### 统计代写|似然估计作业代写Probability and Estimation代考|Introduction to Ridge Regression

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## 统计代写|似然估计作业代写Probability and Estimation代考| Introduction to Ridge Regression

Consider the common multiple linear regression model with the vector of coefficients, $\boldsymbol{\beta}=\left(\beta_{1}, \ldots, \beta_{p}\right)^{\top}$ given by
$$Y=X \boldsymbol{\beta}+\boldsymbol{\epsilon},$$
where $Y=\left(y_{1}, \ldots, y_{n}\right)^{\top}$ is a vector of $n$ responses, $X=\left(\boldsymbol{x}{1}, \ldots, \boldsymbol{x}{n}\right)^{\top}$ is an $n \times p$ design matrix of rank $p(\leq n), x_{i} \in \mathbb{R}^{p}$ is the vector of covariates, and $\boldsymbol{e}$ is an $n$-vector of independently and identically distributed (i.i.d.) random variables (rv.).

The least squares estimator (LSE) of $\boldsymbol{\beta}$, denoted by $\tilde{\boldsymbol{\beta}}{n}$, can be obtained by minimizing the residual sum of squares (RSS), the convex optimization problem, $$\min {\beta}\left{(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})\right}=\min _{\boldsymbol{\beta}}{S(\boldsymbol{\beta})},$$
where $S(\boldsymbol{\beta})=\boldsymbol{Y}^{\top} \boldsymbol{Y}-2 \boldsymbol{\beta}^{\top} \boldsymbol{X}^{\top} \boldsymbol{Y}+\boldsymbol{\beta}^{\top} \boldsymbol{X}^{\top} \boldsymbol{X} \boldsymbol{\beta}$ is the RSS. Solving
$$\frac{\partial S(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}=-2 \boldsymbol{X}^{\top} \boldsymbol{Y}+2 \boldsymbol{X}^{\top} \boldsymbol{X} \boldsymbol{\beta}=\mathbf{0}$$

with respect to (w.r.t.) $\beta$ gives
$$\tilde{\boldsymbol{\beta}}{n}=\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{Y} \text {. }$$ Suppose that $\mathbb{E}(\boldsymbol{\varepsilon})=0$ and $\mathbb{E}\left(\boldsymbol{\varepsilon} \boldsymbol{e}^{\top}\right)=\sigma^{2} \boldsymbol{I}{n}$ for some $\sigma^{2} \in \mathbb{R}^{+}$. Then, the variance-covariance matrix of LSE is given by
$$\operatorname{Var}\left(\tilde{\boldsymbol{\beta}}_{n}\right)=\sigma^{2}\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}$$
Now, we consider the canonical form of the multiple linear regression model to illustrate how large eigenvalues of the design matrix $\boldsymbol{X}^{\top} \boldsymbol{X}$ may affect the efficiency of estimation.

Write the spectral decomposition of the positive definite design matrix $\boldsymbol{X}^{\top} \boldsymbol{X}$ to get $\boldsymbol{X}^{\top} \boldsymbol{X}=\boldsymbol{\Gamma} \boldsymbol{\Lambda} \boldsymbol{\Gamma}^{\top}$, where $\boldsymbol{\Gamma}(p \times p)$ is a column orthogonal matrix of eigenvectors and $\boldsymbol{\Lambda}=\operatorname{Diag}\left(\lambda_{1}, \ldots, \lambda_{p}\right)$, where $\lambda_{j}>0, j=1, \ldots, p$ is the ordered eigenvalue matrix corresponding to $\boldsymbol{X}^{\top} \boldsymbol{X}$. Then,
$$Y=T \xi+\epsilon, \quad T=X \Gamma, \quad \xi=\Gamma^{\top} \beta$$
The LSE of $\xi$ has the form,
\begin{aligned} \tilde{\xi}{n} &=\left(\boldsymbol{T}^{\top} \boldsymbol{T}^{-1} \boldsymbol{T}^{\top} \boldsymbol{Y}\right.\ &=\boldsymbol{\Lambda}^{-1} \boldsymbol{T}^{\top} \boldsymbol{Y} \end{aligned} The variance-covariance matrix of $\tilde{\xi}{n}$ is given by
$$\operatorname{Var}\left(\tilde{\xi}{n}\right)=\sigma^{2} \boldsymbol{\Lambda}^{-1}=\sigma^{2}\left[\begin{array}{cccc} \frac{1}{\lambda{1}} & 0 & \cdots & 0 \ 0 & \frac{1}{\lambda_{2}} & 0 & 0 \ \vdots & \cdots & \ddots & \vdots \ 0 & 0 & \cdots & \frac{1}{\lambda_{p}} \end{array}\right]$$
Summation of the diagonal elements of the variance-covariance matrix of $\tilde{\xi}{n}$ is equal to $\operatorname{tr}\left(\operatorname{Var}\left(\tilde{\xi}{n}\right)\right)=\sigma^{2} \sum_{j=1}^{p} \lambda_{j}^{-1}$. Apparently, small eigenvalues inflate the total variance of estimate or energy of $X^{\top} X$. Specifically, since the eigenvalues are ordered, if the first eigenvalue is small, it causes the variance to explode. If this happens, what must one do? In the following section, we consider this problem. Therefore, it is of interest to realize when the eigenvalues become small.

