### 统计代写|时间序列分析代写Time-Series Analysis代考|STATS 7058

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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|时间序列分析代写Time-Series Analysis代考|Determining the Order of Autoregressive Models

Seemingly, the availability of an exact formula for the spectral density means that the frequency resolution of autoregressive and other parametric spectral estimates is infinitely high because it can be calculated at any frequency. However, as follows from Eq. (3.13), the number of peaks, troughs, and inflection points in autoregressive spectral density estimates cannot exceed the model’s order $p$. Therefore, the AR order is the key parameter that defines the features of an $\mathrm{AR}$ (or MEM) spectral estimate. Similar considerations are true for the moving average and mixed ARMA models.

The role of the order $p$ is convenient to characterize with the following simple example. Let $x_{t}, t=1, \ldots 100$, be a sample record of a dimensionless white noise process with a unit variance and the sampling interval $\Delta t$ (e.g., 1 year, 1 day, etc.). According to Eq. (3.13), the true spectrum is a constant: $s(f)=2 \sigma_{a}^{2} \Delta t$. As the true model of the time series is not known at the initial stage of analysis, the spectral estimates should be sought for several values of $p$, say, from $p=0$ through $p=10$ for a time series of length $N=100$. To select higher values of the AR order would be unreasonable because of common sense considerations: it is not possible to obtain reliable estimates if the number of quantities to be estimated is comparable to the number of observations. The results of analysis of the white noise sequence are shown in Fig. 3.1. Obviously, if one were to choose the AR order $p=10$ arbitrarily, the conclusion would be that the time series contains “cycles” or quasi-periodic components at frequencies about $0.11$ cycles per $\Delta t(\mathrm{cp} \Delta t$ ) and $0.39 \mathrm{cp} \Delta t$. However, the correctly defined confidence limits for the estimates given in Fig. $3.1$ show that such conclusions would have been false because one can draw a very smooth or even a horizontal line within the confidence interval for the estimate. Obviously, a high order (e.g., $p \geq 10$ ) does not necessarily mean that the spectral density contains significant peaks. An AR model having a high order may have a very smooth spectrum, while a low-order model can have very sharp peaks (see Fig. $2.8$ and the well-known AR(4) example given in the Percival and Walden book published in 1993, pp. 46, 148). Similar examples can be given for more complicated spectra, but the major conclusion is that determining the order of parametric models and showing a confidence interval constitute the absolutely necessary element of parametric spectral analysis.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Comparison of Autoregressive and Nonparametric Spectral Estimates

The statement that the spectral density is the most important statistics of any scalar time series is true for Gaussian and non-Gaussian data. The spectral density can be estimated with parametric and nonparametric methods, and if the time series is long, the estimates will be reliable and similar to each other. If the time series is short, which happens regularly in climatology and its branches and in other Earth sciences, the task of proper spectral analysis becomes vital. In this section, we will show the advantages of the parametric approach to the task of spectral estimation, which exist, in particular, due to the fact that many geophysical processes can be well approximated with stochastic difference equations of relatively low order. The examples given here include time series whose spectra are typical for climatic processes, including the

annual global surface temperature. The time series will always be short: the total number of its terms is just 50 . The autoregressive spectral estimates will be compared with respective nonparametric estimates obtained according to Blackman and Tukey (1958). Four time series of length $N=50$ have AR orders from 1 to 4 , and they are quite common for climate and for other geophysical phenomena. The true AR models are known in all cases, and the initial data for the analysis were obtained through simulation. Because of the small length of the time series, the sample estimates of AR coefficients may differ rather significantly from the true values.

The true model of the first time series is an AR(1) with the AR coefficient $\varphi=0.5$. It can be the annual river streamflow, daily temperature, sea level variations in coastal areas with no tides, etc. The sample AR (or MEM) estimate of the spectrum is shown in Fig. 3.3a along with the true spectrum.

According to Fig. 3.3a, the $90 \%$ confidence interval for the AR spectral estimate contains the true spectrum; this estimate can be regarded as satisfactory. The shape of the spectrum is reproduced accurately, and the bias occurs due to the sampling variability of the variance estimate. The nonparametric estimate (Fig. 3.3b) has several peaks, but the peaks are statistically insignificant. The $90 \%$ confidence interval is wide and asymmetric with respect to the spectral estimate due to the asymmetry of $\chi^{2}$ distribution at low degrees of freedom in this and the other three examples below.

The autoregressive approach provides quantitative information about statistical properties of time series in both time and frequency domains. This can also be achieved with other parametric models, but the moving average operator is less reasonable physically and is difficult to deal with. The nonparametric methods of analysis do not produce explicit information about the time series behavior in the time domain and, if the time series is short, the nonparametric spectral estimates are less reliable than the AR estimates.

The autoregressive stochastic difference equation possesses a number of useful features for analysis of stationary time series:

• it presents a ready-to-use tool for linear extrapolation of time series (Chap. 6);
• it allows one to get quantitative estimates of time series dependence upon its previous values (up to the AR order $p$ ) and upon the innovation sequence;
• it can be used to determine the natural frequencies and damping coefficients of the time series, that is, the frequencies, at which the spectral density may have peaks;
• it provides an analytical expression for the time series spectral density;
• the autoregressive approach to spectral estimation satisfies the requirements of the maximum entropy method (MEM), and it is capable of producing satisfactory estimates of the spectrum when the time series is short.

This author is not aware of any disadvantages in applying the autoregressive time and frequency domain approach to time series research in climatology and other Earth and solar sciences.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Comparison of Autoregressive and Nonparametric Spectral Estimates

• 它为时间序列的线性外推提供了一个现成的工具（第 6 章）；
• 它允许人们根据其先前的值（直到 AR 顺序）获得时间序列依赖的定量估计p) 和创新序列；
• 它可用于确定时间序列的固有频率和阻尼系数，即频谱密度可能出现峰值的频率；
• 它提供了时间序列谱密度的解析表达式；
• 谱估计的自回归方法满足最大熵法（MEM）的要求，并且能够在时间序列较短的情况下产生令人满意的谱估计。

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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。