统计代写|时间序列分析代写Time-Series Analysis代考|STATS 3005

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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|Nonparametric Spectral Analysis

The nonparametric spectral analysis means that a spectral estimate is obtained directly from the time series without making any assumptions about its structure, except for its stationarity or, strictly speaking, its ergodicity. It can be obtained through a Fourier transform of the time series or of its parts with subsequent averaging and smoothing, through a Fourier transform of the covariance function estimate, or by applying a number of filters (windows, or tapers). Such methods are widely used in engineering, where the amount of data is often large and the experiments that generate data can often be repeated at will. The necessary software is easily available in R, MATLAB, and other packages for time series analysis. A thorough review of methods of spectral analysis with practical examples is given in Percival and Walden (1993). The traditional methods of nonparametric spectral analysis used in science and engineering include

• Blackman-Tukey method based upon the Fourier transform of the covariance function estimate plus some tapering (Blackman and Tukey 1958; Bendat and Piersol 1966),
• Bendat-Piersol method of nonoverlapping segments (Bendat and Piersol 2010),
• Welch’s overlapped segment averaging method (Welch 1967), and
• Thomson’s multitaper method (Thomson 1982).
The first three methods are good for long time series and, with a few exceptions later in this chapter, they will rarely be applied here for analysis of time series, in particular, because so many of them are short. The fourth nonparametric method suggested by David Thomson, can produce, according to the author of the method as well as to this author’s experience, unbiased and consistent spectral estimates with short time series; it has high-frequency resolution and can be useful for detecting periodic and quasi-periodic components. In his original publication, the author of the method gave an example of successful analysis of a short $(N=100)$ time series with a rather complicated spectral density typical for sample records in communication systems. A brief explanation for the Thomson’s multitaper method (MTM) is that the spectral estimate is obtained as an average of several squared Fourier transforms of the time series which are smoothed with tapers (the so-called discrete prolate spheroidal sequences). This method of spectral analysis will be used in this book along with the autoregressive approach. A review of MTM can be found in Babadi and Brown (2014).

统计代写|时间序列分析代写Time-Series Analysis代考|Parametric Models of Time Series

The reliability problem with the nonparametric estimation of climate spectra exists due to the necessity to estimate many quantities in order to obtain a detailed and at the same time dependable estimate of the spectrum. When the Blackman-Tukey method is used, one needs to calculate the covariance function at many lags; otherwise, the spectral estimate will have low resolution in the frequency domain. And having many lags means poorer reliability. The Bendat and Piersol and Welch methods require splitting the original time series in as many shorter time series as possible and, at the same time, each subseries should be as long as possible. Therefore, even with a long time series, one has to find a compromise solution between the mutually contradicting desires to get a statistically reliable and, at the same time, high-resolution estimate. Obviously, this difficulty would have been less serious if the number of quantities to be estimated was small in comparison with the number of available observations. This improvement becomes possible with the parametric time series analysis.

The parametric approach arises mostly from the works of Yule (1927) and Wold (1938), who developed the concept of parametric models and introduced the general notion of a random process generated by a linear transformation of a white noise sequence-a linearly regular random process. In the autoregressive model, the current value of the process presents a linear combination of a finite number of its past values plus a “disturbance” consisting of the current value of the white noise sequence. Eventually, it gave rise to several types of parametric models, which are studied in detail in the classical book by Box and Jenkins (1970) and in its four subsequent editions. In this book, only the autoregressive models will be used as the means for obtaining parametric estimates of spectral density.

Though the spectral density function contains some information about the time series behavior in the time domain, it is the parametric approach, which allows one to obtain such information explicitly in the form of stochastic difference equations, with the simplest model being the white noise. In accordance with the definition given above, the time domain model of a stationary Markov chain is described with the following stochastic difference equation of order one:
$$x_{t}=\varphi_{1} x_{t-1}+a_{t},$$

统计代写|时间序列分析代写Time-Series Analysis代考|Parametric Spectral Analysis

The parametric approach to time series analysis allows one to obtain information about both time and frequency domain properties. By definition, the time series spectrum is the squared modulus of its Fourier transform: $s(f)=\left|\mathrm{FT}\left(x_{n}\right)\right|^{2}$. This way of spectrum estimation is inacceptable because the estimate is not efficient: its variance does not diminish as the time series length increases; however, it can be applied to the time series model, such as $\operatorname{AR}(p), \operatorname{MA}(q)$, or $\operatorname{ARMA}(p, q)$ given with Eqs. (3.4)-(3.6).

