FREE AREA ESTIMATION IN A PARTIALLY OBSERVED DYNAMIC GERM-GRAIN MODEL 1

The estimation problem of the expected local fraction of free area function for a W partially observed dynamic germ-grain model is presented. Properties of the estimators are proved by martingale and product integral methods. Confidence bounds are provided. Furthermore, an estimator of the hazard rate αÐ>Ñ œ  .WÐ>ÑÎÐWÐ>Ñ.>Ñ is obtained by the kernel function method and asymptotic properties of the estimator are proved and used to find confidence intervals. By a simulated illustrative example, the qualitative behavior of the estimators is shown. Dynamic Germ-Grain Model, Fraction of Free Area, Survival


Introduction
In this paper we study the estimation problem for the expected local fraction of free area (LFFA) function of a partially observed (DGGM) as defined dynamic germ-grain model below.
We start with a model and problem outline.Suppose disks of random area are dropped at random times on the plane in such a way that their centers are random points of a convex ' # Borel region .The time-rate of the dropping-times process at time is assumed C § >  !' # to be of the form , where is a time-dependent parameter and is the 2---Ð>Ñ † jÐ Ñ jÐ † Ñ C dimensional Lebesgue measure.
For any , denotes the random set obtained as the union of the disks at time .A >  !Ð>Ñ > @ random set such as is called a germ-grain model (see, for example, [14]).Because of its @Ð>Ñ evolution in time, we call the family DGGM (see Figure 1, parts A1 and @ @ oe Ö Ð>Ñà >  !× A2).
For a fixed convex Borel set , the LFFA process in is the stochastic process B C B § ??oe Ö Ð>Ñà >  !× defined by where is the set difference.B B Ï Ð>Ñ oe ∩ Ð>Ñ @ @ - Suppose an observer without control of the generation of the DGGM is interested in @ evaluating the realization of the LFFA process .If the falling times, areas, and center ?positions of all disks are known at any time, then is (more or less easily) computable for ?Ð>Ñ any .>  !Now imagine that the region is, unfortunately, obscured by a shadowing element so that C a survey of the positions of the fallen disks covering is not possible.In  where denotes the unitary extended area (see (1) below) and is used to estimate .

E E s
We prove properties of the estimators and by extensive use of the martingale and E W s s product integral theories.Properties of are proved using properties of .W E s s Note that we suppose the observer is able to record the random areas and times of the falling disks.To use our method, these areas and times must be recorded by the observer as necessary data.We provide the following example of an application of our method.
(Bombing Problem) Suppose a bombing activity is taking place on a region Example: C C § ' # .Bombs of random destructive power are dropped at random times on .Each bomb will strike a random point in and destroy a circular region with its center at the struck C point and its area proportional to the bomb's destructive power.
An observer would like to know the fraction of non-destroyed area for a Borel set B C § that is the realization of .He is able to record the landing times and the destructive power ? of each bomb.Because of the presence of clouds obscuring , he cannot observe the point C struck by each bomb.So an estimate of is required.W Note that for a fixed point , ! − F (See the proof of Proposition 1.) Thus, if denotes the time at which the DGGM hits the X !@ point for the first time, then where may be interpreted as a survival function of the point .W ! By analogy with standard results of survival analysis, we observe that So , a hazard rate function of the point , is worth estimating in its own right.α !In this paper, an estimator of is defined as a kernel-function smoothing of and its α α s E s asymptotic properties are proved.
An outline of the rest of the paper is as follows.In Section 2, the problem and mathematical model are described in further detail.In Section 3, the statistical problem is described, the estimators and are defined, and their preliminary properties are proved.

E W s s
Section 4 contains the asymptotic results for and .In Section 5, the results of Section 4 E W s s are used to find confidence intervals and bounds for and .Section 6 is devoted to the E W estimation of .Finally, in Section 7, results of numerical simulations are provided.α

The Model, Notation and Preliminary Results
Let Ð>Ñ oe Ð>ß Ñ − @ @ = = H the union of the random disks dropped up to time , > Because of its evolution time, we call the family a DGGM.@ @ oe Ö Ð>Ñà >  !× Formally, all random variables considered in this paper are defined on the same probability space , .
The history of the process will be represented by a filtration , where for any , is the -field generated by all the events which have occurred up to time .That is, > where and are the Borel -fields on and , respectively.
We suppose that the following assumption holds for the distributions of the areas.
From Assumption A1, it follows that Remark 1: where Given the uniformity of the disk centers, does not T ÒB Â Ð>ÑÓ @ depend on , so B WÐ>Ñ oe T Ò! Â Ð>ÑÓß @ for a fixed point .! − B If, at a time , a disk drops on with random center uniformly distributed on >  !HÐ>Ñ \ ' # C and random area independent of , then it will cover the point 0 with probability Z Ð>Ñ \ The process counting, for any , the number of disks placed in the time interval and covering the point , is a -thinning of the Poisson process .Ð!ß >Ó !: R Ð>Ñ Given the continuity of (see Remark 1), is a Poisson process with mean measure .Ð>Ñ R 7 ! !(see, for example [6]), given by

