markov chain markov field dynamics: models and statistics

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Markov Chain Markov Field dynamics:Models and statisticsX. Guyon a & C. Hardouin aa SAMOS , Universite´ Paris 1 , FrancePublished online: 29 Oct 2010.

To cite this article: X. Guyon & C. Hardouin (2002) Markov Chain Markov Field dynamics: Modelsand statistics, Statistics: A Journal of Theoretical and Applied Statistics, 36:4, 339-363, DOI:10.1080/02331880213192

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Statistics, 2002, Vol. 36(4), pp. 339–363

MARKOV CHAIN MARKOV FIELD DYNAMICS:MODELS AND STATISTICS

X. GUYON* and C. HARDOUIN

SAMOS – Universite Paris 1, France

(Received 14 March 2000; In final form 6 March 2001)

This study deals with time dynamics of Markov fields defined on a finite set of sites with state space E, focussing onMarkov Chain Markov Field (MCMF) evolution. Such a model is characterized by two families of potentials: theinstantaneous interaction potentials, and the time delay potentials. Four models are specified: auto-exponentialdynamics (E ¼ Rþ), auto-normal dynamics (E ¼ R), auto-Poissonian dynamics (E ¼ N) and auto-logisticdynamics (E qualitative and finite). Sufficient conditions ensuring ergodicity and strong law of large numbers aregiven by using a Lyapunov criterion of stability, and the conditional pseudo-likelihood statistics are summarized.We discuss the identification procedure of the two Markovian graphs and look for validation tests usingmartingale central limit theorems. An application to meteorological data illustrates such a modelling.

Keywords: Markov field; Markov Chain dynamics; Auto-model; Lyapunov stability criterion; Martingale CLTtheorem; Model diagnostic

AMS Classification Numbers: 62M40, 62M05, 62E20

1. INTRODUCTION

The purpose of this paper is to study time dynamics of Markov fields defined on a measur-

able state space E and a finite set of sites S. We present a semi causal parametric model called

MCMF, Markov Chain of Markov Field, defined as follows: X ¼ ðX ðtÞ; t 2 N�Þ is a Markov

chain on ES and X ðtÞ ¼ ðXiðtÞ; i 2 SÞ is a Markov field on ES conditionally to the past. We

study the properties of this model characterized by two families of potentials: instantaneous

interaction potentials and time-delay potentials.

Space time modelling has been considered in the literature by Preston (1974) for birth and

death processes, Durrett (1995) for particles systems, K€uunsch (1984) and Koslov and

Vasilyev (1980) for the study of reversible and synchronous dynamics, Pfeifer and

Deutsch (1980) for ARMA models in space. Among statistical studies and their applications,

some important contributions are those of Pfeifer and Deutsch (1980); Besag (1974, 1977);

* E-mail: [email protected] author. SAMOS, Universite Paris 1, 90 rue de Tolbiac 75634 Paris Cedex 13, France. E-mail:

[email protected]

ISSN 0233-1888 print; ISSN 1029-4910 online # 2002 Taylor & Francis LtdDOI: 10.1080/0233188021000010803

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Keiding (1975) for birth and death process, Bennett and Haining (1985) for geographical

data, Chadoeuf et al. (1992) for plant epidemiology. The asymptotic properties of these meth-

ods are obtained in a standard way (see Amemiya, 1985; Dacunha-Castelle and Duflo, 1986;

Guyon, 1995; Bayomog et al., 1994).

The aim of this paper is to specify the structure of conditional Gibbs models according to

the kind of state space considered as well as to study ergodicity, identification, estimation and

validation of such models. For simplicity, we consider only time homogeneous dynamics (but

spatial stationarity is not assumed) and the one step Markovian property. Our results can be

generalized to a larger dependency with respect to the past as well as to time inhomogeneous

chains.

After the description of the probability transition Pðx; yÞ of the model in Section 2, we

study in Section 3 some properties about time reversibility, invariant probability measure,

and marginal distributions and we show that the MCMF dynamics is equivalent to a time

homogeneous space � time non causal Markov field representation.

Four examples are depicted in Section 4: the auto-normal dynamics ðE ¼ RÞ, the auto-ex-

ponential dynamics ðE ¼ RþÞ, the auto-Poissonian dynamics ðE ¼ NÞ, and finally, the auto-

discrete dynamics (E qualitative and finite). For each specification, we give sufficient condi-

tions ensuring the ergodicity of the chain with the help of a Lyapunov criterion. Further, we

specify the results for weak-reversible models, i.e., models such that the reverse (in time)

transition Qðx; yÞ belongs to the same auto-model family as Pðx; yÞ.

In Section 5, we summarize the main results on Conditional Pseudo-Likelihood (CPL) es-

timation: consistency and asymptotic (in time) normality of the estimator, tests of nested hy-

potheses based on the CPL-ratio.

Section 6 is devoted to the identification problem i.e., to the determination of the

dependency graphs G ¼ fG;Gg related to the MCMF dynamics. Graph G is undirected

and determines the instantaneous dependency, while graph G is directed and associated

to the time-delay dependency. In the Gaussian case, the characterization of G is given

through the partial-autocorrelations; for non-Gaussian models, we suggest a stepwise proce-

dure based on the CPL.

In Section 7, we propose two validation tests based on CLT for martingales. We conclude

this paper in Section 8 with the study of a real set of meteorological data to which we fit an

auto-logistic MCMF model. The data consist of daily pluviometric measures on a network of

16 stations in the Mekong Delta (Vietnam) during a three-month period (Tang, 1991). The

auto-logistic model allows for information to be gathered on spatial and temporal dependen-

cies, and the forecasting is relatively accurate for some sites. This study has to be considered

as a first illustrative attempt; there is no doubt that it will have to be refined in a precise

investigation and then compared to other space time models, like hidden Markov chains

or threshold models. It will, of course, be interesting to take into account the dual feature

of such data (it is not raining or the precipitation level is observed in Rþ�) and integrate it

in a Gibbs model. Such a generalization and comparisons to other models are subjected to

another study actually in progress.

2. NOTATIONS AND DESCRIPTION OF THE MCMF MODEL

Let S ¼ f1; 2; . . . ; ng be a finite set of sites. We call X an MCMF model if X is a Markov

Chain of Markov Fields (conditionally to the past). The latter are defined on a measurable

state space ðE; eÞ equipped with a positive s-finite measure v (v is usually the counting

measure if E is discrete, and the Lebesgue one if E � Rp). The product space is

340 X. GUYON AND C. HARDOUIN

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(O;OÞ ¼ ðES; e�S) with the product measure vS ¼ v�S . We shall consider the case O ¼ ES

but all the following still holds for O ¼ Pi2SEi, with measures vi on measurable spaces

ðEi; eiÞ; i 2 S.

We use the following notations. Let A be a subset of S; we denote xA ¼ ðxi; i 2 AÞ and

xA ¼ ðxj; j 2 SnAÞ. Let G be a symmetric graph on S without loop: hi; ji denotes that i

and j are neighbours (in particular, i 6¼ j). The neighbourhood of A is @A ¼ fj 2 S : j=2A

s.t. 9i 2 A with hi; jig. We shall write xi ¼ xfig and @i ¼ @fig. With these notations, our

MCMF model X is the following:

� X ¼ ðX ðtÞ; t 2 N�) is a homogeneous Markov chain on O,

� X ðtÞ ¼ ðXiðtÞ; i 2 SÞ is, conditionally to X ðt 1Þ, a Markov field on S, with XiðtÞ 2 E.

We suppose that the transition probability measure of X has a density Pðx; yÞ w.r.t vS .

If E is discrete, Pðx; yÞ ¼ PðX ðtÞ ¼ yjX ðt 1Þ ¼ xÞ. We look at models for which Pðx; yÞ

is defined by an ðxÞ – a.s. admissible conditional energy U ðyjxÞ, i:e:, such that

Pðx; yÞ ¼ Z1ðxÞ exp U ðyjxÞ and

ZðxÞ ¼

ðO

exp U ðyjxÞ vsðdyÞ < 1 ð1Þ

ðZðxÞ ¼P

O exp U ðyjxÞ < 1 if E is discrete). Let X ðt 1Þ ¼ x. From the Moebius inver-

sion formula (Besag, 1974; Prum, 1986; Guyon, 1995), there exists a minimal family Cx

of non empty subsets of S, and conditional potentials F�W ð:jxÞ for each W 2 Cx such that

U ðyjxÞ ¼P

W2CxF�

W ðyjxÞ.

Throughout the paper, we suppose that Cx ¼ C does not depend on x. If we denote 0 a re-

ference state in O ð0 is 0 when E ¼ N;R or RþÞ, then, almost surely in x, we can choose (in a

unique way) potentials according to the ‘‘identifiability conditions’’: F�W ðyjxÞ ¼ 0 if for some

i 2 W ; yi ¼ 0. Besides, for each W in C, there exists a family CW of non-empty parts of S

such that the potential F�W can be written as:

F�W ðyjxÞ ¼ FW ðyÞ þ

XW 02CW

FW 0;W ðx; yÞ:

This means that the energy U is linked to two families of potentials: instantaneous interaction

potentials fFW ;W 2 Cg and ‘‘conditional’’ interaction potentials fFW 0;W ;W 2 C;W 0 2 CW g.

The semi-causal representation is associated to G ¼ fG;Gg where G and G are defined

respectively by the instantaneous and time-delay dependencies:

hi; jiG , hðt; iÞ; ðt; jÞiG , 9W 2 C s:t: fi; jg � W and FW 6¼ 0

hj; iiG, hðt 1; jÞ; ðt; iÞiG , 9W 2 C;W 0 2 CW s:t:FW 0;W 6¼ 0;

i 2 W ; j 2 W 0

Note that G is a directed graph while G is not. Let us define C¼ [W2CCW ; then

U ðyjxÞ ¼P

W2C FW ðyÞ þP

W2C;W 02C FW 0;W ðx; yÞ with the understanding that FW 0;W � 0

if W 0=2CW .

