# Reduced Rank Regression In sBIC: Computing the Singular BIC for Multiple Models

which.max(results$sBIC) -1  ## Simulation study of rank selection The asymptotic properties of BIC and sBIC can be further studied by simulating multiple data sets from the reduced rank regression model and comparing the results of rank selection (i.e. choosing the optimal model according to BIC or sBIC). Here we set up the simulation parameters. We consider sample sizes ranging from 50 up to 1000 and simulate sims=200 data sets for each sample size. set.seed(1234) n = c(50, 100, 200, 300, 500, 1000) sims = 200  The code below carries out the rank selection and produces a list of frequency tables tt for the optimal model chosen by BIC and sBIC. tt <- vector("list", length(n)) for (i in seq_along(n)) { rank.bic = rank.sbic = factor(rep(0, sims), levels = 0:min(M, N)) for (j in 1:sims) { rreg = ReducedRankRegressions(N, M, min(M, N)) XY = simRR(N, M, n[i], r) results = sBIC(XY, rreg) rank.bic[j] = which.max(results$BIC) - 1
rank.sbic[j] = which.max(results\$sBIC) - 1
}

tt[[i]] = rbind("BIC"=table(rank.bic),"sBIC"=table(rank.sbic))
}


The code below visualizes the frequency tables in tt as a series of bar plots. The output is shown in Figure 1.

par(mfrow=c(2,3), mar=c(2,1,3,1)+0.1, oma=c(0,2,1,0))
for (i in 1:6) {
barplot(tt[[i]], beside=TRUE, ylim=c(0, sims), col=c("white","black"),
legend=c("BIC", "sBIC"), main=paste("n=",n[i]))
}


Figure 1 partly reproduces Figure 2 of Drton and Plummer (2017). The latter also includes a comparison with the Widely Applicable Bayesian Information Criterion (WBIC) of Watanabe (2013).

## Bibliography

• Aoyagi, M. and Watanabe, S. (2005) Stochastic complexities of reduced rank regression in Bayesian estimation. Neurl Netwrks; 18: 924-933.
• Drton M. and Plummer M. (2017), A Bayesian information criterion for singular models. J. R. Statist. Soc. B; 79: 1-38.
• Watanabe, S. (2013) A widely applicable Bayesian information criterion. J. Mach. Learn. Res.; 14: 867-897

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sBIC documentation built on May 2, 2019, 4:16 a.m.