### 统计代写|时间序列分析代写Time-Series Analysis代考|ECOM30004

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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|Covariance and Correlation Functions

In contrast to random variables, the random functions (time series) are timedependent and, therefore, their properties in the time domain need to be described with statistical moments that reflect the dependence of the time series upon time. Such a mixed statistical moment obtained by averaging products of time series values

separated from each other by $k$ time intervals is the covariance function $R(k)$ :
$$R(k)=\lim {N \rightarrow \infty} \frac{1}{(N-k)} \sum{t=1}^{N-k}\left(x_{t}-\bar{x}\right)\left(x_{t+k}-\bar{x}\right), k=1,2, \ldots, K$$
Divided by the time series variance $\sigma_{x}^{2}$, it produces the correlation function $r(k)$ that presents a sequence of correlation coefficients between values of the time series $x_{t}$ and $x_{t+k}$ separated by $k$ time intervals:
$$r(k)=R(k) / \sigma_{x}^{2} .$$
The covariance and correlation functions are even functions of argument $k$ so that $R(k)=R(-k)$ and $r(k)=r(-k)$. The covariance function’s dimension is the square of the time series’ dimension; if the time series is measured in Kelvin (K), the covariance function’s dimension is $\mathrm{K}^{2}$. The correlation function is dimensionless.
In what follows, it will be assumed that the mean values of all time series in this book are equal to zero $(\bar{x}=0)$. The exceptions to this rule include Eqs. (4.1) and (4.2), the test for stationarity in Chap. 4 , and examples of extrapolation in Chap. $6 .$

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

The most essential statistical characteristic of a scalar time series is its spectral density $s(f)$; it defines how the energy of time series variations changes over the frequency $f$ (the number of cycles per unit time), that is, how it varies in the frequency domain. There are different ways to define the spectral density (also called the spectrum), and one of them is to present the spectral density of a stationary random process as a Fourier transform of the covariance function:
$$s(f)=\sum_{k=-\infty}^{\infty} R(k) \mathrm{e}^{-i 2 \pi k f \Delta t}$$
where $i=\sqrt{-1}$ and $f$ is the cyclic frequency defined from $f=0$ through the Nyquist frequency $f_{N}=1 / 2 \Delta t$. If $\Delta t=1$ year, the frequency is measured in cycles per year (cpy) or year ${ }^{-1}$.

The covariance function can be presented as the inverse Fourier transform of the spectral density:
$$R(k)=\int_{-f_{N}}^{f_{N}} s(f) \mathrm{e}^{i 2 \pi k f \Delta t} \mathrm{~d} f$$

According to Eq. $(2.11)$, the spectral density dimension is the square of the time series dimension divided by frequency. Thus, if $x_{t}$ is measured in millimeters and frequency in cycles per year, the dimension of the spectral density is $\mathrm{mm}^{2} / \mathrm{cpy}$, or $\mathrm{mm}^{2} \times$ year. Equations $(2.10)$ and $(2.11)$ constitute the Wiener-Khinchin theorem for discrete random processes. The argument $k=0, \pm 1, \ldots$ of the covariance function is discrete, while the spectral density is a continuous function of frequency $f$. The covariance and correlation functions as well as the spectral density do not exist for sets of random variables (random vectors) because random variables do not depend upon time and, consequently, upon frequency.

An important special case of random processes is the white noise: a sequence $a_{t}$ of identically distributed and mutually independent random variables. The white noise concept allows one to introduce the class of linearly regular, or regular, random process, which is defined as the process at the output of a linear system (linear filter) with a white noise at the input (the Wold decomposition):
$$x_{t}=\sum_{j=0}^{\infty} \psi_{j} a_{t-j}$$
where $\psi_{j}$ are filter’s coefficients. Normally, it is assumed that $\psi_{0}=1$. Thus, a regular random process presents a linear transformation of a white noise. The upper limit in the sum in Eq. (2.12) can be finite while the lower limit is zero because if it is less than zero the process will be unrealizable physically. If the coefficients $\psi_{j}$ do not change with time and if $\sum_{j=0}^{\infty}\left|\psi_{j}^{2}\right|<\infty$, the output $x_{t}$ of the filter belongs to a stationary random process. The quantity $a_{t}$ is also called the innovation sequence. The mean value $\bar{a}$ of the white noise in Eq. $(2.12$ ) is always zero, so that the mean value of the process $x_{t}$ is zero as well. All time series in this book, with the exception of tides, belong to regular stationary processes.

The spectrum $s(f)$ of a regular random process is an absolutely continuous function of frequency $f$, which means, in particular, that a regular random process cannot be presented as a finite or countable infinite set of periodic functions. In other words, a regular random process does not contain any harmonics. If one assumes the presence of harmonic oscillations in the process, the process loses the property of linear regularity, which may have negative consequences for its analysis and forecasting. However, the process stays regular if its spectrum contains sharp peaks that take an arbitrarily narrow but finite interval of frequencies. An example will be given in Chap. $4 .$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Examples of Geophysical Time Series and Their Statistics

Consider several characteristic examples of climatic or other geophysical processes represented here with short ( $N=100)$ simulated dimensionless time series.

The random processes of a white noise type (Fig. 2.3) are quite common and may include atmospheric pressure, precipitation, and climate indices, for example, the North Atlantic Oscillation. As seen from the figure, the estimates obtained from sample records of white noise do not coincide with the true values of the correlation function and spectral density. This happens due to the sampling variability phenomenon, which occurs whenever the length of the time series that is being analyzed is finite, that is, always. The confidence intervals for the estimated functions in this and other figures in this chapter are not shown intentionally to illustrate the phenomenon of sampling variability.

Another process often encountered in climate and geophysical data in general is the Markov chain, that is, a discrete random process whose future state is independent of its past under the condition that the state of the process at the current time is known. The correlation function of a Gaussian Markov process is defined by its first value $r_{1} \equiv r(1):$
$$r(k)=r_{1}^{|k|}$$

The correlation function and spectrum of a Markov process with a positive coefficient $r_{1}$ decrease monotonically as shown in Fig. 2.4. This Markov model is common for many climatic time series. Typical examples are the annual river streamflow and sea level variations. In $1976, \mathrm{~K}$. Hasselmann suggested that the behavior of the climate system can be described with a Markov process (also see Dobrovolski 2000).

A Markov model is often used in climatology to determine statistical reliability of peaks in spectral estimates calculated from time series of geophysical observations. This approach is erroneous because it can only show whether the time series can be regarded as belonging to a Markov process.

Theoretically, the value of $r_{\mathrm{I}}$ can be negative; then, the correlation function will be changing its sign at each lag and the spectrum will be growing with frequency. However, such processes (example b in Fig. 2.2) do not seem to exist in geophysical phenomena.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Covariance and Correlation Functions

R(ķ)=林ñ→∞1(ñ−ķ)∑吨=1ñ−ķ(X吨−X¯)(X吨+ķ−X¯),ķ=1,2,…,ķ

r(ķ)=R(ķ)/σX2.

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

s(F)=∑ķ=−∞∞R(ķ)和−一世2圆周率ķFΔ吨

R(ķ)=∫−FñFñs(F)和一世2圆周率ķFΔ吨 dF

X吨=∑j=0∞ψj一个吨−j

r(ķ)=r1|ķ|

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