### 统计代写|时间序列分析作业代写time series analysis代考| Determining the Order of Autoregressive Models

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## 统计代写|时间序列分析作业代写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 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.

Several methods can be used to determine the optimal order of an autoregressive model that is being fitted to a time series, but the best approach is to use the order selection criteria (OSC) developed in information theory. Such criteria recommend an optimal order by finding a compromise solution for the following dilemma: a higher order may reveal more details in the spectral density estimate, but at the same time a higher order means that the estimates of AR coefficients are less reliable. The order recommended by order selection criteria is optimal in the sense that it minimizes the variance of innovation sequence-the absolutely unpredictable component of the time series and takes into account the loss of reliability of estimates with the growing AR order. Several such OSCs are known and considered reliable for determining the AR order.

## 统计代写|时间序列分析作业代写time series analysis代考|Comparison of Autoregressive and Nonparametric

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 $\operatorname{AR}(1)$ 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. $3.3 \mathrm{a}$, 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.
More complicated AR models are shown in Figs. 3.4, 3.5, and 3.6. In the first case, the true model is $\operatorname{AR}(2)$ with AR coefficients $\varphi_{1}=0.5$ and $\varphi_{2}=-0.4$. The true spectrum is close to the spectrum of the Southern Oscillation Index. The characteristic equation of this model is

cases, the natural frequency of the system is approximately $0.23 \Delta t^{-1}$. The estimate for the AR(3) time series, which simulates the Palmer Draught Severity Index over the contiguous USA, stays close to the true spectral density for the AR(3) time series (Fig. 3.5a) and becomes less accurate for the $\mathrm{AR}(4)$ time series (Fig. 3.6a). However, the shape of the spectrum is reproduced rather accurately including the hump at intermediate frequencies. This simulated time series is similar to the actual annual global surface temperature (Privalsky and Yushkov 2018). In all cases, the nonparametric estimates given in Figs. $3.5 \mathrm{~b}$ and $3.6 \mathrm{~b}$ are less accurate, but one has to remember that the time series are very short for the nonparametric spectral estimation to be efficient. The incorrect behavior of the spectral estimates at low frequencies in Fig. $3.6$ is the results of a poor estimate of the time series variance due to its short length.

These examples are given here to illustrate the following argument: when the optimal autoregressive model of a geophysical or any other time series indicated by order selection criteria is low, the autoregressive approach allows one to obtain statistically reliable estimates of AR coefficients and, consequently, of the spectral density even when the time series is very short. Also, a low AR order does not necessarily mean that the spectrum of the time series does not contain any sharp peaks.

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|时间序列分析作业代写time series analysis代考|Comparison of Autoregressive and Nonparametric

• 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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