### 统计代写|时间序列分析作业代写time series analysis代考|Statistical Forecasting of Geophysical Processes

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|时间序列分析作业代写time series analysis代考|Example 1. Global Annual Temperature

According to Table $5.2$, the higher predictability occurs for the annual surface temperature averaged over very large areas, up to the entire surface of the planet. This happens because respective time series contain most of their energy within the low-frequency part of the spectrum.

The global annual temperature (notated here as GLOBE) from 1850 (#1 in Appendix below and Fig. 6.la) shows two intervals with a definite positive trend; the trend is longer and slightly faster during the latest several decades. Similar to the earlier interval from 1911 through 1944 , the trend that happened during the years from 1974 through 2010 (the initial year for our extrapolation test below) may have been caused by natural factors (Privalsky and Fortus 2011 ) so that its higher predictability could have been the result of regular variations of climate. As for the higher frequencies, the spectral density estimate for the detrended time series (Fig. 6.1b) proves

that detrending the time series does not affect the spectrum at frequencies above $0.02$ cpy (at time scales 50 years and shorter).

The goal of this test is to get an idea of extrapolation efficiency for the original and detrended time series. With year 2010 as the initial time for extrapolation, one has eight observed values of temperature anomalies that can be compared with predictions for 2011-2018.

The entire time series of GLOBE from 1850 through 2018 can be regarded as a sample of a stationary random process and extrapolated in accordance with its best fitting AR model. The second approach regards the time series as nonstationary: the sum of a stationary process plus trend (linear, in our case). The first version means that the trend is a part of the low-frequency variations caused by the natural climate variability; in the second version, the climate variability is regarded as stationary while the trend is caused by some external factors, including possible anthropogenic effects.

In both versions, the time series with the trend present or deleted behave as stationary autoregressive sequences of order $p=4$. Judging by the shape of its spectral density (Fig. 6.1b), the original time series may be close to being nonstationary due to the dominant role of low-frequency variations but the roots of its characteristic equation significantly exceed 1 so that it should be treated as stationary. It also passes the test for stationarity recommended in Chap. 4 though the number of independent observations in the entire time series with the trend present and in its halves are small.
The predictability properties of the time series for both cases are shown in Fig. 6.2. In the first (stationary) case, the RPC increases to $0.6$ at the lead time 5 years; that is, the time series with the trend regarded as a nature caused phenomenon has a high statistical predictability (Fig. 6.2a). In the second case (stationarity plus trend), the predictability limit $\tau_{0.7}$ is just one year (Fig. $6.2 \mathrm{~b}$ ).

The results of the GLOBE extrapolation test in accordance with the two versions of the time series are given in Fig. 6.3. The values of GLOBE observed from 2011 through 2018 and lying outside the $90 \%$ confidence limits (the dashed lines) mean unsatisfactory extrapolation. In the purely stationary case when the trend is not deleted (Fig. 6.3a), there are three or even two such failures. In the case of a sum of stationary time series with a linear trend, the extrapolation is less successful: five values of GLOBE observed after 2010 lie outside the confidence interval. Seemingly, the first approach produces better results. The transition from the stationary model to a mixture of a stationary model with trend results in two important changes: the extrapolation function tends to the mean value of the time series faster than in the stationary case and the time series variance diminishes from $0.070\left({ }^{\circ} \mathrm{C}\right)^{2}$ to $0.024$ $\left({ }^{\circ} \mathrm{C}\right)^{2}$.

## 统计代写|时间序列分析作业代写time series analysis代考|Example 2. Quasi-Biennial Oscillation

The Quasi-Biennial Oscillation will be discussed here at $\Delta t=1$ month. The time series used for this example is $\mathrm{QBO}$ at the atmospheric pressure level $20 \mathrm{hPa}$, which corresponds to the altitude of about $26 \mathrm{~km}$ above mean sea level (Fig. $6.5 \mathrm{a}$ and $# 2$ in Appendix). The spectral density estimate is shown in Fig. $6.5 \mathrm{~b}$ with the frequency axis given in a linear scale.

At the time when this text was being written, monthly observations of QBO were available from January 1953 through April of 2019 . The test extrapolation for the entire 2018 and the next six months of 2019 , from May through November, which have been added in December 2019, is based upon the part of the time series that ends in December $2017(N=780)$.

