Implementation via 2-Stage Procedure

WhenC consists of connected nodes (cf. Section 3.7.1) the regions {EAi; i= 1, . . . , N} may be geometrically quite irregular. Therefore even though the conditional distribution on each cell is rel- atively straightforward (density given byfi in Eq. 5.9), the irregular shape of the cells may pose

implementational difficulties. Drawing from the densityfi via Algorithm 5.2 would require addi-

tional (geometric) tests, which may be both difficult to implement and computationally burdensome. On the other hand if the regionsEAi are disks of radiusr, thenU(EAi)is just a disk of radius2r. In this case it is relatively straightforward to sample fromfi. Therefore the practical implementation

of the exact Gibbs Algorithm 5.4 is carried out via a 2-Stage procedure: a subset A∗ ⊆ C (cf. Definition 3.2) which contains the maximum number of disconnected nodes is chosen. Algorithm 5.4 is applied to A∗; if the output also covers C then this sample is returned. If not then another sample is drawn independently via Algorithm 5.4 applied to A∗. This is continued until a sample which coversCis returned. For similar implementational reasons the implementation of the modified Cai & Kendall Algorithm 5.1 is also carried out via the above 2-Stage procedure.

5.7

Simulation Results

In this section the simulation results for the conditional (attractive) area-interaction process are pre- sented. The experiments aim to evaluate the performance (in terms of actual run times in seconds) of the three perfect algorithms: 2-Stage Rejection (cf. Section 5.3), the modified Cai & Kendall Al- gorithm 5.1, and the Gibbs Algorithm 5.4. Recall from Section 3.7.1 thatC can be viewed as a graph where two nodesci andcj areconnectedifU(ci)∩ U(cj) 6= ∅. The sampling window isWδr, for

someδ, andA∗ is as in Definition 3.2. Recall from Section 5.6.2 above that all three algorithms are implemented via a 2-Stage procedure. For convenience abbreviate the 2-Stage Rejection algorithm by ‘2Stg’, the modified Cai & Kendall algorithm by ‘CK’ and the exact Gibbs algorithm by ‘Gibbs’.

In the experiments for the conditional Boolean model (Section 3.7.2) the sampling window was always taken to beWr. The reason being that the germs of the Boolean model are independent in

disjoint regions. Therefore, as long asU(C) ⊂ W, the size of the sampling window W does not matter. However for the area-interaction model the presence of interactions between the germs vio- lates this independence property. In this case the size ofW influences the strength of the interaction. Therefore run times are compared against size ofW, underlying Poissonian intensityλ, interaction parameterβ and size of the conditioning setk.

Remark 3.2 in Section 3.7.2 comments about the notion oflocal changeandglobal changealgo- rithms. CK is a local change algorithm since it employs birth-death processes while 2Stg and Gibbs are global change algorithms. The pros and cons of either type of algorithm are also discussed in Remark 3.2. For extreme model parameters (eg. highλ,β,kor a largeW) local change algorithms are expected to perform better; for moderate parameters global ones are. The simulation results sup- port this intuition. As parameter values get extreme, CK performs better than the other two, while for moderate parameters 2Stg and Gibbs do well. Notice also that Gibbs here is very competitive to 2Stg for all parameter values, unlike in the case of the conditional Boolean model (cf. Section 3.7). The simulations were carried out on a PC (Pentium 4 2.67GHz, 248MB RAM) running Windows XP; the implementations of the algorithms were programmed inPython(version 2.3).

5.7.1

Experiment 1: Run Times Versus Intensityλ

The evolution of run times as the underlying intensityλ varies is first explored. Holding the other model parameters fixed, as λ decreases the coverage conditioning becomes increasingly stringent. On the other hand asλdecreases so does the mean number of germ-grain pairs; for 2Stg and Gibbs this means that global moves are more likely to be accepted since the acceptance weighte−βψ in- creases asλ decreases. The computational burden of 2Stg and Gibbs should nevertheless increase asλdecreases since coverage ofC will still be a rare event. The results of Experiment 3.7.3 suggest that the Cai & Kendall algorithm performs better than 2Stg and Gibbs asλdecreases. Whether sim- ilar results should hold here requires some careful thought. The modified Cai & Kendall algorithm (CK) involves, additionally, censoring birth transitions. Moreover there is an extra computational burden: that of simulating the marksZξattached to each birthξwhich enable the correct censoring

of births. A priori it was difficult to predict the relative behaviour of the respective run times as

λgets smaller, except that those for 2Stg and Gibbs should increase.

As λ increases the coverage conditioning will be easily satisfied. On the other hand the area- interaction between the germs will increase and so global moves in 2Stg and Gibbs are less likely to be accepted. Therefore their respective run times will also increase withλ; thus the graphs for these two should look somewhat like parabolas. Moreover single births would be readily accepted in CK since the acceptance probability increases with the number of germ-grain pairs. Thus CK should have lower run times for highλ.

Figures 5.2 & 5.3 depict the run times for two different simulations. The results do illustrate parabolic-like curves for 2Stg and Gibbs and support the expectation that CK is more efficient for largeλ. For smallλ, it appears that the added complexity of CK (as compared to the Cai & Kendall algorithm for conditional Boolean models) means that its runs times get very large asλdecreases.