### 统计代写|时间序列分析代写Time-Series Analysis代考|STAT4102

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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|Frequency Resolution of Autoregressive Spectral Analysis

The AR (or MEM) spectral estimation provides an analytical formula for the estimated spectrum. It means that the spectral resolution in the formula is such that the value of spectral density can be calculated at any frequency. This is true, but the actual resolution is defined by the AR order: the number of extrema and inflection points in the spectral curve corresponding to an $\operatorname{AR}(p)$ model cannot be higher than $p$ (see Sect. 4.3). Therefore, a high resolution requires a high AR order, but a high-order model cannot be obtained with a short time series.

By definition, a linearly regular random process does not contain any strictly periodic components. This feature may cause some doubts about the ability of parametric time series analysis designed for regular processes to detect sharp peaks at frequencies which are close to each other, for example, when the data contain harmonic oscillations. Actually, the ability of autoregressive spectral analysis in this respect is very high under just one condition: getting accurate results requires having enough data for analysis. (Certainly, this requirement holds for all nonparametric method of spectral analysis such as Blackman and Tukey’s, MTM, Welch’s, etc.)

A unique case of harmonic oscillations with perfectly known frequencies within the Earth system is tides. The frequencies of tidal constituents are known precisely from astronomy; the amplitudes are determined from observations. The autoregressive analysis in the frequency domain provides a convenient tool for estimating frequencies of harmonic oscillations that are contained in time series of tidal phenomena. If the frequencies are determined correctly in sea level observations, one may hope that they will also be determined correctly in any other stationary data.
The example below is designed to verify how accurately the maximum entropy spectral analysis can determine the frequencies of tidal constituents by analyzing the time series of sea level at station 9414317 , Pier $221 / 2$, San Francisco, USA, using $10^{5}$ hourly sea level observations starting from January 28,2000 . The data source is $# 3$ in Appendix. A part of the record (about 50 days) is shown in Fig. 4.7. The tides obviously dominate the record.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Example of AR Analysis in Time and Frequency Domains

Consider the entire process of time series analysis using as an example the annual values of Tripole Index (TPI) for the Interdecadal Pacific Oscillation (Henley et al. 2015). The time series shown in Fig. $4.9$ extends from 1854 through $2018(N=165)$; it is closely related to other El Niño-Southern Oscillation indices but differs from them in some respects. The data source is taken from the Web site #5 in Appendix to this chapter. The time series does not contain any statistically significant trend, and its behavior allows one to assume, without any further analysis, that it can be treated as a sample of a stationary random process. The test for Gaussianity showed that the probability density function of this time series can be regarded as normal.

The time series has been analyzed in the time domain by fitting to it $\mathrm{AR}(p)$ models of orders from $p=0$ through $p=16$ (one-tenth of the time series length). Three of the five order selection criteria used in this book have chosen the order $p=3$ :
$$x_{t} \approx 0.46 x_{t-1}-0.29 x_{t-2}+0.15 x_{t-3}+a_{t}$$
The RMS error of all estimated AR coefficients equals to approximately $0.08$ so that the coefficients are statistically significant at the confidence level $0.9$ used in this book.

The estimates of the mean value and standard deviation are $\bar{x} \approx-0.15$ and $\hat{\sigma}_{x} \approx 0.61$. The respective confidence intervals for the mean value and variance estimates obtained for the TPI time series expressed with model (4.11) are [ $-0.25$,

$-0.04]$ and $[0.55,0.68]$. These confidence intervals are determined in accordance with Eqs. (4.1) -(4.4) using estimates of the numbers of independent observations $\bar{N}=93$ and $\hat{N}=130$ obtained for the AR(3) model (4.11). These values are calculated through the correlation function estimate under the assumption that the correlation function $r(k)$ at lags $k=1,2,3$ coincides with the sample estimates while its further values behave in the maximum entropy mode. This correlation function obtained according to Eq. (4.5) diminishes very fast so that the numbers of independent observations $\bar{N}$ and $\hat{N}$ do not differ drastically from the total number of observations $N$.

