saABC fits parameter estimators based on simulated data to be
used as summary statistics within ABC. Fitting is by linear
regression. Some simple diagnostics are provided for assistance.
saABC(theta, X, plot = TRUE)
Vector of constant terms from fitted regressions.
Matrix of weights from fitted regressions.
Vector of BIC values for each fitted regression.
A n x d matrix or data frame of simulated parameter values.
theta[i,j] is the ith simulated value of parameter j.
A n x p matrix or data frame of simulated data and/or associated
transformations. X[i,] is a vector of the data for parameter
values theta[i,]. A constant term should not be included.
When plot==TRUE, a plot of parameter values against fitted
values is produced for each parameter as a side-effect.
Dennis Prangle
The semi-automatic ABC method of Fearnhead and Prangle (2012) is as follows:
1) Simulate parameter vectors \(\theta_i\) and corresponding data sets \(x_i\) for i=1,2,...,N.
2) Use the simulations to fit an estimator of each parameter as a linear combination of f(x), where f(x) is a vector of transformations of x (including a constant term).
3) Run ABC using these simulations.
The saABC function automates step 2 of this process. The user
must supply simulated parameter values theta and corresponding
f(x) values x (n.b. excluding the constant term). The function
returns weights for the linear combinations which can easily be used
for step 3. In particular, fitted weights are returned as a matrix
of weights for the columns of x and a vector of constants. The
vector can usually be discarded, as it is not needed to find
differences between summary statistics.
The function also returns BIC values for each parameter so that the user can judge the quality of the fits, and compare different choices of f(x). Diagnostic plots of supplied parameter values against fitted values are also optionally provided. These are useful for exploratory purposes when there are a small number of parameters, but provide less protection from overfitting than BIC values.
Blum, M. G. B, Nunes, M. A., Prangle, D. and Sisson, S. A. (2013) A
comparative review of dimension reduction methods in approximate
Bayesian computation. Stat. Sci. 28, Issue 2, 189--208.
Fearnhead, P. and Prangle, D. (2012) Constructing summary statistics for approximate Bayesian computation:
semi-automatic approximate Bayesian
computation. J. R. Stat. Soc. B 74, Part 3, 1--28.
Nunes, M. A. and Prangle, D. (2016) abctools: an R package for tuning
approximate Bayesian computation analyses. The R Journal
7, Issue 2, 189--205.
set.seed(1)
theta <- matrix(runif(2E3),ncol=2)
colnames(theta) <- c("Mean", "Variance")
X <- replicate(5, rnorm(1E3, theta[,1], theta[,2]))
saABC(theta, X)$BICs
saABC(theta, cbind(X, X^2))$BICs ##Variance parameter estimated better
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