Bayesian and Frequentist Methods in change point ...

Bayesian and Frequentist Methods in change point problems by

Nader Ebrahimi Division of Statistics Northern Illinois University DeKalb, IL 60115

Sujit K. Ghosh and Department of Statistics University of North Carolina Raleigh, NC 27695

Abstract The change-point problem is one of the important problems of statistical inference in which one tries to detect abrupt change in a given sequence of random variables. This problem, which originally started with statistical control theory (see Page (1955)), has now been applied to different fields, including but not restricted to survival analysis and reliability studies. The literature about change point problem, by now, is quite extensive. In this paper, our goal is to review recent developments in this area. In particular, statistical procedures to estimate discrete change point as well as continuous change point are reviewed.

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Introduction There are two types of change-point problems: (i) continuous change point problem and

(ii) discrete change-point problem. In the continuous change point problem we assume that a continuous random variable T representing a survival time or a failure time has the hazard function


h(t) =

h0 (t) if 0 ≤ t ≤ t0 h1 (t) if t > t0 ,

(1.1)

where h0 (t0 ) = h1 (t) as t → t0 . In other words h(·) has a discontinuity at t0 . The problem is to estimate the change point t0 . In the discrete change point problem we assume that 1

X1 , · · · , Xn are observed as a sequence of independent random variables with an abrupt change at
0 {1, · · · , n}. In other words, there exists two probability distribution functions F0 and F1 such that P (X1 ≤ x1 , · · · , Xn ≤ xn ) =


0  i=1

F0 (xi )

n 

F1 (xi ).

(1.2)

i=
0 +1

The problem is to estimate
0 . A vast amount of published literature on this area clearly signifies the importance of this topic. As a result, it is rather impossible to review this topic that would include all possible results and landmarks on change-point problems. However we have tried to bring in the recent developments in this area. The change-point problem, originally started with the question: is the sequence of observations in a sample obtained from the same identical distribution? If not, is it possible to detect several segments of the observed data that came from identical distributions. Parametric and nonparametric methods have been developed for discrete change point problem. Also, parameteric methods have been developed for continuous change points problem. However nonparametric methods are yet fully developed and the field is quite open. Section 2 of this paper presents various Frequentist and Bayesian approaches to estimate a continuous change point. Section 3, describes methods developed during recent years to estimate a discrete change point.

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Continuous change point problem In reliability theory, a widely accepted procedure is to apply “burn in” techniques to

screen out defective items and improve the lifetimes of surviving items. More specifically, suppose the lifetimes of items are independent but that early failures appear to occur at one rate and late failures (after some threshold) appear to occur at another rate. Because survivors actually have greater resistance to fatal stresses than do newer ones, if the threshold parameter were known, the screening process could entail testing up to this threshold and 2

selling only survivors. However, in practice the threshold parameter will not be known. One helpful tool is to model aging process by the model (1.1). Here the change point plays the role of the threshold parameter. The model (1.1) is also applicable in analyzing biomedical data. For example, suppose patients in the clinical trial receive a treatment at time 0. The event times may represent the time until undesirable side effects occur, in which case we would have an initial hazard rate say h0 (t) and expect a lower hazard rate say h1 (t) after the treatment has been in place for some time. The problem gets quite involved if the patients get several treatments over time. In that case, there may exist more than one change point. Another application is in medical follow up studies after a major operation, e.g. bone marrow transplantation. There is usually high initial risk and then risks settles down to a lower constant long term risk. Both frequentist and Bayesian solutions have been proposed to detect a single change point for a parametric family.

2.1

Frequentist approach: The literature to date focuses primarily on examination of the model (1.1) for specific

forms of h0 (x) and h1 (x). Nguyen et al. (1984) consider the estimation of change point t0 when h0 (x) = λ0 , h1 (x) = λ1 , λ1 > λ0 . Under their assumptions the density has the form f (t) = (λ0 exp(−λ0 t))I(0 ≤ t ≤ t0 ) + (λ1 exp(−λ0 t0 − λ1 (t − t0 )))I(t > t0 ),

(2.1)

where


I(A) =

1 , if xA 0 , otherwise.

Using the n ordered observations t(1) ≤ t(2) ≤ · · · ≤ t(n), they construct a kernel Xn (t) such that the solution of Xn (t) provides a consistent estimate of t0 . The construction of Xn (t)is as follows.

