### 统计代写|时间序列分析作业代写time series analysis代考|Stochastic Models and Spectra of Climatic and Related Time Series

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析代写方面经验极为丰富，各种代写时间序列分析相关的作业也就用不着说。

• 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|时间序列分析作业代写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 $A R(1)$ model. The prevalence of the 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 $\mathrm{AR}(1)$ 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.

Most of the 13 climate indices in Table $5.1$ do not show any response to the global warming in the form of a trend; the reason for this feature is not clear. In seven cases, the names include the term “oscillation,” that is, regularly repeating deviations from some equilibrity level. Visually, none of those time series contains oscillations. The presence of oscillations can also be seen from the spectral density and/or from the roots of the characteristic equation corresponding to a given AR model: at least, some of the roots must be complex-valued and the damping coefficient should not be too large. For example, the Antarctic Oscillation (or the Antarctic Oscillation Index) is defined as the difference of mean zonal atmospheric pressure at sea level between $40^{\circ} \mathrm{S}$ and $65^{\circ} \mathrm{S}$ and can be regarded as similar to NAO (Gong and Wang 1999). The optimal model for both NAO and AAO time series is a white noise, that is, a sequence of identically distributed and mutually independent random variables; it cannot contain oscillations and hardly carries any useful probabilistic information. The AR orders 0 and 1 exclude the presence of oscillations in respective time series because their spectral densities are frequency independent when $p=0$ or decrease monotonically with growing frequency when $p=1$. Oscillations may exist in time series whose AR orders exceed 1. In this case, there are six such indices: AMO, SOI, NINO3.4, GPCC, MEI, and TPI. (This TPI time series differs from the time series analyzed in Chap. 4.)

## 统计代写|时间序列分析作业代写time series analysis代考|Properties of Time Series of Spatially Averaged Surface Temperature

With the exception of ENSO-related phenomena, the results of analysis of geophysical time series listed in Table $5.1$ do not contradict the hypothesis of the Markovian behavior of climate (Hasselmann 1976). Out of the 13 time series in Table $5.1$, seven have orders not higher than 1 , which can be regarded as a confirmation of the hypothesis. The other six samples have low predictability, which does not differ much from predictability of the remaining seven time series. The predictability of AMO is better than in all other cases, and it may be high enough for practical applications. The AMO time series differs from other time series in the table in the sense that it is obtained by averaging SST over a large area of the North Atlantic; therefore, one can assume that the comparatively high rate of spectral density decrease and the higher predictability criterion $r_{e}(1) \approx 0.62$ for $\mathrm{AMO}$ could be the result of that averaging.

The global climate is better characterized with data obtained by averaging over large parts of the globe. The AMO time series is just a specific example of such averaging, but we have nine time series that show the surface temperature over the entire globe, its hemispheres, and oceanic and terrestrial parts. Those time series have been analyzed in Privalsky and Yushkov (2018) and found to have a more complicated structure and a higher predictability than the other time series studied in that work.

The data used in the above publication include the complete time series given by the University of East Anglia; most of the time series begin in 1850 . The authors of the data files show that the degree of coverage during the XIX Century was poor. Following the example given in Dobrovolski $(2000)$, we will study the same time series starting from 1920 , when the coverage with observations generally increases to $50 \%$ and higher for the global, hemispheric and oceanic data.

The results given in Table $5.2$ confirm one of the previous conclusions: the annual surface temperature averaged over large parts of the globe is best described with relatively complicated models having AR orders $p=3$ or $p=4$ and a relatively high statistical predictability. The results for the southern hemisphere as a whole and for its land follow a Markov model and have lower statistical predictability; they agree with our results obtained from the data given by the Goddard Institute of Space Studies (GISS). According to the GISS data for the southern hemisphere (#14 in Appendix), the autoregressive order $p=1$ and the criterion $r_{e}(1) \approx 0.55$.

The data sets show that spatial averaging on the global scale and over the northern hemisphere including its oceans and land produces time series whose properties

differ quite significantly from what is shown in Table $5.1$ for individual climate indices. The optimal AR orders increase up to four, and the predictability criterion grows up to $0.82$ for the north hemispheric ocean. The reason for the behavior of temperature over the southern hemisphere for the time series which begin in 1920 is not clear, but it may be related to the change is statistical properties of the trivariate system consisting of the time series of global, land, and terrestrial time series. For example, the predictability criterion $r_{e}(1)$ for the entire time series is $0.74$ (Privalsky and Yushkov 2018) and $0.44$ for the time series that begins in 1920. A more detailed description of the change is given in Chap. $14 .$

The predictability criterion used by Hasselmann (1976) in his Eq. (6.3) can be given as
$$s^{2}=\delta^{2} /\left(\varepsilon^{2}+\delta^{2}\right)$$
where $\delta^{2}$ and $\varepsilon^{2}$ are the variances of the predictable and unpredictable parts of the time series (signal and noise, according to $\mathrm{K}$. Hasselmann). In our notations, $\varepsilon^{2}$ coincides with $\sigma_{a}^{2}$ and $\delta^{2}=\sigma_{x}^{2}-\sigma_{a}^{2}$. Therefore, $s^{2}=r_{e}^{2}(1)$. In Hasselmann’s opinion, the statistical predictability criterion for stationary climate systems generally does not exceed $0.5$ and actually is always much less than unity. The results in Table $5.1$ agree with that statement but spatial averaging seems to lead to more complicated models and to better statistical predictability (Table 5.2). All these time series successfully pass the test for Gaussianity.

