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Chiu et al. (2013). In this paper we focus on the latter type of conditional

distributions which are formally dened in terms of so-called Palm distribu-tions, rst introduced by Palm (1943) for stationary point processes on thereal line. Rigorous denitions and generalizations of Palm distributions to R dand more abstract spaces have mainly been developed in probability theory,see Jagers (1973) for references and an historical account. Palm distributionsare, at least among many applied statisticians and among most students, con-sidered one of the more difficult topics in the eld of spatial point processes.This is partly due to the general denition of Palm distributions which re-lies on measure theoretical results, see e.g. Mller and Waagepetersen (2004)and Daley and Vere-Jones (2008). The account of conditional distributionsfor point processes in Last (1990) is mainly intended for probabilists and isnot easily accessible due to an abstract setting and extensive use of measuretheory.

This tutorial provides an introduction to Palm distributions for spatialpoint processes. Our setting and background material on point processes aregiven in Section 2. Section 3, in the context of nite point processes, providesan explicit denition of Palm distributions in terms of their density functionswhile Section 4 reviews Palm distributions in the general case. Section 5discusses examples of Palm distributions for specic models and Section 6considers applications of Palm distributions in the statistical literature.

2 Prerequisites

2.1 Setting and notation

We view a point process as a random locally nite subset X of a Borel setS R d; for measure theoretical details, see e.g. Mller and Waagepetersen(2004) or Daley and Vere-Jones (2003). Denoting X B = X B the restrictionof X to a set B S , and N (B) the number of points in X B , local nitenessof X means that N (B) < almost surely (a.s.) whenever B is bounded.We denote by B 0 the family of all bounded Borel subsets of S and by N thestate space consisting of the locally nite subsets (or point congurations)of S . Section 3 considers the case where S is bounded and hence N is allnite subsets of S , while Section 4 deals with the general case where S isarbitrary, i.e., including the case S = R d .

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2.2 Poisson process

The Poisson process is of its own interest and also used for constructing otherpoint processes as demonstrated in Section 2.3 and Section 5.

Suppose : S [0, ) is a locally integrable function, that is, (B) :=

B (x) dx < whenever B B 0. Then X is a Poisson process with intensityfunction if for any B B 0, N (B) is Poisson distributed with mean (B),and conditional on N (B) = n, the n points are independent and identicallydistributed, with a density proportional to (if (B) = 0, then N (B) = 0).In fact, this denition is equivalent to that for any B B 0 and any non-negative measurable function h on {x B |x N } , letting |B | denote theLebesgue measure of B ,

Eh(X B ) =

n =0

exp(| B |)n!

B B h({x1, . . . , x n })(x1) (xn ) dx1 dxn , (1)where for n = 0 the term is exp( | B |)h( ), where is the empty pointconguration.

Note that the denition of a Poisson process only requires the existenceof the intensity measure , since a point of the process restricted to B B 0has probability distribution ( B)/ (B) provided (B) > 0. We shall usethis extension of the denition in Section 5.3.2.

2.3 Finite point processes specied by a density

Assume S is bounded, let Z be a unit rate Poisson process on S , and assumethe distribution of X is absolutely continuous with respect to the distributionof Z (in short with respect to Z ) with density f . Thus, for any non-negativemeasurable function h on N ,

Eh(X ) = E {f (Z )h(Z )}. (2)

Moreover, by ( 1),

Eh(X ) =

n =0

exp(| S |)n!

S S h({x1, . . . , x n })f ({x1, . . . , x n }) dx1 dxn . (3)3

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This motivates considering probability statements in terms of exp( | S |)f ().

For example, with h(x ) = 1( x = ), where 1() denotes indicator function,we obtain that P (X = ) is exp(| S |)f ( ). Further, for n 1,

exp(| S |)f ({x1, . . . , x n }) dx1 dxn

is the probability that X consists of precisely n points with one point in eachof n innitesimally small sets B1, . . . , B n around x1, . . . , x n with volumesdx1, . . . dxn , respectively. Loosely speaking this event is X = {x1, . . . , x n }.

