统计代写|linear regression代写线性回归代考|Simple Linear Regression Models

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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|linear regression代写线性回归代考|Simple Linear Regression Models

Chapter 2 describes a conceptual model as an abstract representation of anticipated associations among concepts or ideas designed to represent broader ideas (such as self-esteem, political ideology, or education). Ideally, statistical models are guided by conceptual models, which are used to delineate hypotheses or research questions. Statistical models outline probabilistic relationships among a set of variables, with the goal of estimating whether there are nonrandom patterns among them. Like conceptual models, these models tend to be simplifications of the complexity that occurs in nature but offer enough detail to predict or understand patterns in the data. A useful way of thinking about statistical models is that they assess ways that a set of data may have been produced, or, in statistical parlance, a data generating process (DGP).

A regression model is a type of statistical model that aims to estimate the association between one or more explanatory variables $(x \mathrm{~s})$ and a single outcome variable $(y)$. An outcome variable is presumed to depend on or to be predicted by the explanatory variables. But the explanatory variables are seen as independent predictors of the outcome variable; hence, they are often called independent variables. Later chapters discuss why this term can be misleading because these variables may, if the model is set up correctly, relate to one another. Many researchers therefore prefer to call those included in a regression model explanatory and outcome variables (used in this book), predictor and response variables, exogenous and endogenous variables, or similar terms. The response or endogenous variable is synonymous with the outcome variable.

An LRM seeks to account for or explain differences in values of the outcome variable with information about values of the explanatory variables. The LRM also seeks, to varying degrees, answers to the following questions:

1. What are the predicted mean levels of the outcome variable for particular values of the explanatory variables?
2. What is the most appropriate equation for representing the association between each explanatory variable and the outcome variable? This includes assessing the direction (positive? negative?) and magnitude of each association.Which explanatory variables are good predictors of the outcome and which are not? The answer is based on several results from the LRM, including the size of the coefficients, differences in predicted means, the $p$-values, and the CIs, but each has limitations. ${ }^{1}$

统计代写|linear regression代写线性回归代考|Assumptions of Simple LRMs

The LRM rests on several assumptions that dictate how well it operates. Most of these concern characteristics of the population data and focus on the errors of prediction $\left(\varepsilon_{i}\right)$. But having access to information from a population is unusual, so we must assess, roughly or indirectly, the assumptions of LRMs with information from a sample. In other words, since we do not have information from the $Y$, we cannot compute $\varepsilon_{i}$ directly. The sample includes only the $x$ s and $y$, so we must use an estimate of $\varepsilon_{i}$. This estimate, depicted as the error term $\left(\hat{\varepsilon}{i}\right)$ in Equation $3.5$, is represented by the residuals ${ }^{6}$ from the model, which are computed as $\left(y{i}-\hat{y}{i}\right)$. Rather than distinguishing the errors of prediction from the population and the sample, however, we’ll take for granted that the sample provides a good estimate of $\bar{Y}{i}$ with $\hat{y}{i}$ so that $\left(y{i}-\hat{y}{i}\right) \cong\left(y{i}-\bar{Y}_{i}\right)$.
Here are the key assumptions of simple LRMs:

1. Independence: the errors of prediction $\left(\varepsilon_{i}\right)$ are statistically independent of one another. Using the example from the Nations2018 dataset, we assume that the errors in predicting public expenditures across nations are independent. In practice, this often implies that the observations are independent. One way to (almost) guarantee this is to use simple random sampling. (However, in this example we should ask ourselves: are the economic conditions of these nations likely to be independent?) Chapters 8 and 15 outline additional ways to understand the independence assumption.
2. Homoscedasticity (constant variance): the errors of prediction have equivalent variance for all possible values of $X$. In other words, the variance of the errors is assumed to be constant across the distribution of X. At this point it may be simpler, yet imprecise, to think about the $Y$ values and ask whether their variability is equivalent at different values of $X$. Chapter 9 discusses the homoscedasticity assumption.

