贝叶斯时间序列分析：理论与实践

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"BayesianAnalysis of Time Series" 是一本由Lyle D. Broemeling撰写的书籍，由CRC Press（Taylor & Francis Group）出版。这本书详细介绍了贝叶斯方法在时间序列分析中的应用，旨在提供理论与实践相结合的知识。贝叶斯分析是一种统计学方法，它在处理不确定性问题时，通过结合先验信息和观测数据来更新对模型参数的理解。在时间序列分析中，这种方法特别有用，因为通常我们需要根据过去的观测值来预测未来的趋势或者识别数据中的周期性和结构变化。时间序列是一系列按时间顺序排列的观测值，广泛应用于经济学、金融、气象学、工程等多个领域。分析时间序列通常涉及以下步骤： 1. **数据预处理**：包括数据清洗、缺失值处理、异常值检测等，确保数据质量。 2. **描述性统计**：计算均值、标准差、自相关函数（ACF）和偏自相关函数（PACF），以理解数据的统计特性及序列间的依赖关系。 3. **模型选择**：在贝叶斯框架下，这包括选择合适的概率模型，如ARIMA（自回归整合滑动平均模型）、GARCH（广义自回归条件异方差模型）或状态空间模型等。 4. **参数估计**：使用贝叶斯方法，通过马尔科夫链蒙特卡洛（MCMC）算法，如吉布斯采样或Metropolis-Hastings算法，来估计模型参数的后验分布。 5. **模型验证**：使用残差分析、预测误差以及诊断图来检查模型的适用性。 6. **预测与解释**：基于后验分布，可以进行点预测和区间预测，并对结果进行解释，理解模型如何捕捉时间序列的关键特征。 7. **模型比较**：通过贝叶斯因子或预测后验密度等度量，对比不同模型的优劣。在书中，作者可能深入讨论了这些概念，并通过实例展示了如何实际应用贝叶斯方法。此外，书中可能还包括了如何使用特定的统计软件（如R语言的`bayesplot`、`rstanarm`包或Python的`PyMC3`库）来实现这些分析。对于那些希望将贝叶斯方法应用于时间序列分析的读者来说，这本书提供了宝贵的理论指导和实践经验。尽管书中可能没有涵盖所有可能的贝叶斯时间序列模型，但它无疑会帮助读者理解这一领域的核心概念，并有能力进一步探索更复杂的问题。

Angeles rainfall data using the appropriate R command (e.g. mean(), std(),

autocorrelation or acf(), and partial autocorrelation or pacf()).

1.4 Chapter 4: Basic Random Models

This chapter describes the building blocks for constructing more complex

time series models, beginning with white noise, a sequence of independent

normally identically distributed random variables. This sequence will

serve as the error terms for the autoregressive time series models. We

now progress from white noise to the random walk time series, and

random walk with drift. For the autoregressive time series, the likelihood

function is explicated, a prior distribution for the parameters is speciﬁed,

and via Bayes theorem, the posterior distribution of the autoregressive

parameters and precision of the error terms are derived. R is employed to

generate observations from an AR(1) model with known parameters; then

using those observations as data, WinBUGS is executed for the posterior

analysis which gives Bayesian point and interval estimates of the unknown

parameters. Since the parameter values used to generate the data are

known, the posterior means and medians can be compared to them

giving one some idea of how ‘close’ these estimates are to the so-called

‘true’ values. Another example of an autoregressive time series is pre-

sented, followed by a Bayesian approach to goodness-of-ﬁt, and ﬁnally the

Bayesian predictive density for a future observation is derived.

1.5 Time Series and Regression

The ﬁrst time series to be presented is the linear regression model with AR

(1) errors, where R is employed to generate observations from that linear

model with known parameters followed by an execution with WinBUGS

that produces point and interval estimates of those parameters. The sce-

nario is repeated for a quadratic regression with seasonal effects and AR(1)

errors. In all scenarios, R code generates the data and using that as

information for the Bayesian analysis, the posterior distribution of the

parameters is easily provided. Various generalizations to more compli-

cated situations include a nonlinear regression model (with exponential

trend) with AR(1) errors, a linear regression model with AR(2) errors

which contains ﬁve parameters, two linear regression coefﬁcients, two

autoregressive parameters, plus the precision of the error terms. An inter-

esting time series model is one which is continuous time, which is the

solution to a stochastic differential equation, where the posterior distribu-

tion of the autoregressive parameter is derived assuming a noninformative

Introduction to the Bayesian Analysis of Time Series 3

prior distribution for the autoregressive parameter and the variance of the

errors. The chapter concludes with a section on comments and conclusions

followed by eight problems and eight references.

