### 统计代写|时间序列分析作业代写time series analysis代考|Basics of Scalar Random Processes

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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|时间序列分析作业代写time series analysis代考|Basic Statistical Characteristics

The notation used in this book for time series is $x_{t}, t=\Delta t, \ldots, N \Delta t$, where $N$ is the number of terms in the time series and $\Delta t$ is the time interval between consecutive terms. The sampling rate is the number of samples per unit time and, as a rule, the sampling rate here is one sample per year or one sample per month, that is, $\Delta t=1$ year or $\Delta t=1$ month. The notation for the time series can also be given as $x_{t}, t=1, \ldots, N$ having in mind that $\Delta t=1$. In what follows, the time series is considered as long if its length exceeds the largest time scale of interest by orders of magnitude. Otherwise, the time series has to be treated as short.

A major characteristic of a time series is its probability density function $p(x)$, which defines the probability of encountering different numerical values of $x_{t}$. The common abbreviation for $p(x)$ is PDF. The PDF of a scalar time series is characterized with its statistical moments, or statistics, such as mean value and variance (central moments), covariance function and spectral density (mixed moments), and with higher central moments such as skewness and kurtosis.

There are many different types of PDFs, but the function which is most important for practical time series analysis (if its application can be justified for a given time series) is the Gaussian or normal, probability density function
$$p\left(x_{t}\right)=\frac{1}{\sigma_{x} \sqrt{2 \pi}} \exp \left[-\left(x_{t}-\bar{x}\right)^{2} / 2 \sigma_{x}^{2}\right]$$
where the mean value
$$\bar{x}=\lim {N \rightarrow \infty} \frac{1}{N} \sum{t=1}^{N} x_{t}$$
and

$$\sigma_{x}^{2}=\lim {N \rightarrow \infty} \frac{1}{N} \sum{t=1}^{N}\left(x_{t}-\bar{x}\right)^{2}$$
is the variance of the time series. The positive square root $\sigma_{x}$ of the variance is called the root mean square (RMS) value or standard deviation. The variance (and RMS) describes the variability of the process: a larger variance means a larger dynamic range of numerical values of the process. These definitions are true for random functions of time and for sequences of time-invariant random variables. The Gaussian PDF of random variables is completely described with the mean value $\bar{x}$ and the variance $\sigma_{x}^{2}$. By default, it is generally assumed here that time series are generated by Gaussian (normally distributed) random processes. This assumption does not limit or degrades the properties and abilities of the methods used in this book for time series analysis because the Gaussian probability distribution means the best possible results in all cases when the method is linear. The analysis of non-Gaussian time series requires estimation of the same statistical characteristics as in the Gaussian case and then some higher statistical moments. The methods applied here are linear, and they cover all traditional tasks of time series analysis, including spectral estimation and statistical (probabilistic) forecasting. The normal hypothesis should always be tested for the actual time series that is being analyzed.

The skewness and kurtosis can be helpful for analyzing geophysical and solar time series as measures of PDF’s asymmetry and tail properties, respectively. They are mostly used for determining whether the PDF is close enough to a Gaussian (normal) distribution. Specifically, if the absolute values of standardized skewness and standardized kurtosis do not exceed 2, the time series can be generally regarded as Gaussian. (Standardized skewness and kurtosis are found by dividing respective estimate by $\sqrt{6 / N}$ and by $\sqrt{24 / N}$ ).

The PDFs and their central statistical moments completely describe properties of sets of random variables, which do not depend upon the time argument. In contrast to random vectors, time series present samples of random processes and their description is not possible without mixed statistical moments such as covariance and correlation functions in the time domain and the spectral density in the frequency domain. These functions are defined in Sect. 2.4.

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

In contrast to time series consisting of random variables, one can imagine a time-dependent process that follows a specific mathematical law, which excludes any randomness. Respective data present a deterministic process. Deterministic phenomena or processes can be predicted at any lead time without an error.