Suppose we have observed X B = xB and we wish to predict theremaining point process X S \ B . Then it is natural to consider the conditionaldistribution of X S \ B given X B = xB . By denition of a Poisson process,

Z = Z B Z S \ B where ZB and ZS \ B are each independent unit rate Poissonprocesses on respectively B and S \ B . Thus, in analogy with conditionaldensities for multivariate data, this conditional distribution can be speciedin terms of the conditional density

f S \ B (x S \ B |x B ) = f (x B x S \ B )

f B (x B )

with respect to ZS \ B and where

f B (x B ) = E f (Z S \ B x B )

is the marginal density of X B with respect to Z B . Thus the conditional distri-bution given a realization of X on some prespecied region B is conceptuallyquite straightforward. Conditioning on that some prespecied points belongto X is more intricate but an explicit account of this is provided in the nextsection where it is still assumed that X is specied in terms of a density.

3 Palm distributions in the nite case

For understanding the denition of a Palm distribution, it is useful to assumerst that S is bounded and that X has a density as introduced in Section 2.3with respect to a unit rate Poisson process Z . We make this assumption inthe present section, while the general case will be treated in Section 4.

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3.1 Conditional intensity and joint intensities

Suppose f is hereditary , i.e., for any pairwise distinct x0, x1, . . . , x n S ,f ({x1, . . . , x n }) > 0 whenever f ({x0, x1, . . . , x n }) > 0. We can then denethe so-called nth order Papangelou conditional intensity by

(n ) (x1, . . . , x n , x ) = f (x { x1, . . . , x n })/f (x ) (4)

for pairwise distinct x1, . . . , x n S and x N \{ x1, . . . , x n }, setting 0 / 0 = 0.By the previous interpretation of f , (n )(x1, . . . , x n , x ) dx1 dxn can beconsidered as the conditional probability of observing one point in each of theabovementioned innitesimally small sets Bi conditional on that X outside

ni=1 B i agrees with x .

For any n = 1, 2, . . ., we dene for pairwise distinct x1, . . . , x n S thenth order joint intensity function (n ) by

(n ) (x1, . . . , x n ) = E f (Z { x1, . . . , x n }) (5)

provided the right hand side exists. Particularly, = (1) is the usual inten-sity function. If f is hereditary, then (n ) (x1, . . . , x n ) = E (n ) (x1, . . . , x n , X )and by the interpretation of (n ) it follows that (n )(x1, . . . , x n ) dx1 dxncan be viewed as the probability that X has a point in each of n innites-imally small sets around x1, . . . , x n with volumes dx1, . . . dxn , respectively.Loosely speaking, this event is x1, . . . , x n X .

Combining (2) and (5) with either ( 3) or the extended Slivnyak-Meckeformula for the Poisson process given later in ( 17), it is straightforwardlyseen that

E=

x 1 ,...,x n X

h(x1, . . . , x n )

= S S h(x1, . . . , x n )(n )(x1, . . . , x n ) dx1 . . . dxn (6)for any non-negative measurable function h on S n , where = over the summa-tion sign means that x1, . . . , x n are pairwise distinct. Denoting N = N (S )the number of points in X , the left hand side in ( 6) with h = 1 is seen to bethe factorial moment E {N (N 1) (N n + 1) }.

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3.2 Denition of Palm distributions in the nite case

Now, suppose x1, . . . , x n S are pairwise distinct and (n )(x1, . . . , x n ) > 0.Then we dene the reduced Palm distribution of X given points at x1, . . . , x nas the distribution P !x 1 ,...,x n for the point process X

!x 1 ,...,x n with density

f x 1 ,...,x n (x ) = f (x { x1, . . . , x n })

(n )(x1, . . . , x n ) , x N , x {x1, . . . , x n } = , (7)

with respect to Z . If x1, . . . , x n S are not pairwise distinct or (n ) (x1, . . . , x n )is zero, the choice of X !x 1 ,...,x n and its distribution P

!x 1 ,...,x n is not of any im-

portance for the results in this paper. Furthermore, the (non-reduced) Palm distribution of X given points at x1, . . . , x n is simply the distribution of the

union X !x 1 ,...,x n { x1, . . . , x n }.

