### 统计代写|时间序列分析代写Time-Series Analysis代考|STAT3040

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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|STOCHASTIC PROCESSES AND STATIONARITY

3.1 The concept of a stationary time series was introduced informally in Chapter 1, Time Series and Their Features, but to proceed further it is necessary to consider the concept rather more rigorously. To this end, it is often useful to regard the observations $x_1, x_2, \ldots, x_T$ on the series $x_t$ as a realization of a stochastic process. In general, such a stochastic process may be described by a $T$-dimensional probability distribution, so that the relationship between a realization and a stochastic process is analogous, in classical statistics, to that between a sample and the population from which it has been drawn from.
Specifying the complete form of the probability distribution, however, will typically be too ambitious a task, so attention is usually concentrated on the first and second moments; the $T$ means:
$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$ variances:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$
and $T(T-1) / 2$ covariances:
$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$
If the distribution could be assumed to be (multivariate) normal, then this set of expectations would completely characterize the properties of the stochastic process. As has been seen from the examples in Chapter 2, Transforming Time Series, however, such an assumption will not always be appropriate, but if the process is taken to be linear, in the sense that the current value $x_t$ is generatē by a linear combination of previous valuess $x_{t-1}, x_{t-2}, \ldots$ of the process itself plus current and past values of any other related processes, then again this set of expectations would capture its major properties.

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD’S DECOMPOSITION AND AUTOCORRELATION

3.6 A fundamental theorem in time series analysis, known as Wold’s decomposition, states that every weakly stationary, purely nondeterministic, stochastic process $x_t-\mu$ can be written as a linear combination (or linear filter) of a sequence of uncorrelated random variables. ${ }^2$ “Purely nondeterministic” means that any deterministic components have been subtracted from $x_t-\mu$. Such components are those that can be perfectly predicted from past values of themselves and examples commonly found are a (constant) mean, as is implied by writing the process as $x_t-\mu$, periodic sequences (e.g., sine and cosine functions), and polynomial or exponential sequences in $t$.
This linear filter representation is given by:
$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

The $a_t, t=0, \pm 1, \pm 2, \ldots$ are a sequence of uncorrelated random variables, often known as innovations, drawn from a fixed distribution with:
$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$
and
$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$
Such a sequence is known as a white noise process, and occasionally the innovations will be denoted as $a_t \sim \mathrm{WN}\left(0, \sigma^2\right) .^3$ The coefficients (possibly infinite in number) in the linear filter (3.2) are known as $\psi$-weights.
3.7 It is easy to show that the model (3.2) leads to autocorrelation in $x_t$. From this equation it follows that:
$$E\left(x_t\right)=\mu$$
and
\begin{aligned} \gamma_0 &=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \ &=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2 \ &=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E\left(a_{t-2}^2\right)+\cdots \ &=\sigma^2+\psi_1^2 \sigma^2+\psi_2^2 \sigma^2+\cdots \ &=\sigma^2 \sum_{j=0}^{\infty} \psi_j^2 \end{aligned}
by using the white noise result that $E\left(a_{t-i} a_{t-j}\right)=0$ for $i \neq j$.

## 统计代写|时间序列分析代写Time-Series Analysis代考|STOCHASTIC PROCESSES AND STATIONARITY

$3.1$ 平稳时间序列的概念在第 1 章“时间序列及其特征”中非正式地介绍过，但为了进一步深入，有必要更严格地考 虑这个概念。为此，考虑观察结果通常是有用的 $x_1, x_2, \ldots, x_T$ 在系列上 $x_t$ 作为随机过程的实现。一般来说，这 样的随机过程可以用 $T$ 维概率分布，因此在经典统计中，实现与随机过程之间的关系类似于样本与从中抽取样本的 总体之间的关系。

$$E\left(x_1\right), E\left(x_2\right), \ldots, E\left(x_T\right)$$
$T$ 差异:
$$V\left(x_1\right), V\left(x_2\right), \ldots, V\left(x_T\right)$$

$$\operatorname{Cov}\left(x_i, x_j\right), \quad i<j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|WOLD’S DECOMPOSITION AND AUTOCORRELATION

$3.6$ 时间序列分析中的一个基本定理，称为 Wold 分解，指出每个弱平稳、纯非确定性、随机过程 $x_t-\mu$ 可以写成 一系列不相关的随机变量的线性组合 (或线性滤波器) 。 ${ }^2$ 纯非确定性”意味看已从其中减去任何确定性组件 $x_t-\mu$. 此类组件是可以从其过去的值中完美预测的组件，并且常见的示例是 (恒定的) 均值，正如将过程编写为 所暗示的那样 $x_t-\mu$ ，周期序列（例如，正弦和余弦函数），以及多项式或指数序列 $t$. 该线性滤波器表示由下式给出：
$$x_t-\mu=a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots=\sum_{j=0}^{\infty} \psi_j a_{t-j} \quad \psi_0=1$$

$$E\left(a_t\right)=0 \quad V\left(a_t\right)=E\left(a_t^2\right)=\sigma^2<\infty$$

$$\operatorname{Cov}\left(a_t, a_{t-k}\right)=E\left(a_t a_{t-k}\right)=0, \text { for all } k \neq 0$$

$3.7$ 很容易证明模型 (3.2) 导致自相关 $x_t$. 从这个等式可以得出:
$$E\left(x_t\right)=\mu$$

$$\gamma_0=V\left(x_t\right)=E\left(x_t-\mu\right)^2 \quad=E\left(a_t+\psi_1 a_{t-1}+\psi_2 a_{t-2}+\cdots\right)^2=E\left(a_t^2\right)+\psi_1^2 E\left(a_{t-1}^2\right)+\psi_2^2 E$$

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