统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Covariance matrix and correlation matrix

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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|Covariance matrix and correlation matrix

Since a main focus in multivariate analysis is to incorporate the correlation between variables, the covariance matrix or the correlation matrix play an important role in multivariate analysis for continuous data or variables. Suppose that $X$ and $Y$ are two contimuous random variables, and suppose that $\left{x_{1}, x_{2}, \cdots, x_{n}\right}$ and $\left{y_{1}, y_{2}, \cdots, y_{n}\right}$ are data collected on $X$ and $Y$ respectively. The correlation between $X$ and $Y$ can be measured by the covariance $\operatorname{Cov}(X, Y)$ or the correlation coefficient $r$ :
\begin{aligned} \operatorname{Cov}(X, Y) &=E[(X-E(X))(Y-E(Y))], \ r &=\frac{\operatorname{Cov}(X, Y)}{\sigma_{X} \sigma_{Y}} \end{aligned}
where $\sigma_{X}=\sqrt{\operatorname{Var}(X)}$ and $\sigma_{Y}=\sqrt{\operatorname{Var}(Y)}$ are the standard deviations of $X$ and $Y$ respectively. The correlation coefficient $r$ is a number between $-1$ and 1 , and it measures the linear correlation between two variables. If $r>0, X$ and $Y$ are positively correlated. If $r<0, X$ and $Y$ are negatively correlated. The larger the value $|r|$, the stronger the correlation. When $r=0, X$ and $Y$ are not linearly correlated.

Note that the correlation coefficient $r$ only measures a linear relationship, not other relationships, i.e., the observed values of $X$ and $Y$ roughly fall on a straight line. For example, when $r=0, X$ and $Y$ may still have a nonlinear relationship.
More generally, let $\mathbf{X}=\left(X_{1}, X_{2}, \cdots, X_{p}\right)^{\mathrm{T}}$ be a random vector with $p$ continuous random variables. The mean vector of $\mathbf{X}$ is
$$E(\mathbf{X})=\mu=\left(\mu_{1}, \mu_{2}, \cdots, \mu_{p}\right)^{\mathrm{T}}$$
The covariance matrix of $\mathbf{X}$ is given by
$$\Sigma=\operatorname{Cov}(\mathbf{X})=\left(\sigma_{i j}\right){p \times p}, \quad \text { where } \quad \sigma{i j}=E\left(X_{i}-E\left(X_{i}\right)\right)\left(X_{j}-E\left(X_{j}\right)\right)$$

and the correlation between $X_{i}$ and $X_{j}$ is
$$r_{i j}=\frac{\sigma_{i j}}{\sqrt{\sigma_{i i} \sigma_{j j}}}, \quad i, j=1,2, \cdots, p .$$

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Multivariate Normal Distribution

In the previous sections, we do not assume any parametric distributions for the random variables or data. When we perform statistical inference such as hypothesis testing, however, we need to assume that the multivariate data follow some parametric distributions. The most common distribution for continuous multivariate data is the multivariate normal distribution. There are several reasons. First, by the Central Limit Theorem, many statistics (such as sample means) asymptotically follow normal distributions even if the original data do not follow normal distributions. In other words, when the sample size is large, a normality assumption may be reasonable for these statistics. Second, the normal distributions have many attractive properties. For example, a normal distribution is completely determined by its mean and variance (covariance), which are the two most important characteristics of data. Third, many continuous data, even if they may not be normally distributed, may be transformed into data which are roughly normally distributed. For example, we may consider a log-transformation for data which are positive (such as age) or skewed (such as survival time). Lastly, for multivariate continuous data, there are not many reasonable multivariate distributions to choose. Therefore, for multivariate continuous data, if a distributional assumption is required for statistical inference, we often assume that the data follow a multivariate normal distribution. However, note that this is only an assumption, so it needs to be checked based on the data for its validity.

The random vector $\mathbf{x}=\left(x_{1}, \cdots, x_{p}\right)^{\mathrm{T}}$ follows a $p$-dimensional multivariate normal distribution, denoted by $N_{p}(\mu, \Sigma)$, if any linear combination of the components of the random vector $x$ follows a univariate normal distribution. That is, for any constant

