1 SA method : general description
The SA algorithm is a global optimisation method. Assume that the probability density is to be maximised. This is achieved by sampling while . If the temperature parameter goes to in an appropriate way, then the SA algorithm converges asymptotically towards the global optimum. This method is rather general. Under smooth assumptions, the algorithm can be generalised to minimise any criteria that can be written as .
SA algorithms for maximising probability densities for random fields and marked point process such as (1) are presented in [1, 4, 7]. The obtained cooling schedules for the temperature parameter are of the form
. The solution guaranteed by the method converges towards the uniform distribution over the sub-space of configurations that maximises (1).
The difficulty of solving (3) is due to the fact that the normalising constant is not available in analytic closed form. Hence, special strategies are required to sample from the posterior distribution (2). The present paper use for thie purpose, the ABC Shadow simulation dynamics .
The ABC Shadow dynamics is an approximate algorithm able to sample posteriors. Its main steps are presented below :
Algorithm ABC Shadow : fix and . Assume the observed pattern is and the current state is .
Generate according to .
For to do
Generate a new candidate following the density defined by
representing the uniform probability density over the ball of volume .
The new state is accepted with probability given by
If another sample is needed, go to step with .
The SA Shadow algorithm is applied here to the statistical analysis of patterns which are simulated from a Strauss model [3, 9]. This model describes random patterns made of points exhibiting repulsion. Its probability density is
Here is a point pattern in the window , while and are the sufficient statistic and the model parameter vectors, respectively. The sufficient statistics components and represent respectively, the number of points in and the number of pairs of points at a distance closer than .
The Strauss model on the unit square and with density parameters , and , was considered. This gives for the parameter vector of the exponential model . The CFTP algorithm (see Chapter 11 in ) was used to get samples from the model and to compute the empirical means of the sufficient statistics . The SA based on the ABC Shadow algorithm was run using as observed data.
The prior density was the uniform distribution on the interval . Each time, the auxiliary variable was sampled using steps of a MH dynamics [5, 6]. The and parameters were set to and , respectively. The algorithm was run for iterations. The intial temperature was set to . For the cooling schedule a slow polynomial scheme was chosen
with . A similar scheme was chosen for the parameters, with . Samples were kept every steps. This gave a total of samples.
Figure 1 shows the results obtained after running the SA ABC Shadow based algorithm. The final values for and were and , respectively. These values are close to the true model parameters.
SA results for computing the MAP estimates for the Strauss model parameters.
3 Conclusions and perspectives
The numerical results obtained are satisfactory. Actually, the algorithm is applied on real astronomical data, and the obtained models are tested and validated. Since the ABC Shadow is an approximate algorithm, the theoretical convergence of the SA procedure based on it, needs further investigation .
Part of the work of the first author was supported by a grant of the Romanian Ministry of National Education and Scientific Research, RDI Programme for Space Technology and Avanced Research - STAR, project number 513.
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