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Define B1 (T, t) = T, B2 (T, t) = I(T > t), B3 (T, t) = T I(T > t), B4 (T, t) = T 2 I(T > t), n  ¯j (t) = 1 = (B , B , B , B ), B Bj (ti , t), j = 1, 2, 3, 4, B 1 2 3 4 ∼ n i=1

and ¯ ¯22 (t)) 21 (B ¯2 (t) − B ¯2 (t) log B ¯2 (t) − 1) ¯3 (t)/B ¯ (t)) = ( B4 (t) − B H(B ∼ ¯ B2 (t) ¯ ¯2 (t)) B3 (t) + B ¯2 (t). ¯1 (t) log B +(1 − B ¯2 (t) B The kernel Xn (t) is then Xn (0) = 0 ¯ (t(i)), i = 1, · · · , n − 1 Xn (t(i)) = H(B ∼ Xn (t) = Xn (t(n − 1)), t ≥ t(n − 1). The construction of Xn (t) is ingenious but apart from providing a consistent estimate this estimator does not seem to have any attractive properties. Later, Yao (1986) and Nguyen and Pham (1987, 1989) propose an alternative method for estimating t0 . It is clear that for each n, the likelihood function is Ln (λ0 , λ1 , t0 ) =

 ∞ n 1 log f (t(i)) = (log f (t))dFn (t), n i=1 0

(2.2)

where Fn is the empirical distribution function and f (t) is given by the equation (2.1). Now the procedure to get estimator of t0 will be in the following three steps: i. For each fixed t0 < t(n), maximize Ln with respect to λ0 and λ1 ; ii. Insert the values of λ0 and λ1 from the step (i) into Ln . Denote the resulting function by L∗n (t0 ) and maximize it by varying t0 ; iii. Insert the value of t0 from the step (ii) and maximize the function Ln to get the new values of λ0 and λ1 . Repeat the step (ii) with the new values of λ0 and λ1 to get new 4

value of t0 . Return to the step (i) and continue the remaining steps until the algorithm converges. It should be noted that as t0 tends to t(n) from below, Ln tends to infinity. Thus, we are led to restrict L∗n (t0 ) to some random intervals [An , Bn ] depending on data alone, with 0 ≤ An < Bn < t(n). Examples of An and Bn are given by Nguyen and Pham (1989). Note also that the function L∗n (t0 ) is not defined at data points t(1), · · · , t(n), only its limits at t0 tends to these data points, from below or above exist. Thus, even by restricting t0 to [An , Bn ], we may not be able to achieve a maximium in the step (ii). In order to solve this ∗ problem one can maximize L∗∗ n (t0 ) = Ln (t0 ) if t(r) < t0 < t(r + 1), r = 0, 1, · · · , n − 1 and + ∗ + ∗ − = max{L∗n (t− 0 ), Ln (t0 )} if t0 = t(r), r = 1, · · · , n − 1, where t(0) = 0 and Ln (t0 ) and Ln (t0 )

are the left and right hand limits at t0 respectively. Mathews et al. (1985) also considered the model h0 (t) = λ0 and h1 (t) = λ1 , but they were interested in testing the hypothesis H0 : λ1 = λ0 against Ha : λ0 > λ1 . See also Loader (1991). Basu et al. (1988) extended the model proposed by Nguyen et al. (1984) to allow general h0 (t) keeping h2 (t) = λ1 . They suggested two semi-parametric estimates for t0 based on the assumption that the proportion of population that dies or fails at time t0 or before is known. Let p1 be such that p0 < p1 < 1, where p0 is the known number of items in the population that failed at time t0 or before. Also let k be the number of order statistics between T([np0 ]) and T([np1 ]) , and ˆ1 = λ

t(i) Σ(t(i) log F¯n (t(i))/k + 1) − (Σ k+1 ))(Σ log F¯n (t(i))/k + 1)) 2

(i) t(i) 2 Σ tk+1 − (Σ k+1 )

and the summations range over i = [np0 ] + 1 to i = [np1 ]. Then the proposed estimators by Basu et al. (1988) of t0 are ˆ 1 + n } tˆ0 (1) = Inf {t : yn (t + hn ) − yn (t) ≤ hn λ

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and ˆ 0 (ξˆp − t) + n }, tˆ0 (2) = Inf {t : log F¯n (t) − log(1 − p0 ) ≤ λ 0 n 

