## STAT311 Linear regression课程简介

This course covers various statistical models such as simple linear regression, multiple regression, and analysis of variance. The main focus of the course is to teach students how to use the software package $\mathrm{R}$ to perform the analysis and interpret the results. Additionally, the course emphasizes the importance of constructing a clear technical report on the analysis that is readable by both scientists and non-technical audiences.

To take this course, students must have completed course 132 and satisfied the Entry Level Writing and Composition requirements. This course satisfies the General Education Code W requirement.

## PREREQUISITES

Covers simple linear regression, multiple regression, and analysis of variance models. Students learn to use the software package $\mathrm{R}$ to perform the analysis, and to construct a clear technical report on their analysis, readable by either scientists or nontechnical audiences (Formerly Linear Statistical Models). Prerequisite(s): course 132 and satisfaction of the Entry Level Writing and Composition requirements. Gen. Ed. Code(s): W

## STAT311 Linear regression HELP（EXAM HELP， ONLINE TUTOR）

Proposition 3.1. Suppose that a numerical variable selection method suggests several submodels with $k$ predictors, including a constant, where $2 \leq k \leq p$
a) The model $I$ that minimizes $C_p(I)$ maximizes $\operatorname{corr}\left(r, r_I\right)$.
b) $C_p(I) \leq 2 k$ implies that $\operatorname{corr}\left(\mathrm{r}, \mathrm{r}{\mathrm{I}}\right) \geq \sqrt{1-\frac{\mathrm{p}}{\mathrm{n}}}$. c) As $\operatorname{corr}\left(r, r_I\right) \rightarrow 1$, $$\operatorname{corr}\left(\boldsymbol{x}^{\mathrm{T}} \hat{\boldsymbol{\beta}}, \boldsymbol{x}{\mathrm{I}}^{\mathrm{T}} \hat{\boldsymbol{\beta}}{\mathrm{I}}\right)=\operatorname{corr}(\mathrm{ESP}, \mathrm{ESP}(\mathrm{I}))=\operatorname{corr}\left(\hat{\mathrm{Y}}, \hat{\mathrm{Y}}{\mathrm{I}}\right) \rightarrow 1$$

Remark 3.1. Consider the model $I_i$ that deletes the predictor $x_i$. Then the model has $k=p-1$ predictors including the constant, and the test statistic is $t_i$ where
$$t_i^2=F_{I_i}$$
Using Definition 3.8 and $C_p\left(I_{\text {full }}\right)=p$, it can be shown that
$$C_p\left(I_i\right)=C_p\left(I_{\text {full }}\right)+\left(t_i^2-2\right) .$$
Using the screen $C_p(I) \leq \min (2 k, p)$ suggests that the predictor $x_i$ should not be deleted if
$$\left|t_i\right|>\sqrt{2} \approx 1.414$$
If $\left|t_i\right|<\sqrt{2}$, then the predictor can probably be deleted since $C_p$ decreases. The literature suggests using the $C_p(I) \leq k$ screen, but this screen eliminates too many potentially useful submodels.

Proposition 3.2. Suppose that every submodel contains a constant and that $\boldsymbol{X}$ is a full rank matrix.
Response Plot: i) If $w=\hat{Y}I$ and $z=Y$, then the OLS line is the identity line. ii) If $w=Y$ and $z=\hat{Y}_I$, then the OLS line has slope $b=\left[\operatorname{corr}\left(Y, \hat{Y}_I\right)\right]^2=$ $R^2(I)$ and intercept $a=\bar{Y}\left(1-R^2(I)\right)$ where $\bar{Y}=\sum{i=1}^n Y_i / n$ and $R^2(I)$ is the coefficient of multiple determination from the candidate model.
FF or EE Plot: iii) If $w=\hat{Y}_I$ and $z=\hat{Y}$, then the OLS line is the identity line. Note that $E S P(I)=\hat{Y}_I$ and $E S P=\hat{Y}$.
iv) If $w=\hat{Y}$ and $z=\hat{Y}_I$, then the OLS line has slope $b=\left[\operatorname{corr}\left(\hat{Y}, \hat{Y}_I\right)\right]^2=$ $S S R(I) / S S R$ and intercept $a=\bar{Y}[1-(S S R(I) / S S R)]$ where SSR is the regression sum of squares.
RR Plot: v) If $w=r$ and $z=r_I$, then the OLS line is the identity line.
vi) If $w=r_I$ and $z=r$, then $a=0$ and the OLS slope $b=\left[\operatorname{corr}\left(r, r_I\right)\right]^2$ and
$$\operatorname{corr}\left(r, r_I\right)=\sqrt{\frac{S S E}{S S E(I)}}=\sqrt{\frac{n-p}{C_p(I)+n-2 k}}=\sqrt{\frac{n-p}{(p-k) F_I+n-p}} .$$

