dimensions.

One first scales all variables to $\max =1$ and $\min =0$. The coordinate index $j$ is drawn onto the horizontal axis, and the scaled value of variable $x_{i j}$ is mapped onto the vertical axis. This way of representation is very useful for high-dimensional data. It is however also sensitive to the order of the variables, since certain trends in the data can be shown more clearly in one ordering than in another.

Example 1.5 Take, once again, the observations $96-105$ of the Swiss bank notes. These observations are six dimensional, so we can’t show them in a six-dimensional Cartesian coordinate system. Using the PCP technique, however, they can be plotted on parallel axes. This is shown in Fig. 1.22.

PCP can also be used for detecting linear dependencies between variables: if all the lines are of almost parallel dimensions $(p=2)$, there is a positive linear dependence between them. In Fig. $1.23$ we display the two variables weight and displacement for the car data set in Sect. 22.3. The correlation coefficient $\rho$ introduced in Sect. $3.2$ is $0.9$. If all lines intersect visibly in the middle, there is evidence of a negative linear dependence between these two variables, see Fig. $1.24$. In fact the correlation is $\rho=-0.82$ between two variables mileage and weight: The more the weight, the less the mileage.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Hexagon Plots

This section closely follows the presentation of Lewin-Koh (2006). In geometry, a hexagon is a polygon with six edges and six vertices. Hexagon binning is a type of bivariate histogram with hexagon borders. It is useful for visualising the structure of data sets entailing a large number of observations $n$. The concept of hexagon binning is as follows:

1. The $x y$ plane over the set (range $(x)$, $\operatorname{range}(y)$ ) is tessellated by a regular grid of hexagons.
2. The number of points falling in each hexagon is counted.
3. The hexagons with count $>0$ are plotted by using a colour ramp or varying the radius of the hexagon in proportion to the counts.

This algorithm is extremely fast and effective for displaying the structure of data sets even for $n \geq 10^{6}$. If the size of the grid and the cuts in the colour ramp are chosen in a clever fashion, then the structure inherent in the data should emerge in the binned plot. The same caveats apply to hexagon binning as histograms. Variance and bias vary in opposite directions with bin width, so we have to settle for finding the value of the bin width that yields the optimal compromise between variance and bias reduction. Clearly, if we increase the size of the grid, the hexagon plot appears to be smoother, but without some reasonable criterion on hand it remains difficult to say which bin width provides the “optimal” degree of smoothness. The default number of bins suggested by standard software is 30 .

Applications to some data sets are shown as follows. The data is taken from ALLBUS (2006) [ZA No.3762]. The number of respondents is 2,946 . The following nine variables have been selected to analyse the relation between each pair of variables.

A quadratic form $Q(x)$ is built from a symmetric matrix $\mathcal{A}(p \times p)$ and a vector $x \in \mathbb{R}^{p}:$
$$Q(x)=x^{\top}, \mathcal{A} x=\sum_{i=1}^{p} \sum_{j=1}^{p} a_{i j} x_{i} x_{j}$$
Definiteness of Quadratic Forms and Matrices
$$\begin{array}{ll} Q(x)>0 \text { for all } x \neq 0 & \text { positive definite } \ Q(x) \geq 0 \text { for all } x \neq 0 & \text { positive semidefinite } \end{array}$$
A matrix $\mathcal{A}$ is called positive definite (semidefinite) if the corresponding quadratic form $Q(.)$ is positive definite (semidefinite). We write $\mathcal{A}>0(\geq 0)$.
Quadratic forms can always be diagonalised, as the following result shows.
Theorem 2.3 If $\mathcal{A}$ is symmetric and $Q(x)=x^{\top} \mathcal{A} x$ is the corresponding quadratic form, then there exists a transformation $x \mapsto \Gamma^{\top} x=y$ such that
$$x^{\top} \mathcal{A} x=\sum_{i=1}^{p} \lambda_{i} y_{i}^{2}$$
where $\lambda_{i}$ are the eigenvalues of $\mathcal{A}$.
Proof $\mathcal{A}=\Gamma \Lambda \Gamma^{\top}$. By Theorem $2.1$ and $y=\Gamma^{\top} \alpha$ we have that $x^{\top} \mathcal{A} x=$ $x^{\top} \Gamma \Lambda \Gamma^{\top} x=y^{\top} \Lambda y=\sum_{i=1}^{p} \lambda_{i} y_{i}^{2}$.

Positive definiteness of quadratic forms can be deduced from positive eigenvalues.
Theorem $2.4 \mathcal{A}>0$ if and only if all $\lambda_{i}>0, i=1, \ldots, p$.
Proof $0<\lambda_{1} y_{1}^{2}+\cdots+\lambda_{p} y_{p}^{2}=x^{\top} \mathcal{A} x$ for all $x \neq 0$ by Theorem 2.3.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|KernelParallel Coordinates Plots

PCP 是一种表示高维数据的方法，参见 Inselberg (1985)。PCP 不是在正交坐标系中绘制观测值，而是在平行轴上绘制坐标并用直线连接它们。此方法有助于表示具有四个以上维度的数据。

PCP 也可用于检测变量之间的线性依赖关系：如果所有线的维度几乎平行(p=2)，它们之间存在正线性相关。在图。1.23我们显示了 Sect 中汽车数据集的两个变量权重和位移。22.3. 相关系数ρ节中介绍。3.2是0.9. 如果所有线在中间明显相交，则有证据表明这两个变量之间存在负线性相关性，见图。1.24. 实际上相关性是ρ=−0.82里程和重量两个变量之间：重量越大，里程越少。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Hexagon Plots

1. 这X是平面上的集合（范围(X), 范围⁡(是)) 由六边形的规则网格镶嵌。
2. 计算落在每个六边形中的点数。
3. 有计数的六边形>0通过使用色带或与计数成比例地改变六边形的半径来绘制。

X⊤一个X=∑一世=1pλ一世是一世2

有限元方法代写

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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。