To obtain an equation for the spectral density through the autoregressive model, Eq. (3.4) can be rewritten as
$$\left(1-\varphi_{1} B-\varphi_{2} B^{2}-\cdots-\varphi_{p} B^{p}\right) x_{t}=a_{t}$$
The Fourier transform of this equation is obtained by substituting $\mathrm{e}^{-i 2 \pi j f \Delta t}$ for $B^{j}$ which leads to the following expression for the spectral density of an autoregressive process of order $p$ :
$$s(f)=\frac{2 \sigma_{a}^{2} \Delta t}{\left|1-\sum_{j=1}^{p} \varphi_{j} \mathrm{e}^{-i 2 \pi j f \Delta t}\right|^{2}}, 0 \leq f \leq f_{N}$$

where $i=\sqrt{-1}$. This equation means that, up to a multiplier, the spectral density of time series $x_{t}$ given with an autoregressive model $\operatorname{AR}(p)$ of order $p$ is defined with the autoregressive coefficients $\varphi_{j}, j=1, \ldots p$.

Applying the same technique to Eqs. (3.5) and (3.6) leads to the following expressions for spectral densities of the $\mathrm{MA}(q)$ and mixed $\operatorname{ARMA}(p, q)$ models of time series:
$$s(f)=2 \sigma_{a}^{2} \Delta t\left|1-\sum_{j=1}^{q} \theta_{j} \mathrm{e}^{-i 2 \pi j f \Delta t}\right|^{2}$$
and
$$s(f)=\frac{2 \sigma_{a}^{2} \Delta t\left|1-\sum_{j=1}^{q} \theta_{j} \mathrm{e}^{-i 2 \pi j f \Delta t}\right|^{2}}{\left|1-\sum_{j=1}^{p} \varphi_{j} \mathrm{e}^{-i 2 \pi j f \Delta t}\right|^{2}}$$
within the frequency range from 0 to $f_{N}$. Thus, the shape of the $\operatorname{ARMA}(p, q)$ spectrum is completely defined with $p+q$ parameters.

统计代写|时间序列分析代写Time-Series Analysis代考|Nonparametric Spectral Analysis

• Blackman-Tukey 方法基于协方差函数估计的傅里叶变换加上一些锥度（Blackman 和 Tukey 1958；Bendat 和 Piersol 1966），
• 非重叠段的 Bendat-Piersol 方法（Bendat 和 Piersol 2010），
• Welch 的重叠分段平均方法 (Welch 1967)，以及
• Thomson 的多锥度法（Thomson 1982）。
前三种方法适用于长时间序列，除了本章后面的少数例外，它们很少用于时间序列分析，特别是因为它们中的很多都是短的。根据该方法的作者以及作者的经验，David Thomson 提出的第四种非参数方法可以产生具有短时间序列的无偏和一致的谱估计；它具有高频分辨率，可用于检测周期性和准周期性分量。在他的原始出版物中，该方法的作者给出了一个成功分析一个简短的例子(ñ=100)具有相当复杂的频谱密度的时间序列，对于通信系统中的样本记录而言是典型的。对 Thomson 的多锥体方法 (MTM) 的简要解释是，光谱估计是作为时间序列的几个平方傅里叶变换的平均值获得的，这些变换用锥度进行平滑（所谓的离散长椭球体序列）。本书将使用这种光谱分析方法以及自回归方法。可以在 Babadi 和 Brown (2014) 中找到对 MTM 的评论。

X吨=披1X吨−1+一个吨,

统计代写|时间序列分析代写Time-Series Analysis代考|Parametric Spectral Analysis

(1−披1乙−披2乙2−⋯−披p乙p)X吨=一个吨

s(F)=2σ一个2Δ吨|1−∑j=1p披j和−一世2圆周率jFΔ吨|2,0≤F≤Fñ

s(F)=2σ一个2Δ吨|1−∑j=1qθj和−一世2圆周率jFΔ吨|2

s(F)=2σ一个2Δ吨|1−∑j=1qθj和−一世2圆周率jFΔ吨|2|1−∑j=1p披j和−一世2圆周率jFΔ吨|2

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MATLAB代写

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