Proof: Recall that $\boldsymbol{H}$ and $\boldsymbol{H}_I$ are symmetric idempotent matrices and that $\boldsymbol{H}_I=\boldsymbol{H}_I$. The mean of OLS fitted values is equal to $\bar{Y}$ and the mean of OLS residuals is equal to 0 . If the OLS line from regressing $z$ on $w$ is $\hat{z}=a+b w$, then $a=\bar{z}-b \bar{w}$ and $$b=\frac{\sum\left(w_i-\bar{w}\right)\left(z_i-\bar{z}\right)}{\sum\left(w_i-\bar{w}\right)^2}=\frac{S D(z)}{S D(w)} \operatorname{corr}(z, w) .$$
Also recall that the OLS line passes through the means of the two variables $(\bar{w}, \bar{z})$
$\left(^\right)$ Notice that the OLS slope from regressing $z$ on $w$ is equal to one if and only if the OLS slope from regressing $w$ on $z$ is equal to $[\operatorname{corr}(z, w)]^2$. i) The slope $b=1$ if $\sum \hat{Y}{I, i} Y_i=\sum \hat{Y}{I, i}^2$. This equality holds since $\hat{\boldsymbol{Y}}I^T \boldsymbol{Y}=$ $\boldsymbol{Y}^T \boldsymbol{H}_I \boldsymbol{Y}=\boldsymbol{Y}^T \boldsymbol{H}_I \boldsymbol{H}_I \boldsymbol{Y}=\hat{\boldsymbol{Y}}_I^T \hat{\boldsymbol{Y}}_I$. Since $b=1, a=\bar{Y}-\bar{Y}=0$. ii) By $\left(^\right)$, the slope
$$b=\left[\operatorname{corr}\left(Y, \hat{Y}_I\right)\right]^2=R^2(I)=\frac{\sum\left(\hat{Y}{I, i}-\bar{Y}\right)^2}{\sum\left(Y_i-\bar{Y}\right)^2}=\operatorname{SSR}(I) / \operatorname{SSTO} .$$
The result follows since $a=\bar{Y}-b \bar{Y}$.
iii) The slope $b=1$ if $\sum \hat{Y}{I, i} \hat{Y}_i=\sum \hat{Y}{I, i}^2$. This equality holds since $\hat{\boldsymbol{Y}}^T \hat{\boldsymbol{Y}}_I=\boldsymbol{Y}^T \boldsymbol{H} \boldsymbol{H}_I \boldsymbol{Y}=\boldsymbol{Y}^T \boldsymbol{H}_I \boldsymbol{Y}=\hat{\boldsymbol{Y}}_I^T \hat{\boldsymbol{Y}}_I$. Since $b=1, a=\bar{Y}-\bar{Y}=0$.
iv) From iii),
$$1=\frac{S D(\hat{Y})}{S D\left(\hat{Y}_I\right)}\left[\operatorname{corr}\left(\hat{Y}, \hat{Y}_I\right)\right] .$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供metrostate.edu STAT311 Linear regression线性回归课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## STAT311 Linear regression课程简介

This course covers various statistical models such as simple linear regression, multiple regression, and analysis of variance. The main focus of the course is to teach students how to use the software package $\mathrm{R}$ to perform the analysis and interpret the results. Additionally, the course emphasizes the importance of constructing a clear technical report on the analysis that is readable by both scientists and non-technical audiences.

To take this course, students must have completed course 132 and satisfied the Entry Level Writing and Composition requirements. This course satisfies the General Education Code W requirement.

## PREREQUISITES

Covers simple linear regression, multiple regression, and analysis of variance models. Students learn to use the software package $\mathrm{R}$ to perform the analysis, and to construct a clear technical report on their analysis, readable by either scientists or nontechnical audiences (Formerly Linear Statistical Models). Prerequisite(s): course 132 and satisfaction of the Entry Level Writing and Composition requirements. Gen. Ed. Code(s): W

## STAT311 Linear regression HELP（EXAM HELP， ONLINE TUTOR）

2.10. In the above table, $x_i$ is the length of the femur and $y_i$ is the length of the humerus taken from five dinosaur fossils (Archaeopteryx) that preserved both bones. See Moore (2000, p. 99).
a) Complete the table and find the least squares estimators $\hat{\beta}_1$ and $\hat{\beta}_2$.
b) Predict the humerus length if the femur length is 60 .

2.11. Suppose that the regression model is $Y_i=7+\beta X_i+e_i$ for $i=$ $1, \ldots, n$ where the $e_i$ are iid $N\left(0, \sigma^2\right)$ random variables. The least squares criterion is $Q(\eta)=\sum_{i=1}^n\left(Y_i-7-\eta X_i\right)^2$.
a) What is $E\left(Y_i\right)$ ?
b) Find the least squares estimator $\beta$ of $\beta$ by setting the first derivative $\frac{d}{d \eta} Q(\eta)$ equal to zero.
c) Show that your $\hat{\beta}$ is the global minimizer of the least squares criterion $Q$ by showing that the second derivative $\frac{d^2}{d \eta^2} Q(\eta)>0$ for all values of $\eta$.

2.12. The location model is $Y_i=\mu+e_i$ for $i=1, \ldots, n$ where the $e_i$ are iid with mean $E\left(e_i\right)=0$ and constant variance $\operatorname{VAR}\left(e_i\right)=\sigma^2$. The least squares estimator $\hat{\mu}$ of $\mu$ minimizes the least squares criterion $Q(\eta)=\sum_{i=1}^n\left(Y_i-\eta\right)^2$. To find the least squares estimator, perform the following steps.

a) Find the derivative $\frac{a}{d \eta} Q$, set the derivative equal to zero and solve for
$\eta$. Call the solution $\hat{\mu}$.
b) To show that the solution was indeed the global minimizer of $Q$, show that $\frac{d^2}{d \eta^2} Q>0$ for all real $\eta$. (Then the solution $\hat{\mu}$ is a local min and $Q$ is convex, so $\hat{\mu}$ is the global min.)

2.14. Suppose that the regression model is $Y_i=10+2 X_{i 2}+\beta_3 X_{i 3}+e_i$ for $i=1, \ldots, n$ where the $e_i$ are iid $N\left(0, \sigma^2\right)$ random variables. The least squares criterion is $Q\left(\eta_3\right)=\sum_{i=1}^n\left(Y_i-10-2 X_{\mathrm{i} 2}-\eta_3 X_{i 3}\right)^2$. Find the least squares estimator $\hat{\beta}_3$ of $\beta_3$ by setting the first derivative $\frac{d}{d \eta_3} Q\left(\eta_3\right)$ equal to zero. Show that your $\hat{\beta}_3$ is the global minimizer of the least squares criterion $Q$ by showing that the second derivative $\frac{d^2}{d \eta_3^2} Q\left(\eta_3\right)>0$ for all values of $\eta_3$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供metrostate.edu STAT311 Linear regression线性回归课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## STAT311 Linear regression课程简介

This course covers various statistical models such as simple linear regression, multiple regression, and analysis of variance. The main focus of the course is to teach students how to use the software package $\mathrm{R}$ to perform the analysis and interpret the results. Additionally, the course emphasizes the importance of constructing a clear technical report on the analysis that is readable by both scientists and non-technical audiences.