1.6 Time Series and Stationarity

Stationarity of a time series is deﬁned including stationary in the mean and

stationary in the variance. The ﬁrst time series to be considered is the

moving average model and the ﬁrst and second moments are derived. It is

important to note that moving average processes are always stationary. R

is employed to generate observations from an MA(1) series with known

parameters (the moving average coefﬁcient and the precision of the white

noise); then using those observations, WinBUGS is executed for the poster-

ior analysis which provides point and credible intervals for the two

parameters. The MA(1) series is generalized to quadratic regression

model with MA(1) errors, where again R generates observations from it

where the parameters are known.

Bayesian analysis via WinBUGS performs the posterior analysis. An

additional generalization focuses on a quadratic regression model with an

MA(2) series for the errors with R used to generate observations from the

model but with known values for the parameters. It is interesting to

compare the results of the posterior estimates of the quadratic regression

with MA(1) errors to that with MA(1) errors. Generalizing to a regression

model with a quadratic trend, but also including seasonal effects, presents

a challenge to the Bayesian way of determining the posterior analysis. The

predictive density for forecasting future observations is derived, and the

last section of the chapter presents the Bayesian technique of testing

hypotheses about the moving average parameters. There are 16 exercises

and 5 references.

1.7 Time Series and Spectral Analysis

It is explained how spectral analysis gives an alternative approach to

studying time series, where the emphasis is on the frequency of the

series, instead of the time. Chapter 7 continues with a brief history of the

role of the spectral density function in time series analysis, and the review

includes an introduction to time series with trigonometric components

(sine and cosine functions) as independent variables. Next to be explained

are time series with trigonometric components whose coefﬁcients are Four-

ier frequencies and their posterior distributions are revealed with a Baye-

sian analysis. It is important to remember that frequency is measured in

4 Bayesian Analysis of Time Series

hertz (one cycle per second) or in periods measured as so many units of pi

radians per unit time. It can be shown that for the basic time series

(autoregressive and moving average series), the spectral density function

is known in terms of the parameters of the corresponding model. It is

important to remember the spectrum is related to the Fourier line spec-

trum which is a plot of the fundamental frequencies versus m, where m is

the number of harmonics in the series. Section 10 continues with the

derivation of the spectral density function for the basic series, AR(1), AR

(2), MA(1), MA(2), ARMA(1,1), etc. Remember that the spectral density

function is a function of frequency (measured in hertz or multiples of pi in

radians per cycle). For each model, R is used to generate observations from

that model (with known parameters); then, using those observations as

data, the Bayesian analysis is executed with WinBUGS and as a conse-

quence the posterior distribution of the spectral density at various frequen-

cies is estimated. For the last example, the spectral density of the sunspot

cycle is estimated via a Bayesian analysis executed with WinBUGS. There

are 17 problems followed by y references.

1.8 Dynamic Linear Model

The dynamic linear model consists of two equations: the observation

equation and the system equation.

The ﬁrst expresses the observations as a linear function of the states of the

system, while the second expresses the current state as a linear function of

previous states. Usually, the system equation is a ﬁrst-order autoregressive

series, and various expressions of the dynamic linear model depend on what

is assumed about the parameters of the model. First to be considered is the

dynamic model in discrete time and the various phases of inference are

explained. The various phases are as follows: (1) estimation, (2) ﬁltering, (3)

smoothing, and (4) prediction. Filtering is the posterior distribution of the

present state at time t induced by the prior distribution of the previous state at

time t – 1 in a sequential manner, and this recursive Bayesian algorithm is

called the Kalman Filter. An important generalization of the dynamic linear

model is where the system equation is amended to include a control variable,

which is quite useful in navigation problems. Brieﬂy, the control problem is as

follows: at time t, the position of the object (satellite, missile, etc.) is measured

as the observation at t, then the next position of the object is predicted so that

the posterior mean of the next state is equal to the target value (the desired

position) at time t + 1. These ideas are illustrated with various examples

beginning with a model where the observation and system vectors are

univariate with known coefﬁcients. For this case, the Kalman ﬁlter estimates

the current state in a recursive manner as described earlier. As usual, R

generates the observations and states of the system via the observation and

Introduction to the Bayesian Analysis of Time Series 5

system equations of the dynamic linear model, which is followed by a

Bayesian analysis via WinBUGS code that estimates the current state (the

Kalman ﬁlter) of the system. R includes a powerful package dlm that will

generate the observations and states of the process. One must download the

TSA package and dlm package into R! The last section of the chapter deals

with the control problem, where the choice of the control variable is described

by a six-step algorithm. Finally, the problem of adaptive estimation is

explained; this a generalization of the dynamic linear model, where the

precision parameters of the observation and system equations are unknown.