3.3 Remarks

By the previous innitesimal interpretations of f and (n ) , we can viewexp(| S |)f x 1 ,...,x n (x ) as the joint probability that X equals the union x{x1, . . . , x n } divided by the probability that x1, . . . , x n X . Thus P !x 1 ,...,x nhas an interpretation as the conditional distribution of X \{ x1, . . . , x n } giventhat x1, . . . , x n X . Conversely,

exp(| S |)f ({x1, . . . , x n }) = (n )(x1, . . . , x n )P X !{x 1 ,...,x n } = (8)

provides a factorization into the probability of observing {x1, . . . , x n } timesthe conditional probability of not observing further points.

We obtain immediately from ( 5) and (7) that for any pairwise distinctx1, . . . , x n S and m = 1, 2, . . ., X !x 1 ,...,x n has mth order joint intensityfunction

(m )x 1 ,...,x n (u1, . . . , u m ) = ( m + n ) (u 1 ,...,u m ,x 1 ,...,x n )

( n ) (x 1 ,...,x n ) if (n ) (x1, . . . , x n ) > 0

0 otherwise(9)

for pairwise distinct u1, . . . , u m S \ { x1, . . . , x n }. Moreover, the so-calledpair correlation function is for u, v S dened as

g(u, v) = (2) (u, v)/ {(u)(v)}

provided (u)(v) > 0 (otherwise we set g(u, v) = 0). If (u)(v) > 0, then

g(u, v) = v(u)/ (u) = u (v)/ (v), (10)

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cf. (9). Thus, g(u, v) > 1 (g(u, v) < 1) means that the presence of a point at

u yields an elevated (decreased) intensity at v and vice versa.For later use, notice that

E=

x 1 ,...,x n X

h(x1, . . . , x n , X \ { x1, . . . , x n })

= S S Eh(x1, . . . , x n , X !x 1 ,...,x n )(n )(x1, . . . , x n ) dx1 dxn (11)for any non-negative measurable function h on S n N . This is straightfor-wardly veried using (3) and (7). Assuming f is hereditary and rewritingthe expectation in the right hand side of ( 11) in terms of

f x 1 ,...,x n (x ) = f (x ) (n )(x1, . . . , x n , x )/ (n ) (x1, . . . , x n ) ,

the nite point process case of the celebrated Georgii-Nguyen-Zessin (GNZ) formula

E=

x 1 ,...,x n X

h(x1, . . . , x n , X \ { x1, . . . , x n })

= S S Eh(x1, . . . , x n , X ) (n )(x1, . . . , x n , X ) dx1 dxn (12)is obtained ( Georgii, 1976; Nguyen and Zessin, 1979). We return to the GNZformula in connection to Gibbs processes in Section 5.2.

4 Palm distributions in the general case

The denitions and results in Section 3 extend to the general case where S is any Borel subset of R d . However, if |S | = , the unit rate Poisson processon S will be innite and we can not in general assume that X is absolutelycontinuous with respect to the distribution of this process. Thus we do notlonger have the direct denitions ( 5) and (7) of (n ) and X !x 1 ,...,x n in terms of density functions.

4.1 Denition of Palm distributions in the general case

In fact (6) is usually taken as the denition of the nth order joint intensityfor X , provided there exists such a non-negative measurable function (n ) .

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Technically speaking, viewing the left hand side in ( 6) as an integral

h d (n ) ,

where (n )

is called the nth order factorial moment measure on S n

, (n )

isassumed to be a density for (n ) with respect to Lebesgue measure on S n .Here, (n ) is required to be a locally nite measure, i.e.,

B B (n )(x1, . . . , x n ) dx1 dxn < for all B B 0. (13)Thereby ( 11) can be used as the denition of the reduced Palm distribu-tions, where their existence follows by measure theoretical arguments, see e.g.Mller and Waagepetersen (2004). This denition thus extends in a math-ematically sound manner the previous denition of Palm distributions topoint processes in general but seems intuitively less appealing. Furthermore,the (non-reduced) Palm distribution of X given points at x1, . . . , x n is thedistribution of X !x 1 ,...,x n { x1, . . . , x n }.

