維納過程單變點(diǎn)模型的貝葉斯參數(shù)估計(jì)

何朝兵

(安陽師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,中國 安陽　455000)

?

維納過程單變點(diǎn)模型的貝葉斯參數(shù)估計(jì)

何朝兵

(安陽師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,中國 安陽455000)

摘要通過引入潛在變量,利用正態(tài)分布的重要性質(zhì)得到了維納過程單變點(diǎn)模型比較簡單的似然函數(shù).結(jié)合Metropolis-Hastings算法對參數(shù)進(jìn)行Gibbs抽樣,基于Gibbs樣本對參數(shù)進(jìn)行估計(jì).隨機(jī)模擬的結(jié)果表明估計(jì)的精度較高.

關(guān)鍵詞潛在變量;可加性;滿條件分布;Gibbs抽樣;Metropolis-Hastings算法

變點(diǎn)問題成為近年來比較熱的研究方向,它在經(jīng)濟(jì)、質(zhì)量控制和醫(yī)學(xué)等領(lǐng)域應(yīng)用廣泛[1-5].變點(diǎn)分析方法主要有非參數(shù)方法、最小二乘法和貝葉斯方法等.而隨著統(tǒng)計(jì)計(jì)算技術(shù)的發(fā)展,貝葉斯變點(diǎn)分析方法越來越受到人們的歡迎,而復(fù)雜性的計(jì)算是貝葉斯方法的難點(diǎn).貝葉斯計(jì)算方法中的Markov chain Monte Carlo (MCMC) 方法是最近發(fā)展起來的一種簡單有效的計(jì)算方法.MCMC方法中的Gibbs抽樣和Metropolis-Hastings算法使變點(diǎn)分析變得非常方便[6-9].Gibbs抽樣可以簡化變點(diǎn)問題,例如,未知參數(shù)的滿條件分布可轉(zhuǎn)化為無變點(diǎn)的后驗(yàn)分布,變點(diǎn)的滿條件分布可轉(zhuǎn)化為分布參數(shù)已知的后驗(yàn)分布.維納過程是具有平穩(wěn)獨(dú)立增量的二階矩過程,是一種特殊的擴(kuò)散過程,它在純數(shù)學(xué)、應(yīng)用數(shù)學(xué)、經(jīng)濟(jì)學(xué)與物理學(xué)中都有重要應(yīng)用.維納過程不只是布朗運(yùn)動(dòng)的數(shù)學(xué)模型,在應(yīng)用數(shù)學(xué)中,維納過程可以描述高斯白噪聲的積分形式;在電子工程中,維納過程是建立噪音的數(shù)學(xué)模型的重要部分;控制論中,維納過程可以用來表示不可知因素.對擴(kuò)散過程變點(diǎn)模型的研究較多[10-13],雖然維納過程是特殊的擴(kuò)散過程,但對維納過程變點(diǎn)模型的研究卻較少[14-15],并且這些文獻(xiàn)都是基于隨機(jī)微分方程的求解來進(jìn)行參數(shù)估計(jì),計(jì)算比較繁瑣,但基于似然函數(shù)并且利用MCMC方法研究此模型還不多見.

本文主要利用MCMC方法研究維納過程單變點(diǎn)模型的參數(shù)估計(jì)問題.通過添加潛在變量得到了比較簡單的似然函數(shù),結(jié)合Metropolis-Hastings算法對參數(shù)進(jìn)行Gibbs抽樣,基于Gibbs樣本對參數(shù)進(jìn)行估計(jì).隨機(jī)模擬的結(jié)果表明估計(jì)的精度較高.

1維納過程單變點(diǎn)模型

定義1隨機(jī)過程W(t)如果滿足:

1)W(0)=0,具有獨(dú)立增量;

2)對任意s,t>0,W(s+t)-W(s)服從正態(tài)分布N(0,σ2t),σ>0,則稱W(t)為以σ2為參數(shù)的維納過程.

當(dāng)維納過程的參數(shù)σ2在某個(gè)時(shí)刻改變時(shí),有如下定義.

定義2隨機(jī)過程W(t)如果滿足:

1)W(0)=0,具有獨(dú)立增量,

(1)

在n個(gè)時(shí)刻t1

W(ti)-W(ti-1)=zi,t0=0,i=1,2,…,n.

假設(shè)已知在觀察時(shí)間區(qū)域(0,tn]內(nèi)有一個(gè)變點(diǎn),即0<τ

τ∈(tm,tm+1],0≤m≤n-1,實(shí)際上m是τ的函數(shù).

由式(1)知

由正態(tài)分布的可加性得

W(tm+1)-W(tm)的觀察值為zm+1.

