### 统计代写|实验设计作业代写experimental design代考|FITTING A MODEL TO DATA

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling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。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design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|实验设计作业代写experimental design代考|FITTING A MODEL TO DATA

The title of this chapter could well be the title of this book. In the first four chapters, we consider problems associated with fitting a regression model and in the last four we consider experimental designs. Mathematically, the two topics use the same model. The term regression is used when the model is fltted to observational data, and experimental design is used when the data is carefully organized to give the model special properties. For some data, the distinction may not be at all clear or, indeed, relevant, he shall consider sets of data consisting of observations of a variable of Interest which we shall call $y$, and we shall assume that these observations are a random sample from a population, usually infinite, of possible values. It is this population which is of primary interest, and not the sample, for in trying to fit models to the data we are really trying to flt models to the population from which the sample is drawn. For each observation, $y$, the model will be of the fork
observed $y=$ population mean + deviation
$(1.1 .1)$
The population mean may depend on the corresponding values of a prem dictor variable which we often label as $x$. For this reason, y is

called the dependent variable. The deviation term indicates the individual peculiarity of the observation, $y$, which makes it differ from the population mean.

As an example, $\$ y$could be the price paid for a house in a certain oity. The population mean could be thought of as the mean price paid for houses in that city, presumably in a given time period. In this case the deviation term could be very large as house prices would vary greatly depending on a number of factors such as the size and condition of the house as well as its position in the oity. In New Zealand, each house is given a government valuation, GV, which is reconsidered on a five year cycle. The price paid for a house will depend to some extent on its GV. The regression model could then be written in terms of$\$x$, the $\mathrm{GV}$, as:

## 统计代写|实验设计作业代写experimental design代考|HON TO FIT A LINE

As the deviation term involves the unexplained variation in $y$, we try to minimise this in some way. Suppose we postulate that the mean value of $y$ is a function of $x$. That is
$$E(y)=f(x)$$
Then for a sample of $n$ pairs of $y^{\prime} s$ with their corresponding $x^{\prime} s$ we have

The above notation assumes that the $x^{\prime} s$ are not random variables but are fixed in advance. If the $x^{\prime} s$ were in fact random variables we should write

$$f\left(x_{1}\right)=E\left(y_{1} \mid x_{1}=x_{1}\right)$$
= mean of $Y_{i}$ given that $X_{i}=x_{i}$
which gives the same results. We wil1 therefore assume in future that the $x^{\prime} s$ are $f$ ixed.
The simplest example of a function $f$ would arise if $y$ was proportional to $x$. We could imagine a situation where an inspector of weights and measures set out to test the scales used by shopkeepers. In this case, the $x^{\prime} s$ would be the welghts of standand measures while $y^{\prime} s$ would be the corresponding weights indicated by the shopkeeper’s scales. The model would be
The mean value of $y$ when $x=x_{i}$ is given by
$$E\left(y_{i}\right)=B x_{i}=f\left(x_{i}\right)$$
This is called a regression curve. In this simple example we would expect the parameter $B$ to be 1 , or at least close to 1 . We think of the parameters as being fixed numbers which describe some attributes of the population.

## 统计代写|实验设计作业代写experimental design代考|Other Ways of Fitting a Curve

The main mroblem with the approach of least squares is that a large deviation will have an even larger square and this deviation may have an unduly large influence on the $f$ itted curve. To guard against such distortions we could try to isolate large deviations. We consider this in more detail in Chapter 4 under outliers and sensitive points. Alternatively, we could seek estimates which minimize a different function of the deviations.

If the model is expressed in terms of the population median of $y$, rather than its mean, another method of fitting a curve would be by minimizing $T$, the sum of the absolute values of deviations, that is
$$T=\sum_{i=1}^{n}\left|\varepsilon_{1}\right|$$
Although this is a sensible approach which works well, the actual mathematics is difficult when the distributions of estimates are sought. Hogg (1974) suggests minimizing
$T=\sum\left|\varepsilon_{i}\right|^{p} \quad$ with $\quad 1<p<2$
and $p=1.5$, in particular, may be a reasonable compromise. Again it is difficult to determine the exact distributions of the resulting estimates. If we are not so much interested in testing hypothesis as

estimating coefficients then this method provides estimates which are robust in the sense that they are not unduly affected by large dev1ations.

Notice that the deviations are the vertical distances from the regression line. It might, perhaps, seem more logical, or at least more symmetrical, to consider the perpendicular (orthogonal) distances from the regression line. However when our major concern is predicting $y$ from $x$ the vertical distances are more relevant because they represent the prediction error.

(1.1.1)

## 统计代写|实验设计作业代写experimental design代考|HON TO FIT A LINE

= 的平均值是的一世鉴于X一世=X一世

## 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。