# 统计代写|统计推断代写Statistical inference代考|MAST20005

#### Doug I. Jones

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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
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## 统计代写|统计推断代写Statistical inference代考|The t-Plot and Independence

The bridge between theoretical concepts and observed data has two components. The first component establishes a connection between theoretical concepts, such as independence, non-correlation, identical distribution, and Normality on the one hand, and the ideal data on the other. The ideal data come in the form of pseudo-random numbers generated by computer algorithms so as to artificially satisfy the restrictions we impose upon them; see Devroye (1986). Generating pseudo-random numbers enables the modeler to create a pictionary of ideal data plots which can be used as reference for assessing the features of real data.

The second component is concerned with comparing these ideal plots with actual data plots, in an attempt to relate the purposefully generated patterns with those in real data. The pictionary of simulated t-plots will provide a reference framework for assessing the features of actual data plots. In this chapter we concentrate on the first component. In the meantime, we take the generation of the pseudo-random numbers for granted and proceed to provide a pictionary of simulated series designed to teach the reader how to discern particular patterns of chance regularity.
t-plot. A t-plot is a display of data with the values of the variable $Z$ measured on the $y$-axis and the index $t$, that represents the ordering of interest, on the $x$-axis:

Note that the original term for a t-plot, introduced by Playfair, was line graph. Although “time” is the obvious ordering for time series, it is no different from other deterministic orderings for cross-section data such as “gender,” marital status, age, geographical position, etc.; only the scale of measurement might differ. The aim of the discussion that follows is to compile a pictionary of t-plots of simulated data exhibiting not only IID data, but also various departures from that.
When reading t-plots one should keep a number of useful hints in mind.
First, it is important to know what exactly is being measured on each axis, the units of measurement used, and the so-called aspect ratio: the physical length of the vertical axis divided by that of the horizontal axis. A number of patterns associated with dependence and heterogeneity can be hidden by choosing the aspect ratio non-intelligently! In Figures 5.2 and 5.2 we can see the same data series with different aspect ratios. The regularity patterns are more apparent in Figure 5.2.

To enhance the ability to discern patterns over the $t$ index, it is often advisable to use lines to connect the observations; see Figure 5.3. In what follows we employ this as the default option.

## 统计代写|统计推断代写Statistical inference代考|The t-Plot and Homogeneity

The third important feature exhibited by Figure 5.3 comes in the form of a certain apparent homogeneity over $t$ exhibited by the plot. With the mind’s eye we can view $t$-homogeneity by imagining a density function cutting the $x$-axis at each observation point and standing vertically across the t-plot with its support parallel to the $y$-axis. Under complete homogeneity all such density functions are identical in shape and location and create a dome-like structure over the observations. That is, for each observation we have a density over it and we view the observation as that realized from the particular density hanging over it. Naturally, if the relevant distribution is Normal we expect more observations in the middle of the density but if the distribution is uniform we expect the observations to be dispersed uniformly over the relevant area.

This $t$-homogeneity can be assessed in two different but equivalent ways. The first way to assess the $t$-homogeneity exhibited by the data in Figure 5.3 is to use the first two data moments evaluated via a thought experiment. The mean of the data can be imagined by averaging the values of $\left{Z_t, t=1,2, \ldots, n\right}$ moving along the $t$-axis. As can be seen, such averaging will give rise to a constant mean close to zero. The variance of the data can be imagined using the virtual bands on either side of the mean of the data, which will cover almost all observations. In the case where the bands appear to be parallel to the mean axis, there appears to exist some sort of second-order homogeneity. In the case of the observed data in Figure 5.3 it seems that the data exhibit both mean and variance constancy (homogeneity); see assumptions [2] and [3] in Table 5.1.
Another way to assess $t$-homogeneity is the following thought experiment.
Thought experiment 2 Choose a frame high enough to cover the values on the $y$-axis but smaller than half of the $x$-axis and slide this frame along the latter axis keeping an eye on the picture inside the frame. If the picture does not change drastically, then the observed data exhibit homogeneity along the dimension $t$.

In the case of the data in Figure 5.9 we can see that this thought experiment suggests that the data do exhibit complete homogeneity because the pictures in the three frames shown do not differ in any systematic way. The chance regularity pattern of homogeneity, as exhibited by the data in Figure 5.9 , corresponds to the probabilistic notion of identical distribution (ID).

In contrast to Figures 5.3 and 5.9, the mean of the data in Figure 5.10 is no longer constant; it increases with $t$. The thought experiment of sliding a frame along the $x$-axis, shown in Figure 5.11, indicates that the picture in each window changes drastically, a clear indication of $t$-heterogeneity. When the change looks like a polynomial function of the index $t$, we call it a trend.

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|The t-Plot and Independence

t 图。t 图是数据的显示，其中变量Z的值在yZ轴上测量，索引t表示感兴趣的排序，在x上ytx-轴：

## 统计代写|统计推断代写Statistical inference代考|The t-Plot and Homogeneity

yx- 轴并沿后一个轴滑动此框架，同时注意框架内的图片。如果图片没有发生剧烈变化，则观察到的数据在维度上表现出同质性。t

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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。

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