Before discussing this problem, a very primitive understanding is that if we enlarge the eigenvalues from $\lambda_{j}$ to $\lambda_{j}+k$, for some positive value, say, $k$, then we can prevent the total variance from exploding. Of course, the amount of recovery depends on the correct choice of the parameter, $k$.

## 统计代写|似然估计作业代写Probability and Estimation代考| Multicollinearity Problem

Multicollinearity or collinearity is the existence of near-linear relationships among the regressors, predictors, or input/exogenous variables. There are terms such as exact, complete and severe, or supercollinearity and moderate collinearity. Supercollinearity indicates that two (or multiple) covariates are linearly dependent, and moderate occurs when covariates are moderately correlated. In the complete collinearity case, the design matrix is not invertible. This case mostly occurs in a high-dimensional situation (e.g. microarray measure) in which the number of covariates $(p)$ exceeds the number of samples $(n)$.

Moderation occurs when the relationship between two variables depends on a third variable, namely, the moderator. This case mostly happens in structural equation modeling. Although moderate multicollinearity does not cause the mathematical problems of complete multicollinearity, it does affect the

interpretation of model parameter estimates. According to Montgomery et al. (2012), if there is no linear relationship between the regressors, they are said to be orthogonal.

Multicollinearity or ill-conditioning can create inaccurate estimates of the regression coefficients, inflate the standard errors of the regression coefficients, deflate the partial $t$-tests for the regression coefficients, give false and nonsignificant $p$-values, and degrade the predictability of the model. It also causes changes in the direction of signs of the coefficient estimates. According to Montgomery et al. (2012), there are five sources for multicollinearity: (i) data collection, (ii) physical constraints, (iii) overdefined model, (iv) model choice or specification, and (v) outliers.

There are many studies that well explain the problem of multicollinearity. Since theoretical aspects of ridge regression and related issues are our goal, we refer the reader to Montgomery et al. (2012) for illustrative examples and comprehensive study on the multicollinearity and diagnostic measures such as correlation matrix, eigen system analysis of $\boldsymbol{X}^{\top} \boldsymbol{X}$, known as condition number, or variance decomposition proportion and variance inflation factor (VIF). To end this section, we consider a frequently used example in a ridge regression, namely, the Portland cement data introduced by Woods et al. (1932) from Najarian et al. (2013). This data set has been analyzed by many authors, e.g. Kaciranlar et al. (1999), Kibria (2003), and Arashi et al. (2015). We assemble the data as follows.