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

The “rule of no significant sharp peaks” in climate spectra has at least one exception which is supported with decades of direct observations. At least one atmospheric process-the Quasi-Biennial Oscillation, or QBO-does not follow this rule. The QBO phenomenon exists in the equatorial stratosphere at altitudes from about 16 $\mathrm{km}$ to $50 \mathrm{~km}$, and it is characterized with quasi-periodic variations of the westerly and easterly wind speed. The period of oscillations is about 28 months, which corresponds to the frequency of about $0.43$ cpy. It has been discovered in the 1950 ‘s and investigated in a number of publications, in particular, in Holton and Lindzen (1972) who proposed a physical model for QBO. In the review of QBO research by Baldwin et al (2001), QBO is called “a fascinating example of a coherent, oscillating mean flow that is driven with propagating waves with periods unrelated to the resulting oscillation.” Some effects of QBO upon climate are discussed by Anstey and Shepherd (2014).

The statistical properties of QBO such as its spectra and statistical predictability do not seem to have been analyzed within the framework of theory of random processes; this section (along with Chaps. 6 and 10$)$ is supposed to fill this gap in the part related to $\mathrm{QBO}$ as a scalar and bivariate (Chap. 10) phenomenon. It will be analyzed here using the set of monthly observational data provided by the Institute of Meteorology of the Free University of Berlin for the time interval from 1953 through December 2018 (see #15 in Appendix and Naujokat 1986). The set includes monthly wind speed data in the equatorial stratosphere at seven atmospheric pressure levels, from 10 to $70 \mathrm{hPa}$; these levels correspond to altitudes from $31 \mathrm{~km}$ to $18 \mathrm{~km}$.

If the goal of the study were to analyze $\mathrm{QBO}$ as a scalar random process, the data could have been taken at the sampling interval $\Delta t=6$ months or even 1 year. As QBO’s statistical predictability at a monthly sampling rate will also be studied in Chap. 6, the sampling interval $\Delta t=1$ month is taken in this section as well. Examples of $\mathrm{QBO}$ variations are shown in Fig. 5.3.

The basic statistical characteristics of $\mathrm{QBO}$ are shown in Table 5.3. The average wind speed is easterly (negative), and it decreases below the $20 \mathrm{hPa}$ level turning eastward at the lowest level. The variance increases from the $10 \mathrm{hPa}$ level by about $10 \%$ to $15 \mathrm{hPa}$ and $20 \mathrm{hPa}$ and then gradually decreases downward by an order of magnitude. These facts are well known (e.g., Baldwin et al. 2001). The optimal AR models have orders from $p=11$ to $p=29$; such orders are too high for individual time domain analysis.

The typical shape of the spectrum shows an almost periodic random function of time at $f \approx 0.43$ cpy (Fig. $5.4 \mathrm{a}$ ). The maximum is very narrow and completely dominates the spectrum so that a more detailed picture can only be seen when the scale is logarithmic along both axes (Fig. 5.4b). This seems to be an absolutely

unique phenomenon at climatic time scales. At higher frequencies, the spectral density diminishes rather quickly with all other peaks being statistically insignificant. Having this in mind, the spectra will be shown in what follows at frequencies not exceeding 1 cpy.

The AR spectral estimates of $\mathrm{QBO}$ at all other levels are given in Fig. 5.5. In all seven cases (including the $10 \mathrm{~Pa}$ level in Fig. 5.4), the frequency of the major spectral peak is found at $f=0.432$ cpy. A tenfold increase in the frequency resolution from $0.012$ cpy to $0.0012$ cpy showed that the period of oscillation determined through the frequency of the maximum spectral density in all seven estimates varies by less than $2 \%$ and stays between $27.7$ and $28.1$ months. The average over the seven spectral estimates frequency of the spectral peak corresponds to the average period equal to $27.99$ month. This very clear-cut stability can be regarded as another distinctive property of the QBO phenomenon.

According to the model suggested by Holton and Lindzen (1972), the QuasiBiennial Oscillation is forced by vertically propagating planetary waves with periods of $5-15$ days. This explanation means that at altitudes where $\mathrm{QBO}$ occurs the equatorial stratosphere works as a filter that transforms the upward propagation of this high-frequency noise into a downward moving oscillation whose frequency is up to two orders of magnitude lower that the frequency of the forcing.

As seen from these descriptions and from the figures, the QBO system has a complicated structure and it should be studied as a multivariate random process. This will be done in Chap. 10 .

## 统计代写|时间序列分析作业代写time series analysis代考|Properties of Time Series of Spatially Averaged Surface Temperature

Hasselmann (1976) 在他的方程式中使用的可预测性标准。(6.3) 可以表示为
s2=d2/(e2+d2)

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

QBO 的统计特性，如光谱和统计可预测性，似乎没有在随机过程理论的框架内进行分析；本节（连同第 6 章和第 10 章）)应该在与问乙这作为标量和双变量（第 10 章）现象。此处将使用柏林自由大学气象研究所提供的 1953 年至 2018 年 12 月期间的月度观测数据集进行分析（见附录中的 #15 和 Naujokat 1986）。该集合包括赤道平流层在七个大气压水平下的每月风速数据，从 10 到70H磷一种; 这些级别对应于从31 ķ米到18 ķ米.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。