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# 数学代写|时间序列分析代写Time Series Analysis代考|State Space Models for Time Series

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## 数学代写|时间序列分析代写Time Series Analysis代考|State Space Models: Pluses and Minuses

State space models can be used for both deterministic and stochastic applications, and they can be applied to both continuously sampled data and discretely sampled data. $^{1}$

This alone gives you some inkling of their utility and tremendous flexibility. The flexibility of state space models is what drives both the advantages and disadvantages of this class of models.

There are many strengths of a state space model. A state space model allows for modeling what is often most interesting in a time series: the dynamic process and states producing the noisy data being analyzed, rather than just the noisy data itself. With a state space model, we inject a model of causality into the modeling process to explain what is generating a process in the first place. This is useful for cases where we have strong theories or reliable knowledge about how a system works, and where we want our model to help us suss out more details about general dynamics with which we are already familiar.

A state space model allows for changing coefficients and parameters over time, which means that it allows for changing behavior over time. We did not impose a condition of stationarity on our data when using state space models. This is quite different from the models we examined in Chapter 6, in which a stable process is assumed and modeled with only one set of coefficients rather than time-varying coefficients.

Nonetheless, there are also some disadvantages to a state space model, and sometimes the strength of the state space model is also its weakness:

• Because state space models are so flexible, there are many parameters to be set and many forms a state space model can take. This means that the properties of a particular state space model have often not been well studied. When you formulate a state space model tailored to your time series data, you will rarely have statistical textbooks or academic research papers in which others have studied the model too. This leaves you in less certain territory as far as understanding how your model performs or where you may have committed errors.
• State space models can be very taxing computationally because there are many parameters. Also, the very high number of parameters for some kinds of state space models can leave you vulnerable to overfitting, particularly if you don’t have much data.

## 数学代写|时间序列分析代写Time Series Analysis代考|The Kalman Filter

The Kalman filter is a well-developed and widely deployed method for incorporating new information from a time series and incorporating it in a smart way with previously known information to estimate an underlying state. One of first uses of the Kalman filter was on the Apollo 11 mission-the filter was chosen when NASA engineers realized that the onboard computing elements would not allow other, more memory-intensive techniques of position estimation. As you will see in this section, the benefits of the Kalman filter are that it is relatively easy to compute and does not require storage of past data to make present estimates or future forecasts.
Overview
The mathematics of the Kalman filter can be intimidating to a newcomer, not because it is especially difficult but because there are a fair number of quantities to keep track of, and it’s an iterative, somewhat circular process with many related quantities. For this reason, we will not derive the Kalman filter equations here, but instead go through a high-level overview of those equations to get a sense of how they work. $^{2}$

We begin with a linear Gaussian model, positing that our state and our observations have the following dynamics:
\begin{aligned} &x_{t}=F \times x_{t-1}+B \times u_{t}+w_{t} \ &y_{t}=A \times x_{t}+v_{t} \end{aligned}
That is, the state at time $\mathrm{t}$ is a function of the state at the previous time step $\left(F \times x_{t-1}\right)$, an external force term $\left(B \times u_{t}\right)$, and a stochastic term $\left(w_{t}\right)$. Likewise, the measurement at time $t$ is a function of the state at time $t$ and a stochastic error term, measurement error.

## 数学代写|时间序列分析代写TIME SERIES ANALYSIS代考|Hidden Markov Models

Hidden Markov Models (HMMs) are a particularly useful and interesting way of modeling a time series because it is a rare instance of unsupervised learning in time series analysis, meaning there is no labeled correct answer against which to train. An HMM is motivated by an intuition similar to what we used when experimenting with the Kalman filter earlier in this chapter, namely the idea that the variables we are able to observe may not be the most descriptive variables of the system. As with the Kalman filter applied to a linear Gaussian model, we posit the idea that the process has a state, and our observations give information about this state. And again, as before, we need to have some opinion as to how the state variables influence what we can observe. In the case of an HMM, what we posit is that the process is a nonlinear one characterized by jumps between discrete states.
How the Model Works
An HMM posits a system in which there are states that are not directly observable. The system is a Markov process, which means that it is “memoryless” in the sense that the probabilities of future events can be fully calculated given only the system’s current state. That is, knowing the system’s current state and its previous states is no more useful than simply knowing the system’s current state.

Markov processes are often described in terms of matrices. For example, suppose there was a system that fluctuated between state $A$ and state $B$. When in either state, the system was statistically more likely to remain in the same state than to flip to the other state at any distinct time step. One such system would be described by the following matrix:
A B
A 0.7 0.3
B 0.2 0.8
Let’s imagine that our system is in state $A$, namely $(1,0)$. (State B would be $(0,1)$.) In such a case, the probability that the system would remain in state A is $0.7$, whereas the probability that the system would flip is $0.3$. We don’t need to know what states the system was in before its most recent moment in time. This is what it means to be a Markov process.

## 数学代写|时间序列分析代写TIME SERIES ANALYSIS代考|STATE SPACE MODELS: PLUSES AND MINUSES

• 由于状态空间模型非常灵活，因此需要设置许多参数，并且状态空间模型可以采用多种形式。这意味着特定状态空间模型的属性通常没有得到很好的研究。当您制定适合您的时间序列数据的状态空间模型时，您将很少有其他人也研究过该模型的统计教科书或学术研究论文。就了解您的模型如何执行或您可能在哪里犯了错误而言，这使您处于不太确定的领域。
• 状态空间模型在计算上可能非常繁重，因为有很多参数。此外，某些状态空间模型的大量参数可能会使您容易受到过度拟合的影响，尤其是在您没有太多数据的情况下。

## 数学代写|时间序列分析代写TIME SERIES ANALYSIS代考|THE KALMAN FILTER

X吨=F×X吨−1+乙×在吨+在吨 是吨=一种×X吨+在吨

AB
A 0.7 0.3
B 0.2 0.8

## 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。