4.2 Remarks

In the general setting, (n ) (x1, . . . , x n ) and P !x 1 ,...,x n are then clearly onlydetermined up to a Lebesgue nullset of S n . For simplicity and since usuallythere are natural choices of (n ) (x1, . . . , x n ) and P !x 1 ,...,x n , such nullsets areoften ignored. Further, as in the nite case, (n ) (x1, . . . , x n ) and P !x 1 ,...,x n areinvariant under permutations of the points x1, . . . , x n , and

X !x 1 ,...,x m!x m +1 ,...,x n

= X !x 1 ,...,x n (14)

whenever 0 < m < n and x1, . . . , x n are pairwise distinct.Suppose that X is stationary , i.e., its distribution is invariant under trans-

lations in R d and so S = R d (unless X = which is not a case of our interest).This is a specially tractable case, which makes an alternative description of Palm distributions possible. Let denote the constant intensity of X and leto denote the origin in R d . First, we dene

P !o(F ) = 1|B |

E

x X B

1(X \ { x} x F ) (15)

for any B B 0 with |B | > 0, where by stationarity of X the right hand sidedoes not depend on the choice of B . Second, we dene

P !x (F ) = P!o(F x) (16)

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Jagers (1973). Further, it makes it possible to calculate various useful func-

tional summaries, see e.g. Mller and Waagepetersen (2004), and construc-tions such as stationary Poisson-Voronoi tessellations become manageable,see Mller (1989, 1994).

5.2 Gibbs processes

Gibbs processes play an important role in statistical physics and spatialstatistics, see Mller and Waagepetersen (2004) and the references therein.Below, for ease of presentation, we consider rst a nite Gibbs process.

A nite Gibbs process on a bounded set S R d is usually specied interms of its density or equivalently in terms of the Papangelou conditionalintensity, where the density is of the form

f (x ) = exp y x

(y )

for a so-called potential function on N . It follows that the nth orderPalm distribution of a Gibbs process with respect to x1, . . . , x n is itself aGibbs process with potential function x 1 ,...,x n (y ) = ( {x1, . . . , x n } y ).Moreover, for pairwise distinct u1, . . . , u m , x1, . . . , x n S and x N \{u1, . . . , u m , x1, . . . , x n }, the mth order Papangelou conditional intensity of X !x 1 ,...,x n is simply

!(m )x 1 ,...,x n (u1, . . . , u m , x ) = (m ) (u1, . . . , u m , x { x1, . . . , x n }).

For instance, a rst order inhomogeneous pairwise interaction Gibbs pointprocess has rst order potential ( {u}) = 1(u), second order potential({u, v}) = 2(v u), and (y ) = 0 whenever the cardinality of y is largerthan two; see Mller and Waagepetersen (2004) for conditions on the func-tions 1 and 2 ensuring that the model is well-dened. The Strauss model(Strauss , 1975; Kelly and Ripley, 1976) is a particular case with 1(u) =1 R and 2(u v) = 21( u v R), for 2 0 and 0 < R < . ThePalm process X !x 1 ,...,x n becomes again an inhomogeneous pairwise interac-

tion Gibbs process with inhomogeneous rst order potential x 1 ,...,x n ({u}) =1(u) + ni=1 2(u xi ) and second order potential identical to that of X .

In the general case, a Gibbs process can be dened (Nguyen and Zessin,1979) in terms of the GNZ formula ( 12) briey discussed at the end of Sec-tion 3: X is a Gibbs point process with Papangelou conditional intensity

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and the reduced Palm distributions satisfy

E h x1, . . . , x n , X !x 1 ,...,x n (n ) (x1, . . . , x n )

= E h(x1, . . . , x n , X )n

i=1

(xi ) . (20)

In the sequel, we consider distributions of , where (19)-(20) become useful.

5.3.1 Log Gaussian Cox processes

Let (x) = exp {Y (x)}, where Y = {Y (x)}x S is a Gaussian process withmean function and covariance function c so that is locally integrablea.s. (simple conditions ensuring this are given in Mller et al., 1998). ThenX is a log Gaussian Cox process (LGCP) as introduced by Coles and Jones(1991) in astronomy and independently by Mller et al. (1998) in statistics.By Mller et al. (1998, Theorem 1), for pairwise distinct x1, . . . , x n S ,

(n )(x1, . . . , x n ) = n

i=1

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5.3.2 Shot noise Cox processes

For a shot noise Cox process (Mller, 2003),

(x) = j

j k(c j , x),

where k(c j , ) is a kernel (i.e., a density function for a continuous d-dimensionalrandom variable) and the ( c j , j ) are the points of a Poisson process onR d (0, ) with intensity measure so that becomes locally integrablea.s. It can be viewed as a cluster process X = j Y j , where conditional on , the cluster Y j is a Poisson process with intensity function j k(c j , ) andthe clusters are independent.