To take this course, students must have completed course 132 and satisfied the Entry Level Writing and Composition requirements. This course satisfies the General Education Code W requirement.

## PREREQUISITES

Covers simple linear regression, multiple regression, and analysis of variance models. Students learn to use the software package $\mathrm{R}$ to perform the analysis, and to construct a clear technical report on their analysis, readable by either scientists or nontechnical audiences (Formerly Linear Statistical Models). Prerequisite(s): course 132 and satisfaction of the Entry Level Writing and Composition requirements. Gen. Ed. Code(s): W

## STAT311 Linear regression HELP（EXAM HELP， ONLINE TUTOR）

11.1. Suppose $Y_i=\boldsymbol{x}_i^T \boldsymbol{\beta}+\epsilon_i$ where the errors are independent $N\left(0, \sigma^2\right)$. Then the likelihood function is
$$L\left(\boldsymbol{\beta}, \sigma^2\right)=\left(2 \pi \sigma^2\right)^{-n / 2} \exp \left(\frac{-1}{2 \sigma^2}|\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta}|^2\right) .$$
a) Since the least squares estimator $\hat{\beta}$ minimizes $|\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta}|^2$, show that $\hat{\boldsymbol{\beta}}$ is the MLE of $\boldsymbol{\beta}$.
b) Then find the MLE $\hat{\sigma}^2$ of $\sigma^2$.

11.3. Suppose $Y_i=\boldsymbol{x}i^T \boldsymbol{\beta}+\epsilon_i$ where the errors are independent $N\left(0, \sigma^2 / w_i\right)$ where $w_i>0$ are known constants. Then the likelihood function is $$L\left(\boldsymbol{\beta}, \sigma^2\right)=\left(\prod{i=1}^n \sqrt{w_i}\right)\left(\frac{1}{\sqrt{2 \pi}}\right)^n \frac{1}{\sigma^n} \exp \left(\frac{-1}{2 \sigma^2} \sum_{i=1}^n w_i\left(y_i-\boldsymbol{x}_i^T \boldsymbol{\beta}\right)^2\right) .$$ a) Suppose that $\hat{\boldsymbol{\beta}}W$ minimizes $\sum{i=1}^n w_i\left(y_i-\boldsymbol{x}_i^T \boldsymbol{\beta}\right)^2$. Show that $\hat{\boldsymbol{\beta}}_W$ is the MLE of $\boldsymbol{\beta}$.
b) Then find the MLE $\hat{\sigma}^2$ of $\sigma^2$.

11.2. Suppose $Y_i=\boldsymbol{x}i^T \boldsymbol{\beta}+e_i$ where the errors are iid double exponential $(0, \sigma)$ where $\sigma>0$. Then the likelihood function is $$L(\boldsymbol{\beta}, \sigma)=\frac{1}{2^n} \frac{1}{\sigma^n} \exp \left(\frac{-1}{\sigma} \sum{i=1}^n\left|Y_i-\boldsymbol{x}i^T \boldsymbol{\beta}\right|\right) .$$ Suppose that $\tilde{\boldsymbol{\beta}}$ is a minimizer of $\sum{i=1}^n\left|Y_i-\boldsymbol{x}i^T \boldsymbol{\beta}\right|$. a) By direct maximization, show that $\tilde{\boldsymbol{\beta}}$ is an MLE of $\boldsymbol{\beta}$ regardless of the value of $\sigma$. b) Find an MLE of $\sigma$ by maximizing $$L(\sigma) \equiv L(\tilde{\boldsymbol{\beta}}, \sigma)=\frac{1}{2^n} \frac{1}{\sigma^n} \exp \left(\frac{-1}{\sigma} \sum{i=1}^n\left|Y_i-\boldsymbol{x}_i^T \tilde{\boldsymbol{\beta}}\right|\right) .$$

11.8. Let $Y \sim N\left(\mu, \sigma^2\right)$ so that $E(Y)=\mu$ and $\operatorname{Var}(Y)=\sigma^2=E\left(Y^2\right)-$ $[E(Y)]^2$. If $k \geq 2$ is an integer, then
$$E\left(Y^k\right)=(k-1) \sigma^2 E\left(Y^{k-2}\right)+\mu E\left(Y^{k-1}\right) .$$
Let $Z=(Y-\mu) / \sigma \sim N(0,1)$. Hence $\mu_k=E(Y-\mu)^k=\sigma^k E\left(Z^k\right)$. Use this fact and the above recursion relationship $E\left(Z^k\right)=(k-1) E\left(Z^{k-2}\right)$ to find a) $\mu_3$ and b) $\mu_4$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供metrostate.edu STAT311 Linear regression线性回归课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## STAT501 Linear regression课程简介

This course covers various statistical models such as simple linear regression, multiple regression, and analysis of variance. The main focus of the course is to teach students how to use the software package $\mathrm{R}$ to perform the analysis and interpret the results. Additionally, the course emphasizes the importance of constructing a clear technical report on the analysis that is readable by both scientists and non-technical audiences.

To take this course, students must have completed course 132 and satisfied the Entry Level Writing and Composition requirements. This course satisfies the General Education Code W requirement.

## PREREQUISITES

Covers simple linear regression, multiple regression, and analysis of variance models. Students learn to use the software package $\mathrm{R}$ to perform the analysis, and to construct a clear technical report on their analysis, readable by either scientists or nontechnical audiences (Formerly Linear Statistical Models). Prerequisite(s): course 132 and satisfaction of the Entry Level Writing and Composition requirements. Gen. Ed. Code(s): W

## STAT501 Linear regression HELP（EXAM HELP， ONLINE TUTOR）

(4) Table 5 contains data for the number of dairy cows (thousands) in the U.S. in various years.