Thus we finally write:

Pðx; yÞ ¼ Z1ðxÞ expnX

W

FW ðyÞ þX

W1;W2

FW1;W2ðx; yÞ

oð2Þ

where we have put:P

W forP

W2C andP

W1;W2for

PW12C

;W22C, with FW ðyÞ ¼ 0 (resp.

FW1;W2ðx; yÞ ¼ 0) if for some i 2 W (resp. i 2 W2), yi ¼ 0.

MARKOV CHAIN MARKOV FIELD DYNAMICS: MODELS AND STATISTICS 341

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There are three components in the neighbourhood of a site i:

� @i ¼ fj 2 Snfig; hi; jiGg: the t-instantaneous neighbourhood of i

� @i ¼ fj 2 S; hj; iiGg: the ðt 1Þ-antecedent neighbourhood of i

� @iþ ¼ fj 2 S; hi; jiGg: the ðt þ 1Þ-successor neighbourhood of i

The semi-causal representation is related to @i and @i, while the non causal representation

that we will present in the next section depends on @i; @i and @iþ.

Therefore, for each t � 1 and A � SðA 6¼ ;Þ, the conditional distribution of XAðtÞ given the

past and X AðtÞ ¼ yA depends on X@Aðt 1Þ ¼ x@A and X@AðtÞ ¼ y@A only, where

@A ¼ fi 2 S : 9j 2 A s:t: hi; jiGg. The corresponding conditional energy is:

UAðyAjyA; xÞ ¼

XW :W\A6¼;

FW ðyÞ þX

W2:W2\A6¼;

nXW1

FW1;W2ðx; yÞ

o

3. SOME PROPERTIES OF AN MCMF

3.1. Time Reversibility, Invariant and Marginal Distributions

In this section we only consider potentials FW1;W2such that FW1;W2

ðx; yÞ ¼ 0 if for an i 2 W1,

xi ¼ 0 or for a j 2 W2; yj ¼ 0. When there is no ambiguity, 0 denotes also the layout with 0 in

any site of S.

The transition P is synchronous if we can write Pðx; yÞ ¼Q

s2S psðx; ysÞ: the values at all

sites s are independently and synchronously relaxed with distributions psðx; :Þ in s.

PROPOSITION 1

(i) The chain is time-reversible if and only if for all W1; W2; x; y : FW1;W2ðx; yÞ ¼

FW2;W1ðy; xÞ. In this case, P has an unique invariant probability measure given by:

pðyÞ ¼ pð0ÞPð0; yÞ

Pðy; 0Þ¼ pð0ÞZ1ð0ÞZðyÞ exp

XW

FW ðyÞ:

(ii) This invariant measure p is usually not a Markov field. If the transition P is synchronous

and reversible, then p has a Markov property.

Proof

(i) This can be derived from K€uunsch (1984), and Guyon (1995, Theorem 2.2.3.).

(ii) We give in Appendix 1 two examples of non Markovian p.

If P is synchronous, only singletons occur for W in FW and for W2 in FW1;W2. As the chain

is reversible, FW1;W2� 0 if either jW1j > 1 or jW2j > 1. This means that

PðxiðtÞjxiðtÞ; xðt 1ÞÞ ¼ PðxiðtÞjx@iðt 1ÞÞ, so that,

Pðx; yÞ ¼ Z1ðxÞ expnX

s2S

FsðysÞ þXhs0;siG

Ffs0g;fsgðxs0 ; ysÞ

o¼Ys2S

psðx; ysÞ


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with

psðx; ysÞ ¼ Z1ðxÞ expnFsðysÞ þ

Xs0

Ffs0g;fsgðxs0 ; ysÞg

o:

Then we have:

pðyÞ ¼ pð0Þ Z1ð0Þ ZðyÞYs2S

expFsðysÞ:

Besides, if Usðzs; yÞ ¼ FsðzsÞ þP

s02@s Ffs0g;fsgðys0;zsÞ; ZðyÞ ¼Ð

expP

s2S

U ðZs; yÞgvSðdzÞ ¼

Ps2S

Ðexp U ðzs; yÞnðdzsÞ ¼ exp

Ps2S C@sðy@sÞ with C@sðy@sÞ ¼ ln

nPzs

exp U ðzs; yÞo

. u

Example 1 A synchronous and reversible transition.

Let E ¼ f0; 1g, S is the one dimensional torus; the transition Pðx; yÞ ¼

Z1ðxÞ expfaP

i2S yiðxi1 þ xiþ1Þg is reversible with invariant law pðyÞ ¼ pð0ÞZ1ð0Þ

expP

i2S Fi1;iþ1ðyÞ,where Fi1;iþ1ðyÞ ¼ ln 1 þ exp aðyi1 þ yiþ1Þ

. The conditional distri-

bution at site l depends on yl2 and ylþ2.

Marginal Distributions For A � S, A 6¼ S, the marginal distributions ðyAjxÞ, conditionally to

x, are generally not local in x, and not explicit and not local in y, except in specific cases as

the Gaussian case. This is illustrated in Appendix 2.

3.2. Non Causal Markov Field Representation

Let X be an MCMF with the semi-causal representation (2). We are going to show that there

is a unique equivalent time homogeneous space � time non-causal Markov field representa-

tion given by the bilateral transitions: for X ðt 1Þ ¼ x and X ðt þ 1Þ ¼ z,

Pðyjx; zÞ ¼ Z1ðx; zÞ exp

� XW1;W2

FW1;W2ðx; yÞ þ FW1;W2

ðy; zÞ

þX

W

FW ðyÞ

�ð3Þ

where the normalizing constant Zðx; zÞ is finite a.s. in ðx; zÞ. The time translation invariant

potentials on S ¼ S � Z are ~FFW�f0gðy; 0Þ ¼ FW ðyÞ and ~FFW1�f0g;W2�f1gððx; 0Þ; ðy; 1ÞÞ ¼

FW1;W2ðx; yÞ. The non-causal representation depends on all the three neighbourhoods @i,

@i and @iþ.

PROPOSITION 2 The representation (2) of MCMF dynamics with the neighbourhood system

f@i; @i�; i 2 Sg is equivalent to the S Space � Time Marko-Field representation (3) with the

neighbourhood system f@i; @i�; @iþ; i 2 Sg.

Proof

(i) It is easy to see that the chain is also a two-nearest neighbours Markov field in time.

Let t be the density of X ð0Þ; the likelihood of ðxð0Þ; xð1Þ; . . . ; xðT ÞÞ is tðxð0ÞÞQt¼1;T Pðxðt 1Þ; xðtÞÞ For 1 ! t ! T 1. the conditional density is:

PðxðtÞjxðt 1Þ; xðt þ 1ÞÞ ¼Pðxðt 1Þ; xðtÞÞ PðxðtÞ; xðt þ 1ÞÞÐ

O Pðxðt 1Þ; aðtÞÞ PðaðtÞ; xðt þ 1ÞvsðdaÞÞ:


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Let us denote x ¼ xðt 1Þ, y ¼ xðtÞ, z ¼ xðt þ 1Þ. As ðX ðt þ 1Þj X ðt 1Þ ¼ xÞ admits an

a.s. finite density, Zðx; zÞ ¼ PðX ðt þ 1Þ ¼ zjX ðt 1Þ ¼ xÞ ¼ÐO Pðx; aÞ Pða; zÞvSðdaÞ, is

finite. We obtain from (2):

Pðyjx; zÞ ¼

exp

�XW1;W2

FW1;W2

ðx; yÞ þ FW1;W2ðy; zÞ

þX

W

fW ðyÞ þ FW ðzÞ

�ðO

exp

�XW1;W2

FW1;W2

ðx; aÞ þ FW1;W2ða; zÞ

þX

W

FW ðaÞ þ FW ðzÞ

�v sðdaÞ

:

This is nothing other than (3).

(ii) Conversely, let X be the space � time Markov field on S (3) with the neighbourhood

system f@i; @i; @iþg. The field being time-homogeneous, a direct computation shows

that its semi-causal representation is (2). u

We can derive easily from (2) or (3) the semi-causal or non-causal conditional distributions

at any point (i; t). Note that we have for each i; j : i 2 @j()j 2 @iþ.

Figure 1 shows an example of both representations. For the causal representation, we have

@i ¼ fj; kg; @i ¼ fi; jg (note that i 2 @lÞ; while for the non-causal representation,

@i ¼ fj; kg; @i ¼ fi; jg; @iþ ¼ fi; j; lg and now i 2 @l and l 2 @iþ.

The semi-causal conditional distribution at (i; t) is:

Pðyijy@i; x@iÞ ¼ Z1i ðy@i; x@iÞ exp

n XW1;W23i

FW1;W2ðx; yÞ þ

XW3i

FW ðyÞo: ð4Þ

The non-causal conditional distribution Pðyijy@i; x@i ; z@iþÞ at (i; t) has conditional energy:XW1;W23i

FW1W2

ðx; yÞ þ FW1;W2ðy; zÞ

þXW3i

FW ðyÞ:

FIGURE 1 AN EXAMPLE OF SEMI-CAUSAL AND NON-CAUSAL REPRESENTATIONS OF AN MCMF.


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A time inhomogeneous Markov field on S ¼ S � T is not reducible to a semi-causal re-

presentation because @i and @iþ are strongly related: the Markov field (3) is very specific.

For the example considered in Figure 1, the non-causal representation without the dotted

arrow ði; t 1Þ ! ðl; tÞ cannot be reducible to a causal representation. In all the following,

we consider the time homogeneous framework.

Reversed Dynamics We give in Appendix 3 an example where the time reversed process of

an MCMF is no more an MCMF.