## 统计代写|似然估计作业代写Probability and Estimation代考| Ridge Regression Estimator: Ridge Notion

If the regression coefficients $\beta_{j} s$ are unconstrained, then they can explode (become large); this results in high variance. Hence, in order to control the variance, one may regularize the regression coefficients and determine how large the coefficient grows. In other words, one may impose a constraint on them so as not to get unboundedly large or penalized large regression coefficients. One type of constraint is the ridge constraint given by $\sum_{j=1}^{p} \beta_{j}^{2} \leq t$ for some positive value $t$. Hence, the minimization of the penalized residual sum of squares (PRSS) is equivalent to solving the following convex optimization problem,
$$\min {\beta}\left{(Y-X \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})\right} \quad \text { such that } \sum{j=1}^{p} \beta_{j}^{2} \leq t$$
for some positive value $t$.
In general, the PRSS is defined by
$$(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})+k|\boldsymbol{\beta}|^{2}, \quad|\boldsymbol{\beta}|^{2}=\sum_{j=1}^{p} \beta_{j}^{2}$$
Since the PRSS is a convex function w.r.t. $\boldsymbol{\beta}$, it has a unique solution. Because of the ridge constraint, the solution is termed as the ridge regression estimator (RRE).

To derive the RRE, we solve the following convex optimization problem
$$\min {\boldsymbol{\beta}}\left{(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})+k|\boldsymbol{\beta}|^{2}\right}=\min {\boldsymbol{\beta}}{\operatorname{PS}(\boldsymbol{\beta})},$$
where $\operatorname{PS}(\boldsymbol{\beta})=\boldsymbol{Y}^{\top} \boldsymbol{Y}-2 \boldsymbol{\beta}^{\top} \boldsymbol{X}^{\top} \boldsymbol{Y}+\boldsymbol{\beta}^{\top} \boldsymbol{X}^{\top} \boldsymbol{X} \boldsymbol{\beta}+k \boldsymbol{\beta}^{\top} \boldsymbol{\beta}$ is the PRSS. Solving
$$\frac{\partial \mathrm{PS}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}=-2 \boldsymbol{X}^{\top} \boldsymbol{Y}+2 \boldsymbol{X}^{\top} \boldsymbol{X} \boldsymbol{\beta}+2 k \boldsymbol{\beta}=\mathbf{0}$$
w.r.t. $\beta$ gives the RRE,
$$\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k)=\left(\boldsymbol{X}^{\top} \boldsymbol{X}+k \boldsymbol{I}{p}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{Y} .$$
Here, $k$ is the shrinkage (tuning) parameter. Indeed, $k$ tunes (controls) the size of the coefficients, and hence regularizes them. As $k \rightarrow 0$, the RRE simplifies to the LSE. Also, as $k \rightarrow \infty$, the RREs approach zero. Hence, the optimal shrinkage parameter $k$ is of interest.

One must note that solving the optimization problem (1.13) is not the only way of yielding the RRE. It can also be obtained by solving a RSS of another data, say augmented data. To be specific, consider the following augmentation approach. Let
$$\boldsymbol{X}^{}=\left[\begin{array}{c} \boldsymbol{X} \ \sqrt{k} \boldsymbol{I}{p} \end{array}\right], \quad \boldsymbol{Y}^{}=\left[\begin{array}{c} \boldsymbol{Y} \ \mathbf{0} \end{array}\right]$$
Assume the following multiple linear model,
$$Y^{}=X^{} \beta+\epsilon^{},$$ where $\boldsymbol{e}^{}$ is an $(n+p)$-vector of i.i.d. random variables. Then, the LSE of $\boldsymbol{\beta}$ is obtained as
\begin{aligned} \boldsymbol{\beta}{n}^{} &=\min {\boldsymbol{\beta}}\left{\left(\boldsymbol{Y}^{}-\boldsymbol{X}^{} \boldsymbol{\beta}\right)^{\top}\left(\boldsymbol{Y}^{}-\boldsymbol{X}^{} \boldsymbol{\beta}\right)\right} \ &=\left(\boldsymbol{X}^{} \boldsymbol{X}^{}\right)^{-1} \boldsymbol{X}^{ \top} \boldsymbol{Y}^{*} \ &=\left(\boldsymbol{X}^{\top} \boldsymbol{X}+k \boldsymbol{I}{p}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{Y} \ &=\hat{\boldsymbol{\beta}}_{n}^{\mathrm{RR}}(k) . \end{aligned}
Thus, the LSE of the augmented data is indeed the RRE of the normal data.