The intensity function is

(x) = k (c, x) d(c, ),provided the integral is nite for all x S . Making this assumption, it canbe veried that for x S with (x) > 0, X !x is a Cox process with randomintensity function ( ) + x (), where x () = x k(cx , ), and where (cx , x )is a random variable independent of and dened on S (0, ) such thatfor any Borel set B S (0, ),

P {(cx , x ) B} = B k(c, x) d(c, )(x) ,

cf. Mller (2003, Proposition 2). In other words, X !x is distributed as X Y x ,where Y x is independent of X and conditional on ( cx , x ), the extra clusterY x is a nite Poisson process with intensity function x k(cx , ). Thus, as foran LGCP with positive covariance function, P !x has a higher intensity thanX !x .

For instance, if d (c, ) = d c d ( ), where is a locally nite measure on(0, ), then (x) = f (x), where it is assumed that =

d ( ) < and

f (x) =

k(c, x) dc < , and furthermore, for (x) > 0, cx and x are inde-

pendent, cx follows the density k(, x)/f (x), and P( x A) = 1 A d ( ).The special case of a Neyman-Scott process (Neyman and Scott , 1958) oc-curs when S = R d , is concentrated at a given value > 0, ({ }) < ,and k(c, ) = ko( c), where ko is a density function. Then X is stationary, = = ({ }), cx has density ko(x ), and conditional on cx , Y x is a

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nite Poisson process with intensity function ko( cx ). Examples include

a (modied) Thomas process, where ko is a zero-mean normal density, anda Matern cluster process, where ko is a uniform density on a ball centeredat the origin. For n > 1, the nth order reduced Palm distributions becomemore complicated.

In a Neyman-Scott process, the number of points in the clusters are in-dependent and identically Poisson distributed. A stationary Poisson clusterprocess is obtained by replacing the Poisson distribution by any discrete dis-tribution on the non-negative integers. Finally, we notice that the Palmdistribution for stationary Poisson cluster processes and more generally in-nitely divisible point processes can also be derived, see Chiu et al. (2013)and the references therein.

6 Examples of applications

In this section we review a number of applications of Palm distributions inspatial statistics.

6.1 Functional summary statistics

Below we briey consider two popular functional summary statistics, whichare used for exploratory purposes as well as model tting and model assess-ment.

First, suppose X is stationary, with intensity > 0. The nearest-neighbour distribution function G is dened by G(t) = P !o{X b(o, t) = } ,where b(o, t) is the ball centered at o and of radius t > 0. Thus G(t) is inter-preted as the probability of having a point within distance t from a typicalpoint. Moreover, Ripleys K -function (Ripley, 1976) times is dened byK (t) = E v X !o 1( v t), that is, the expected number of further pointswithin distance t of a typical point.

Second, if the pair correlation function g(u, v) = g0(u v) only dependson v u (see (10)), the denition of the K -function can be extended: Theinhomogeneous K -function (Baddeley et al. , 2000) is dened by

K (t) = v t g0(v) dv.

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By (10), it follows that

K (t) = Ev X !u

1( v u t)(v)

for any u S with (u) > 0. If for v u t, (v) is close to (u), weobtain (u)K (t) E v X !u 1( v u t). This is a local version of theinterpretation of K (t) in the stationary case.

Nonparametric estimation of K and G is based on empirical versions ob-tained from ( 15). For some parametric Poisson and Cox process models, K or G are expressible on closed form and may be compared with correspondingnonparametric estimates when nding parameter estimates or assessing a t-

ted model. See Mller and Waagepetersen (2007) and the references therein.