4. ERGODICITY OF AUTOMODELS

Here we examine several examples of auto-models (see Besag, 1974; Guyon, 1995), and we

give conditions ensuring their ergodicity. We will use the Lyapunov Stability Criterion (see

e.g., Duflo, 1997, 6.2.2).

For n � 0, let F n ¼ sðX ðsÞ; s ! nÞ be the s-algebra generated by the X ðsÞ; s ! n. The Lya-

punov Stability Criterion is the following. Let us assume that a Markov chain defined on a

closed subset of Rd is strongly Feller, and that there exists a Lyapounov function V such that,

for n > 0, and for some 0 ! a < 1 and b < 1

E

V ðXnÞjF n1

�! aV ðXn1Þ þ b:

Then if there is at most one invariant probability measure, the chain is positive recurrent.

Besides, the following law of large numbers applies: 1=ðn þ 1ÞP

k¼0;n jðXkÞ !a:s:

mðjÞ for

any m-integrable function j, where m is the invariant measure of the chain. A difficulty is to

check whether j is m-integrable. Another useful result is available: we get the same law of

large numbers for any function j which is m-a.s. continuous and s.t. jjj ! aV þ b for

some constants a, b.

4.1. The Autoexponential Dynamics

Let us consider S ¼ f1; 2; . . . ; ng, and E ¼ Rþ. We suppose that for each i, conditionally to

ðX iðtÞ ¼ yi;X ðt 1Þ ¼ x) (later, we shall note this condition ðyi; xÞ), the distribution of XiðtÞ

is exponential with parameter liðyi; xÞ (in fact liðy@i; x@i )). From Arnold and Strauss (1988)

we can write:

U ðyjxÞ ¼Xi2S

aiðxÞyi þX

W :jW j�2

lW ðxÞyW

with yW ¼Q

i2W yi; aiðxÞ > 0 for any x and lW ðxÞ � 0. Then, the parameters are equal to

liðyi; xÞ ¼ aiðxÞ þ

PW3i lW ðxÞyWnfig. The ergodicity of the chain is obtained through the fol-

lowing assumption:

E1 (i) 8i 2 S; 8W ; x ! aiðxÞ and x ! lW ðxÞ are continuous.

(ii) 9a 2 ð0;1Þ; s.t. 8x 2 ðRþÞS; 8i 2 S : aiðxÞ � a:

PROPOSITION 3 Under assumption E1, the autoexponential dynamics is positive recurrent.

The strong law of large numbers holds for any integrable function and particularly for

functions x 7!f ðxÞ such that jf ðxÞj aVrðxÞ þ b (for some finite constants a and b) with

VrðxÞ ¼P

i2S xri ; r being any positive integer.


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Proof The proof uses the Lyapounov stability criterion.

� The lower bound condition E1 (ii) says that exp U ðyjxÞ ! expaP

i2S yi, which is Le-

besgue integrable. Then, the chain is strongly Feller.

� As P is strictly positive, the chain is irreducible and there exists no more than one invariant

distribution (see Duflo, 1997, Proposition 6.1.9).

� On the other hand, for any positive integer r, we have for all x; y 2 E; i 2 S,

E�

XiðtÞrjx; yi

�¼

Gðr þ 1Þ

liðyi; xÞr!

Gðr þ 1Þ

ar:

It follows:

E�Vr

�X ðtÞ

��X ðt 1Þ ¼ x�!

Gðr þ 1Þ

arjSj < 1 ð5Þ

where jSj is the cardinal of S. u

Example 2 A case of ‘‘weak-reversibility’’

We assume that, conditionally to ðX iðt 1Þ ¼ xi;X ðtÞ ¼ yÞ, the reversed transition Qðy; xÞ

is also exponential with parameter miðxi; yÞ: Then, from Arnold and Strauss (1988), the joint

density for ðX ðt 1Þ ¼ x;X ðtÞ ¼ yÞ must be:

f ðx; yÞ ¼ C exp U ðx; yÞ;

with U ðx; yÞ ¼ X

W¼W1�W2�S2;W 6¼;

lW xW1yW2

with lW > 0 if jW j ¼ jW1j þ jW2j ¼ 1 and lW � 0 if jW j � 2. So E1 is satisfied.

Example 3 Besag’s conditional auto-models

We consider the case of conditional auto-models, i.e., for W � S, lW ¼ 0 if jW j > 2.

Therefore:

U ðyjxÞ ¼Xi2S

aiðxÞyi þXhi;ji

bijðxÞyiyj:

E1-(ii) is satisfied if aiðxÞ � a and bijðxÞ � 0 for all x, i, j. For example, if

U ðyjxÞ ¼Xi2S

diyi þXhi;jiG

bijyiyj þXhj;iiG

ajixjyi ð6Þ

with di > 0, bij and aij � 0, the distribution of XiðtÞ conditionally to ðyi; xÞ is exponential with

parameter liðyi; xÞ ¼ di þ

Pj2@i bijyj þ

Pl2@i alixl. The condition E1 is fulfilled.

4.2. The Autonormal Dynamics

Let E ¼ R. We assume that the conditional distribution of XiðtÞ given X iðtÞ ¼ yi and

X ðt 1Þ ¼ x is Gaussian with mean miðyi; xÞ and variance s2

i ðyi; xÞ. The principle of compat-


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ibility requires (see Arnold and Press, 1989; Arnold et al., 1991) that the conditional energy

is of the following feature:

U ðyjxÞ ¼Xi2S

aiðxÞyi þ biðxÞy

2i

þ

XW s:t:jW j�2

gW ðxÞylWW

where ylWW ¼

Qi2W y

lii , li ¼ 1; 2; and the functions a; b; g ensuring that all the conditional var-

iances are positive and U ð(jxÞ is admissible.

Let us consider now Besag’s automodels. Then U ðyjxÞ ¼ P

i2SfaiðxÞyi þ biðxÞy2i g þP

hi;jiGfgijðxÞyiyj þ dijðxÞy

2i yj þ vijðxÞyiy

2j þ wijðxÞy

2i y2

j g. A typical example is:

Pðx; yÞ ¼ Z1ðxÞ exp

(Xi2S

yi

di þ

Xl2@i

alixl þ giyi

!þXhi;ji

bijyiyj

)ð7Þ

with gi > 0; bij ¼ bji such that the matrix Q ¼ ðQijÞ defined by Qii ¼ 2gi and Qij ¼ bij is

definite positive. The conditional distribution of XiðtÞ given ðyi; xÞ is Gaussian with mean

ðdi þP

l2@i alixl þP

j2@i bijyjÞ�

2gi and variance ð2giÞ1.

Ergodicity We consider the model (7). Let us note D ¼ ðDilÞ;Dil ¼ ali; i; l 2 S, and

d ¼ ðdiÞ; a direct identification of the distribution of

ðY jxÞ ¼�X ðtÞ

��X ðt 1Þ ¼ x�

gives

ðY jxÞ ) N Sðm þ Ax;GÞ; with

m ¼ Q1d;A ¼ Q1D; and G ¼ Q1:

This can be written as an AR(1) process X ðtÞ ¼ m þ AX ðt 1Þ þ eðtÞ with a Gaussian white

noise e having covariance matrix G. Define t ¼ ðI AÞ1m when ðI AÞ is regular. Then the

zero-mean variable X �ðtÞ ¼ X ðtÞ t verifies X �ðtÞ ¼ AX �ðt 1Þ þ eðtÞ. Let rðAÞ be the

spectral radius of A (i.e., the greatest modulus of its eigen values).

PROPOSITION 4 If rðAÞ < 1, then (I � A) is regular and the chain is ergodic with a

Gaussian stationary measure m.

Proof The result is classic, given for example in Duflo (1997), Theorem 2.3.18. As e is

Gaussian, the stationary distribution for X � is the one ofP

k�0 AkeðkÞ, i:e:, N Sð0;SÞ;S being

the unique solution of G ¼ S ASðtAÞ, that is S ¼P

k�0 AkGðtAÞk . For X ðtÞ, we add the

mean t. As e has finite moments of all orders, we get the strong law of large numbers for any

continuous integrable function. u

4.3. The Auto-Poissonian Dynamics

Now E ¼ N (v is the counting measure) and we consider the dynamics associated with the

following conditional energy:

U ðyjxÞ ¼Xi2S

aiðxÞyi lnðyi!Þ

þXi 6¼j

bijðxÞyiyj

where bijðxÞ ! 0 for all i, j in order to make U admissible. Conditionally to ðyi; xÞ;XiðtÞ has a

Poisson distribution with parameter liðyi; xÞ ¼ expfaiðxÞ þ

Pj:j 6¼1 bijðxÞyjg. The ergodicity of

the chain is obtained through the following hypothesis:

P1: 9M < 1 such that 8i 2 S; supx aiðxÞ ! M


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PROPOSITION 5 Under P1, the auto-Poissonian chain is positive recurrent. Besides, the

strong law of large numbers holds for any m-integrable function and for any functions f such

that jf ðxÞj aGuðxÞ þ b (for some finite constants a and b) where GuðxÞ ¼Q

i2S eui;xi , for

any fixed u 2 ðRþÞS.

Proof The proof follows the same lines as the one given for the auto-exponential dynamics.

P1 implies that liðyi; xÞ ! eM . Then the transition is strongly Feller. As this transition is

strictly positive, the invariant measure is unique.

