### 统计代写|广义线性模型代写generalized linear model代考|MAST30025

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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|广义线性模型代写generalized linear model代考|Challenger Disaster Example

In January 1986, the space shuttle Challenger exploded shortly after launch. An investigation was launched into the cause of the crash and attention focused on the rubber O-ring seals in the rocket boosters. At lower temperatures, rubber becomes more brittle and is a less effective sealant. At the time of the launch, the temperature was $31^{\circ} \mathrm{F}$. Could the failure of the O-rings have been predicted? In the 23 previous shuttle missions for which data exists, some evidence of damage due to blow by and erosion was recorded on some O-rings. Each shuttle had two boosters, each with three O-rings. For each mission, we know the number of $\mathrm{O}$-rings out of six showing some damage and the launch temperature. This is a simplification of the problem-see Dalal, Fowlkes, and Hoadley (1989) for more details.

Let’s start our analysis with R. For help in obtaining R and installing the necessary add-on packages and datasets, please see Appendix B. First we load the data. To do this, you will first need to load the faraway package using the library command as seen in here. You will need to do this in every session that you run examples from this book. If you forget, you will receive a warning message about the data not being found. We then plot the proportion of damaged O-rings against temperature in Figure 2.1:
$>$ library (faraway)
$>$ data (orings)
$>$ plot (damage/ $6 \sim$ temp, orings, $x l i m=c(25,85)$, ylim $=$
$c(0,1)$,
$\quad x l a b=$ “Temperature”, ylab=”Prob of damage”)
We are interested in how the probability of failure in a given O-ring is related to the launch temperature and predicting that probability when the temperature is $31^{\circ} \mathrm{F}$. A naive approach, based on linear models, simply fits a line to this data:
\begin{aligned} & >\text { lmod }<-1 \mathrm{~lm} \text { (damage } / 6 \sim \text { temp, orings) } \\ & >\text { abline (lmod) } \end{aligned}
The fit is shown in Figure 2.1. There are several problems with this approach. Most obviously from the plot, it can predict probabilities greater than one or less than zero. One might suggest truncating predictions outside the range to zero or one as appropriate, but it does not seem eredible that these probabilities would be exactly zero or one, in this particular example or many others.

## 统计代写|广义线性模型代写generalized linear model代考|Binomial Regression Model

Suppose the response variable $Y_i$ for $i=1, \ldots, n_i$ is binomially distributed $B\left(n_i, p_i\right)$ so that:
$$P\left(Y_i=y_i\right)=\left(\begin{array}{c} n_i \ y_i \end{array}\right) p_i^{y_i}\left(1-p_i\right)^{n_i-y_i}$$
We further assume that the $Y_i$ are independent. The individual trials that compose the response $Y_i$ are all subject to the same $q$ predictors $\left(x_{i 1}, \ldots, x_{i q}\right)$. The group of trials is known as a covariate class. We need a model that describes the relationship of $x_1, \ldots, x_q$ to $p$. Following the linear model approach, we construct a linear predictor:
$$\eta_i=\beta_0+\beta_1 x_{i 1}+\ldots+\beta_q x_{i q}$$

Since the linear predictor can accommodate quantitative and qualitative predictors with the use of dummy variables and also allows for transformations and combinations of the original predictors, it is very flexible and yet retains interpretability. This notion that we can express the effect of the predictors on the response solely through the linear predictor is important. The idea can be extended to models for other types of response and is one of the defining features of the wider class of generalized linear models (GLMs) discussed in Chapter 6.

We have already seen above that setting $\eta_i=p_i$ is not appropriate because we require $0 \leq p_i \leq 1$. Instead we shall use a link function $g$ such that $\eta i=g\left(p_i\right)$. For this application, we shall need $g$ to be monotone and be such that $0 \leq \mathrm{g}^{-1}(\eta) \leq 1$ for any $\eta$. There are three common choices:

1. Logit: $\eta=\log (p /(1-p))$.
2. Probit: $\eta=\Phi^{-1}(p)$ where $\Phi^{-1}$ is the inverse normal cumulative distribution function.
3. Complementary $\log -\log : \eta=\log (-\log (1-p))$.
The idea of the link function is also one of the central ideas of generalized linear models. It is used to link the linear predictor to the mean of the response in the wider class of models.

We will compare these three choices of link function later, but first we estimate the parameters of the model. We shall use the method of maximum likelihood; see Appendix A for a brief introduction to this method. The log-likelihood is given by:
$$l(\beta)=\sum_{i=1}^n\left[y_i \eta_i-n_i \log \left(1+e_i^\eta\right)+\log \left(\begin{array}{l} n_i \ y_i \end{array}\right)\right]$$
We can maximize this to obtain the maximum likelihood estimates $\hat{\beta}$ and use the standard theory to obtain approximate standard errors. An algorithm to perform the maximization will be discussed in Chapter 6 .

# 广义线性模型代考

## 统计代写|广义线性模型代写generalized linear model代考|Challenger Disaster Example

1986 年 1 月，挑战者号航天飞机在发射后不久就爆炸了。对坠机原因展开了调查，并将注意 力集中在火箭助推器中的橡胶 $O$ 形密封圈上。在较低的温度下，橡胶变得更脆并且是一种效 果较差的密封剂。发射时的温度是 $31^{\circ} \mathrm{F}$. 是否可以预测 $\mathrm{O}$ 形圈的失效? 在现有数据的 23 次航 天飞机任务中，一些 $O$ 形环上记录了由于吹过和侵蚀造成的损坏证据。每架航天飞机都有两 个助推器，每个助推器都有三个 $O$ 形圈。对于每个任务，我们知道有多少 0 -六分之一的环显 示了一些损坏和发射温度。这是问题的简化一一有关详细信息，请参阅 Dalal、Fowlkes 和 Hoadley (1989)。

$>$ 图书馆 (遥远)

$>$ 情节 (伤害 $6 \sim$ 温度, orings, $x$ lim $=c(25,85)$ ， 优越的 $=$ $c(0,1)$,
$x l a b=$ “Temperature”, ylab=”Prob of damage”)

\begin{aligned} & >\operatorname{lmod}<-1 \operatorname{lm}(\text { damage } / 6 \sim \text { temp, orings) } \\ & >\text { abline }(\operatorname{lmod}) \end{aligned}

## 统计代写|广义线性模型代写generalized linear model代考|Binomial Regression Model

$$P\left(Y_i=y_i\right)=\left(n_i y_i\right) p_i^{y_i}\left(1-p_i\right)^{n_i-y_i}$$

$$\eta_i=\beta_0+\beta_1 x_{i 1}+\ldots+\beta_q x_{i q}$$

1. 登录: $\eta=\log (p /(1-p))$.
2. 概率: $\eta=\Phi^{-1}(p)$ 在哪里 $\Phi^{-1}$ 是逆正态累积分布函数。
3. 补充 $\log -\log : \eta=\log (-\log (1-p))$.
链㢺函数的思想也是广义线性模型的核心思想之一。它用于将线生预测变量链㢺到更广 泛的模型类别中的响应均值。

## 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。