### 机器学习代写|聚类分析作业代写clustering analysis代考|Time series features and 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

## 机器学习代写|聚类分析作业代写clustering analysis代考|Time series features and models

The topic of time series analysis is the subject of a large number of books and journal articles. In this chapter, we highlight fundamental time series concepts, as well as features and models that are relevant to the clustering and classification of time series in subsequent chapters. Much of this material on time series analysis, in much greater detail, is available in books by authors such as Box et al. (1994), Chatfield (2004), Shumway and Stoffer (2016), Percival and Walden (2016) and Ord and Fildes (2013).

## 机器学习代写|聚类分析作业代写clustering analysis代考|Stochastic processes

A stochastic process is defined as a collection of random variables that are ordered in time and defined as a set of points which may be discrete or continuous. We denote the random variable at time $t$ by $X(t)$ if time is continuous or by $X_{t}$ if time is discrete. A continuous stochastic process is described as ${X(t),-\infty<t<\infty}$ while a discrete stochastic process is described as $\left{X_{t}, t=\ldots,-2,-1,0,1,2, \ldots\right}$.

Most statistical problems are concerned with estimating the properties of a population from a sample. The properties of the sample are typically determined by the researcher, including the sample size and whether randomness is incorporated into the selection process. In time series analysis there is a different situation in that the order of observations is determined by time. Although it may be possible to increase the sample size by varying the length of the observed time series, there will be a single outcome of the process and a single observation on the random variable at time $t$. Nevertheless, we may regard the observed time series as just one example of an infinite set of time series that might be observed. The infinite set of time series is called an ensemble. Every member of the ensemble is a possible realization of the stochastic process. The observed time series can be thought of as one possible realization of the stochastic process and is denoted by ${x(t),-\infty<t<\infty}$ if time is continuous or $\left{x_{t}, t=0,1,2, . . T\right}$ if time is discrete. Time series analysis is essentially concerned with evaluating the properties of the underlying probability model from this observed time series. In what follows, we will be working with mainly discrete time series which are realizations of discrete stochastic processes.

Many models for stochastic processes are expressed by means of algebraic expressions relating the random variable at time $t$ to past values of the process, together with values of an unobservable error process. From one such model we may be able to specify the joint distribution of $X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{k}}$, for any set of times $t_{1}, t_{2}, \ldots, t_{k}$ and any value of $k$. A simple way to describe a stochastic process is to examine the moments of the process, particularly the first and second moments, namely, the mean and autocovariance function.
$$\begin{gathered} \mu_{t}=E\left(X_{t}\right) \ \gamma_{t_{1}, t_{2}}=E\left[\left(X_{t_{1}}-\mu_{t}\right)\left(X_{t_{2}}-\mu_{t}\right)\right] \end{gathered}$$
The variance is a special case of the autocovariance function when $t_{1}=t_{2}$, that is,
$$\sigma_{t}^{2}=E\left[\left(X_{t}-\mu_{t}\right)^{2}\right] .$$
An important class of stochastic processes is that which is stationary. A time series is said to be stationary if the joint distribution of $X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{k}}$

is the same as that of $X_{t_{1}+\tau}, X_{t_{2}+\tau}, \ldots, X_{t_{k}+\tau}$, for all $t_{1}, t_{2}, \ldots, t_{k}, \tau$. In other words, shifting the time origin by the amount $\tau$ has no effect on the joint distribution which must therefore depend only on the intervals between $t_{1}, t_{2}, \ldots, t_{k}$.
This definition holds for any value of $k$. In particular, if $k=1$, strict stationarity implies that the distribution of $X_{t}$ is the same for all $t$, provided the first two moments are finite and are both constant, that is, $\mu_{t}=\mu$ and $\sigma_{t}^{2}=\sigma^{2}$. If $k=2$, the joint distribution of $X_{t_{1}}$ and $X_{t_{2}}$ depends only on the time difference $t_{1}-t_{2}=\tau$ which is called a lag. Thus the autocovariance function which depends only on $t_{1}-t_{2}$ may be written as
$$\gamma_{\tau}=\operatorname{COV}\left(X_{t}, X_{t+\tau}\right)=E\left[\left(X_{t}-\mu\right)\left(X_{t+\tau}-\mu\right)\right]$$

## 机器学习代写|聚类分析作业代写clustering analysis代考|Autocorrelation and partial autocorrelation functions

Autocovariance and autocorrelation measure the linear relationship between various values of an observed time series that are lagged $k$ periods apart, that is, given an observed time series $\left{x_{t}, t=0,1,2, . T\right}$, we measure the relationship between $x_{t}$ and $x_{t-1}, x_{t}$ and $x_{t-2}, x_{t}$ and $x_{t-3}$, etc.. Thus, the autocorrelation function is an important tool for assessing the degree of dependence in observed time series. It is useful in determining whether or not a time series is stationary. It can suggest possible models that can be fitted to the observed time series and it can detect repeated patterns in a time series such as the presence of a periodic signal which has been buried by noise. The sample autocorrelation function (ACF), $r_{k}, k=1,2, \ldots$ is typically plotted for at least a quarter of the number of lags or thereabouts. The plot is supplemented with $5 \%$ significance limits to enable a graphical check of whether of not dependence is statistically significant at a particular lag.

Partial autocorrelations are used to measure the relationship between $x_{t}$ and $x_{t-k}$, with the effect of the other time lags, $1,2, \ldots, k$-1 removed. It is also useful to plot the partial autocorrelation function (PACF) because it, together with the plot of the ACF, can help inform one on a possible appropriate model that can be fitted to the time series. Refer to any of the references mentioned in Section $2.1$ for more details on the ACF and PACF including their sampling distributions which enable the determination of the significance limits.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Stochastic processes

μ吨=和(X吨) C吨1,吨2=和[(X吨1−μ吨)(X吨2−μ吨)]

σ吨2=和[(X吨−μ吨)2].

Cτ=冠状病毒⁡(X吨,X吨+τ)=和[(X吨−μ)(X吨+τ−μ)]

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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。