For the conditional moment generating function, we have, for all s > 0, CXiðsÞ ¼

E½esXiðtÞjyi; x, ¼ eliðyi;xÞðes1Þ ! expðeM ðes 1ÞÞ. Let us set u ¼ ðui; i 2 SÞ 2 ðRþ

ÞS; Ku ¼

maxi2S expðeM ðeui 1ÞÞ and GuðxÞ ¼Q

i2S euixi . Then, conditionally to X ðt 1Þ ¼ x,

EhGu

�X ðtÞ

�jxi¼ E

" Yi2Snfjg

euiXiðtÞE�eujXjðtÞ

��y j; x��x#

! KuE

" Yi2Snfjg

euiXiðtÞ��x#:

Taking successive conditional expectations, one obtains:

E

"Yi2S

euiXiðtÞ��X ðt 1Þ

#! K jSj

u < 1:

u

For example, P1 is satisfied for

aiðxÞ ¼ di þXl2@i

alixl and bijðxÞ ¼ bij ¼ bji ð8Þ

with ali ! 0 for any i, l, and bij < 0 if hi; ji and 0 else.

4.4. The Auto-discrete Dynamics

Let E be a finite qualitative set. The conditional energy of the auto-model is:

U ðyjxÞ ¼Xi2S

aiðyi; xÞ þXhi;ji

bijðyi; yj; xÞ:

As aið(Þ and bijð(Þ are finite conditional potentials, the ergodicity is ensured without any

restrictions on the parameters. For instance, the autologistic dynamics is defined for

E ¼ f0; 1g, and U ðyjxÞ ¼P

i2S aiðxÞyi þP

hi;ji bijyiyj; XiðtÞ has a conditional Bernouilli

distribution

with parameter

pi yi; x� �

¼exp di yi; x

� �1 þ exp di yi; xð Þ

with di yi; x� �

¼ aiðxÞ þXj2@i

bijyj: ð9Þ


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For E ¼ f0; 1; . . . ;mg, the autobinomial dynamics is given by U ðyjxÞ ¼Pi2S aiðxÞyi þ

Phi;ji bijyiyj. Conditionally to yi; x

� �;XiðtÞ has a Binomial distribution

B m; yi yi; x� ��

with parameter yi yi; x� �

¼ ð1 þ expfaiðxÞ þP

j2@i bijyjgÞ1 if we take the

counting measure with densitym

yi

� �as reference measure.

5. CONDITIONAL PSEUDO-LIKELIHOOD STATISTICS

5.1. Parametric Estimation

We suppose that the transition probabilities of the MCMF depend on an unknown parameter

y, y lying in the interior of Y, a compact subset of Rd . When ergodicity holds, we can obtain

the asymptotic properties of the estimators derived from the classic estimation methods (max-

imum likelihood, pseudo maximum likelihood, maximum of another ‘‘nice’’ objective func-

tion) in a standard way. An analytical and numerical difficulty inherent to the maximum

likelihood procedure is the complexity of the normalizing constant ZyðxÞ in the likelihood;

one can use stochastic gradient algorithms to solve the problem (see Younes, 1988); another

numerical option is to compute the log likelihood and its gradient by simulations via a Monte

Carlo algorithm (see Geyer 1999, Chapter 3; and Geyer and Thompson, 1992). A third alter-

native (see Besag, 1974) is to consider the Conditional Pseudo-Likelihood (CPL); in the pre-

sence of strong spatial autocorrelation this method performs poorly, and we then have to use

the previous procedures. In the absence of strong dependency, it has good asymptotic proper-

ties, the same rate of convergence as the maximum likelihood estimator with a limited loss in

efficiency (see Besag, 1977; Guyon, 1995; Guyon and K€uunsch, 1992). The asymptotic beha-

viour follows in a standard way (see Amemiya, 1985; Dacunha-Castelle and Duflo, 1986, for

general theory – Besag, 1984; Guyon and Hardouin, 1992; Guyon, 1995, for Markov field

estimation – Bayomog, 1994; Bayomog et al., 1996, for field dynamics estimation). We

briefly recall the main results.

We assume that the chain is homogeneous and that for all i 2 S; x; y 2 E; y 2 Y, the

conditional distribution of XiðtÞ given X iðtÞ ¼ yi and X ðt 1Þ ¼ x is absolutely continuous

with respect to v, with positive conditional density fiðyi; yi; x; yÞ (which is in fact

fiðyi; y@i; x@i; yÞ).Let y0 be the true value of the parameter, and P0 be the associated transition. The process

is observed at times t ¼ 0; . . . ; T . Let us denote yyT ¼ arg miny2YUT ðyÞ the conditional pseu-

do-conditional likelihood estimator (CPLE) of y, a value minimizing the opposite of the Log-

CPL:

UT ðyÞ ¼ 1

T

XT

t¼1

Xi2S

ln fiðxiðtÞ; xiðtÞ; xðt 1Þ; yÞ:

The following conditions C and N ensure the consistency and the asymptotic normality of yyT

respectively.

Conditions for consistency (C)

C1: For y ¼ y0, the chain X is ergodic with a unique stationary measure m0.

C2: (i) For all i 2 S; x; y 2 E; y 7! fiðyi; yi; x; yÞ is continuous.


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(ii) There exists a measurable m0 � P0-integrable function h on E � E such that for

all i 2 S; y 2 Y; x; y 2 E; j ln fiðyi; yi; x; yÞ� ln fiðyi; yi; x; y0Þj hðy; xÞ.C3: Identifiability: if y 6¼ y0 then

Pi2S m0 fx s:t: fið�; �; x; y0Þ 6¼ð fið�; �; x; yÞgÞ > 0.

Let fð1Þ

i ðyÞ and fð2Þ

i ðyÞ stand for the gradient and the Hessian matrix of fiðyi; yi; x; yÞ with

respect to y. We define the following conditional (pseudo) information matrices: for

x; y 2 E; i; j 2 S; i 6¼ j; y 2 V0, a neighbourhood of y0;V0 � Y:

Iijðyfi;jg; x; yÞ ¼ Ey0

"fð1Þ

i ðyÞf ð1Þj ðyÞ0

fiðyÞfjðyÞ

��X fi;jgðtÞ

¼ yfi;jg;X ðt 1Þ ¼ x

#;

Iijðx; yÞ ¼ Ey0Iij X fi;jgðtÞ;X ðt 1Þ; y� �

X ðt 1Þ ¼ x��

:

If Zi ¼ ð@=@yÞ ln fi XiðtÞ;X iðtÞ;X ðt 1Þ; y� �

jy¼y0, then Iij yfi;jg; x; y

� �is the covariance matrix

of ðZi; ZjÞ given X fi;jg ¼ yfi;jg and X ðt 1Þ ¼ x.

Conditions for asymptotic normality (N)

N1: For some V0 � Y., a neighbourhood of y0; y 7! fi yi; yi; x; y� �

is two times continuously

differentiable on V0 and there exists a measurable, m0 � P0-square integrable function H

on E � E such that for all y 2 V0, x; y 2 E; 1 ! u; n ! d:

1

fi

@

@yu

fi yi; yi; x; y� �� and

1

fi

@2

@yu@ynfi yi; yi; x; y� �� ! Hðy; xÞ:

N2: I0 ¼P

i2S Em0IiiðX ðt 1Þ; y0Þ½ , and J0 ¼

Pi;j2S Em0

IijðX ðt 1Þ;�

y0Þ, are positive definite. >

PROPOSITION 6 (Bayomog et al., 1996; Guyon and Hardouin, 1992; Guyon, 1995)

(i) Under assumptions C; yyT

P0

T !1��! y0

(ii) Under assumptions C and N,ffiffiffiffiT

pðyyT y0Þ

Dd

T !1��!N dð0; I1

0 J0I10 Þ

Identifiability of the model and regularity of I0

We give here sufficient conditions ensuring both (C3) and the regularity of the matrix I0. We

suppose that each conditional density belongs to an exponential family

fi yi; yi; x; y� �

¼ KiðyiÞ exp tygi yi; yi; x� �

Ci y; yi; x� �

; ð10Þ

and set the hypotheses:

(H): there exists ðiðkÞ; xðkÞ; yðkÞ; k ¼ 1; dÞ s.t. g ¼ ðg1; g2; . . . ; gdÞ is of rank d, where

we denote gk ¼ giðkÞ yiðkÞðkÞ; yiðkÞðkÞ; xðkÞ� �

. We strengthen (H) in (H0):

ðH01Þ: for each i 2 S; giðyi; yi; xÞ ¼ hiðyiÞGiðyi; xÞ with hið�Þ 2 R

ðH02Þ: 9a > 0 s.t. for each i; x; yi;VarfhiðYiÞjyi; xÞg � a > 0


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ðH03Þ: 9fðiðkÞ; xðkÞ; yiðkÞðkÞÞ; k ¼ 1; dg s.t. G ¼ ðG1;G2; . . . ;GdÞ is of rank d where

Gk ¼ GkðyiðkÞðkÞ; xðkÞÞ. Obviously, ðH0Þ implies ðHÞ.

PROPOSITION 7 We suppose that the conditional densities ffi �; yi; x; yð Þ; i 2 Sg belong to the

exponential family (10).

(i) Under ðHÞ, the model related to this family of conditional densities is identifiable.

(ii) I0 is regular under ðH0Þ.

The proof is given in Appendix 4. A sufficient condition ensuring that J0 is positive

definite can also be obtained using a strong coding set; this idea is developed in Jensen

and K€uunsch (1994). It is sketched in the same Appendix 4. Many models fulfil ðHÞ or

ðH0Þ, for instance log-linear models or automodels. We give explicit conditions ðCÞ and

ðNÞ for the autoexponential dynamics in Section 5.3.

5.2. Testing Submodels

It is now possible to test the submodel ðHqÞ : y ¼ jðaÞ; a 2 Rq; q < d;j : Rq! Rd such

that:

� j is twice continuously differentiable in a bounded open set L of Rq, with jðLÞ � Y, and

there exists a0 2 L such that jða0Þ ¼ y0.

� R ¼ ð@j=@aÞja¼a0is of rank q.

Let �yyT ¼ arg mina2LUT ðjðaÞÞ be the CPLE of y under ðHqÞ, and �II0 ¼ �II ða0Þ the associated

information matrix I . If A is a positive definite matrix, with spectral decomposition

A ¼ PDP0, P orthogonal, we take Að1=2Þ ¼ PDð1=2ÞP0 as a square root of A.

