COMPOSITE LIKELIHOOD ESTIMATION FOR MULTIVARIATE MIXED POISSON DISTRIBUTIONS Florent Chatelain and Jean-Yves Tourneret IRIT/ENSEEIHT/T´eSA, 2 rue Charles Camichel, BP 7122, 31071 Toulouse cedex 7, France [email protected], [email protected],

ABSTRACT This paper addresses the problem of estimating the parameters of multivariate mixed Poisson distributions. The classical maximum likelihood approach cannot be used for such distributions since they cannot be expressed in simple closedform. This paper studies an estimation strategy based on the maximization of a so-called composite likelihood criterion. This strategy is compared to a more classical estimator based on the method of moments.

1. INTRODUCTION Univariate mixed Poisson distributions have received much attention in image processing applications (see for instance [1], [2] and references therein). These applications include active imaging, where the image is obtained from a scene illuminated with laser light [3], or astronomy, where low-flux images are recorded by using photocounting cameras [2]. A univariate mixed Poisson distribution is the distribution of a random variable N such that the conditional distribution of N |λ is a Poisson distribution with parameter λ (denoted as N |λ ∼ P(λ)). The parameter λ is also a random variable (called intensity) whose distribution is referred to as structure distribution [1, p. 3] or mixing distribution. When λ has an absolutely continuous distribution defined on R+ (whose probability density function is denoted as p(λ)), the probability masses of N can be written: Z

∞

Pr(N = k) =

Pr(N = k|λ)f (λ)dλ, 0

Z = 0

∞

λk exp (−λ)f (λ)dλ. k!

(1)

Multivariate extensions of mixed Poisson distributions are naturally constructed from a joint intensity distribution p(λ1 , ..., λd ) defined on Rd+ . The corresponding masses can This work was supported by the CNRS under MathSTIC Action No. 80/0244.

be computed as follows: Z Z Y d (λ` )k` Pr(N = k) = · · · exp (−λ` )f (λ)dλ, k` ! Rd +

`=1

(2) where k = (k1 , ..., kd ) and λ = (λ1 , ..., λd ). Some properties of multivariate mixed Poisson distributions (MMPDs) have been recently reported in [4]. In particular, conditions ensuring that MMPDs belong to an exponential family have been derived. This result is important since the maximum likelihood estimator is known to have interesting properties when the observations have a distribution belonging to an exponential family. Unfortunately, conditions on the mixing density ensuring that MMPDs belong to an exponential family are generally too restrictive. This paper studies a composite likelihood approach to estimate the parameters of MMPDs when the mixing distribution is a multivariate Gamma distribution. This paper is organized as follows. Section 2 recalls some important results on multivariate mixed Poisson distributions. The main properties of the composite likelihood estimator are explained in section 3. Simulation results and conclusions are presented in sections 4 and 5. 2. MMPDS GENERATED BY GAMMA INTENSITIES 2.1. Multivariate Gamma Distributions A polynomial P (z) with respect to z = (z1 , . . . , zd ) is said to be affine if the one variable polynomial zj 7→ P (z) can be written Azj + B (for any j = 1, . . . , d), where A and B are polynomials with respect to the zi ’s with i 6= j. For any q ≥ 0 and for any affine polynomial P (z), a multivariate Gamma distribution on Rd+ with shape parameter q and scale parameter P (z) (denoted as γq,P ) is defined by its Laplace transform [5]: ψγq,P (z) = [P (z)]−q ,

(3)

on an appropriate domain of existence (note that the affine polynomial has to satisfy the condition P (0) = 1). De-

termining necessary and sufficient conditions on the pair (q, P ) such that γq,P exist is a difficult problem. The reader is invited to look at [5] for more details. This distribution has been used intensively in optics [2] or image processing [3]. Indeed, the complex wave-front amplitude is generally modeled as a zero mean circular Gaussian vector in these applications. Consequently, the vector containing the square modulus of the complex amplitudes is distributed according to multivariate Gamma distribution (with q = −1). 2.2. Negative multinomial distributions For any q ≥ 0 and for any affine polynomial P (z), a negative multinomial distribution N ∼ NMq,P on Nd is defined by its generating function [6]: ! d Y Nk E zk = [P (z)]−q , (4) k=1

with the obvious condition P (1) = 1. Determining necessary and sufficient conditions on the pair (q, P ) such that NMq,P do exist is a difficult problem (see [6] for more details). Note that, for any affine polynomial P , P1 (z1 , . . . , zd ) = P (a1 z1 + b1 , . . . , ad zd + bd ) is also an affine polynomial, for any real numbers ai ’s and bi ’s. 2.3. MMPDs generated by multivariate Gamma distributions The moment generating function of an MMPD N expresses as: ! ! d d Y Y Nk Nk E zk =E E(zk |λk ) , k=1

k=1

= ψλ (z1 − 1, . . . , zd − 1), = ψλ (z − 1),

(5)

3. COMPOSITE LIKELIHOOD ESTIMATOR A composite likelihood is a combination of valid likelihood associated to marginal or conditional events. The concept of composite likelihood has been widely studied in the literature (see [7], [8] and references therein) since the seminal paper of Lindsay [9]. Usual composite likelihoods include the marginal likelihood, the pairwise likelihood [9] and the Besag’s pseudolikelihood [10]. The composite likelihood estimator is obtained by maximizing the corresponding composite likelihood. The advantage of using composite likelihood instead of standard likelihood is to reduce the computational complexity of the optimization procedure. As a consequence, it allows to handle very complex models, even if the full likelihood cannot be expressed in closed form. This is the case when multivariate mixed Poisson distributions are studied since the joint masses Pr(N = k) cannot be generally computed easily by using (2). This section studies a composite likelihood estimator based on the pairwise likelihood of an MMPD governed by a multivariate Gamma distribution. 3.1. Definition Consider n time series N = (N (1) , . . . , N (k) ), where N (i) is distributed according to a MMPD defined on Rd . The maximum likelihood estimator of the unknown parameters θ = (q, pij ) (where pij are the coefficients of the polynomial P ) requires to optimize the masses of a multinomial distribution defined by its moment generating function (5). This problem is complicated since it is difficult to obtain a tractable expression of the masses Pr(N = k) from (5). As an alternative, we consider the log-likelihood associated with pairwise (Nji , Nli ) (corresponding to the ith time series) lij,l (θ) = log Pr(Nji = kji , Nli = kli ). The composite log-likelihood for the ith time series is defined as follows X li (N (i) ; θ) = wj,k lij,k (θ), 1≤j