vector $\mathbf{a}=\left(a_{1}, a_{2}, \cdots, a_{p}\right)^{\mathrm{T}}$, the univariate random variable $y=\mathbf{a}^{\mathrm{T}} \mathbf{x}=\sum_{i=1}^{p} a_{i} x_{i}$ follows a univariate normal distribution
$$\mathbf{a}^{\mathrm{T}} \mathbf{x} \sim N\left(\sum_{i=1}^{p} a_{i} E\left(x_{i}\right), \quad \operatorname{var}\left(\sum_{i=1}^{p} a_{i} x_{i}\right)\right)$$
Thus, if $\mathrm{x}$ follows $N_{p}(\boldsymbol{\mu}, \Sigma)$, each component $x_{k}$ follows $N\left(\mu_{k}, \sigma_{k k}\right), k=1, \cdots, p$. Note that the reverse may not be true: if each $x_{k}$ follows $N\left(\mu_{k}, \sigma_{k k}\right)$, x may or may not follow a normal distribution. The probability density function of $\mathbf{x} \sim N_{p}(\boldsymbol{\mu}, \Sigma)$ can be written as
\begin{aligned} f(\mathrm{x})=\frac{1}{(2 \pi)^{p / 2}|\Sigma|^{1 / 2}} \exp [&\left.-\frac{1}{2}(\mathrm{x}-\boldsymbol{\mu})^{\mathrm{T}} \Sigma^{-1}(\mathrm{x}-\boldsymbol{\mu})\right] \ &-\infty<x_{j}<\infty, \quad j=1,2, \cdots, p \end{aligned}
which reduces to a univariate normal density when $p=1$.
As an example, consider a bivariate normal random vector $\mathrm{x}=\left(x_{1}, x_{2}\right)^{\mathrm{T}}$ with the mean vector and covariance matrix given by
$$\boldsymbol{\mu}=\left(\begin{array}{l} 2 \ 1 \end{array}\right), \quad \Sigma=\left(\begin{array}{ll} 4 & 3 \ 3 & 3 \end{array}\right)$$
Then, we have $x_{1} \sim N(2,4), x_{2} \sim N(1,3)$, and the correlation between $x_{1}$ and $x_{2}$ is
$$r=\frac{\operatorname{Cov}\left(x_{1}, x_{2}\right)}{\sqrt{\operatorname{Var}\left(x_{1}\right) \operatorname{Var}\left(x_{2}\right)}}=\frac{3}{\sqrt{4 \times 3}}=0.866 .$$

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Properties of multivariate normal distributions

Since the multivariate normal distribution is the most important distribution for multivariate continuous data, we list some of its important properties, which are useful in multivariate analysis. Note that properties (i) and (ii) below do not require that the distribution is normal.

Let $\mathbf{x}$ and $\mathbf{y}$ be two random vectors, and let $B$ and $\mathbf{b}$ be a constant matrix and a constant vector respectively, Let $\mathbf{y}=\boldsymbol{B x}+\mathbf{b}$ be a linear transformation. Then, we have the following properties:
(i) $E(\mathbf{y})=\boldsymbol{B E}(\mathbf{x})+\mathbf{b}$;
(ii) $\operatorname{Cov}(\mathbf{y})=\boldsymbol{B C o v}(\mathbf{x}) \boldsymbol{B}^{\mathrm{T}}$;
(iii) If $\mathbf{x} \sim N_{p}(\boldsymbol{\mu}, \Sigma)$, then
$$\mathbf{y}=\boldsymbol{B x}+\mathbf{b} \sim N\left(\boldsymbol{B} \boldsymbol{\mu}+\mathbf{b}, \boldsymbol{B} \Sigma \boldsymbol{B}^{\mathrm{T}}\right)$$
i.e., a linear transformation of a normal random vector is still normally distributed;
(iv) If $\mathrm{x} \sim N_{p}(\boldsymbol{\mu}, \Sigma)$ and
$$\mathbf{x}=\left(\begin{array}{l} \mathbf{x}{1} \ \mathbf{x}{2} \end{array}\right), \quad \mu=\left(\begin{array}{l} \mu_{1} \ \mu_{2} \end{array}\right), \quad \Sigma=\left(\begin{array}{ll} \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} \end{array}\right)$$
then
$$\mathbf{x}{1} \sim N\left(\mu{1}, \Sigma_{11}\right), \quad \mathbf{x}{2} \sim N\left(\mu{2}, \Sigma_{22}\right),$$
and the conditional distribution of $\mathbf{x}{1}$ given $\mathbf{x}{2}$ is still normally distributed and is given by
$$\mathbf{x}{1} \mid \mathbf{x}{2} \sim N\left(\mu_{1}+\Sigma_{12} \Sigma_{22}^{-1}\left(\mathbf{x}{2}-\mu{2}\right), \Sigma_{11.2}\right),$$ where $\Sigma_{11,2}=\Sigma_{11}-\Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}$. In other words, for a multivariate normal distribution, its components’ distributions and conditional distributions are still normal.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Covariance matrix and correlation matrix


\Sigma=\operatorname{Cov}(\mathbf{X})=\left(\sigma_{ij}\right) {p \times p}, \quad \text { where } \quad \sigma 给出{ij}=E\left(X_{i}-E\left(X_{i}\right)\right)\left(X_{j}-E\left(X_{j}\right)\right)


r一世j=σ一世jσ一世一世σjj,一世,j=1,2,⋯,p.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Multivariate Normal Distribution

F(X)=1(2圆周率)p/2|Σ|1/2经验⁡[−12(X−μ)吨Σ−1(X−μ)] −∞<Xj<∞,j=1,2,⋯,p

μ=(2 1),Σ=(43 33)

r=这⁡(X1,X2)曾是⁡(X1)曾是⁡(X2)=34×3=0.866.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Properties of multivariate normal distributions

(i)和(是)=乙和(X)+b;
(二)这⁡(是)=乙C○在(X)乙吨;
(iii) 如果X∼ñp(μ,Σ)， 然后

(iv) 如果X∼ñp(μ,Σ)和

X=(X1 X2),μ=(μ1 μ2),Σ=(Σ11Σ12 Σ21Σ22)

X1∼ñ(μ1,Σ11),X2∼ñ(μ2,Σ22),

X1∣X2∼ñ(μ1+Σ12Σ22−1(X2−μ2),Σ11.2),在哪里Σ11,2=Σ11−Σ12Σ22−1Σ21. 换句话说，对于一个多元正态分布，它的分量分布和条件分布仍然是正态的。

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