1 where hn = (n)− 4 , yn (t) = − log F¯n (t), F¯n (t) = ( 1

and n = c(log n)n− 2 .

i=1

I(ti > t)/n), ξˆp0 is p0 -th sample quantile

Basu et al. (1988) proved consistency of their estimators. Later Ghosh and Joshi (1992) investigated asymptotic distribution of tˆ0 (2). Modeling the aging process by the change point mean residual life function was initiated by Ebrahimi (1991). He considered the model m(t) = m0 (t)I(0 < t < t0 ) + m1 I(t > t0 ), where m(t) is the mean residual life function or the remaining life expectancy function at age t and is defined as ⎧ ⎪ ⎪ ⎪ ⎨

m(t) = E(X − t|X > t) = ⎪ ⎪ ⎪ ⎩

1 ∞ ¯ F (x)dx, F¯ (t) 0 0,

if if

F¯ (t) > 0 F¯ (t) = 0.

He proposed an estimator for t0 and also studied its asymptotic properties.

2.2

Bayesian methods None of the classical methods described in Section 2.1 are quite satisfactory and puts

stringent restrictions in order to obtain asymptotic normality. Moreover, simulation studies show that asymptotics are poor for small to moderate sample sizes. Bayesian approach on the other hand avoids asymptotics and provide more reliable inference conditional only upon the data actually observed. However, Bayesian methods are also susceptible to impropriety of posterior distribution of t0 if one is not careful in specifying the prior for t0 . In fact, Ebrahimi et al. (1999) showed that an improper prior on t0 , necessarily leads to an improper posterior for t0 . Achkar and Bolfarine (1989) consider the model h0 (t) = λ0 and h1 (t) = λ1 and avoid this problem by using a discrete uniform prior for t0 . However, such a choice 6

could tremendously limit the scope of application. A nice review on problems that arises within Bayesian framework is presented in Ghosh et al (1993) which concentrates on the case λ0 ≥ λ1 . In fact Ghosh et al. (1993) shows that one needs the restriction λ0 ≥ λ1 for some known λ0 in order to make the posterior proper. Ebrahimi et al. (1999) gave general Bayesian formulation of the change point problem and they discussed the case of h(t) = λ0 > h1 (t) = λ1 which yields particularly simple fitting and the Weibull case. Returning to (1.1), Ebrahimi et al. (1999) adopt fully parametric modeling assuming h0 (t) = h0 (t; θ0 ) and h1 (t) = h1 (t; θ1 ). Thus, h0 and h1 are two, possibly distinct, parametric families of hazard functions indexed by θ0 and θ1 , where θ0 and θ1 could be vector valued. To capture the order restriction on h0 and h1 they let S = {(θ0 , θ1 ) : h0 (t; θ0 ) ≥ h1 (t; θ1 ) for all t ≥ 0}. The likelihood takes the form L(θ0 , θ1 , t0 ; t) =

n  j=1

h(tj ) exp(−H(tj )),

(2.3)

where t = (t1 , · · · , tn ) denotes the observed values of the lifetimes. Let vj (t0 ) = 1 if tj ≤ t0 , = 0 if tj > t0 . Then (2.3) becomes L (θ0 , θ1 , t0 ; t) = × exp{−

n  j=1

n  j=1

h0 (tj ; θ0 )vj (t0 ) h1 (tj ; θ1 )1−vj (t0 )

(H0 (min(tj , t0 ); θ0 ) + [H1 (tj ; θ1 ) − H1 (t0 ; θ1 )]+ )}I(S).

(2.4)

For their model they restrict the likelihood so that t0 ≥ t(1) . Certainty of a change-point during the period of observation would then add the further restriction t0 < t(n) . In order to complete probability specifications Ebrahimi et al. (1999) require a prior distribution for θ0 , θ1 and t0 . They assume that it takes the general form f (θ0 , θ1 ) · I(S) · f (t0 )

(2.5)

and that it is proper which assures that the posterior f (θ0 , θ1 , t0 |t) is proper. The prior information on t0 places it on the interval (0, b) with b possibly ∞. The actual support for t0 7

is truncated according to the restrictions imposed by the likelihood. When t0 is not bounded above they argue that if f (t0 ) is improper the posterior must necessarily be improper. Combining (2.4) and (2.5) provides the complete Bayesian specification and thus the posterior f (θ0 , θ1 , t0 |t) which is proportional to L(θ0 , θ1 , t0 ; t) · f (θ0 , θ1 )I(S) · f (t0 ).