PROPOSITION 8 CPL ratio test (Bayomog, 1994; Bayomog et al., 1996; Guyon, 1995) If

UT ðyÞ and UT ðjðaÞÞ satisfy assumptions C and N, then, under ðHqÞ, we have as T ! 1:

DT ¼ 2T UT�yyT

� � UT yyT

! "! "!D Xdq

i¼1

liw21;i

where the w21;i’s are independent w2

1 variables and the li; i ¼ 1; d q, are the ðd qÞ strictly

positive eigenvalues of G0 ¼ J1=20 I1

0 R�II10 R0

� �J

1=20 .

Let C be a coding subset of S, i.e., for all i; j 2 C; i 6¼ j; i and j are not (instantaneous)

neighbour sites. We can define coding estimators as previously, but in the definition of the

coding contrast UCT ðyÞ, the summation in i is then restricted to C. For those estimators, we

have IC0 ¼ J C0 ; so the asymptotic variance forffiffiffiffiT

pðyyC

T y0Þ is ðIC0 Þ1, and the former statistic

has a w2dq asymptotic distribution.

5.3. An Example: The Autoexponential Model

Here we look at these various conditions for the model given in Section 4.1. The assumption

E1 ensures the ergodicity. For a positive integer r and VrðxÞ ¼P

i2S xri , we define the follow-

ing property for a function f :

There exist two finite constants a and b such that

f ðxÞ ! aVrðxÞ þ b: ð11Þ


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We add the following hypotheses to obtain the consistency and the asymptotic normality of

the CPL estimator.

E2: For all i 2 S;W 2 C; y 2 Y and x 2 E; aiðx; yÞ and lW ðx; yÞ satisfy (11).

E3: If y 6¼ y0, then there exists A � ðRþÞS; lðAÞ > 0, such that for one i 2 S and all

x 2 A; lðfy 2 Ejliðyi; x; yÞ 6¼ liðy

i; x; y0ÞgÞ > 0.

E4: The functions y ! aiðx; yÞ; y ! lW ðx; yÞ are twice continuously differentiable for all

x; i; j;W , and for all 1 ! u; n ! d, and the absolute values of their first and second order

derivatives satisfy (11).

E5: I0 and J0 are positive definite.

E6: ðH0Þ is satisfied and J0 is positive definite.

PROPOSITION 9 Let yyT be the CPLE of y for the autoexponential model. Then:

(i) under assumptions E1, E2, and E3, byyT is consistent. If we add E4 and E5, yyT is

asymptotically normal.

(ii) under assumptions E1, E2, E4 and E6, yyT is asymptotically normal.

Proof E1 implies C1 and E4 implies C2-(i). E3 implies C3. Then we just have to show that

E2 implies C2-(ii). The conditional density of XiðtÞ is given here by

ln fiðyi; yi; x; yÞ ¼ ln liðyi; x; yÞ � yiliðyi; x; yÞ. Then j ln fiðyi; yi; x; yÞ � ln fiðyi; yi; x; y0Þj j lnðliðyi; x; yÞ=liðyi; x; y0ÞÞjþ yijliðyi; x; yÞ� liðyi; x; y0Þj. As j lnðx=yÞj jx � yjðx�1 þ y�1Þfor all x; y > 0, we obtain j ln fiðyi; yi; x; yÞ � ln fiðyi; yi; x; y0Þj ðð2=aÞ þ yiÞ jliðyi; x; yÞ �liðyi; x; y0Þj. Finally, there exist two integers r and r0 and four constants a1; a2; a3; a4 such

that we can take hðy; xÞ ¼P

i2S 2ðð2=aÞ þ yiÞ fa1VrðxÞ þ a2 þ ða3Vr0 ðxÞ þ a4ÞP

j 6¼i yjg.

On the other hand, E4 implies that the modulus of the first and second derivatives of

liðyi; x; yÞ with respect to y are bounded by a square integrable function of x and y,

and this ensures N1. Finally, E5 is N2. In another hand, E6 ensures E3 and E5 (see

Proposition 7). u

Example 4 The conditional-exponential dynamics.

For the particular model (6), all conditions E are fulfilled without any assumption on the

parameters. Besides, E6 is satisfied with d ¼ n þ ð1=2ÞP

i2S j@ij þP

i2S j@ij; liðyi; x; yÞ ¼t

ygiðyi; xÞ;t y ¼t ððdiÞi2S; ðbijÞhi;jiG ; ðajiÞhj;iiG Þ 2 Rd .

6. MODEL IDENTIFICATION

We suppose that we want to fit a semi-causal MCMF dynamics model; first, we have to

determine the graphs G and G; then, we will estimate the parameters, and lastly validate

the model. There are two possible strategies for the identification procedure, the choice

depending on the complexity of the problem and on the number of sites. The first one is

global: we could try to determine globally the graphs by CPL maximization, joined with a

convenient ‘‘AIC’’ penalization criterion.

On the other hand, a less expensive procedure is to work locally site by site, providing

estimations for @i and @i for each site i 2 S. For this, we maximize the likelihood of

XiðtÞ; t ¼ 1; T , conditionally to X iðtÞ ¼ yi (jSj 1 sites) and X ðt 1Þ ¼ x (jSj sites) (the

conditional distribution of XiðtÞ depends on (2jSj 1) sites) using a backward procedure;


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we choose a small signification level for the adequate statistics in order to keep only the

most significant variables. This can be associated with a forward procedure if we want to

take into account a particular information on the geometry of S. Further, we have to harmo-

nize the instantaneous neighbourhood relation to get a symmetric graph G: if j 2 @@i, we

decide i 2 b@@j. Thus we generally get an overfitted model, and the next step is to reduce it

by progressive elimination using a descending stepwise procedure. Finally, if we have got

two or more models, we choose the one which minimizes the AIC (or BIC) criterion.

In a Gaussian context, the computation is particularly easy and fast, because it is linear and

explicit. Using the partial correlation rP, we have the characterizations (see Garber, 1981;

Guyon, 1995 x1.4.): rPðXj;XijXLiÞ ¼ 0 , j=2Li ¼ @i [ @i. This equation allows us to deter-

mine Li by a fast linear stepwise procedure of the regression of XiðtÞ on the ð2jSj 1Þ other

variables.

For other log linear models, the conditional log likelihood is concave so we can get the

CPLE using a gradient algorithm. We then consider the general procedure given previously.

An alternative is to follow the Gaussian approach, even if the model is not Gaussian. This

can be done when the dimension of the parameter becomes large enough to make the

results suspicious and slow progressing. This procedure is used in Section 8.

7. MODEL VALIDATION

7.1. Some Central Limit Theorems

In the case E � R, validation tests are based on the estimated conditional residuals. Let us

denote y0 and yyT the true value of the parameter and its CPLE respectively. Let us set:

eit ¼ XiðtÞ mit and eeit ¼ XiðtÞ mmit ð12Þ

where mitðyÞ ¼ E½XiðtÞjXiðtÞ;X ðt 1Þ; y,; mit ¼ mitðy0Þ; mmit ¼ mitðyyT Þ is explicit in y@i; x@i

and yyT .

For an autodiscrete dynamics on a K-states space E, we will have an expression equivalent

to (12), but with the ðK 1Þ-dimensional encoded variable ZiðtÞ related to XiðtÞ (see Section

7.2.4).

From (12), we propose a validation statistic; we derive its limit distribution using a Central

Limit Theorem for martingales (Duflo, 1997; Hall and Heyde, 1980).

We build up two tests, based on the estimated residuals eeit and on the squared residuals ee2it.

The latter allows for the detection of possible variance deviation. Anyway, those tests are

more useful to reject a model than to select the best one.

7.1.1. CLT for the Residuals eit

We denote A ? B if the variables A and B are independent. Let C be a coding subset of S (for

G), eit ¼ ðeit=sðeitÞ and ~eet ¼P

i2C eit :

(i) eit is zero mean and of variance 1;

(ii) eit ? XjðtÞ if i 6¼ j and eit ? Xjðt 1Þ for all j 2 S.

Besides, for all i and j which are not neighbour sites, XiðtÞ and XjðtÞ are independent,

conditionally to the past and the neighbourhood values on @i; @j. Then eit ? ejs if t 6¼ s,

for all i; j 2 S, and conditionally to X ðt 1Þ and X �CCðtÞ; eit ? ejt for all i; j 2 C; i 6¼ j;


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(iii) as E eitjF t1j ¼ E E eitjyi; x

� �F t1

� ��¼ 0; ðeitÞt�0 is a square integrable martingale dif-

ference sequence w.r.t. the filtration ðF t ¼ sðX ðsÞ; s ! tÞ; t � 0Þ. Then ~eet ¼P

i2C eit is

also a square integrable martingale difference sequence.

We can thus apply a martingale’s CLT (Duflo, 1996; Corollary 2.1.10). For a square integ-

rable martingale ðMtÞ, let ðhMitÞ be its increasing process defined by:

hMit ¼ hMit1 þ E jjMt Mt1jj2jF t1

� �for t � 1; and hMi0 ¼ 0:

The first condition to check is:

hMiT

T

P

T !1��! G; for some positive G: ð13Þ

We set Ms ¼Ps

t¼1 ~eet; we then have:

hMit hMit1 ¼Xi2C

E e2itjF t1

� �¼Xi2C

E E e2itjy

i; x� �

jF t1

� �¼ jCj:

The second condition is the Lindeberg condition. It is implied by the following one:

9 a > 2 such that E j~eetjajF t1½ , is bounded: ð14Þ

Therefore, (13) is fulfilled with G ¼ jCj, whatever the model, while (14) has to be checked in

each particular case.