## 统计代写|似然估计作业代写Probability and Estimation代考| Introduction to Ridge Regression

∂小号(b)∂b=−2X⊤是+2X⊤Xb=0

$$\tilde{\boldsymbol{\beta}} {n}=\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{ \top} \boldsymbol{Y} \text {. }$$ 假设和(e)=0和 $\mathbb{E}\left(\boldsymbol{\varepsilon} \boldsymbol{e}^{\top}\right)=\sigma^{2} \boldsymbol{I} {n}F这rs这米和\sigma^{2} \in \mathbb{R}^{+}.吨H和n,吨H和在一种r一世一种nC和−C这在一种r一世一种nC和米一种吨r一世X这F大号小号和一世sG一世在和nb是曾是⁡(b~n)=σ2(X⊤X)−1ñ这在,在和C这ns一世d和r吨H和C一种n这n一世C一种lF这r米这F吨H和米在l吨一世pl和l一世n和一种rr和Gr和ss一世这n米这d和l吨这一世ll在s吨r一种吨和H这在l一种rG和和一世G和n在一种l在和s这F吨H和d和s一世Gn米一种吨r一世X\boldsymbol{X}^{\top} \boldsymbol{X}$ 可能会影响估计的效率。

X~n=(吨⊤吨−1吨⊤是 =Λ−1吨⊤是的方差-协方差矩阵X~n是（谁）给的

## 统计代写|似然估计作业代写Probability and Estimation代考| Ridge Regression Estimator: Ridge Notion

\min {\beta}\left{(YX \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})\right} \quad \text {这样 } \sum{j=1}^{p} \beta_{j}^{2} \leq t\min {\beta}\left{(YX \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})\right} \quad \text {这样 } \sum{j=1}^{p} \beta_{j}^{2} \leq t

(是−Xb)⊤(是−Xb)+ķ|b|2,|b|2=∑j=1pbj2

\min {\boldsymbol{\beta}}\left{(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \ boldsymbol{\beta})+k|\boldsymbol{\beta}|^{2}\right}=\min {\boldsymbol{\beta}}{\operatorname{PS}(\boldsymbol{\beta})}，\min {\boldsymbol{\beta}}\left{(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \ boldsymbol{\beta})+k|\boldsymbol{\beta}|^{2}\right}=\min {\boldsymbol{\beta}}{\operatorname{PS}(\boldsymbol{\beta})}，

∂磷小号(b)∂b=−2X⊤是+2X⊤Xb+2ķb=0

b^nRR(ķ)=(X⊤X+ķ一世p)−1X⊤是.

X=[X ķ一世p],是=[是 0]

\begin{对齐} \boldsymbol{\beta}{n}^{} &=\min {\boldsymbol{\beta}}\left{\left(\boldsymbol{Y}^{}-\boldsymbol{X}^ {} \boldsymbol{\beta}\right)^{\top}\left(\boldsymbol{Y}^{}-\boldsymbol{X}^{} \boldsymbol{\beta}\right)\right} \ & =\left(\boldsymbol{X}^{} \boldsymbol{X}^{}\right)^{-1} \boldsymbol{X}^{ \top} \boldsymbol{Y}^{*} \ &= \left(\boldsymbol{X}^{\top} \boldsymbol{X}+k \boldsymbol{I}{p}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{Y } \ &=\hat{\boldsymbol{\beta}}_{n}^{\mathrm{RR}}(k) 。\end{对齐}\begin{对齐} \boldsymbol{\beta}{n}^{} &=\min {\boldsymbol{\beta}}\left{\left(\boldsymbol{Y}^{}-\boldsymbol{X}^ {} \boldsymbol{\beta}\right)^{\top}\left(\boldsymbol{Y}^{}-\boldsymbol{X}^{} \boldsymbol{\beta}\right)\right} \ & =\left(\boldsymbol{X}^{} \boldsymbol{X}^{}\right)^{-1} \boldsymbol{X}^{ \top} \boldsymbol{Y}^{*} \ &= \left(\boldsymbol{X}^{\top} \boldsymbol{X}+k \boldsymbol{I}{p}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{Y } \ &=\hat{\boldsymbol{\beta}}_{n}^{\mathrm{RR}}(k) 。\end{对齐}

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