(2.6)

The posteriors f (θ0 |t) and f (θ1 |t) enables us to learn about the pre and post threshold hazards. In fact for each t, since h0 (t; θ0 ), H0 (t; θ0 ), h1 (t; θ1 ), H1 (t; θ1 ), h(t; θ0 , θ1 , t0 ) and H(t; θ0 , θ1 t0 ) are all random variables, they all have posterior distributions which would be of interest as well. However, primary interest is in the posterior for t0 , f (t0 |t) and when a change is not certain, P (t0 > t(n) |t). The expression in (2.6) is not analytically tractable so they turn to simulation based approaches for fitting such a model and use Markov chain Monte Carlo techniques. For more details see Ebrahimi et al (1999). Even if the proposed models are all parametric, they can be quite rich if one incorporates finite mixture distributions. The semi-parametric approach is proposed in Ebrahimi and Ghosh (1999) where the problem is formulated in terms of dynamic weight mixture model. Also, the methodologies described by Ebrahimi et al. (1999) could be extended to detect more than one change-point. However one must be careful in specifying the joint prior distribution of change-points.

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Discrete change point problem Discrete change points problem occurs as a result of nonhomogeneity in a sample. In some

cases we have a priori knowledge about the physical nature of the process that generates data and we propose some parametric family to characterize such knowledge. However, one must construct a statistical test based on initial data to check the validity of the model. In this situation a vicious circle arises: in order to create such a model one must guarantee the 8

statistical homogeneity of data, but it is just the same model which is used to check the statistical homogeneity hypothesis of the data obtained. Thus nonparametric methods are the only satisfactory way to tackle this kind of problem. Between opposing parametric and nonparametric methods there is a large area of semi-parametric methods of change-point detection. The broad applicability of discrete change-point problem in various areas makes this area an attractive field of research. In this section we give two applications. Suppose we are monitoring the rate of occurance of a rare health event, for example a specific congenital malformation. Since the number of malformed births is small, one can assume that the malformed births occur according to a realization of a Poisson process with parameter say λ1 . Suppose an epidemic occurs at an unknown instant of time and the normal rate is subject to increase. (Environmental risk factors such as toxic spills, contaminated drinking water and radiation may also increase the normal rate.) Let V be this change. Since the inter-arrival times for a Poisson process are i.i.d. exponential, X1 , X2 , · · · , XV will be independent having common exponential distribution with parameter λ1 and XV +1 , XV +2 , · · · will be independent having common exponential distribution with parameter λ2 , where λ2 ≥ λ1 . The goal is to estimate V. Another important application is on-line quality control of a manufacturing process. Imagine a machine that produces some product. The machine might break down at some point. The purpose of an on-line quality control scheme is to determine, based on the observation of the manufacturing process, whether the machine is functioning properly or not. In this setting, it is assumed that the observations are independent and their common distribution function before the change is F0 and after the change is F1 . Clearly, F0 = F1 implies that the machine is functioning properly. In this section we discuss several Frequentist as well as Bayesian approaches to the discrete change point problem.

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3.1

Frequentist approach

Assume that there exists 1 ≤
0 ≤ n − 1 such that the joint probability density function of X1 , · · · , Xn is


0  i=1

f (xi ; θ1 )

n 

f (xi , θ2 ),

i=
0 +1

where f is known and θ1 = θ2 . If we assume that both θ1 and θ2 are known, then the maximum likelihood estimator of
0 is
ˆ0 = arg max

1≤r≤n−1

where Wi = log[

r  i=1

Wi ,

f (xi , θ1 ) ] and any non-uniqueness in maximization is resolved by suitable f (xi , θ2 )

convention. For unknown θ1 and θ2 the maximum likelihood estimator of (θ1 , θ2 ,
0 ) is the maximizer of
(θ1 , θ2 ,
0 ) =


0  i=1

log f (xi , θ1 ) +

n 

log f (xi , θ2 ).

i=
0 +1

If the conditional maximum likelihood estimator of (θ1 , θ2 ), (θˆ1
0 , θˆ2
0 ) given
0 is available in closed form, then the maximum likelihood estimator of
0 ,
ˆ0 is the maximizer of
(θˆ1
0 , θˆ2
0 ,
0 ). Asymptotic properties of the maximum likelihood estimator of
0 was derived by Bhattacharya (1987), see also Rukhin (1994). Worsley (1986) constructed a confidence interval of
0 in exponential families. Some other approaches to estimation as well as confidence intervals for
0 have been discussed by Siegmund (1988) and references therein. For more details see the book by Sinha et al. (1995). The non-parametric estimator of
0 was proposed by Darkovsky (1976). This estimator is based on the Mann-Whitney discrepancy measure. Later Carlstein (1988) generalized this estimator by considering different measures of discrepancy. Methods based on U -statistic was proposed by Ferger (1991). For more details about different non-parametric approaches we refer you to Brodsky and Darkhovsky (1993) and reference therein.

