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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|AUTOREGRESSIVE-MOVING AVERAGE MODELS

3.25 We may also entertain combinations of autoregressive and moving average models. For example, consider the natural combination of the $\operatorname{AR}(1)$ and MA(1) models, known as the first-order autoregressive-moving average, or $\operatorname{ARMA}(1,1)$, process:
$$x_t-\phi x_{t-1}=a_t-\theta a_{t-1}$$
or
$$(1-\phi B) x_t=(1-\theta B) a_t$$
The $\psi$-weights in the MA( $(\infty)$ representation are given by:
$$\psi(B)=\frac{(1-\theta B)}{(1-\phi B)}$$
so that
$$x_t=\psi(B) a_t=\left(\sum_{i=0}^{\infty} \phi^i B^i\right)(1-\theta B) a_t=a_t+(\phi-\theta) \sum_{i=1}^{\infty} \phi^{i-1} a_{t-i}$$
Likewise, the $\pi$-weights in the $\operatorname{AR}(\infty)$ representation are given by:
$$\pi(B)=\frac{(1-\phi B)}{(1-\theta B)}$$
so that
$$\pi(B) x_t=\left(\sum_{i=0}^{\infty} \theta^i B^i\right)(1-\phi B) x_t=a_t$$
or
$$x_t=(\phi-\theta) \sum_{i=1}^{\infty} \theta^{i-1} x_{t-i}+a_t$$
The ARMA(1,1) process, thus, leads to both MA and autoregressive representations having an infinite number of weights. The $\psi$-weights converge for $|\phi|<1$ (the stationarity condition) and the $\pi$-weights converge for $|\theta|<1$ (the invertibility condition). The stationarity condition for the $\operatorname{ARMA}(1,1)$ process is, thus, the same as that for an $\operatorname{AR}(1)$.

统计代写|时间序列分析代写Time-Series Analysis代考|ARMA MODEL BUILDING AND ESTIMATION

3.29 An essential first step in fitting ARMA models to observed time series is to obtain estimates of the generally unknown parameters $\mu, \sigma_x^2$, and the $\rho_k$. With the stationarity and (implicit) ergodicity assumptions, $\mu$ and $\sigma_x^2$ can be estimated by the sample mean and sample variance, respectively, of the realization $x_1, x_2, \ldots, x_T$, that is, by Eqs. (1.2) and (1.3). An estimate of $\rho_k$ is then provided by the lag $k$ sample autocorrelation given by Eq. (1.1), which, because of its importance, is reproduced here:
$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2} \quad k=1,2, \ldots$$
Recall from $\S \mathbf{1 . 2}$ that the set of $r_k \mathrm{~s}$ defines the sample ACF (SACF), which is sometimes referred to as the correlogram.
3.30 Consider a time series generated as independent observations drawn from a fixed distribution with finite variance (i.e., $\rho_k=0$ for all $k \neq 0$ ). Such a series is said to be independent and identically distributed or i.i.d. For such a series the variance of $r_k$ is approximately given by $T^{-1}$. If $T$ is large as well, $\sqrt{T} r_k$ will be approximately standard normal, so that $r_k \stackrel{a}{\sim} N\left(0, T^{-1}\right)$, implying that an absolute value of $r_k$ in excess of $2 / \sqrt{T}$ may be regarded as “significantly” different from zero at the $5 \%$ significance level. More generally, if $\rho_k=0$ for $k>q$, the variance of $r_k$, for $k>q$, is:
$$V\left(r_k\right)=T^{-1}\left(1+2 \rho_1^2+\cdots+2 \rho_q^2\right) .$$
Thus, by successively increasing the value of $q$ and replacing the $\rho_k$ s by their sample estimates, the variances of the sequence $r_1, r_2, \ldots, r_k$ can be estimated as $T^{-1}, T^{-1}\left(1+2 r_1^2\right), \ldots, T^{-1}\left(1+2 r_1^2+\ldots+2 r_{k-1}^2\right)$, and, of course, these will be larger for $k>1$ than those calculated using the simple formula $T^{-1}$. Taking the square root of $V\left(r_k\right)$ gives the standard error to be attached to $r_k$ and these are often referred to as Bartlett standard errors, as (3.12) was derived in Bartlett (1946).

时间序列分析代考

统计代写|时间序列分析代写时间序列分析代考|AUTOREGRESSIVE-MOVING – AVERAGE MODELS

$$x_t-\phi x_{t-1}=a_t-\theta a_{t-1}$$

$$(1-\phi B) x_t=(1-\theta B) a_t$$
MA($(\infty)$表示中的$\psi$ -权重由:
$$\psi(B)=\frac{(1-\theta B)}{(1-\phi B)}$$

$$x_t=\psi(B) a_t=\left(\sum_{i=0}^{\infty} \phi^i B^i\right)(1-\theta B) a_t=a_t+(\phi-\theta) \sum_{i=1}^{\infty} \phi^{i-1} a_{t-i}$$

$$\pi(B)=\frac{(1-\phi B)}{(1-\theta B)}$$

$$\pi(B) x_t=\left(\sum_{i=0}^{\infty} \theta^i B^i\right)(1-\phi B) x_t=a_t$$

$$x_t=(\phi-\theta) \sum_{i=1}^{\infty} \theta^{i-1} x_{t-i}+a_t$$

统计代写|时间序列分析代写Time-Series Analysis代考|ARMA MODEL BUILDING AND estimate

$$r_k=\frac{\sum_{t=k+1}^T\left(x_t-\bar{x}\right)\left(x_{t-k}-\bar{x}\right)}{T s^2} \quad k=1,2, \ldots$$

3.30考虑一个时间序列，它是由具有有限方差的固定分布(即，$\rho_k=0$为所有$k \neq 0$)。我们称这样的一个级数为独立同分布的或i.i.d。对于这样的一个级数，$r_k$的方差近似由$T^{-1}$给出。如果$T$也很大，$\sqrt{T} r_k$将近似于标准正态，因此$r_k \stackrel{a}{\sim} N\left(0, T^{-1}\right)$，这意味着在$5 \%$显著性水平上，$r_k$超过$2 / \sqrt{T}$的绝对值可能被视为“显著”不同于零。更一般地，如果$\rho_k=0$对于$k>q$, $r_k$对于$k>q$的方差为:
$$V\left(r_k\right)=T^{-1}\left(1+2 \rho_1^2+\cdots+2 \rho_q^2\right) .$$因此，通过依次增加$q$的值并用它们的样本估计替换$\rho_k$，序列$r_1, r_2, \ldots, r_k$的方差可以估计为$T^{-1}, T^{-1}\left(1+2 r_1^2\right), \ldots, T^{-1}\left(1+2 r_1^2+\ldots+2 r_{k-1}^2\right)$，当然，$k>1$的方差将大于使用简单公式$T^{-1}$计算的方差。取$V\left(r_k\right)$的平方根会得到附加在$r_k$上的标准误差，这些通常被称为巴特利特标准误差，如(3.12)在巴特利特(1946)中得到

有限元方法代写

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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。

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