• Enter the data into a spreadsheet so that $x$ represents the number of years since 1940. e.g enter $x=10$ for 1950 , enter $x=20$ for 1960 , etc.
• Create the scatter plot for the number of cows $y$ (thousands) as a function of $x$ (years since 1940).
• Adjust the minimum and maximum of the axes of each plot to slightly below and slightly above the data values.
• Compute the regression equation using logarithmic regression. The trendline will be $y=a \ln (x)+b$ for some values of $a$ and $b$. Round $a$ and $b$ to the nearest whole number.
• Use your regression equation to estimate the number of dairy cows in 2020 $(x=2020-1940=80)$.

To create a scatter plot in a spreadsheet:

1. Enter the data into two columns, with the year values in one column (let’s say column A) and the number of dairy cows values in another column (let’s say column B).
2. Select both columns of data.
3. Click on the “Insert” tab and then on the “Scatter” chart icon.
4. Choose a scatter plot with markers only.

After creating the scatter plot, adjust the minimum and maximum of the axes by right-clicking on each axis and choosing “Format Axis.” In the “Format Axis” panel, choose “Fixed” for the minimum and maximum values and adjust them to slightly below and slightly above the data values.

Using logarithmic regression, we can find the equation of the line of best fit for this data. In Excel, we can add a trendline to the scatter plot by right-clicking on one of the data points and selecting “Add Trendline.” In the “Add Trendline” panel, select “Logarithmic” as the Trend/Regression type. This will add a trendline to the scatter plot with an equation in the form of $y = a\ln(x) + b$, where $a$ and $b$ are the coefficients of the regression equation.

The regression equation for this data is $y = -327\ln(x) + 10629$. Rounding $a$ and $b$ to the nearest whole number, we get $a=-327$ and $b=10629$.

To estimate the number of dairy cows in 2020, we substitute $x=80$ into the equation and get $y = -327\ln(80) + 10629 \approx 9317$. Therefore, the estimated number of dairy cows in 2020 is about 9317 thousand (or 9.317 million) cows.

Know how to tell whether the experiment is a fixed or random effects one way Anova. (Were the levels fixed or a random sample from a population of levels?)

In a one-way ANOVA, we are comparing the means of multiple groups or levels on a single variable or outcome. The distinction between a fixed effects and random effects one-way ANOVA depends on whether the levels being compared are considered fixed or random.

Fixed effects one-way ANOVA: The levels being compared are considered fixed, meaning that they are chosen in advance and are of specific interest to the researcher. The goal of the analysis is to make inferences about the specific levels that were included in the study. For example, if we want to compare the performance of students in three different schools, and those three schools were chosen specifically for the study, we would use a fixed effects one-way ANOVA.

Random effects one-way ANOVA: The levels being compared are considered a random sample from a larger population of possible levels. The goal of the analysis is to make inferences about the population of levels from which the sample was drawn. For example, if we want to compare the effectiveness of three different brands of fertilizer on plant growth, and those three brands were chosen at random from a larger population of possible brands, we would use a random effects one-way ANOVA.

To determine whether a one-way ANOVA is a fixed or random effects design, we need to know how the levels were selected for the study. If the levels were chosen in advance and are of specific interest to the researcher, it is a fixed effects design. If the levels are a random sample from a larger population, it is a random effects design.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供psu.edu STAT501 Linear regression线性回归课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 统计代写|线性回归代写linear regression代考|MATH5386

statistics-lab™ 为您的留学生涯保驾护航 在代写线性回归linear regression方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写线性回归linear regression代写方面经验极为丰富，各种代写线性回归linear regression相关的作业也就用不着说。

• 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|线性回归代写linear regression代考|Checking Goodness of Fit

It is crucial to realize that an MLR model is not necessarily a useful model for the data, even if the data set consists of a response variable and several predictor variables. For example, a nonlinear regression model or a much more complicated model may be needed. Chapters 1 and 13 describe several alternative models. Let $p$ be the number of predictors and $n$ the number of cases. Assume that $n \geq 5 p$, then plots can be used to check whether the MLR model is useful for studying the data. This technique is known as checking the goodness of fit of the MLR model.

Notation. Plots will be used to simplify regression analysis, and in this text a plot of $W$ versus $Z$ uses $W$ on the horizontal axis and $Z$ on the vertical axis.

Definition 2.10. A scatterplot of $X$ versus $Y$ is a plot of $X$ versus $Y$ and is used to visualize the conditional distribution $Y \mid X$ of $Y$ given $X$.

Definition 2.11. A response plot is a plot of a variable $w_i$ versus $Y_i$. Typically $w_i$ is a linear combination of the predictors: $w_i=\boldsymbol{x}_i^T \boldsymbol{\eta}$ where $\boldsymbol{\eta}$ is a known $p \times 1$ vector. The most commonly used response plot is a plot of the fitted values $\widehat{Y}_i$ versus the response $Y_i$.

Proposition 2.1. Suppose that the regression estimator $\boldsymbol{b}$ of $\boldsymbol{\beta}$ is used to find the residuals $r_i \equiv r_i(\boldsymbol{b})$ and the fitted values $\widehat{Y}_i \equiv \widehat{Y}_i(\boldsymbol{b})=\boldsymbol{x}_i^T \boldsymbol{b}$. Then in the response plot of $\widehat{Y}_i$ versus $Y_i$, the vertical deviations from the identity line (that has unit slope and zero intercept) are the residuals $r_i(b)$.

Proof. The identity line in the response plot is $Y=\boldsymbol{x}^T \boldsymbol{b}$. Hence the vertical deviation is $Y_i-\boldsymbol{x}_i^T \boldsymbol{b}=r_i(\boldsymbol{b})$.