The optimal, according to three of the five order selection criteria used here, is the AR model of order $p=10$ :
$$x_{t}=\varphi_{1} x_{t-1}+\varphi_{2} x_{t-2}+\cdots+\varphi_{10} x_{t-10}+a_{t} .$$
It means that the extrapolation equation is
$$\hat{x}{t}(\tau)=\varphi{1} \hat{x}{t}(\tau-1)+\varphi{2} \hat{x}{t}(\tau-2)+\cdots+\varphi{10} \hat{x}{t}(\tau-10) .$$ The white noise variance corresponding to the $\operatorname{AR}(10)$ model is $\sigma{a}^{2} \approx 21(\mathrm{~m} / \mathrm{s})^{2}$ while the total variance of wind speed at $20 \mathrm{hPa}$ is $\sigma_{x}^{2} \approx 389(\mathrm{~m} / \mathrm{s})^{2}$. Therefore, the predictability criterion $\rho(1) \approx 0.05$ and the correlation coefficient ( $6.13$ ) between the unknown true and predicted values of wind speed at lead time $\tau=1$ month is $0.97$. As seen from Fig. 6.6, the statistical predictability of $\mathrm{QBO}$ at the $20 \mathrm{hPa}$ level described with the predictability criteria $r_{e}(\tau)$ and $\rho(\tau)$ is quite high.

The results of prediction test with the initial time in December 2017 (Fig. 6.7a) show that the AR method of extrapolation is working quite well with this time series: 19 of the 20 monthly forecasts stay within the $90 \%$ confidence limits. More predictions are given from December 2018 through January 2021 for future verification (Fig. 6.7b). The data used for the AR models were from January 1953 through December 2017 and through December 2018, respectively. By the time when the book was ready for the publisher, more observations became available and they are included into Fig. 6.7b. The quality of extrapolation seems to be high, but one should have in mind that the $90 \%$ confidence intervals shown in the figure are wide.

## 统计代写|时间序列分析作业代写time series analysis代考|Example 3. ENSO Components

Predicting the behavior of the oceanic ENSO component-sea surface temperature in equatorial Pacific-is regarded as a very important task in climatology and oceanography (e.g., #3 and #4 in Appendix). Attempts to predict ENSO’s atmospheric component-the Southern Oscillation Index-do not seem to be numerous (e.g., Kepenne and Ghil 1992). In this section, both tasks will be treated within the KWT framework.

At the annual sampling rate, the ENSO components behave similar to white noise (Chap. 5); their predictions through any probabilistic method would be practically useless. In the current example, the statistical forecasts of sea surface temperature in the ENSO area NINO3 $\left(5^{\circ} \mathrm{N}-5^{\circ} \mathrm{S}, 150^{\circ} \mathrm{W}-90^{\circ} \mathrm{W}\right)$ and the Southern Oscillation Index are executed at a monthly sampling rate using the data from January 1854 through February 2019 and from 1876 through February 2019 , respectively. The data are available at websites #5 and #6 given in Appendix below. The NINO3 time series is shown in Fig. 6.8a. It can be treated as a sample of a stationary process.
The autoregressive analysis of this time series showed an AR(5) model as optimal. Its spectral density estimates are shown in Fig. $6.8 \mathrm{~b}$. The low-frequency part of the spectrum up to $0.5$ cpy contains about $70 \%$ of the NINO3 variance and the ratio of the white noise RMS to the NINO3 RMS is $0.39$. In contrast to the annual global

temperature with the trend present, the predictability of NINO3 diminishes quite fast, but, as seen from Fig. 6.9, it still extends to several months.

A KWT prediction from the end of 2017 through January 2019 is given in Fig. $6.10 \mathrm{a}$. The result of the test turned out to be satisfactory but one has to remember that the $90 \%$ confidence limits for the extrapolated values are rather wide. Only the first four or five predicted values lie within the relatively narrow interval not exceeding $\pm \sigma_{x}$.

## 统计代写|时间序列分析作业代写time series analysis代考|Example 1. Global Annual Temperature

GLOBE 从 1850 年到 2018 年的整个时间序列可以看作是一个平稳随机过程的样本，并根据其最佳拟合 AR 模型进行外推。第二种方法将时间序列视为非平稳的：平稳过程加上趋势的总和（在我们的例子中是线性的）。第一个版本意味着趋势是自然气候变率引起的低频变化的一部分；在第二个版本中，气候变率被认为是静止的，而趋势是由一些外部因素引起的，包括可能的人为影响。

## 统计代写|时间序列分析作业代写time series analysis代考|Example 2. Quasi-Biennial Oscillation

X吨=披1X吨−1+披2X吨−2+⋯+披10X吨−10+一种吨.

X^吨(τ)=披1X^吨(τ−1)+披2X^吨(τ−2)+⋯+披10X^吨(τ−10).白噪声方差对应于和⁡(10)模型是σ一种2≈21( 米/s)2而风速的总方差为20H磷一种是σX2≈389( 米/s)2. 因此，可预测性准则ρ(1)≈0.05和相关系数（6.13) 在提前期风速的未知真实值和预测值之间τ=1月份是0.97. 从图 6.6 可以看出，问乙这在20H磷一种用可预测性标准描述的水平r和(τ)和ρ(τ)相当高。

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