The innovation sequence variance $\sigma_{a}^{2} \approx 0.31$ and the predictability (persistence) criterion $r_{e}(1)=\sqrt{1-\sigma_{a}^{2} / \sigma_{x}^{2}}$ equals $0.17$ meaning that the unpredictable innovation sequence $a_{t}$ plays a dominant role in the time series of Tripole Index. This time series is quite close to a white noise sequence, and the variance of its prediction errors will be high.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Properties of Climate Indices

At climatic time scales, the basic statistical properties of a large number of geophysical time series-about 3000 -has been summarized in the fundamental work by Dobrovolski $(2000)$ dealing with stochastic models of scalar climatic data. The time series in that book include surface temperature, atmospheric pressure, precipitation, sea level, and some other geophysical variables observed at individual stations; the data set includes 195 time series of sea surface temperature averaged within $5^{\circ} \times 5^{\circ}$ squares. Most of those time series are best approximated with either a white noise or a Markov process (Dobrovolski 2000 , p. 135). The white noise model AR(0) can be justly regarded as a specific case of the $\mathrm{AR}(1)$ model. The prevalence of the $\mathrm{AR}(1)$ model for climatic time series obtained without large-scale spatial averaging has been noted recently in Privalsky and Yushkov (2018), but the results given in Dobrovolski $(2000)$ are based upon a much larger observation base.

In this section, we will complement the available information by studying first a number of geophysical time series that are often used as climate indicators or indices; their names usually contain the term “oscillation” or “index.” The list is given in Table 5.1, and the data sources are shown in the Appendix to this chapter. In all cases, the value of $\Delta t$ is one year. Along with the optimal AR orders $p$ for the time series, the table contains the values of statistical predictability criterion (3.7): $r_{e}(1)=\sqrt{1-\sigma_{a}^{2} / \sigma_{x}^{2}}$, where $\sigma_{x}^{2}$ and $\sigma_{a}^{2}$ are the time series variance and the variance of its innovation sequence.

The sources of data listed in the table are given in Appendix to this chapter: the numbers in the first column of the table coincide with the numbers in the Appendix.
Two characteristic features are common for the time series in Table 5.1: all of them can be regarded as Gaussian, and, with one exception, all of them have low statistical predictability. This means that they present sample records of random processes similar to a white noise; that is, their behavior in the time domain is very irregular, and, consequently, none of them contains oscillations as the term is understood in physics. The exception is the relatively high predictability of the Atlantic Multidecadal Oscillation. Thus, judging by the low optimal AR orders and the low predictability, one may say that though the optimal model for most of these time series is not AR(1) their behavior does not contradict the assumption of the Markov character of climate variability and that the value of the autoregressive coefficient is significantly smaller than one.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Frequency Resolution of Autoregressive Spectral Analysis

AR（或 MEM）谱估计为估计的谱提供了一个分析公式。这意味着公式中的光谱分辨率使得光谱密度的值可以在任何频率下计算。这是真的，但实际分辨率由 AR 阶定义：光谱曲线中极值和拐点的数量对应于和⁡(p)型号不能高于p（见第 4.3 节）。因此，高分辨率需要高AR阶数，而时间序列短却无法得到高阶模型。

## 统计代写|时间序列分析代写Time-Series Analysis代考|Example of AR Analysis in Time and Frequency Domains

X吨≈0.46X吨−1−0.29X吨−2+0.15X吨−3+一个吨

−0.04]和[0.55,0.68]. 这些置信区间是根据方程式确定的。(4.1) -(4.4) 使用独立观察次数的估计ñ¯=93和ñ^=130为 AR(3) 模型 (4.11) 获得。这些值是在相关函数的假设下通过相关函数估计计算得出的r(ķ)在滞后ķ=1,2,3与样本估计值一致，而其进一步的值表现为最大熵模式。该相关函数根据方程式获得。(4.5) 减少得非常快，因此独立观察的数量ñ¯和ñ^与观察总数没有太大差异ñ.

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