Posted on Categories:Multivariate Statistical Analysis, 多元统计分析, 统计代写, 统计代考

# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Chernoff-Flury Faces

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## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Chernoff-Flury Faces

If we are given data in numerical form, we tend to display it also numerically. This was done in the preceding sections: an observation $x_1=(1,2)$ was plotted as the point $(1,2)$ in a two-dimensional coordinate system. In multivariate analysis we want to understand data in low dimensions (e.g., on a 2D computer screen) although the structures are hidden in high dimensions. The numerical display of data structures using coordinates therefore ends at dimensions greater than three.

If we are interested in condensing a structure into $2 \mathrm{D}$ elements, we have to consider alternative graphical techniques. The Chernoff-Flury faces, for example, provide such a condensation of high-dimensional information into a simple “face”. In fact faces are a simple way to graphically display high-dimensional data. The size of the face elements like pupils, eyes, upper and lower hair line, etc., are assigned to certain variables. The idea of using faces goes back to Chernoff (1973) and has been further developed by Bernhard Flury. We follow the design described in Flury and Riedwyl (1988) which uses the following characteristics.
\begin{aligned} 1 & \text { right eye size } \ 2 & \text { right pupil size } \ 3 & \text { position of right pupil } \ 4 & \text { right eye slant } \ 5 & \text { horizontal position of right eye } \ 6 & \text { vertical position of right eye } \ 7 & \text { curvature of right eyebrow } \ 8 & \text { density of right eyebrow } \ 9 & \text { horizontal position of right eyebrow } \ 10 & \text { vertical position of right eyebrow } \ 11 & \text { right upper hair line } \ 12 & \text { right lower hair line } \ 13 & \text { right face line } \ 14 & \text { darkness of right hair } \ 15 & \text { right hair slant } \ 16 & \text { right nose line } \ 17 & \text { right size of mouth } \ 18 & \text { right curvature of mouth } \ 36 & \text { like } 1-18 \text {, only for the left side. }\end{aligned}
1 right eye size
2 right pupil size
3 position of right pupil
4 right eye slant
5 horizontal position of right eye
6 vertical position of right eye
7 curvature of right eyebrow
8 density of right eyebrow
9 horizontal position of right eyebrow
10 vertical position of right eyebrow
11 right upper hair line
12 right lower hair line
13 right face line
14 darkness of right hair
15 right hair slant
16 right nose line
17 right size of mouth
18 right curvature of mouth
19-36 like 1-18, only for the left side.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Andrews’ Curves

The basic problem of graphical displays of multivariate data is the dimensionality. Scatterplots work well up to three dimensions (if we use interactive displays). More than three dimensions have to be coded into displayable 2D or 3D structures (e.g., faces). The idea of coding and representing multivariate data by curves was suggested by Andrews (1972). Each multivariate observation $X_i=\left(X_{i, 1}, . ., X_{i, p}\right)$ is transformed into a curve as follows:
$$f_i(t)= \begin{cases}\frac{X_{i, 1}}{\sqrt{2}}+X_{i, 2} \sin (t)+X_{i, 3} \cos (t)+\ldots+X_{i, p-1} \sin \left(\frac{p-1}{2} t\right)+X_{i, p} \cos \left(\frac{p-1}{2} t\right) & \text { for } p \text { odd } \ \frac{X_{i, 1}}{\sqrt{2}}+X_{i, 2} \sin (t)+X_{i, 3} \cos (t)+\ldots+X_{i, p} \sin \left(\frac{p}{2} t\right) & \text { for } p \text { even }\end{cases}$$
such that the observation represents the coefficients of a so-called Fourier series $(t \in[-\pi, \pi])$.
Suppose that we have three-dimensional observations: $X_1=(0,0,1), X_2=(1,0,0)$ and $X_3=(0,1,0)$. Here $p=3$ and the following representations correspond to the Andrews’ curves:
\begin{aligned} f_1(t) & =\cos (t) \ f_2(t) & =\frac{1}{\sqrt{2}} \text { and } \ f_3(t) & =\sin (t) . \end{aligned}
These curves are indeed quite distinct, since the observations $X_1, X_2$, and $X_3$ are the 3D unit vectors: each observation has mass only in one of the three dimensions. The order of the variables plays an important role.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Chernoff-Flury Faces

\begin{aligned} 1 & \text { right eye size } \ 2 & \text { right pupil size } \ 3 & \text { position of right pupil } \ 4 & \text { right eye slant } \ 5 & \text { horizontal position of right eye } \ 6 & \text { vertical position of right eye } \ 7 & \text { curvature of right eyebrow } \ 8 & \text { density of right eyebrow } \ 9 & \text { horizontal position of right eyebrow } \ 10 & \text { vertical position of right eyebrow } \ 11 & \text { right upper hair line } \ 12 & \text { right lower hair line } \ 13 & \text { right face line } \ 14 & \text { darkness of right hair } \ 15 & \text { right hair slant } \ 16 & \text { right nose line } \ 17 & \text { right size of mouth } \ 18 & \text { right curvature of mouth } \ 36 & \text { like } 1-18 \text {, only for the left side. }\end{aligned}
1右眼大小
2瞳孔大小合适
3 .右瞳孔位置
4 .右眼倾斜
5 .右眼水平位置
6 .右眼垂直位置

8 .右眉密度
9 .右眉水平位置
10 .右眉垂直位置
11号右上发际线
12号右下发际线
13右面线
14黑色的右头发
15 .右头发倾斜
16右鼻线
17 .嘴巴大小合适
18 .嘴巴右弯曲
19-36和1-18一样，只在左边。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Andrews’ Curves

$$f_i(t)= \begin{cases}\frac{X_{i, 1}}{\sqrt{2}}+X_{i, 2} \sin (t)+X_{i, 3} \cos (t)+\ldots+X_{i, p-1} \sin \left(\frac{p-1}{2} t\right)+X_{i, p} \cos \left(\frac{p-1}{2} t\right) & \text { for } p \text { odd } \ \frac{X_{i, 1}}{\sqrt{2}}+X_{i, 2} \sin (t)+X_{i, 3} \cos (t)+\ldots+X_{i, p} \sin \left(\frac{p}{2} t\right) & \text { for } p \text { even }\end{cases}$$

\begin{aligned} f_1(t) & =\cos (t) \ f_2(t) & =\frac{1}{\sqrt{2}} \text { and } \ f_3(t) & =\sin (t) . \end{aligned}

## 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。