Definition 2.12. A residual plot is a plot of a variable $w_i$ versus the residuals $r_i$. The most commonly used residual plot is a plot of $\hat{Y}_i$ versus $r_i$.
Notation: For MLR, “the residual plot” will often mean the residual plot of $\hat{Y}_i$ versus $r_i$, and “the response plot” will often mean the plot of $\hat{Y}_i$ versus $Y_i^*$

If the unimodal MLR model as estimated by least squares is useful, then in the response plot the plotted points should scatter about the identity line while in the residual plot of $\hat{Y}$ versus $r$ the plotted points should scatter about the $r=0$ line (the horizontal axis) with no other pattern. Figures $1.2$ and $1.3$ show what a response plot and residual plot look like for an artificial MLR data set where the MLR regression relationship is rather strong in that the sample correlation $\operatorname{corr}(\tilde{Y}, Y)$ is near 1 . Figure $1.4$ shows a response plot where the response $Y$ is independent of the nontrivial predictors in the model. Here $\operatorname{corr}(\hat{Y}, Y)$ is near 0 but the points still scatter about the identity line. When the MLR relationship is very weak, the response plot will look like the residual plot.

## 统计代写|线性回归代写linear regression代考|Checking Lack of Fit

The response plot may look good while the residual plot suggests that the unimodal MLR model can be improved. Examining plots to find model violations is called checking for lack of fit. Again assume that $n \geq 5 p$.

The unimodal MLR model often provides a useful model for the data, but the following assumptions do need to be checked.
i) Is the MLR model appropriate?
ii) Are outliers present?
iii) Is the error variance constant or nonconstant? The constant variance assumption $\operatorname{VAR}\left(e_i\right) \equiv \sigma^2$ is known as homoscedasticity. The nonconstant variance assumption $\operatorname{VAR}\left(e_i\right)=\sigma_i^2$ is known as heteroscedasticity.
iv) Are any important predictors left out of the model?
v) Are the errors $e_1, \ldots, e_n$ iid?
vi) Are the errors $e_i$ independent of the predictors $\boldsymbol{x}_i$ ?
Make the response plot and the residual plot to check i), ii), and iii). An MLR model is reasonable if the plots look like Figures 1.2, 1.3, 1.4, and 2.1. A response plot that looks like Figure $13.7$ suggests that the model is not linear. If the plotted points in the residual plot do not scatter about the $r=0$ line with no other pattern (i.e., if the cloud of points is not ellipsoidal or rectangular with zero slope), then the unimodal MLR model is not sustained.
The $i$ th residual $r_i$ is an estimator of the $i$ th error $e_i$. The constant variance assumption may have been violated if the variability of the point cloud in the residual plot depends on the value of $\hat{Y}$. Often the variability of the residuals increases as $\hat{Y}$ increases, resulting in a right opening megaphone shape. (Figure $4.1 \mathrm{~b}$ has this shape.) Often the variability of the residuals decreases as $\hat{Y}$ increases, resulting in a left opening megaphone shape. Sometimes the variability decreases then increases again, and sometimes the variability increases then decreases again (like a stretched or compressed football).

# 线性回归代写

## 统计代写|线性回归代写线性回归代考|检查不符合

. . . .

i) MLR模型是否合适?
ii)是否存在异常值?
iii)误差方差是常数还是非常数?恒定方差假设 $\operatorname{VAR}\left(e_i\right) \equiv \sigma^2$ 被称为同异性。非常数方差假设 $\operatorname{VAR}\left(e_i\right)=\sigma_i^2$
iv)模型中是否遗漏了任何重要的预测因子?
v)是否存在误差 $e_1, \ldots, e_n$
vi)错误 $e_i$ 独立于预测因子 $\boldsymbol{x}_i$ ?

$i$ 残差 $r_i$ 的估计值 $i$ 错误 $e_i$。如果残差图中点云的可变性取决于的值，则可能违反恒定方差假设 $\hat{Y}$。通常残差的可变性会随着 $\hat{Y}$ 增加，导致右开口扩音器形状。(图 $4.1 \mathrm{~b}$ 有这个形状。)残差的变异性往往随着 $\hat{Y}$ 增加，导致左侧开口扩音器形状。有时可变性会减小然后再次增大，有时可变性会增大然后再次减小(就像一个拉伸或压缩的足球)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|线性回归代写linear regression代考|MATH839

statistics-lab™ 为您的留学生涯保驾护航 在代写线性回归linear regression方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写线性回归linear regression代写方面经验极为丰富，各种代写线性回归linear regression相关的作业也就用不着说。

• 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|线性回归代写linear regression代考|Variable Selection

A standard problem in $1 \mathrm{D}$ regression is variable selection, also called subset or model selection. Assume that the 1D regression model uses a linear predictor
$$Y \Perp \boldsymbol{x} \mid\left(\alpha+\boldsymbol{\beta}^T \boldsymbol{x}\right),$$
that a constant $\alpha$ is always included, that $\boldsymbol{x}=\left(x_1, \ldots, x_{p-1}\right)^T$ are the $p-1$ nontrivial predictors, and that the $n \times p$ matrix $\boldsymbol{X}$ with $i$ th row $\left(1, \boldsymbol{x}_i^T\right)$ has full rank $p$. Then variable selection is a search for a subset of predictor variables that can be deleted without important loss of information.

To clarify ideas, assume that there exists a subset $S$ of predictor variables such that if $x_S$ is in the $1 \mathrm{D}$ model, then none of the other predictors are needed in the model. Write $E$ for these (‘extraneous’) variables not in $S$, partitioning $\boldsymbol{x}=\left(\boldsymbol{x}_S^T, \boldsymbol{x}_E^T\right)^T$. Then
$$S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}=\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S+\boldsymbol{\beta}_E^T \boldsymbol{x}_E=\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S .$$
The extraneous terms that can be eliminated given that the subset $S$ is in the model have zero coefficients: $\boldsymbol{\beta}_E=\mathbf{0}$.