6.2 Prediction given partial observation of point pro-cess

Suppose S is bounded and we observe a point process Y contained in a nitepoint process X specied by some density f with respect to the unit ratePoisson process Z . If B S with |B | > 0 and Y = X B , then predictionof X S \ B given Y = y can be based on the conditional density f S \ B (|y )introduced in Section 2.3. On the other hand, if we just know that y X ,then it could be tempting to try to predict X \ y using X !y . This wouldin general be incorrect. For instance, for an LGCP with positive covariancefunction, the intensity of X !y can be much larger than the one of X , cf.(22). Thus on average X !y y would contain more points than X . The issuehere is that the reduced Palm distribution is concerned with the conditionaldistribution of X conditional on that prespecied points fall in X . Hence thesampling mechanism that leads from X to Y must be taken into account.For instance, if the distribution of Y conditional on X = x is specied bya probability density function p(|x ) (on the set of all subsets of x), thenby Proposition 1 in Baddeley et al. (2000), the marginal density of Y withrespect to Z is

g(y ) = (n )(y ) exp( |S |)E p(y |X !y y ) ,where n = n(y ) is the cardinality of y . Thus the conditional distribution of X \ y given Y = y has density

f (x |y ) = p(y |x y )f (x y ) exp( |S |)/g (y )

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with respect to Z .

6.3 Matern-thinned Cox processes

Some applications of spatial point processes require models that combineclustering at a large scale with regularity at a local scale ( Lavancier and Mller ,2015). Andersen and Hahn (2015) study a class of so-called Matern thinnedCox processes where (clustered) Cox processes are subjected to dependentMatern type II thinning ( Matern , 1986) that introduces regularity in theresulting point processes. The intensity function and second-order joint in-tensity of the Matern-thinned Cox process is expressed in terms of univariateand bivariate inclusion probabilities which in turn are expressed in terms of one- and two-point Palm probabilities for an independently marked version of the underlying Cox process. In case of an underlying shot-noise Cox process,explicit expressions for the univariate inclusion probabilities are obtained us-ing the simple characterization of one-point Palm distributions described inSection 5.3.2.

6.4 Palm likelihood

Minimum contrast estimators based on the K -function or the pair correlationfunction or composite likelihood methods are standard methods to estimateparametric models (see e.g. Jolivet , 1991; Guan, 2006; Mller and Waagepetersen ,2007; Waagepetersen and Guan , 2009; Biscio and Lavancier, 2015). Tanaka et al.(2008) proposed an approach based on Palm intensities to estimate paramet-ric stationary models, which is briey presented below.

Given a parametric model g(u, v) = g0(v u; ) for the pair correlationfunction of X and a location u S , the intensity function of X !u is u (v; ) =g0(v u; ) where is the constant intensity of X assumed here to be known.Following Schoenberg (2005), the so-called log composite likelihood score

v X !u b(u,R )

dd

logu (v; ) b(u,R ) u (v; ) dvforms an unbiased estimating function for , where R > 0 is a user-speciedtuning parameter. Usually X !u is not known. However, suppose that Xis observed on W B 0 and in order to introduce a border correction let

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W R = {u W |b(u, R ) W }. Then, by ( 15),

=

u X W R,v X b(u,R )

dd

log u (v; ) N (W R) b(o,R ) o(v; ) dv (23)is an unbiased estimate of the above composite likelihood score times |W R |. Tanaka et al. (2008) coined the antiderivative of ( 23) the Palm likelihood.Asymptotic properties of Palm likelihood parameter estimates are studiedby Prokesov a and Jensen (2013) who also proposed the border correctionapplied in (23).

Acknowledgments

J. Mller and R. Waagepetersen are supported by the Danish Council forIndependent Research Natural Sciences, grant 12-124675, Mathematicaland Statistical Analysis of Spatial Data, and by the Centre for Stochas-tic Geometry and Advanced Bioimaging, funded by grant 8721 from theVillum Foundation. J.-F. Coeurjolly is supported by ANR-11-LABX-0025PERSYVAL-Lab (2011, project OculoNimbus).

References

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A. Baddeley and G. Nair. Fast approximation of the intensity of Gibbs pointprocesses. Electronic Journal of Statistics , 6:11551169, 2012.

A.J. Baddeley, J. Mller, and R. Waagepetersen. Non- and semi-parametricestimation of interaction in inhomogeneous point patterns. Statistica Neer-landica , 54:329350, 2000.

C.A.N. Biscio and F. Lavancier. Contrast estimation for parametric station-ary determinantal point processes. Submitted for publication. Availableat arXiv:1510.04222, 2015.

S.N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and Its Applications . Wiley, Chichester, third edition, 2013.

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