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# 统计代写|时间序列分析作业代写time series analysis代考|Foundations for Stochastic Spectral Analysis

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## 统计代写|时间序列分析作业代写time series analysis代考|Spectral Representation of Stationary Processes

In this section we motivate and state but do not prove the spectral representation theorem for stationary processes due to Cramér 1942. A rigorous proof is rather involved – the interested reader is referred to Priestley 1981, section 4.11 for details. This theorem is fundamental in the spectral analysis of stationary processes since it allows us to relate the spectrum of such a process directly to a representation for the process itself. Indeed, in the words of Koopmans (1974, p. 36), “One of the essential reasons for the central position held by stationary stochastic processes in time series analysis is the existence of a spectral representation for the process from which the spectrum can be directly computed.”

We motivate the spectral representation theorem for discrete parameter stationary processes by considering the special case of a real-valued discrete time harmonic process that was formulated in Exercise 37 and involves fixed amplitudes and random phases:
$$X_{t}=\sum_{l=1}^{L} D_{l} \cos \left(2 \pi f_{l} t+\phi_{l}\right), \quad t \in \mathbb{Z}$$

## 统计代写|时间序列分析作业代写time series analysis代考|Alternative Definitions for the Spectral Density Function

In the previous section we defined the integrated spectrum and the SDF in terms of the spectral representation of a stationary process. This approach allows us to relate the integrated spectrum and SDF directly to the representation of the process itself. It is sometimes stated that by definition the SDF is the Fourier transform of the ACVS, i.e., that Equation for discrete parameter stationary processes (or Equation (114) for continuous parameter processes) defines $S(\cdot)$ . It is difficult to attach much meaning to $S(\cdot)$ from this definition alone. The usual approach is to appeal to the theory of linear filters in order to establish a physical meaning for the SDF.

Another way of defining the SDF makes use of the Fourier theory for deterministic sequences with finite energy (Section 3.8). Here, instead of just drawing comparisons between square summable sequences and stationary processes as we did in the previous section, we treat portions of realizations of a stationary process as a square summable sequence and apply some limiting arguments. We present this definition here because it is particularly informative (see sections $4.7$ and $4.8$ of Priestley, 1981). Let $\left{x_{t}\right}$ be any realization of the discrete parameter stationary process $\left{X_{t}\right}$ with zero mean. Then we should have
$$\frac{1}{N} \sum_{t=0}^{N-1} x_{t}^{2} \approx \sigma^{2} \stackrel{\text { def }}{=} E\left{X_{t}^{2}\right}$$

## 统计代写|时间序列分析作业代写time series analysis代考|Basic Properties of the Spectrum

In this case, $S^{(1)}(\cdot)$ and $S(\cdot)$ are seen to have the following properties.
1 $S^{(\mathrm{I})}(-1 / 2)=0$ because $S^{(\mathrm{I})}(-1 / 2)=\int_{-1 / 2}^{-1 / 2} S\left(f^{\prime}\right) \mathrm{d} f^{\prime}=0$.

2 $S^{(\mathrm{I})}(1 / 2)=s_{0}$ because $S^{(\mathrm{I})}(1 / 2)=\int_{-1 / 2}^{1 / 2} S\left(f^{\prime}\right) \mathrm{d} f^{\prime}=s_{0}$ (from Equation (111c) with $\tau$ set to zero).

3 $S(f) \geq 0$ because Equation (111b) says that $S(f) \mathrm{d} f=E\left{|\mathrm{~d} Z(f)|^{2}\right}$, and we must have $\mathrm{d} f>0$ and $E\left{|\mathrm{~d} Z(f)|^{2}\right} \geq 0$.

4 $f<f^{\prime}$ implies $S^{(\mathrm{I})}(f) \leq S^{(\mathrm{I})}\left(f^{\prime}\right)$ because $S^{(\mathrm{I})}\left(f^{\prime}\right)-S^{(\mathrm{I})}(f)$ is equal to the integral of a nonnegative function $S(\cdot)$ over the interval $\left[f, f^{\prime}\right]$ (this also follows because the spectral representation theorem states that $S^{(\mathrm{I})}(\cdot)$ is a nondecreasing function of f .

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|SPECTRAL REPRESENTATION OF STATIONARY PROCESSES

$$X_{t}=\sum_{l=1}^{L} D_{l} \cos \left(2 \pi f_{l} t+\phi_{l}\right), \quad t \in \mathbb{Z}$$

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|ALTERNATIVE DEFINITIONS FOR THE SPECTRAL DENSITY FUNCTION

$$\frac{1}{N} \sum_{t=0}^{N-1} x_{t}^{2} \approx \sigma^{2} \stackrel{\text { def }}{=} E\left{X_{t}^{2}\right}$$

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|BASIC PROPERTIES OF THE SPECTRUM

1小号(一世)(−1/2)=0因为小号(一世)(−1/2)=∫−1/2−1/2小号(F′)dF′=0.

2 小号(一世)(1/2)=s0因为小号(一世)(1/2)=∫−1/21/2小号(F′)dF′=s0 Fr这米和q在一种吨一世这n(111C和τ设置为零）。

3 小号(F)≥0因为方程111b说S(f) \mathrm{d} f=E\left{|\mathrm{~d} Z(f)|^{2}\right}S(f) \mathrm{d} f=E\left{|\mathrm{~d} Z(f)|^{2}\right}，我们必须有dF>0和E\left{|\mathrm{~d} Z(f)|^{2}\right} \geq 0E\left{|\mathrm{~d} Z(f)|^{2}\right} \geq 0.

4 F<F′暗示小号(一世)(F)≤小号(一世)(F′)因为小号(一世)(F′)−小号(一世)(F)等于非负函数的积分小号(⋅)在区间内[F,F′] 吨H一世s一种ls这F这ll这在sb和C一种在s和吨H和sp和C吨r一种lr和pr和s和n吨一种吨一世这n吨H和这r和米s吨一种吨和s吨H一种吨$小号(一世)(⋅$ 是 f 的非减函数。

## 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。