# 统计代写|时间序列分析代写Time-Series Analysis代考|Piecewise vector autoregressive model

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Piecewise vector autoregressive model

Davis, Lee, and Rodriguez-Yam (2006) considered modeling nonstationary time series using piecewise VAR processes. The number and locations of the piecewise VAR segments, and the orders of the corresponding VAR process, are assumed to be unknown. Based on the minimum description length (MDL) principle, the method penalizes complexity of the model, and thus provides the criteria to define the best fitting model. We refer readers to Rissanen (1989), Hansen and Yu (2001) for more details about the MDL principle.

Let $\mathbf{Z}t$ be a $m$-dimensional time series with $k$ segments, and assume that there are partitions or changepoints $\boldsymbol{\delta}=\left(\delta_0, \ldots, \delta_k\right)^{\prime}$ with $\delta_0=0$ and $\delta_k=n$. The time series within the $j$ th segment is modeled by $\operatorname{VAR}\left(p_j\right)$ process, such that $$\mathbf{Z}_t^{(j)}=\boldsymbol{\Phi}{j, 1} \mathbf{Z}{t-1}^{(j)}+\cdots+\boldsymbol{\Phi}{j, p_j} \mathbf{Z}{t-p_j}^{(j)}+\boldsymbol{\varepsilon}_t^{(j)}$$ where $\left(\boldsymbol{\Phi}{j, 1}, \ldots, \boldsymbol{\Phi}_{j, p_j}\right)^{\prime}$ are $m \times m$ dimensional coefficient matrices of the VAR process. We define the entire class of piecewise VAR models as $\mathbf{M}$ and a model from this class as $\mathbf{F} \in \mathbf{M}$. The principle of MDL is to find the best fitting model from $\mathbf{M}$ as the one that produces the shortest code length. The code length of an object is the amount of memory space required to store the data $\mathbf{Z}_t$. The MDL has two components, a fitted model $\hat{\mathbf{F}}$ and the portion that is unexplained by $\hat{\mathbf{F}}$. The later component can be defined as the residuals $\hat{\boldsymbol{\varepsilon}}_t=\mathbf{Z}_t-\hat{\mathbf{Z}}_t$.

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

There are two recently proposed Bayesian methods to estimate multivariate time-varying spectrum, including Zhang (2016) and Li and Krafty (2018). In this section, we focus on the method of Li and Krafty (2018). The method is also based on locally stationary time series to estimate the time-varying spectrum
$$\mathbf{f}(u, \omega)=\mathbf{A}(u, \omega) \mathbf{A}(u, \omega)^*$$
The method assumes that for every $u \in(0,1)$, each component of $\mathbf{f}(u, \cdot)$ possesses a squareintegrable first derivative as a function of frequency; for every $\omega$, each component of $\mathbf{f}(\cdot, \omega)$ is continuous as a function of scaled time at all but a possible finite number of points. The assumption on transfer function and time-varying spectrum are slightly different from Definitions 9.1 and 9.2 in two ways. First, Definition 9.1 assumes a series of transfer functions $\mathbf{A}^0(u, \omega)$ that converge to a large-sample transfer function $\mathbf{A}(u, \omega)$ in order to allow for the fitting of parametric models. Since the method considers nonparametric estimation, in a manner similar to the smoothing ANOVA method, $\mathbf{A}(u, \omega)$ is used. Second, Definitions 9.1 and 9.2 require the time-varying spectrum to be continuous in both time and frequency. The method is more flexible and allows for components of spectrum to evolve not only continuously, but also abruptly in time.
The analysis of a locally stationary time series $\left{\mathbf{Z}t: t=1, \ldots, n\right}$ begins by using the piecewise stationary approximation. Consider a partition of the time series into $k$ segments defined by partition points $\boldsymbol{\delta}=\left(\delta_0, \ldots, \delta_k\right)$ with $\delta_0=0$ and $\delta_k=n$ such that $\mathbf{Z}_t$ is approximately stationary within the segments $\left{t: \delta{q-1}<t \leq \delta_q\right}$ for $q=1, \ldots, k$. Then
$$\mathbf{Z}t \approx \sum{q=1}^k \int_{-\pi}^\pi \mathbf{A}q(\omega) \exp (i \omega t) d \mathbf{U}(\omega),$$ where $\mathbf{A}_q(\omega)=\mathbf{A}\left(u_q, \omega\right) I\left(\delta{q-1}<t \leq \delta_q\right), I(\cdot)$ is the indicator function, and $u_q=\left(\delta_q+\delta_{q-1}\right) / 2 n$ is the scaled midpoint of the $q$ th segment. Within the $q$ th segment, the time series is approximately second-order stationary with local power spectrum $\mathbf{f}\left(u_q, \omega\right)=\mathbf{A}_q(\omega) \mathbf{A}_q(\omega)^*$.

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|Piecewise vector autoregressive model

Davis, Lee和Rodriguez-Yam(2006)考虑使用分段VAR过程建模非平稳时间序列。假设分段VAR段的数量和位置以及相应VAR过程的顺序是未知的。该方法基于最小描述长度(MDL)原则，对模型的复杂度进行惩罚，从而为定义最佳拟合模型提供了准则。我们建议读者参考Rissanen(1989)、Hansen和Yu(2001)，了解MDL原则的更多细节。

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

$$\mathbf{f}(u, \omega)=\mathbf{A}(u, \omega) \mathbf{A}(u, \omega)^*$$

## 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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