PROPOSITION 10 (Duflo, 1997; Hall and Heyde, 1980) Let eit be defined by (12), C a coding

subset such that ~eet fulfill (14). Then,

1ffiffiffiffiffiffiffiffiffijCjT

pXT

t¼1

~eetD

T !1��!Nð0; 1Þ

7.1.2. CLT for the Squared Residuals

Let us define

wit ¼e2

it 1

sðe2itÞ

: ð15Þ

The wit’s have the same properties of independency as the eit’s. Hence (wit) is a martingale

difference sequence in t, and so is ~wwt ¼P

i2C wit, for C a coding subset of S. Let

NT ¼P

t¼1;T ~wwt. As hNit hNit1 ¼P

i2C E w2itjF t1

� �¼ jCj, the first condition (13) for

the CLT is fulfilled. The second condition will be ensured under:

9a > 2 such that E j ~wwtjajF t1½ , is bounded : ð16Þ

PROPOSITION 11 Let C be a coding subset of S, wit be defined by (15) s.t. ~wwt fulfills (16).

Then,


pXi2C

XT

t¼1

witD

T !1��!Nð0; 1Þ:

As we do not know y0, we apply the previous results to the residuals calculated with yyT .

When CPLE is convergent, standard manipulations show that Propositions 10 and 11 still

remain valid for the estimated residuals.


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7.2. Applications

We give the explicit results for the auto-models studied in Section 4. The proofs of the con-

ditions (14) or (16) are given in Appendix 5.

7.2.1. The Autoexponential Dynamics

We suppose that we are in the framework of (6), assuming E1;E2;E3. Then yyT is convergent

and we have:


pXi2C

XT

t¼1

beeit

D

T !1��!Nð0; 1Þ and

1

2ffiffiffiffiffiffiffiffiffiffiffi2jCjT

pXi2C

XT

t¼1

bee2it 1

� � D

T !1��!Nð0; 1Þ:

7.2.2. The Autopoissonian Dynamics

We consider the framework of (8) and we suppose that the CPLE yyT is consistent. Then for

li ¼ li X iðtÞ;X ðt 1Þ� �

; g2 ¼P

i2C E½li, and s2 ¼P

i2C E½liðli þ 1Þðl2i þ 5li þ 1Þ,, we

have:

1ffiffiffiffiT

pXi2C

XT

t¼1

eeit

D

T !1��!Nð0; g2Þ and

1ffiffiffiffiT

pXi2C

XT

t¼1

ee2it liðy

i; xÞ� � D

T !1��!Nð0; s2Þ:

7.2.3. The Autologistic Dynamics

For E ¼ f0; 1g, we consider the framework of (9) and we assume:

B1: For y 6¼ y0; 9 x; y 2 E s.t., for an i 2 S; piðyi; x; yÞ 6¼ piðy

i; x; y0Þ.

Under assumption B1, we have:


pXi2C

XT

t¼1

eeit

D

T !1��!Nð0;1Þ and


pXi2C

XT

t¼1

bwwit

D

T !1��!Nð0;1Þ:

7.2.4. The Autodiscrete Dynamics

More generally, if E is qualitative, E ¼ fa0; a1; . . . ; aK1g, we consider the encoded ðK 1Þ-

dimensional variable Zit 2 f0; 1gK1 linked to the XiðtÞ by:

Zitl ¼ 1 if XiðtÞ ¼ al; Zitl ¼ 0 elsewise; 1 ! l ! K 1:


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We suppose that the conditional energy is given by U ðztjzt1Þ ¼P

i2S

Pl¼

1;K 1zitl di;l þP

k¼1;K1 ai;lk ziðt1Þk

� �þP

hi;ji

Pl¼1;K1

Pk¼1;K1 bi;j;klzitlzjtk . We have:

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK 1ÞjCjT

p Xi2C

XT

t¼1

!teitC

1it eit ðK 1Þ

"D

T !1��!Nð0; 1Þ

where Cit ¼ ðCitÞkl; 1 ! k; l ! K 1, with ðCitÞkk ¼ pikðzit; zt1Þ

ð1 pikðzit; zt1ÞÞ; ðCitÞkl ¼ ðCitÞlk ¼ pik zi

t; zt1

� �pil zi

t; zt1

� �if l 6¼ k, and

pil zit; zt1

� �¼

exp yitl

1 þP

l¼1;K1 exp zitlyitl

ð17Þ

where yitl ¼ di;l þP

k¼1;K1 ai;lk ziðt1Þk þP

j2@i

Pk¼1;K1 bi;j;klzjtk :

7.2.5. The Autonormal Dynamics

In the framework of (7), we suppose that the ergodicity condition is fulfilled. We have:


pXi2C

XT

t¼1

eitD

T !1��!Nð0; 1Þ and

Xi2C

XT

t¼1

2gie2it

D

T !1��! w2

jCjT

and so,

1


pXi2C

XT

t¼1

ð2gie2it 1Þ

D

T !1��!Nð0; 1Þ:

A third result holds, which uses the joint distribution on S of all the residuals. The eit’s covar-

iances are Covðeit; ejtÞ ¼ ð1=2giÞ if i ¼ j ¼ ðbij=2gigjÞ if j 2 @i, and 0 else. Let Ce be the

n � n covariance matrix of et ¼ ðeit; i 2 SÞ. ThenP

t¼1;TtetC

1e et ) w2

nT and thus

1ffiffiffiffiffiffinT

pXT

t¼1

tetC1e et

D

T !1��!Nð0; 1Þ:

Let yyT be a consistent estimator of y0 and CCe is the estimate of Ce obtained by replacing the

parameters by their estimates. Then, the three statistics T1; T2; T3 defined below are asymp-

totically Gaussian Nð0; 1Þ and can be used for validation tests of the model:

T1 ¼1ffiffiffiffiffiffiffiffiffijCjT

pXi2C

XT

t¼1

eeit; T2 ¼1


pXi2C

XT

t¼1

ð2ggiee2it 1Þ;

T3 ¼1ffiffiffiffiffiffinT

pXT

t¼1

t eetCC1e eet:

8. MCMF MODELLING OF METEOROLOGICAL DATA

We have tried an MCMF modelling on a real set of meteorological data. The data comes from

the study of Tang (1991) and consists of daily pluviometric measures on a network of 16 sites


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in the Mekong Delta (Vietnam). We have retained a period of 123 consecutive days from July

to October, 1983. Geographically, the 16 meteorological stations, situated at latitude 9.17 to

10.8 and longitude 104.48 to 106.67, are not regularly located (see Tab. I).

The data have been previously studied by Tang (1991); he proposed different models for

each site, each being free of the other sites; this seems to be insufficient and hardly satisfac-

tory.

We study an MCMF autologistic model; we consider the binary f0; 1g data where 1 stands

for rain and 0 for no rain. The results lead to an average of exact prediction of about 77%,

which is rather satisfactory noting that the random part is large in this kind of meteorological

phenomenon. We compare this model with a site-by-site Markov Chain, for which the pre-

diction results are bad.

Of course, we do not pretend that our models are definitive and we know they need to be

refined for effective forecasting. An interesting study would be to fit other competitive

models of more or less the same dimension, as an Hidden Markov Chain (see e:g., Zucchini

and Guttorp, 1991; MacDonald and Zucchini, 1997), the hidden states standing for the under-

lying climatic stage, or threshold models, the dynamics at the time t depending on the posi-

tion of the state at the time t 1 with respect to thresholds (see Tong, 1990 and references

herein). Also, these models should take into account the dual character of such a data, i:e:,the binary feature f0g [ Rþ� of the space state. This is work in progress.

The first task is to identify the two dependency graphs. Considering the large number of para-

meters involved in the CPL procedure, we use the Gaussian linear procedure with the original

data (see Section 6); we fit a regression model on each site, with respect to the 15 other variables

at the same time and all the 16 variables at the previous time. We then select the neighbours as the

variables giving the best multiple correlation coefficient R2 in a stepwise procedure, taking into

account the symmetry of the instantaneous graph together with a principle of parsimony. We

give in Table II the neighbourhoods and the R2 obtained from all the 31 variables (denoted

R231) and the R2 calculated on the selected neighbours (denoted R2

@i;@i ). The instantaneous

graph has 25 links and the time-delay graph 17 links. We see that for each site, the instantaneous

neighbourhood may contain few sites, while there are at most 3 sites making up the time delay

neighbourhood. We note that if we draw the directed graph G on a geographical map, there is a

main direction of the arrows, which is roughly S-N.