At climatic time scales (longer than one year), the only geophysical processes that can be regarded as deterministic are those that are caused by astronomical factors. With one exception (a barely detectible tidal harmonic with period of $18.61$ years), the time scales of such processes extend to millennia and longer (e.g., Monin 1986) and their effects upon climate at the practically important time scales of year-toyear variability, decades and centuries are negligibly small. At smaller time scales, the only deterministic process in the Earth system is tides, which can be predicted practically precisely in the open ocean and along most shorelines. With the exception of solar and lunar tides, all other geophysical and relevant solar processes are random. Moreover, in some coastal areas (e.g., Newlyn, UK), the sea level variations can be better predicted in a different manner, “without astronomical prejudice and fully allowing for the presence of noise” (Munk and Cartwright 1966). An earlier method of tidal prediction without tidal harmonics had been developed by A. Duvanin (1960).

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

A random (stochastic, probabilistic) process is any process running in time and controlled by probabilistic laws (Doob 1953). A random process consists of an infinite set of all possible sample records (time series, random functions of time) generated supposedly as the results of repeated experiments or observations. The random processes discussed here are always discreet, that is, the time argument takes only discrete values.

There are two major types of random processes: stationary and nonstationary. A stationary process is the process whose statistical properties determined by averaging at different time origins over the infinite set of sample records of the process do not depend upon the time origin. A nonstationary process does not possess this property. The time series studied in this book belong to the class of stationary random processes.
Consider this concept using the mean value and variance as examples. Let $x_{i, t}, i=$ $1, \ldots, M ; t=1, \ldots, N$ be an ensemble of sample records of a random process as shown in Fig. 2.1.

The mean value (or mathematical expectation) $\hat{x}(t)$ at time $t$ is defined as the limit of the sum of $M$ values of $x_{i, t}$ at time $t$ divided by the number of sample records $M$ as $M$ tends to infinity:
$$\hat{x}(t)=\lim {M \rightarrow \infty} \frac{1}{M} \sum{i=1}^{M} x_{i, t} .$$
A random process is stationary with respect to the mean value if the mean values $\hat{x}(t)$ are statistically the same for all $t$.
If it is also true that the variance
$$\hat{\sigma}{x}^{2}(t)=\lim {M \rightarrow \infty} \frac{1}{M} \sum_{i=1}^{M}\left[x_{i, t}-\hat{x}(t)\right]^{2}$$

of the process possesses the same property, that is, it takes statistically the same value for all $t$, then we have a second order stationary random process.

According to the above definition, one sample record (time series) is not sufficient for studying a random process. This statement is true even if the sample record is known on the entire time axis from $-\infty$ to $+\infty$. Therefore, studying a random process requires a set of sample records (theoretically, an infinite set), so that all characteristics of the process (i.e., its PDF and its statistical moments) are determined by averaging over the ensemble of sample records. This cannot happen in geophysics in general, in climatology, or in solar physics where many if not all time series are unique. Therefore, by studying a single sample record of a stationary random process and extending the results of analysis to the entire process, one assumes by default that the results obtained by analyzing its properties through averaging over time coincide with the results of averaging over an ensemble of such records at different time origins. A random process that possesses this property is called ergodic. An ergodic process is always stationary.

## 统计代写|时间序列分析作业代写time series analysis代考|Basic Statistical Characteristics

p(X吨)=1σX2圆周率经验⁡[−(X吨−X¯)2/2σX2]

X¯=林ñ→∞1ñ∑吨=1ñX吨

PDF 及其中心统计矩完全描述了随机变量集的属性，这些属性不依赖于时间参数。与随机向量相比，时间序列呈现随机过程的样本，如果没有混合统计矩（例如时域中的协方差和相关函数以及频域中的谱密度），则无法对其进行描述。这些功能在 Sect 中定义。2.4.

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

X^(吨)=林米→∞1米∑一世=1米X一世,吨.

σ^X2(吨)=林米→∞1米∑一世=1米[X一世,吨−X^(吨)]2

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