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3.2

Bayesian approach First we discuss parametric modeling of change point problem. Assume that F0 and

F1 admit densities of f (x|θ0 ) and of f (x|θ1 ) respectively; where θ0 and θ1 could be vector valued but unknown parameters. As we mentioned in the section 1 that the problem is to estimate the discrete-valued parameter
0 from the sampling distribution p(x, · · · xn |
0 , θ0 , θ1 ) =


0  i=1

f (xi |θ0 )

n 

f (xi |θ1 )

i=
0 +1

In order to complete the full probability specification we consider prior distribution for
0 , θ0 and θ1 . As θ0 and θ1 are treated as nuisance parameters we assume that
0 and (θ0 , θ1 ) are independent. In other words, p(
0 , θ0 , θ1 ) = p(
0 )p(θ0 , θ1 ). Then, from a Bayesian point of view one can obtain the posterior distribution of θ0 given the data x = (x1 · · · xn ) as

p(
0 |x) α p(
0 )



p(x|
0 , θ0 , θ1 )p(θ0 , θ1 )dθ0 dθ1 .

(3.1)

Smith (1975) assumes θ0 and θ1 are independent, however such an assumption may not be valid even when the change point is known. Muliere and Scarsini (1984) extends the approach to hierarchial specifications. Typically the discrete uniform prior is used for p(
0 ), however other parametric priors can also be used. If integration in (3.1) can be achieved analytically (e.g. if a conjugate family is used for p(θ0 , θ1 )), then the posterior distribution (which is necessarily discrete) can be enumerated rather straightforwardly. Otherwise Markov Chain Monte Carlo (MCMC) methods are employed to obtain posterior distribution of
0 . For simple parametric families (e.g. exponential, gamma, lognormal etc.) a simple code is easily developed using a generic software called BUGS. We now turn to nonparametric approaches. Notice that given
0 , X1 , X2 , · · · , X
0 are independent and identically distributed random-variables with the common distribution function 11

F0 and X
0 +1 , · · · , Xn are independent and identically distributed random variables with the common distribution function F1 . Again assuming that the random measures F0 and F1 are independent of
0 , nonparametric priors as developed in Ferguson (1973) can be considered for F0 and F1 . Muliere and Scarsini (1985) considers Dirichlet process priors on F0 and F1 . However such a prior process selects a discrete distribution with probability one. Such undesirable feature can be removed by using mixture of Dirichlet process priors. Dykstra and Laud (1981) propose Gamma process prior and extended Gamma process prior on the hazard rate instead of directly priors on F0 and F1 . Other possible priors on F0 and F1 includes a Beta process prior (Hjort, 1990) or Levy process prior. Typically, for such non-parametric classes of priors analytical solutions are rarely available and hence one uses sampling-based method (e.g. Metropolis-Hastigs etc.) to obtain desired posterior distributions. We refer the reader to Domient et al. (1996) for sampling-based methods on nonparametric Bayesian models. In this part we have not considered a third approach which to estimate
0 using sequential procedures. Brodsky and Darkhovsky (1993) presents several nonparametric methods for change-point problems. A brief survey of such parametric and nonparametric problem can be found in Muliere and Scarsini (1993).

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Concluding Remarks In this paper we discussed both Frequentist and Bayesian approaches to different change

point problems. Discrete change point problems are easily handled using sampling based methods. However, when using improper priors one must be careful in verifying the propriety of posterior distributions. On the other hand continuous change point problems are much harder to tackle. Even simple parametric models could lead to very unstable estimates. Bayesian approaches to this problems are flexible. However, they could lead to serious problem of identifiability and impropriety of posterior distributions. Much work is needed

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for the implementation of nonparametric Bayes methods for this problem. As a final remark, it may be worth to extend the idea of a single change point to several (countable) such points when necessary.

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