Abstract
Functional statistics are commonly used to characterize spatial patterns in general and spatial genetic structures in population genetics in particular. Such functional statistics also enable the estimation of parameters of spatially explicit (and genetic) models. Recently, Approximate Bayesian Computation (ABC) has been proposed to estimate model parameters from functional statistics. However, applying ABC with functional statistics may be cumbersome because of the high dimension of the set of statistics and the dependences among them. To tackle this difficulty, we propose an ABC procedure which relies on an optimized weighted distance between observed and simulated functional statistics. We applied this procedure to a simple step model, a spatial point process characterized by its pair correlation function and a pollen dispersal model characterized by genetic differentiation as a function of distance. These applications showed how the optimized weighted distance improved estimation accuracy. In the discussion, we consider the application of the proposed ABC procedure to functional statistics characterizing non-spatial processes.
We thank the reviewers for their useful suggestions and comments. This research was supported by the French research agency ANR (EMILE project).
7 Appendix
A Approximation of the posterior distribution of θ given S
This appendix shows in a simple case that pε(θ|S) given by Eq. (1) approximates the posterior distribution conditional on the statistics p(θ|S) given by Eq. (2). This simple case is based on regularity assumptions about the conditional probability distribution function 
of S given θ which cannot be checked in usual applications of ABC where 
is generally analytically intractable.
Here, (i) θ and S are fixed, (ii) the space 
of statistics is ℝ, (iii) the conditional probability distribution function of S given θ is three times differentiable over ℝ, (iv) the absolute values of its second and third derivatives are π-integrable over Θ, (v) its third derivative is a Lipschitz function 
.
From the Taylor’s theorem,
Using assumption (v), the absolute value of the remainder term R(ε, S, θ) is bounded from above by:
Let r(ε, S, θ)=R(ε, S, θ)/ε3. Using assumption (iv), the upper bound of |R(ε, S, θ)| is π-integrable over Θ as well as θ↦r(ε, S, θ) and, consequently,
Therefore, when ε tends to zero, pε(θ|S) approximates the posterior distribution conditional on the statistics p(θ|S).
B Implementation of algorithms A2-ME, TS and PLS and comparison criteria
In Subsection 5.5, algorithm A2 was applied to three different sets of summary statistics: (i) a subset of the raw statistics obtained with the minimum entropy approach of Nunes and Balding (2010), (ii) a subset of the raw statistics btained with the two-stage procedure of Nunes and Balding (2010) and (iii) a subset of the axes obtained after a PLS regression between the parameters and the statistics like in Wegmann et al. (2009). The three algorithms are denoted A2-ME, A2-TS and A2-PLS. Note that these approaches can be carried out with the abctools R package proposed by Nunes and Prangle and the ABCtoolbox software of (Wegmann et al., 2010).
For the ME (resp. TS) selection of the statistics, the exhaustive search of the subset of statistics which minimizes the entropy (resp. mean square root of the sum of squared errors, denoted MRSSE) over all the possible subsets was not feasible (for 91 statistics there are almost 2.5×1027 possible subsets). Therefore, we replaced the exhaustive search by an iterative search: we used a forward search based on the entropy (resp. MRSSE). At iteration one, the subset is made of the statistic which leads to the lowest entropy (resp. MRSSE). Then, at each of the following iterations, the current subset is completed by the statistic which leads, when it is merged to the current subset of statistics, to the lowest entropy (resp. MRSSE). The iterative search is stopped when the entropy (resp. MRSSE) is no more decreasing by the addition of any of the remaining statistics. For A2-ME, 5 (resp. 7) statistics were selected with τ=10–4 (resp. τ=10–3). For A2-TS, 11 (resp. 14) statistics were selected with τ=10–4 (resp. τ=10–3).
It has to be noted that, in A2-TS, the MRSSE that we used includes a standardization to rescale the components of the parameter vector (as suggested by Nunes and Balding, 2010) and satisfies:
where 
is the set of 100 PODS selected in stage one of the two-stage procedure, 
is the posterior sample (set of accepted parameter vectors) obtained for each PODS in obtained after stage one of the two-stage procedure. The size of the posterior sample is nacc=Iτ=106τ where τ is the acceptance threshold (10–4 or 10–3).
For A2-PLS, we fitted the PLS regression to 104 simulations and kept the minimum number of axes explaing at least 99% of the variance of the parameters. This led us to keep 24 axes among 91 possible axes.
We computed marginal criteria measuring the accuracy of the estimation for each parameter component. The marginal BMSE was computed with 1000 new pseudo-observed data sets (PODS) not used in the implementation of the algorithms which are compared. Among the 1000 new PODS, only 266 were used to compute the marginal PMSE, the marginal coverage of the 95%-posterior intervals of δ and b and the marginal mean square root of the sum of squared errors [MRSSE; criterion used in the two-stage approach of Nunes and Balding (2010)]. The 266 PODS were obtained as follows: For each value of τ (10–4 and 10–3), we selected the 250 PODS with the summary statistics reduced by the ME approach which are the closest from the observed summary statistics reduced by the ME approach (the closeness is quantified with the Euclidean distance). Then we merged the two sets of PODS and obtained a set of 266 different PODS. This selection of PODS is similar to stage one in the TS procedure of Nunes and Balding (2010).
Let 
, j=1,…,103 denote the 1000 new PODS. The marginal BMSE satisfies:
where 
are the marginal posterior medians of the K components of 
, the marginal posterior medians being obtained with either A4, A2-ME, A2-TS or A2-PLS.
Let 
denote the set of 266 selected PODS. The marginal PMSE satisfies:
The marginal coverage of any parameter by the corresponding marginal 95%-posterior interval is:
where 
and 
are the posterior quantiles of order 0.025 and 0.975 of the k-th component of θ*j, the posterior quantiles being obtained with either A4, A2-ME, A2-TS or A2-PLS, and 1 is the indicator function [1(E)=1 if event E holds, zero otherwise]. The marginal MRSSE satisfies:
where 
denotes the posterior sample (set of accepted parameter vectors) obtained by applying either A4, A2-ME, A2-TS or A2-PLS to the j-th PODS (to infer θ*j). The size of the posterior sample is nacc=Iτ=106τ where 
in algorithm A4 and τ=10–4 or 10–3 in algorithms A2-ME, TS and PLS.
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