Now suppose that $I$ is a candidate subset of predictors, that $S \subseteq I$ and that $O$ is the set of predictors not in $I$. Then
$$S P-\alpha+\boldsymbol{\beta}^T \boldsymbol{x}-\alpha+\boldsymbol{\beta}S^T \boldsymbol{x}_S-\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S+\boldsymbol{\beta}{(I / S)}^T \boldsymbol{x}{I / S}+\mathbf{0}^T \boldsymbol{x}_O-\alpha+\boldsymbol{\beta}_I^T \boldsymbol{x}_I,$$ where $\boldsymbol{x}{I / S}$ denotes the predictors in $I$ that are not in $S$. Since this is true regardless of the values of the predictors, $\boldsymbol{\beta}O=\mathbf{0}$ if $S \subseteq I$. Hence for any subset $I$ that includes all relevant predictors, the population correlation $$\operatorname{corr}\left(\alpha+\boldsymbol{\beta}^{\mathrm{T}} \boldsymbol{x}{\mathrm{i}}, \alpha+\boldsymbol{\beta}{\mathrm{I}}^{\mathrm{T}} \boldsymbol{x}{\mathrm{I}, \mathrm{i}}\right)=1 .$$
This observation, which is true regardless of the explanatory power of the model, suggests that variable selection for a $1 \mathrm{D}$ regression model (1.11) is simple in principle. For each value of $j=1,2, \ldots, p-1$ nontrivial predictors, keep track of subsets $I$ that provide the largest values of corr(ESP,ESP $(I))$. Any such subset for which the correlation is high is worth closer investigation and consideration.

## 统计代写|线性回归代写linear regression代考|Other Issues

The $1 \mathrm{D}$ regression models offer a unifying framework for many of the most used regression models. By writing the model in terms of the sufficient predictor $S P=h(\boldsymbol{x})$, many important topics valid for all $1 \mathrm{D}$ regression models can be explained compactly. For example, the previous section presented variable selection, and equation (1.14) can be used to motivate the test for whether the reduced model can be used instead of the full model. Similarly, the sufficient predictor can be used to unify the interpretation of coefficients and to explain models that contain interactions and factors.
Interpretation of Coefficients
One interpretation of the coefficients in a 1D model (1.11) is that $\beta_i$ is the rate of change in the SP associated with a unit increase in $x_i$ when all other predictor variables $x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_p$ are held fixed. Denote a model by $S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}=\alpha+\beta_1 x_1+\cdots+\beta_p x_p$. Then
$$\beta_i=\frac{\partial S P}{\partial x_i} \text { for } \mathrm{i}=1, \ldots, \mathrm{p} .$$
Of course, holding all other variables fixed while changing $x_i$ may not be possible. For example, if $x_1=x, x_2=x^2$ and $S P=\alpha+\beta_1 x+\beta_2 x^2$, then $x_2$ cannot be held fixed when $x_1$ increases by one unit, but
$$\frac{d S P}{d x}=\beta_1+2 \beta_2 x .$$
The interpretation of $\beta_i$ changes with the model in two ways. First, the interpretation changes as terms are added and deleted from the SP. Hence the interpretation of $\beta_1$ differs for models $S P=\alpha+\beta_1 x_1$ and $S P=\alpha+\beta_1 x_1+\beta_2 x_2$. Secondly, the interpretation changes as the parametric or semiparametric form of the model changes.

# 线性回归代写

## 统计代写|线性回归代写线性回归代考|变量选择

$1 \mathrm{D}$回归中的一个标准问题是变量选择，也称为子集或模型选择。假设一维回归模型使用线性预测器
$$Y \Perp \boldsymbol{x} \mid\left(\alpha+\boldsymbol{\beta}^T \boldsymbol{x}\right),$$
，总是包含一个常数$\alpha$, $\boldsymbol{x}=\left(x_1, \ldots, x_{p-1}\right)^T$是$p-1$的非平凡预测器，并且$n \times p$矩阵$\boldsymbol{X}$与$i$的第一行$\left(1, \boldsymbol{x}_i^T\right)$具有满秩$p$。然后，变量选择是搜索一个预测变量的子集，可以删除而不丢失重要信息

$$S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}=\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S+\boldsymbol{\beta}_E^T \boldsymbol{x}_E=\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S .$$

$$S P-\alpha+\boldsymbol{\beta}^T \boldsymbol{x}-\alpha+\boldsymbol{\beta}S^T \boldsymbol{x}_S-\alpha+\boldsymbol{\beta}_S^T \boldsymbol{x}_S+\boldsymbol{\beta}{(I / S)}^T \boldsymbol{x}{I / S}+\mathbf{0}^T \boldsymbol{x}_O-\alpha+\boldsymbol{\beta}_I^T \boldsymbol{x}_I,$$ 哪里 $\boldsymbol{x}{I / S}$ 中的预测器 $I$ 这些都不在 $S$。由于无论预测因子的值是多少，这都是正确的， $\boldsymbol{\beta}O=\mathbf{0}$ 如果 $S \subseteq I$。因此对于任何子集 $I$ 这包括所有相关的预测因子，总体相关性 $$\operatorname{corr}\left(\alpha+\boldsymbol{\beta}^{\mathrm{T}} \boldsymbol{x}{\mathrm{i}}, \alpha+\boldsymbol{\beta}{\mathrm{I}}^{\mathrm{T}} \boldsymbol{x}{\mathrm{I}, \mathrm{i}}\right)=1 .$$这一观察结果，无论模型的解释能力如何，都是正确的，表明对a的变量选择 $1 \mathrm{D}$ 回归模型(1.11)原理简单。的每一个值 $j=1,2, \ldots, p-1$ 非平凡预测器，跟踪子集 $I$ 提供最大的corr值(ESP,ESP $(I))$。任何这种相关性较高的子集都值得仔细研究和考虑

## 统计代写|线性回归代写线性回归代考|其他问题

.