TABLE I 16 Meteorological stations in the Mekong Delta

Site number Name Latitude Longitude

1 Tan Chau 10.80 105.252 My Tho 10.35 106.373 Chau Doe 10.70 105.014 Can Tho 10.03 105.785 Soc Tran 09.60 105.976 Vinh Long 10.25 105.977 Sa Dec 10.30 105.758 Go Cong 10.35 106.679 Ca Mau 09.17 105.1710 Long Xuyen 10.40 105.4211 Rach Gia 10.00 105.0812 Ha Tien 10.38 104.4813 Cao Lanh 10.47 105.6314 Moc Hoa 10.75 105.9215 Vi Thanh 09.77 105.4516 Tan An 10.53 106.40


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Going back to the binary data, we estimate the autologistic model (9): for each site i, XiðtÞ

has a Bernouilli distribution of parameter piðtÞ, conditionally to X@iðtÞ and X@iðtÞ, with

piðtÞ ¼ ð1 þ expliðtÞÞ1 and liðtÞ ¼ di þ

Pj2@i bijXjðtÞ þ

P‘2@i aliXlðt 1Þ. The dimen-

sion of the model is 16 þ 25 þ 17 ¼ 58. We first estimate the parameters site-by-site, max-

imizing the log conditional pseudo-likelihoods; then, we calculate the validation statistics

based on the square residuals (see Section 7.2): Vi ¼ ð1=ffiffiffiffiT

pÞPT

t¼1 wwit for each site i. Sec-

ondly, we proceed to the global estimation on the basis of the global pseudo-likelihood;

we take as initial values the first site-by-site estimates for the parameters di and ali, and

the means of the two previous estimations bij and bji (theoretically equal) for the instanta-

neous interaction parameters. We do not give here the results for the 58 parameters for

sake of place, but we summarize the results: concerning the statistics Vi, each local model

is accepted. The final (global) estimation provides results close to the individual ones. The

TABLE II Neighbourhoods and R2 statistics. R231: regression on all vari-

ables. R2@i;@� : regression on the neighbourhoods variables

Site i @i @i� R231 R2

@i;@�

1 f2; 3; 8; 14g 0.611 0.4962 f1; 5; 6; 7; 8; 15; 16g f14g 0.648 0.5743 f1; 12g f1g 0.741 0.6384 f6g f7; 10g 0.319 0.1695 f2; 9g f4g 0.297 0.1396 f2; 4; 7; 10; 14g f7g 0.589 0.4847 f2; 6; 13; 15; 16g f6g 0.633 0.5318 f1; 2; 15g 0.592 0.3709 f5; 11g f4; 9; 12g 0.589 0.48110 f6; 12; 13g f4; 5g 0.463 0.34811 f9; 15g f11g 0.596 0.45112 f3; 10g f11g 0.648 0.52813 f7; 10g f4g 0.463 0.32114 f1; 6; 16g f4g 0.441 0.26415 f2; 7; 8; 11g f10g 0.568 0.47616 f2; 7; 14g 0.527 0.279

TABLE III Percentages of similarity between the realdata and the predicted values for the autologistic modeland the site by site Markov chain

SiteAutologistic model from

global estimates Markov chain

1 79.59 45.902 84.43 35.253 72.95 49.184 84.43 52.465 82.79 82.796 81.97 81.977 82.79 59.028 71.31 56.569 77.05 57.3810 68.03 54.9211 78.69 51.6412 72.13 57.3813 81.15 50.0014 78.69 51.6415 72.13 50.0016 74.59 57.38


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validation statistic V ¼ ð1=ffiffiffiffiffiffiCT

pÞP

i2C

PTt¼1 wwit computed for the coding set

C ¼ f3; 4; 5; 7; 8; 10; 11; 14g (of maximal size 8) is equal to 0:1186 and we do not reject

the model.

Another way to validate the model is to compare the true data with what is predicted. So

we compute the predicted values (maximizing the local conditional probabilities), first site by

site (with the local estimates) and then globally (with the global estimates). Table III gives the

percentages of similarity which spread out from 68.03% (site 10) to 84.43% (site 2), with a

mean of 77.36% (for the global parameter’s estimate).

We compare our model with a site by site Markov chain. Of course, this alternative is very

poor; as expected, the predictions are bad, spreading out between 35.25% and 59.02% with a

mean equal to 51.84% (see Tab. III). In conclusion, the autologistic model leads to a rela-

tively correct forecasting; it could be better if we increase the number of links or the depen-

dence in time, at the expense of the rising parametric dimension.

References

Amemiya, T. (1985). Advanced Econometrics (Blackwell).Arnold, B. C. and Press, J. (1989). ‘‘Compatible conditional distributions’’, JASA 84(405), 152–156.Arnold, B. C. and Strauss, D. (1988). ‘‘Bivariate distribution with exponential conditionals’’, JASA 83(402), 522–

527.Arnold, B.C., Castillo, E. and Sarabia, J. M. (1991). ‘‘Conditionally specified distributions’’, L.N.S. 74 (Springer).Bayomong, S. (1994). Modelisation et analyse des donnees spatio-temporelles, PhD thesis, Univ. d’Orsay, France.Bayomong, S., Guyon, X., Hardouin, C. and Yao, J. F. (1996). ‘‘Test de difference de constraste et somme ponderee

de khi-deux’’, Can. J. Stat. 24(1), 115–130.Bennett, R. J. and Haining, H. (1985). ‘‘Spatial structure and spatial interaction: Modelling approaches to the

statistical analysis of geographical data’’, JRSS 148, 1–36.Besag, J. (1974). ‘‘Spatial interaction an the statistical analysis of lattice systems’’, JRSS B, 36, 192–236.Besag, J. (1974). ‘‘On spatial temporal models and Markov fields’’, 7th Prague Conference and of European Meeting

of Statisticians, pp. 47–55.Besag, J. (1977). ‘‘Efficiency of pseudo likelihood estimation for simple Gaussian fields’’, Biometrika 64, 616–618.Chadoeuf, J., Nandris, D., Geiger, J.P., Nicole, M. and Piarrat, J. C. (1992). ‘‘Modelisation spatio temporelle d’une

epidemie par un processus de Gibbs: Estimation et tests’’, Biometrics 48, 1165–1175.Dacunha-Castelle, D. and Duflo, M. (1986). Probability and Statistics, Vol. 2 (Springer, Verlag).Duflo, M. (1977). Random Iterative Models (Springer).Durrett, R. (1995). Ten Lectures on Particles Systems, Saint-Flour 1993. L. N. M. 1608 (Springer), pp. 97–201.Garber, D. D. (1981). Computational models for texture analysis and texture synthesis, PhD thesis, Univ South

California.Geyer, C. (1999). ‘‘Likelihood inference for spatial point processes’’, In: Barndorff-Nielsen, Kendall and Van

Lieshout (Eds.), Stochastic Geometry, Likelihood and Computation, Chapter 3 (Chapman and Hall).Geyer, C. J. and Thompson, E. A. (1992). ‘‘Constrained Monte Carlo maximum likelihood for dependent data (with

discussion)’’, JRSS, B 54, 657–699.Guyon, X. (1995). Randon Fields on a Network: Modelling, Statistics and Applications (Springer).Guyon, X. and Hardouin, C. (1992). ‘‘The chi2 difference of coding test for testing Markov Random Field

hypothesis’’, L.N.S. 74 (Springer), pp. 165–176.Guyon, X. and Kunsch, H. R. (1992). ‘‘ Asymptotic comparison of estimators in the Ising model’’, L.N.S. 74

(Springer), pp. 177–198.Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications (Acad. Press).Jensen, J. L. and Kunsch, H. R. (1994). ‘‘On asympotic normality of pseudo-likelihood estimate for pairwise

interaction processes’’, Ann. Inst. Statist. Math. 46, 475–486.Keiding, N. (1975). ‘‘Maximum likelihood estimation in the birth and death process’’, Ann. of Stats. 3, 363–372.Koslov, O. and Vasilyev, N. (1980). ‘‘Reversible Markov chain with local interaction’’, In: Dobrushin and Sinai

(Eds.), Multicomponent Random Systems (Dekker).Kunsch, H. R. (1984). ‘‘Time reversal and stationery Gibbs measures’’, Stoch. Proc. and Applications 17, 159–166.Pfeifer, P. E. and Deutsch, S. J. (1980). ‘‘Identification and interpretation of first order for space–time ARMA

models’’, Technometrics 22, 397–408.Pfeifer, P. E. and Deutsch, S. J. (1980). ‘‘A three stage iterative procedure for space time modelling’’, Technometrics

22, 35–47.Preston, C. (1974). Gibbs States on Countable Set, Cambridge tracts in math 68.Prum, B. (1986). Processus sur un reseau et mesure de Gibbs Applications, (Masson).Ripley, B. (1986). Spatial Statistics (Wiley).


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Tang van Tham (1991) Rainfall greenhouse effect in the delta of Mekong, PhD thesis, Univ. of Cantho., Vietnam.Tong, H. (1990). Non Linear Time Series. A Dynamical System Approach (Clarendon Press, Oxford).Younes, L. (1988). ‘‘Estimation and annealing for Gibssian fields’’, Ann. I.H.P. 2, 269–294.Zucchini, W. and Guttorp, P. (1991). ‘‘A hidden Markov model for space time precipitation’’, Water Resour. Res. 27,

1917–1923.MacDonald, I. and Zucchini, W. (1997). ‘‘Hidden Markov and other models for discrete valued time series’’,

Monographs on Statistic and Applied Probability (Chapman and Hall).

APPENDIX 1

The Invariant Law p of an MCMF is Generally Not Markovian

We illustrate this with two examples.

(1) The one dimensional marginal of a two dimensional Gaussian Markov field is not

anymore a Markov process. Let us consider a centered Gaussian Markov isotropic field

with respect to the four-nearest neighbours over Z2 (see Besag, 1974; Ripley, 1981 and

Guyon, 1995, x1.3.4.)

Xst ¼ aðXs1;t þ Xsþ1;t þ Xs;t1 þ Xs;tþ1Þ þ est; jaj <1

4

with E½Xstes0t0 , ¼ 0 if ðs; tÞ 6¼ ðs0; t0Þ. The spectral density of X is

f ðl; mÞs2eð1 2aðcos lþ cos mÞÞ1. The one index field ðXs;0; s 2 ZÞ has spectral density

FðlÞ ¼ 2

ðp0

f ðl; mÞ dm ¼ 2ps2e ð1 2a cos lÞ2 4a2� �1

2

which cannot be written ð1=QðeilÞÞ or ð1=jPðeilÞj2Þ as for a Markovian or an AR process.

The distribution p of ðXs;0; s 2 ZÞ is not Markovian.

(2) Conditional Ising model: O ¼ f0; 1gS; S ¼ f1; 2; . . . ; ng is the one dimensional torus

with the agreement n þ 1 � 1, and Pðx; yÞ ¼ Z1ðxÞ expfaP

i2S xiyi þ bP

i2S yiyiþ1g.

This chain is reversible and the invariant distribution can be written

pðyÞ ¼ CZðyÞ expfgP

i2S yi þ bP

i2S yiyiþ1g where ZðyÞ is the normalizing constant of

Pð:jyÞ. It is easy to see that log pðyÞ has a non zero potential FS on the whole set S.