An example involving bivariate observations and states and unknown 2 × 2

precision matrices illustrates the Bayesian approach to the Kalman ﬁlter. The

bivariate observations and states are generated with the dlm R package. There

are 14 exercises and 21 references.

1.9 The Shift Point Problem

Shift point models are deﬁned in general, followed by an example invol-

ving a univariate normal sequence of normal random variables with one

shift where the time when the shift occurs is known. For such a case, the

posterior distribution of the parameters of the two sequences is derived,

assuming noninformative prior distributions for the parameters. There are

three parameters: the mean of the ﬁrst and second phases, plus the

common precision parameter of both. Next to be considered is a two-

phase AR(1) model with one shift but unknown time where the shift

occurs. R generates the observations from this model where the time

when the shift occurs is known but where the shift point is a random

variable. A prior distribution is speciﬁed for the distribution of the shift

point (where the shift occurs), the common precision of the errors, and the

two AR(1) parameters. Of course, the parameters of the model are

assumed to be known when generating the observations via R, and the

posterior analysis executed with WinBUGS. The next series to be consid-

ered with one shift is the MA(1) and a similar scenario of generating

observations via R and posterior analysis via WinBUGS. A large part of

the chapter focuses on examples taken from econometrics. For example, a

regression model with AR(1) errors and one gradual shift (modeled with a

transition function) is presented and the posterior distribution of the

continuous shiftpoint is derived. This is repeated for a linear regression

model with MA(1) errors and one unknown shift point, followed by a

section on testing hypotheses concerning the moving average parameter of

the two phases. An important subject in shift point models is that of

threshold autoregressive models, where the model changes parameters

depending on when the observation exceeds a given value, the threshold.

The chapter ends with 10 problems and 28 references.

6 Bayesian Analysis of Time Series

1.10 Chapter 10: Residuals and Diagnostic Tests

Residuals of a time series model are deﬁ ned and their use as diagnostic

checks for how well the model ﬁts the data is described. Procedures for

calculating residuals and the corresponding standardized residuals are

explained followed by many examples that illustrate those procedures. R

generates observations from an AR(1) model with known parameters, and

using these as data, a Bayesian analysis computes the posterior character-

istics of the residuals. Given the posterior means of the residuals, the

sample mean and standard deviation of the residuals are computed.

Based on the mean and standard deviation of the residuals, the posterior

analysis is repeated but this time the posterior means of the standardized

residuals are computed. A normal q–q plot of the standardized residuals is

evaluated to see how well the model ﬁts the data. If the model ﬁts the data

well, the normal q–q plot should appear as linear. The more pronounced

the linearity, the more conﬁdent is one of the models ﬁt. The above

scenario of diagnostic checks is repeated for a linear regression model

with AR(1) errors, and a linear regression model with MA(1) errors.

The chapter concludes with comments and conclusions, four problems,

and eight references.

It is hoped that this brief summary of the remaining chapters will initiate

enough interest in the reader so that they will continue to learn how the

Bayesian approach is used to analyze time series.

References

1. Pole, A. and West, M. (1994). Applied Bayesian Forecasting and Time Series Analysis.

Taylor and Francis, Boca Raton, FL.

2. West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, Second

Edition. Springer, New York, NY.

3. Spall, J. (1988). Bayesian Analysis of Time Series and Dynamic Models. Marcel-

Dekker Inc., New York, NY.

4. Venkatesan, D. and Aramugam, P. (2011). A Bayesian Analysis of Changing Time

Series Models: A Monograph on Time Series Analysis. CRC Press, Taylor & Francis,

Boca Raton, FL.

5. Broemeling, L.D. (1985). Bayesian Analysis of Linear Models. Marcel-Dekker Inc.,

New York, NY.

6. Broemeling, L.D. (1987). Econometric and Structural Change. Marcel-Dekker Inc.,

New York, NY.

Introduction to the Bayesian Analysis of Time Series 7

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