$1 \mathrm{D}$回归模型为许多最常用的回归模型提供了统一的框架。通过将模型写成充分的预测因子$S P=h(\boldsymbol{x})$，可以简洁地解释对所有$1 \mathrm{D}$回归模型有效的许多重要主题。例如，上一节介绍了变量选择，可以使用(1.14)式来激励是否可以使用简化模型代替完整模型的检验。同样，充分预测器可以用来统一系数的解释和解释包含相互作用和因素的模型。对一维模型(1.11)中系数的一种解释是，当所有其他预测变量$x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_p$保持固定时，$\beta_i$是与$x_i$的单位增长相关的SP的变化率。通过$S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}=\alpha+\beta_1 x_1+\cdots+\beta_p x_p$表示一个模型。那么
$$\beta_i=\frac{\partial S P}{\partial x_i} \text { for } \mathrm{i}=1, \ldots, \mathrm{p} .$$

$$\frac{d S P}{d x}=\beta_1+2 \beta_2 x .$$
$\beta_i$的解释随模型有两种变化。首先，随着术语从SP中添加和删除，解释会发生变化。因此，对于模型$S P=\alpha+\beta_1 x_1$和$S P=\alpha+\beta_1 x_1+\beta_2 x_2$, $\beta_1$的解释是不同的。其次，随着模型的参数或半参数形式的变化，解释也随之变化。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|线性回归代写linear regression代考|STAT6450

statistics-lab™ 为您的留学生涯保驾护航 在代写线性回归linear regression方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写线性回归linear regression代写方面经验极为丰富，各种代写线性回归linear regression相关的作业也就用不着说。

• 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|线性回归代写linear regression代考|Some Regression Models

In data analysis, an investigator is presented with a problem and data from some population. The population might be the collection of all possible outcomes from an experiment while the problem might be predicting a future value of the response variable $Y$ or summarizing the relationship between $Y$ and the $p \times 1$ vector of predictor variables $\boldsymbol{x}$. A statistical model is used to provide a useful approximation to some of the important underlying characteristics of the population which generated the data. Many of the most used models for 1D regression, defined below, are families of conditional distributions $Y \mid \boldsymbol{x}=\boldsymbol{x}_o$ indexed by $\boldsymbol{x}=\boldsymbol{x}_o$. A 1D regression model is a parametric model if the conditional distribution is completely specified except for a fixed finite number of parameters, otherwise, the 1D model is a semiparametric model. GLMs and GAMs, defined below, are covered in Chapter 13.

Definition 1.1. Regression investigates how the response variable $Y$ changes with the value of a $p \times 1$ vector $\boldsymbol{x}$ of predictors. Often this conditional distribution $Y \mid \boldsymbol{x}$ is described by a $1 D$ regression model, where $Y$ is conditionally independent of $\boldsymbol{x}$ given the sufficient predictor $S P=h(\boldsymbol{x})$, written
$$Y \Perp x \mid S P \text { or } \mathrm{Y} \Perp \boldsymbol{x} \mid \mathrm{h}(\boldsymbol{x}) \text {, }$$
where the real valued function $h: \mathbb{R}^p \rightarrow \mathbb{R}$. The estimated sufficient predictor $\mathrm{ESP}=\hat{h}(\boldsymbol{x})$. An important special case is a model with a linear predictor $h(\boldsymbol{x})=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}$ where ESP $=\hat{\alpha}+\hat{\boldsymbol{\beta}}^T \boldsymbol{x}$. This class of models includes the generalized linear model (GLM). Another important special case is a generalized additive model (GAM), where $Y$ is independent of $\boldsymbol{x}=\left(x_1, \ldots, x_p\right)^T$ given the additive predictor $A P=\alpha+\sum_{j=1}^p S_j\left(x_j\right)$ for some (usually unknown) functions $S_j$. The estimated additive predictor $\mathrm{EAP}=\mathrm{ESP}=\hat{\alpha}+\sum_{j=1}^p \hat{S}_j\left(x_j\right)$.

Notation: In this text, a plot of $x$ versus $Y$ will have $x$ on the horizontal axis, and $Y$ on the vertical axis.

Plots are extremely important for regression. When $p=1, x$ is both a sufficient predictor and an estimated sufficient predictor. So a plot of $x$ versus $Y$ is both a sufficient summary plot and a response plot. Usually the SP is unknown, so only the response plot can be made. The response plot will be extremely useful for checking the goodness of fit of the 1D regression model.
Definition 1.2. A sufficient summary plot is a plot of the SP versus $Y$. An estimated sufficient summary plot (ESSP) or response plot is a plot of the ESP versus $Y$.

## 统计代写|线性回归代写linear regression代考|Multiple Linear Regression

Suppose that the response variable $Y$ is quantitative and that at least one predictor variable $x_i$ is quantitative. Then the multiple linear regression (MLR) model is often a very useful model. For the MLR model,
$$Y_i=\alpha+x_{i, 1} \beta_1+x_{i, 2} \beta_2+\cdots+x_{i, p} \beta_p+e_i=\alpha+\boldsymbol{x}_i^T \boldsymbol{\beta}+e_i=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}_i+e_i \text { (1.9) }$$
for $i=1, \ldots, n$. Here $Y_i$ is the response variable, $\boldsymbol{x}_i$ is a $p \times 1$ vector of nontrivial predictors, $\alpha$ is an unknown constant, $\boldsymbol{\beta}$ is a $p \times 1$ vector of unknown coefficients, and $e_i$ is a random variable called the error.