APPENDIX 2

Two Examples Where Marginals in y are Not Local in x

(1) Binary state space. Let us consider a binary chain on the one dimensional torus

S ¼ f1; 2; :::; ng (with n þ 1 � 1) with the transition

Pðx; yÞ ¼ Z1ðxÞ exp aXn

i¼1

xiyi þ bXn

i¼1

yiyiþ1 þ gXn

i¼1

yi

( ):

Then PiðyijxÞ ¼P

yi Pðx; yi; yiÞ ¼ ðeyiðaxiþgÞP

yj;j 6¼i ebyiðyj1þyjþ1ÞþAiÞ= ðP

yj;j 6¼i eAi þ eðaxiþgÞPyj;j 6¼i ebðyj1þyjþ1ÞþAiÞ, with Ai ¼ b

Pj 6¼i;i1 yjyjþ1 þ

Pj 6¼iðaxj þ gÞyj. This distribution

depends on x on all sites.


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(2) A Gaussian example. We take Pðx; yÞ ¼ Z1ðxÞ exp U ðyjxÞ with U ðyjxÞ ¼ ðy mðxÞÞ0

�Qðy mðxÞÞ, where Q is symmetric, definite positive, and E½Y jx, ¼ mðxÞ. For

U ðyjxÞ ¼Pn

i¼1ðgiy2i þ diyiÞ þ

Pni¼1 aixiyi þ

Pni;j¼1;hi;ji bijyiyj; mðxÞ ¼ ðQ1=2ÞðdþaxÞ

where d ¼ ðdiÞi2S; ax ¼ ðaixiÞi2S , and Var½Y jx, ¼ ð1=2ÞQ1 : V ðY jxÞ1¼ 2Q is local in x

but it is easy to find parameters for which each component mlðxÞ of mðxÞ depends on all

xi; i 2 S.

APPENDIX 3

An Example of a Reverse MCMF Non MCMF

We consider S ¼ f1; 2; . . . ; ng; X ¼ X ð0Þ; Y ¼ X ð1Þ and we suppose that Z ¼ ðX ; Y Þ is

a 2n-dimensional centered Gaussian variable with covariance S ¼D BtB D

� �. Then,

ðY jxÞ ) N nðAx;QÞ, where A ¼ tBD1 and Q ¼ D tBD1B. We fix D and look at B as a

parameter (s.t. S is definite positive).

If we want conditional independence for fðXið1ÞjX ð0ÞÞ ¼ xÞ; i ¼ 1; ng, we force Q to be

diagonal; this involves ðnðn 1Þ=2Þ conditions on B, for n2 degrees of freedom.

On the other hand, ðX jyÞ ) N nðTy;RÞ where T ¼ BD1 and R ¼ D BD1tB. We can

chose B such that for some i;Rij 6¼ 0 for any j. In such a case, ðXið0Þjxi; yÞ depends on xi

on whole Snfig and the reverse chain is not an MCMF.

For example, this happens for n ¼ 2 if we take D ¼1 1=2

1=2 1

� �, B ¼

1=4 3=4

1=2 0

� �.

APPENDIX 4

Parametric Identifiability; Regularity of I0 and J0

First we take n�S ¼ �i2SKin as the reference measure, where Ki is defined in (10).

Identifiability. We suppose that the conditional densities are equal for y and y0. Then, for

each i 2 S; x and yi we have

t y y0ð Þgi yi; yi; x� �

¼ C y; yi; x� �

C y0; yi; x� �

But the right member is 0 as gið0; yi; xÞ ¼ 0. Identifiability follows because the d � d matrix

g is regular.

Regularity of I0. First we note that for the stationary distribution

I0 ¼ EX ðt1Þ;X iðtÞ

Xi2S

VarXiðtÞ gi XiðtÞ;X iðtÞ;X ðt 1Þ� � " #

:

Then, under (H0), we haveXi2S

Var gi Yi; yi; xÞjyi; x� ��

� aXi2S

Gi x; yi� �

t Gi x; yi� �

:


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As the density of ðX ðt 1Þ;X ðtÞÞ is strictly positive anywhere under C1 (see Duflo (1997),

Proposition 6.1.9), I0 is regular.

Regularity of J0. Let us recall that J0 ¼ VaryfgðX ; Y Þg where ðX ; Y Þ ¼ ðY ð0Þ;Y ð1ÞÞ; gðx; yÞ ¼

Pi2S giðx; yÞ; giðx; yÞ ¼ log fiðyi; yi; x; yÞ,ð1Þy . We follow an idea given by

Jensen and K€uunsch (1994). We suppose that there exists a ‘‘strong coding subset’’ C � S

in the following sense:

(i) there exists a partition fSj; j 2 Cg of S s. t. j 2 Sj.

(ii) For j 2 C, let us define Gj ¼P

i2Sjgi and let F be the s-field generated by X ¼ Y ð0Þ and

fYið1Þ; i =2 Cg. Then, conditionally to F , the variables fGj; j 2 Cg are independent.

A sufficient condition for (ii) is that for each j; l 2 C; j 6¼ l; l=2Sj [ @Sj. For example, for the

2-dimensional torus T ¼ f1; 2; . . . ; 3Kg2 and the 4-nearest neighbours vicinity,

C ¼ f3ðm; nÞ;m; n ¼ 1;Kg is a strong coding subset and Sm;n ¼ fði; kÞ : ji mj þ

jk nj ! 2g.

As g ¼P

j2C Gj, and J0 ¼ Var gðX ; Y Þ � EF ðVarfgðX ; Y ÞjFgÞ, we have, as a conse-

quence of (ii):

J0 �Xj2C

EF ðVarfGjðX ; Y ÞjFgÞ ¼ G0:

Then a sufficient condition ensuring that J0 is p. d. is that G0 is p. d. Such a verification has to

be done. For example, if S is the 1-dimensional torus with n ¼ 3K sites, with energy

U ðyjxÞ ¼ aP

i2S yini; ni ¼ xi þðyi1 þ yiþ1Þ; yi 2 f0; 1g, we can take C ¼ f3j; j ¼ 1;Kg and

Sj ¼ f3j 1; 3j; 3j þ 1g. As the model is homogeneous in the space, and as

ðg2 þ g3 þ g4jF Þ is never constant, G0 � EF ðVarðg2 þ g3 þg4ÞjF Þ > 0.

APPENDIX 5

Validation Tests for MCMF Models

The Autoexponential Dynamics (6)

eit ¼ liðyi; xÞXiðtÞ 1 and wit ¼ 23=2ðe2

it 1Þ. Next, for any a > 2, as ða þ bÞa !

2aðaa þ baÞ, we have

E jeitjajyi; x

� �! 2aliðy

i; xÞaGðaþ 1Þ

liðyi; xÞaþ liðy

i; xÞa� �

! 2aðGðaþ 1Þ þ 1Þ

and (14) is fulfilled.

Besides, E jwitjajyi; x

� �! 2ða=2Þ 22aðGð2aþ 1Þ þ 1Þ þ 1

and we get (16).

The Autopoissonian Dynamics (8)

The conditional distribution of XiðtÞ is Poisson with mean liðyi; xÞ ¼ expfdi þP

l2@i alixl þP

j2@i bijyjg. We look at conditions (13) and (14) for the eit . First we have


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hMit hMit1 ¼P

i2C liðXiðtÞ;X ðt 1ÞÞ ! eM jCj. Next, for any integer a, and some con-

stants Ck;a,

E eaitjyi; x

� �! 2a

Xk¼1;a

Ck;ali yi; x� �k

þli yi; x� �a !

! 2aX

k¼1;a

Ck;aeMk þ eMa

!:

Next, we consider the squared residuals nit ¼ e2it liðy

i; xÞ and ~nnt ¼P

i2C nit. As previously,

we can bound E½je2it liðy

i; xÞjajyi; x, for any a > 2, which implies (14). And we have (16)

with hNit hNit1 ¼ E½j~nntj2jF t1, ! jCjðe4M þ 6e3M þ 6e2M þ eM Þ. We get the announced

result.

The Autologistic Dynamics (9)

eit ¼ ðXiðtÞ piðyi; xÞÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipiðyi; xÞð1 piðyi; xÞÞ

p. We can easily check that E½jeitj

ajyi; x,

as E½jwitjajyi; x, are bounded for any a > 2. For example, if the conditional energy

is U ðyjxÞ ¼P

i2S yiðdi þ aixiÞ þP

hi;ji bijyiyj, then piðyi; xÞ ¼ ð1 þ exp ½di þ aixi þP

j2@i bijyj,Þ1 and B1 is satisfied.

The Autodiscrete Dynamics

For each l; Zitl has a conditional Bernouilli distribution of parameter pilðyi; xÞ given by (17).

So, eit ¼ Zit E½Zitjyi; x; y0, has conditional variance Cit . Conditionally to the past and

X �CCðtÞ; eit ? ejt0 for all i; j 2 S if t 6¼ t0; and eit ? ejt if i; j are elements of a coding subset C

of S; thus feitgt is a martingale’s increment. Besides, E½teitC1it eitjy

i; x, ¼ K 1.

Finally, the state space E being finite and eit belonging to f0; 1gK1, all the moments of eit

are bounded and the condition on the increasing process is fulfilled.

The Autonormal Dynamics

We consider (7). The conditional distribution of XitðtÞ is Gaussian Nðmit; ð1=ð2giÞÞ with

mit ¼ ð1=ð2giÞÞðdi þP

l2@i alixl þP

j2@i bijyjÞ. We immediately deduce: eit ¼ ðXiðtÞ mitÞ=ffiffiffiffiffiffi2gi

p) Nð0; 1Þ, and e2

it ¼ 2gie2it ) w2

1.


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markov chain markov field dynamics: models and statistics

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