下面介紹概率論中一個(gè)很重要的結(jié)果,即下面的引理1.

當(dāng)m=0時(shí),D1=?, 當(dāng)m=n-1時(shí),D2=?.記x為X的取值,添加潛在變量后的似然函數(shù)為

2模型的貝葉斯估計(jì)

下面給出參數(shù)的先驗(yàn)分布.

1) 取τ的先驗(yàn)分布為均勻分布,即π(τ)∝1,0<τ

下面介紹MCMC方法的具體步驟.

3隨機(jī)模擬

下面進(jìn)行隨機(jī)模擬試驗(yàn).

表1　各參數(shù)的貝葉斯估計(jì)

圖1　τ的Gibbs抽樣迭代　　　　　　　　　　　　　　　圖2　τ的兩條迭代鏈　　　　　Fig.1　Gibbs sampling iterations of τ　　　　　　　　　　　　　　　Fig.2　Two iterative chains of τ

參考文獻(xiàn)：

[1]PAGE E S. Continuous inspection schemes[J]. Biometrika, 1954,41(1):100-115.

[2]CHERNOFF H, ZACKS S. Estimating the current mean of a normal distribution which is subjected to changes in time[J]. Ann Math Stat, 1964,35(3):999-1018.

[3]CS?RG? M, HORVTH L. Limit theorems in change-point analysis[M]. New York: Wiley, 1997.

[4]PERREAULT L, BERNIER J, BOBéE B,etal. Bayesian change-point analysis in hydrometeorological time series. Part 1. The normal model revisited[J]. J Hydrol, 2000,235(3):221-241.

[5]FEARNHEAD P. Exact and efficient Bayesian inference for multiple changepoint problems[J]. Stat Comput, 2006,16(2):203-213.

[6]LIANG F, WONG W H. Real-parameter evolutionary Monte Carlo with applications to Bayesian mixture models[J]. J Am Stat Assoc, 2001,96(454):653-666.

[7]LAVIELLE M, LEBARBIER E. An application of MCMC methods for the multiple change-points problem[J]. Sig Pro, 2001,81(1):39-53.

[8]KIM J, CHEON S. Bayesian multiple change-point estimation with annealing stochastic approximation Monte Carlo[J]. Comput Stat, 2010,25(2):215-239.

[9]YUAN T, KUO Y. Bayesian analysis of hazard rate, change point, and cost-optimal burn-in time for electronic devices[J]. IEEE Trans Rel, 2010,59(1):132-138.

[10]ABBAS-TURKI L A, KARATZAS I, LI Q. Impulse control of a diffusion with a change point[J]. Stoch Int J Probab Stoch Process, 2015,87(3):382-408.

[11]MISHRA M N, PRAKASA RAO B L S. Estimation of change point for switching fractional diffusion processes[J]. Stoch Int J Probab Stoch Process, 2014,86(3):429-449.

[12]GAPEEV P V, SHIRYAEV A N. Bayesian quickest detection problems for some diffusion processes[J]. Adv Appl Probab, 2013,45(1):164-185.

[13]NEGRI I, NISHIYAMA Y. Asymptotically distribution free test for parameter change in a diffusion process model[J]. Ann Inst Stat Math, 2012,64(5):911-918.

[14]VOSTRIKOVA, L YU. Detection of a “disorder” in a Wiener process[J]. Theor Probab Appl, 1981,26(2):356-362.

[15]HADJILIADIS O, MOUSTAKIDES V. Optimal and asymptotically optimal CUSUM rules for change point detection in the Brownian motion model with multiple alternatives[J]. Theor Probab Appl, 2006,50(1):75-85.

(編輯HWJ)

DOI:10.7612/j.issn.1000-2537.2016.04.014

收稿日期：2015-11-12

基金項(xiàng)目：國家自然科學(xué)基金(61174099); 河南省高等學(xué)校重點(diǎn)科研項(xiàng)目(16A110001)

*通訊作者,E-mail：chaobing5@163.com

中圖分類號O212.8; O212.4

文獻(xiàn)標(biāo)識碼A

文章編號1000-2537(2016)04-0084-05

Bayesian Parameter Estimation of Wiener Process with a Change-Point

HEChao-bing*

(School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China)

AbstractBy introducing a latent variable, the simple likelihood function of Wiener process with a change-point is obtained according to the important property of the normal distribution. All the parameters are sampled by Gibbs sampler together with Metropolis-Hastings algorithm, and the parameters are estimated based on the Gibbs samples. Random simulation results show that the estimations are fairly accurate.

Key wordslatent variable; additivity; full conditional distribution; Gibbs sampling; Metropolis-Hastings algorithm