统计代写|linear regression代写线性回归代考|An Example of an LRM Using $R$

You may be confused at this point, though let’s hope not. An example using some data should be beneficial. The dataset StateData2018.csv includes a number of variables from all 50 states in the U.S. These data include population characteristics, crime rates, substance use rates, and various economic and social factors. We’ll treat the data as a sample, even though one might argue that they represent a population. Similar to the code that produces Figure 3.2, the following $\mathrm{R}$ code creates a scatter plot and overlays a linear fit line with the number of opioid deaths per 100,000 residents (OpioidoDDeathRate) as the outcome (y) variable and average life satisfaction (LifeSatis), which is based on state-specific survey data ${ }^{8}$ that gauges happiness and satisfaction with one’s family life and health among adult residents, as the explanatory $(x)$ variable.
R code for Figure $3.4$
plot (StateData2018\$LifeSatis, StateData2018 \$OpioidoDdeathRate, xlab=”Average life
satisfaction”, ylab=”Opioid overdose deaths per
100,000 population”, pch=1)
abline (1m (StateData2018\$OpioidodDeathRate StateData2018\$LifeSatis), col=”red”)
R code for Figure $3.4$
plot (StateData2018\$LifeSatis, StateData2018 \$OpioidoDDeathRate, xlab=”Average life
satisfaction”, ylab= “Opioid overdose deaths per
100,000 population”, pch=1)
abline ( $1 \mathrm{~m}$ (StateData2018\$OpioidoDDeathRate$~$StateData2018\$LifeSatis), col= “red”)
Figure $3.4$ displays a negative slope. Yet the points diverge from the line;
only a few are relatively close to it. Do you see any other patterns in the data
relative to the line?
We’ll now estimate a simple LRM using these two variables. As you may
have already determined given R’s abline function that created the linear
fit lines in Figures $3.2$ and 3.4, an LRM is estimated in R using the lm func-
tion. The abbreviation signifies “linear model.”
Figure $3.4$ displays a negative slope. Yet the points diverge from the line; only a few are relatively close to it. Do you see any other patterns in the data relative to the line?

We’ll now estimate a simple LRM using these two variables. As you may have already determined given R’s abline function that created the linear fit lines in Figures $3.2$ and 3.4, an LRM is estimated in R using the $1 \mathrm{~m}$ function. The abbreviation signifies “linear model.”

统计代写|linear regression代写线性回归代考|Simple Linear Regression Models

LRM 旨在利用有关解释变量值的信息来解释或解释结果变量值的差异。LRM 还在不同程度上寻求以下问题的答案：

1. 对于解释变量的特定值，结果变量的预测平均水平是多少？
2. 什么是表示每个解释变量和结果变量之间关联的最合适的方程？这包括评估每个关联的方向（正面？负面？）和幅度。哪些解释变量可以很好地预测结果，哪些不是？答案基于 LRM 的几个结果，包括系数的大小、预测均值的差异、p-values 和 CI，但每个都有局限性。1

统计代写|linear regression代写线性回归代考|Assumptions of Simple LRMs

LRM 依赖于几个假设，这些假设决定了它的运作情况。其中大多数关注人口数据的特征，并关注预测的错误(e一世). 但是从人群中获取信息是不寻常的，因此我们必须粗略或间接地评估 LRM 的假设与来自样本的信息。换句话说，由于我们没有来自是，我们无法计算e一世直接地。该样本仅包括X沙是, 所以我们必须使用一个估计e一世. 这个估计，描述为误差项(e^一世)在方程3.5, 由残差表示6从模型，计算为(是一世−是^一世). 然而，我们不会将预测误差与总体和样本区分开来，而是理所当然地认为样本提供了一个很好的估计是¯一世和是^一世以便(是一世−是^一世)≅(是一世−是¯一世).

1. 独立性：预测的错误(e一世)在统计上相互独立。使用 Nations2018 数据集中的示例，我们假设预测各国公共支出的错误是独立的。在实践中，这通常意味着观察是独立的。（几乎）保证这一点的一种方法是使用简单的随机抽样。（然而，在这个例子中，我们应该问自己：这些国家的经济状况是否可能是独立的？）第 8 章和第 15 章概述了理解独立假设的其他方法。
2. Homoscedasticity（常数方差）：预测的误差对于所有可能的值具有等价的方差X. 换句话说，假设误差的方差在 X 的分布中是恒定的。在这一点上，考虑是并询问它们的可变性在不同的值下是否相等X. 第 9 章讨论了同方差性假设。

统计代写|linear regression代写线性回归代考|An Example of an LRM Using R

plot (StateData2018 $LifeSatis, StateData2018$ OpioidoDdeathRate, xlab=”平均生活

100,000 人口”, pch=1)
abline (1m (StateData2018 $OpioidodDeathRate StateData2018$ LifeSatis), col=”red” )

plot (StateData2018 $LifeSatis, StateData2018$ OpioidoDDeathRate, xlab=”平均生活

100,000 人中阿片类药物过量死亡人数”, pch=1)
abline (1 米（StateData2018 $OpioidoDDeathRate StateData2018$ LifeSatis), col= “red”)

。该缩写表示“线性模型”。

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。