### 统计代写|时间序列分析作业代写time series analysis代考|Frequency Resolution of Autoregressive Spectral

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## 统计代写|时间序列分析作业代写time series analysis代考|Frequency Resolution of Autoregressive Spectral

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.

The frequencies of the main tidal constituents were determined by conducting autoregressive spectral analysis of the entire time series of length $10^{5} \mathrm{~h}$, the AR order $p=10^{4}$, and the frequency resolution of the spectral estimate $10^{-6} \mathrm{cph}$. The results for the diurnal tides are shown in Fig. $4.8$ and in Table 4.2.

As seen from the table, the errors in estimates of the constituents’ periods do not exceed $0.022 \%$. The average error for the entire range of diurnal tides is $0.0077 \%$. This is not a misprint; it is a proof that the autoregressive spectral analysis does allow one to obtain very accurate estimates of periods of tidal constituents and, consequently, of any other periodic or quasi-periodic component. This statement is correct as long as there is a sufficient amount of reliable data. Good estimates can be obtained with shorter time series, for example, one year of hourly data, and the resolution will still be high but not as high as when the time series contains $10^{5}$ hourly observations.

## 统计代写|时间序列分析作业代写time series analysis代考|Example of AR Analysis in Time and Frequency

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 $\operatorname{AR}(p)$ 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 $\operatorname{AR}(3)$ 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.

The characteristic equation of the $\operatorname{AR}(3)$ model of TPI given with Eq. (4.11) is
$$1-0.46 B+0.29 B^{2}-0.15 B^{3}=0$$
and it has a pair of complex-conjugated roots $[(-0.05+1.81 i),(-0.05-1.81 i)]$ where $i=\sqrt{-1}$. The roots correspond to the natural frequency $f_{e} \approx 0.25 \mathrm{cpy}$. An estimate of the spectrum such as shown in Fig. 4.10a and/or b must be included into analysis of any time series. In this case, the spectrum diminishes with frequency almost monotonically and contains a shelf at frequencies close to $0.25$ cpy.

The above described steps are normally required for analysis of any stationary time series.

## 统计代写|时间序列分析作业代写time series analysis代考|Research activities

Abstract Research activities at all stages of analysis constitute preliminary steps for the most important task-time series forecasting. One of such stages includes efforts to understand statistical properties of the processes that are being studied including probability density functions, spectral densities, and the degree of statistical predictability. Climate is often regarded as a Markov process with a small parameter, which means a slowly and monotonically decreasing spectral density without any oscillations and/or quasi-periodic phenomena. Many climatic time series and indices including $\mathrm{AO}$ and $\mathrm{AAO}$, NAO, PDO, AMO, and PNA behave in agreement with that Markov model or even with white noise. The climate indices related to ENSO behave in a different manner: their spectra are nonmonotonic and contain a smooth maximum at about $0.2$ cpy. Yet, none of them contains regular oscillations and their predictability stays low. The annual surface temperature for 1920-2018 averaged over large parts of the globe generally does not follow the Markov model, and its predictability is relatively high. Some other oscillatory processes are studied as well, including a version of $\mathrm{AAO}$ and $\mathrm{MJO}$ – a bivariate random process whose scalar components are shown to possess some statistical predictability.

The final and most important stage of analysis of time series generated by stationary random processes is forecasting. It consists of two parts: determining the achievable quality of forecasting, that is, measuring statistical predictability, and performing the forecast, which means constructing a forecast formula and calculating the future trajectory of the time series with respective confidence intervals.

In order to understand the results of prediction one needs to know what features of the time series have led to its specific forecasts and forecast error variances. This information is implicitly contained in the most essential statistical moment of any stationary time series: its spectral density. If the prediction error variance at the unit lead time coincides with the time series variance, the spectral density will be independent of frequency – a white noise. This random process is unpredictable. If the spectrum is concentrated at low frequencies, the predictability improvement occurs due to the ability to forecast long-term variations of the time series at relatively short lead times. If the spectrum contains a peak whose area composes a significant

part of the total area under the spectral density curve, the better predictability occurs due to the presence of a quasi-periodic or cyclic, component.

Thus, time series analysis must provide a time domain model of the time series and an estimate of the spectral density corresponding to it. This chapter contains results of analysis of time series obtained from observations, mostly at climatic time scales. Specifically, the tasks here are to describe typical time domain models of different types of climatic time series and to characterize the behavior of their spectral densities.

Essentially, Sects. $5.1$ and $5.2$ can be regarded as an attempt to sum up information about the typical behavior of climate as a stationary random process starting from its first stochastic model suggested by Hasselmann (1976) in the form of a Markov process. The early estimates of climate spectra used in particular, to verify the model, include publications by Privalsky $(1976,1977)$ and by Frankingnoul and Hasselmann (1977). The other goal is to see if the concept of low statistical predictability of respective climate models agrees with observation data. The task of time series forecasting will be discussed and illustrated with examples in Chap. 6 .

## 统计代写|时间序列分析作业代写time series analysis代考|Frequency Resolution of Autoregressive Spectral

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：光谱曲线中极值点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|时间序列分析作业代写time series analysis代考|Example of AR Analysis in Time and Frequency

X吨≈0.46X吨−1−0.29X吨−2+0.15X吨−3+一种吨

−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获得的和⁡(3)模型（4.11）。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.

1−0.46乙+0.29乙2−0.15乙3=0

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