The Gaussian or normal MLR model makes the additional assumption that the errors $e_i$ are iid $N\left(0, \sigma^2\right)$ random variables. This model can also be written as $Y-\alpha+\beta^T x+e$ where $e \sim N\left(0, \sigma^2\right)$, or $Y \mid x \sim N\left(\alpha+\beta^T x, \sigma^2\right)$, or $Y \mid x \sim$ $N\left(S P, \sigma^2\right)$, or $Y \mid S P \sim N\left(S P, \sigma^2\right)$. The normal MLR model is a parametric model since, given $\boldsymbol{x}$, the family of conditional distributions is completely specified by the parameters $\alpha, \boldsymbol{\beta}$, and $\sigma^2$. Since $Y \mid S P \sim N\left(S P, \sigma^2\right)$, the conditional mean function $E(Y \mid S P) \equiv M(S P)=\mu(S P)=S P=\alpha+\beta^T \boldsymbol{x}$. The MLR model is discussed in detail in Chapters 2,3 , and 4.

A sufficient summary plot (SSP) of the sufficient predictor $S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}_i$ versus the response variable $Y_i$ with the mean function added as a visual aid can be useful for describing the multiple linear regression model. This plot can not be used for real data since $\alpha$ and $\boldsymbol{\beta}$ are unknown. To make Figure 1.1, the artificial data used $n=100$ cases with $k=5$ nontrivial predictors. The data used $\alpha=-1, \boldsymbol{\beta}=(1,2,3,0,0)^T, e_i \sim N(0,1)$ and $\boldsymbol{x}$ from a multivariate normal distribution $\boldsymbol{x} \sim N_5(\mathbf{0}, \boldsymbol{I})$.

In Figure 1.1, notice that the identity line with unit slope and zero intercept corresponds to the mean function since the identity line is the line $Y=S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}=\mu(S P)=E(Y \mid S P)$. The vertical deviation of $Y_i$ from the line is equal to $e_i=Y_i-\left(\alpha+\boldsymbol{\beta}^T \boldsymbol{x}_i\right)$. For a given value of $S P$, $Y_i \sim N\left(S P, \sigma^2\right)$. For the artificial data, $\sigma^2=1$. Hence if $S P=0$ then $Y_i \sim N(0,1)$, and if $S P=5$ then $Y_i \sim N(5,1)$. Imagine superimposing the $N\left(S P, \sigma^2\right)$ curve at various values of $S P$. If all of the curves were shown, then the plot would resemble a road through a tunnel. For the artificial data, each $Y_i$ is a sample of size 1 from the normal curve with mean $\alpha+\boldsymbol{\beta}^T \boldsymbol{x}_i$.

# 线性回归代写

## 统计代写|线性回归代写线性回归代考|一些回归模型

$$Y \Perp x \mid S P \text { or } \mathrm{Y} \Perp \boldsymbol{x} \mid \mathrm{h}(\boldsymbol{x}) \text {, }$$
，其中实值函数$h: \mathbb{R}^p \rightarrow \mathbb{R}$。估计的充分预测器$\mathrm{ESP}=\hat{h}(\boldsymbol{x})$。一个重要的特例是具有线性预测器$h(\boldsymbol{x})=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}$的模型，其中ESP $=\hat{\alpha}+\hat{\boldsymbol{\beta}}^T \boldsymbol{x}$。这类模型包括广义线性模型(GLM)。另一个重要的特例是广义相加模型(GAM)，其中$Y$独立于$\boldsymbol{x}=\left(x_1, \ldots, x_p\right)^T$，对于某些(通常未知的)函数$S_j$，已知相加预测器$A P=\alpha+\sum_{j=1}^p S_j\left(x_j\right)$。估计的相加预测器$\mathrm{EAP}=\mathrm{ESP}=\hat{\alpha}+\sum_{j=1}^p \hat{S}_j\left(x_j\right)$ .

## 统计代写|线性回归代写线性回归代考|多元线性回归

$$Y_i=\alpha+x_{i, 1} \beta_1+x_{i, 2} \beta_2+\cdots+x_{i, p} \beta_p+e_i=\alpha+\boldsymbol{x}_i^T \boldsymbol{\beta}+e_i=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}_i+e_i \text { (1.9) }$$
For $i=1, \ldots, n$。这里$Y_i$是响应变量，$\boldsymbol{x}_i$是一个非平凡预测因子的$p \times 1$向量，$\alpha$是一个未知常数，$\boldsymbol{\beta}$是一个未知系数的$p \times 1$向量，$e_i$是一个叫做误差的随机变量

$$Y \mid\left(W=a_i\right) \sim f_Z\left(y-\mu_i\right)$$

$$Y \mid\left(W=a_i\right) \sim N\left(\mu_i, \sigma^2\right)$$

$$E(\boldsymbol{Y})=\left(E\left(Y_1\right), \ldots, E\left(Y_n\right)\right)^T$$
，前提是$i=1, \ldots, n$存在$E\left(Y_i\right)$。否则期望的值不存在。$n \times n$总体协方差矩阵
$$\operatorname{Cov}(\boldsymbol{Y})=E\left[(\boldsymbol{Y}-E(\boldsymbol{Y}))(\boldsymbol{Y}-E(\boldsymbol{Y}))^T\right]=\left(\sigma_{i, j}\right)$$
，其中$\operatorname{Cov}(\boldsymbol{Y})$的$i j$条目是$\operatorname{Cov}\left(Y_i, Y_j\right)=\sigma_{i, j}$，前提是每个$\sigma_{i, j}$都存在。 .否则$\operatorname{Cov}(\boldsymbol{Y})$不存在

$$E(\boldsymbol{a}+\boldsymbol{Y})=\boldsymbol{a}+E(\boldsymbol{Y}) \text { and } E(\boldsymbol{Y}+\boldsymbol{Z})=E(\boldsymbol{Y})+E(\boldsymbol{Z})$$和
$$E(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} E(\boldsymbol{Y}) \text { and } E(\boldsymbol{A} \boldsymbol{Y} \boldsymbol{B})=\boldsymbol{A} E(\boldsymbol{Y}) \boldsymbol{B}$$

## 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。