The Statistical Problem
In the statistical problem solved below, the time rate and the distributions of the areas -ÖZ Ð>Ñà >  !× are unknown.The landing times and the area of each disk are the observable data.The centers of the dropped disks (that is, their positions in the plane) are uniformly distributed on but are not observable C Recall that we are looking for an estimator of the expected LFFA function .An W immediate consequence of Proposition 1 is that where denotes the product integral.# Therefore, the natural estimator of is where is a suitable estimator of .

E E s
As an estimator of we choose the process where is the random area of a disk dropped at time .Note that is a stochastic jump Z Ð=Ñ = E s process adapted to with random jumps (see Figure 1, A3).In the sequel, we will Ö à>  !× Y > first obtain results for and use these results to prove properties of .E W s s Proposition 2: The process , defined by is a zero mean square integrable martingale with predictable variation given by - By definition, is -adapted.Moreover, for any , Proof: ` The assumption of independence on , , and implies that is .
Then is a zero-mean martingale and is square integrable because (see Reasoning as above, we have the following equalities (a proof of the second equality can be found in [1, p. 54])

C
The second equality above holds because is a step function.

Confidence Intervals and Bounds for and E W
From Theorems 1 and 2 in the previous section, for any α To find the confidence bounds for and consider the following convergence results E W using the same notation and assumptions as in the previous sections.

Estimation of α
As pointed out in the introduction, the estimation of is important.In this α .-Ð>Ñ oe Ð>Ñ Ð>Ñ section we obtain the estimator of by smoothing with a kernel function (see α α ŝÐ>Ñ Ð>Ñ E s (13)).In the following, is a function of bounded variation that vanishes outside , ^Ò  "ß "Ó such that ' " " ^Ð?Ñ.? oe "ß and is a sequence such that as and For example, we could take ., À oe ÐjÐ ÑÑ 8 8 #Î& C From here on, the interval of estimation will be fixed and will be so We define, for any , the estimator by > − Ò> ß > Ó Ð>Ñ s `? − Ò!ß X Ó×, defined by ' Š ‹ , is a zero-mean square integrable martingale, and It follows that We are now ready to prove the following asymptotic results.(Uniform consistency) Theorem 4: is a uniformly consistent estimator of in , that is, The proof is complete.
(Asymptotic Normality) Theorem 5: Let and suppose that has a bounded > − Ò!ß X Ó α derivative in a neighborhood of , that is, numbers and exist such that In Figure 1, parts A1 and A2, we have plotted the state of at two different times.In part @ A3 of Figure 1, the related realization of the unitary extended area process is shown.
E s To illustrate the qualitative asymptotic behavior of the estimators, we have also simulated the DGGM on regions of different Lebesgue measures.We then computed the estimators C 8 E Ð>Ñ W Ð>Ñ Ò!ß &Ó s s and defined by (5).The 95% confidence bounds on the time interval have also been computed following (11) and (12).We have taken the value of / Ð-Ñ oe / Ð-Ñ from the table in [13] after estimating the unknown by --À oe ß s @ ÐXÑ s "@ ÐX Ñ s 8 8 where .In Figure 2, we have plotted the true function , the computed estimator X oe & E E s 8 and the confidence bounds for .In Figure 3, we have plotted the true function , the E W estimator and the related confidence bounds.W s 8 With the simulated data, the estimator has been obtained using (13).The related α s Ð>Ñ 8 confidence intervals have been calculated following (15).The results obtained are plotted in Figure 4.
A1 and A2: An example of realization of the dynamic germ-grain model observed at two different times.@ A3: Plot of the related realization of the unitary extended area E s

Figure 2 : 8 8 Figure 3 : 8 8 Figure 4 :
Figure 2: Comparison between (continuous line), its estimator , E E s 8 and the 95% confidence bounds for and jÐ Ñ oe $! jÐ Ñ oe (!Þ C C 8 8 as observed in Remark 1, the areas of the disks are a.s.bounded, there are no edge effects to consider if, as in our case, we are interested only in the region .
Using terminology of the General Theory of Stochastic Processes, it follows, Remark 3: from Propositions 2 and 4, that and are, respectively, the compensators of and . # To show the evolution in time of the DGGM , we have simulated it in a time interval .@ Ò!ß &Ó As an example, we have assumed that Èwhere is a continuous random variable having uniform distribution on .