### 统计代写|实验设计作业代写experimental design代考|TRANSFORMATIONS TO OBTAIN LINEARITY

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|TRANSFORMATIONS TO OBTAIN LINEARITY

Two variables, $x$ and $y$, may be closely related but the relationship may not be linear. Ideally, theoretical clues would be present wh1ch point to a particular relationship such as an exponential growth model which is common in blology. Without such clues, we could firstly examine a scatter plot of $y$ against $x$.
Sometimes we may recognize a mathematical model which $f$ its the data well. Otherwise, we try to choose a simple transformation such

as ralsing the variable to a power p as in Table 1.4.1. A power of 1 leaves the variable unchanged, that is as raw data. As we proceed up or down the table from 1 , the strength of the transformation increases; as we move up the table the transformation stretohes larger values relatively more than smaller ones. Although the exponential does not flt in very well, we nave included $1 t$ as $1 t 1 s$ the inverse of the logarithmic transformation. other fractional powers oould be used but they may be difficult to interpret.
It would be feasible to transform either $y$ or $x$, and, indeed, a transformation of $y$ would be equivalent to the inverse transformation of $x$. For example, squaring $y$ would be equivalent to taking the square root of $x$. If there are two or more predictor variables, it is often advisable to transform these in different ways nather than $y$, for if y is transformed to be linearly related to one predictor variable it may then not be 1 inear ly related to another.
In Eigure 1.4.1, it is clear that we should stretoh out the graph by increasing large $x$ values, or, alternatively; reduce large $y$ values. Thus, we could try changing $x$ to $x$-squared, or $y$ to the square root of $y_{*}$ One point to be kept in mind here is that for p>0, $y=0$ when $x=0$ so that it may be advisable before invoking the power transformation to change the origin; in partfoular, we could onange y to $(y-a)$, and a good guess may be a $=30$. In Figure $1.4 .2$, it seems that large $x$ values and large $y$ values should be reduced suggesting that a reciprooal transformation may be appropriate. This would require the $x$ and $y$ axes to be asymptotes which, in particular, would

mean that a constant, perhaps 14 , should be subtracted from $y$. We could try changing
\begin{aligned} &y \text { to }(y-14)^{-1} \ &\text { or } y \text { to }(y=14)^{-0.5} \ &\text { or } x \text { to } x{ }^{-1} \ &\text { or } x \text { to } x=0.5 \end{aligned}

## 统计代写|实验设计作业代写experimental design代考|FITTING A MODEL USING VECTORS AND MATRICES

Appendix A contains a review of vectors, vector spaces and matrices and some readers may wish to refer to that section while reading the following.

In regression, we consider the relationship between a string of values of the dependent variable, $y$, and one or more strings of corresponding values of the predictor variables, the $x$ ‘s. It is useful to think of each string as a vector for it turns out that the relationships of interest between the variables are encapsulated in the lengths of the vectors and the angles between them. For the simple Example $1.3 .1$, the $x$ and $y$ readings can be written as column vectors:

The simplest model for this example would be a line through the origin
$$\mathrm{y}{\mathrm{i}}=B \mathrm{X}{\mathrm{i}}+\mathrm{E}{i}$$ or $\mathbf{y}=\beta \mathbf{x}+\boldsymbol{\varepsilon}$ in vector terms The normal equation $1 \mathrm{~s}$ $b \sum x{1}^{2}=\sum x_{1}^{y_{1}}$
or $\left(x^{T} x\right) b=x^{T} y$
giving $b=\left(x^{T} \mathbf{y}\right) /\left(x^{T} \mathbf{x}\right)=0.973$
$(1.5 .2)$
For each value of $x$ we can calculate the predicted value of $y$ as
$$\mathbf{y}=b x \text { or } x b=\left|\begin{array}{l} 0.5 \ 1.0 \ 1.5 \ 2.0 \ 2.5 \end{array}\right| 0.973=\left|\begin{array}{l} 0.486 \ 0.973 \ 1.459 \ 1.945 \ 2.432 \end{array}\right|$$
The predicted value can also be written as
$$y=x b=x\left(x^{T} x\right)^{-1} x^{T} y=P y$$
The matrix $P=x\left(x^{T} \mathbf{x}\right)^{-1} x^{T}$ is termed the projection matrix. More is said about this in section 1.7. Notice that for this case with $n=5$, $P$ is a $5 \times 5$ matrix, namely.

## 统计代写|实验设计作业代写experimental design代考|DEVIATIONS FROM MEANS

It is common practice when fitting a model to use the original (also called raw) data and to include $1 n$ the model a $y=i n t e r c e p t ~ t e r m ~(a l s o ~$ called a constant, or general mean). Most computer programs would convert the raw data to deviations from the mean, as these are used in such statistics as the correlation coefficient. Converting to deviations has the advantage of removing a parameter from the model, making 1 t easier to man1 pulate. Sometimes an examination of the deviations shows up trends which are not as clearly noticeable in the raw data. Problem $2.1$ is an example where deviations from the mean prove useful. It turns out that the estimated coefficients will be the same for the raw data with constant term as with the data 1 n deviation form.

For this section we change our notation slightly to make 1 t clear whether we are referring to the raw data (which we indicate by capital letters, $X, Y$, etc) or deviations from means (lower case $x$, $y$, etc).
$1.6 .1$ Estimates
Ignoring subsor ipts for simplicity, we can write for the case of one predictor variable,
$$x=X=\bar{X} \text { and } y=Y-\bar{Y}$$
where $\vec{X}$ and $\vec{Y}$ are the sample means. For the model
$$\mathrm{y}=8 x+\varepsilon$$
we saw in $(1.2 .7)$ the least squares estimates are given by
\begin{aligned} b &=\sum_{1}^{y_{1} / 2 x_{1}^{2}} \quad(\text { from } 1.2 .7) \ &=\mathrm{S}{x y} / \mathrm{S}{x x} \end{aligned} \quad (def ined by 1.2.13)
$(1.6 .2)$
Notice that the predicted value of $y$ in this case is $\hat{y}=b x$

(1.5.2)

## 统计代写|实验设计作业代写experimental design代考|DEVIATIONS FROM MEANS

1.6.1估计

X=X=X¯ 和 是的=是的−是的¯

b=∑1是的1/2X12( 从 1.2.7) =小号X是的/小号XX（由 1.2.13 定义）
(1.6.2)

## 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。