ma2 {BSL}R Documentation

An MA(2) model

Description

In this example we wish to estimate the parameters of a simple MA(2) time series model. We provide the data and tuning parameters required to reproduce the results in An et al. (2019). The journal article An et al. (2022) provides a full description of how to use this package for the toad example.

Usage

data(ma2)

ma2_sim(theta, TT)

ma2_sim_vec(n, theta, TT)

ma2_sum(x, epsilon = 0, delta = 1)

ma2_prior(theta)

Arguments

theta

A vector of proposed model parameters, \theta_1 and \theta_2.

TT

The number of observations.

n

The number of simulations to run with the vectorised simulation function.

x

Observed or simulated data in the format of a vector of length TT.

epsilon

The skewness parameter in the sinh-arcsinh transformation.

delta

The kurtosis parameter in the sinh-arcsinh transformation.

Details

This example is based on estimating the parameters of a basic MA(2) time series model of the form

y_t = z_t + \theta_1 z_{t-1} + \theta_2 z_{t-2},

where t=1,\ldots,TT and z_t \sim N(0,1) for t=-1,0,\ldots,TT. A uniform prior is used for this example, subject to the restrictions that -2<\theta_1<2, \theta_1+\theta_2>-1 and \theta_1-\theta_2<1 so that invertibility of the time series is satisfied. The summary statistics are simply the full data.

Functions

A simulated dataset

An example “observed” dataset and the tuning parameters relevant to that example can be obtained using data(ma2). This “observed” data is a simulated dataset with \theta_1 = 0.6, \theta_2=0.2 and TT=50. Further information about this model and the specific choices of tuning parameters used in BSL and BSLasso can be found in An et al. (2019).

Author(s)

Ziwen An, Leah F. South and Christopher Drovandi

References

An Z, South LF, Drovandi CC (2022). “BSL: An R Package for Efficient Parameter Estimation for Simulation-Based Models via Bayesian Synthetic Likelihood.” Journal of Statistical Software, 101(11), 1–33. doi: 10.18637/jss.v101.i11.

An Z, South LF, Nott DJ, Drovandi CC (2019). “Accelerating Bayesian Synthetic Likelihood With the Graphical Lasso.” Journal of Computational and Graphical Statistics, 28(2), 471–475. doi: 10.1080/10618600.2018.1537928.

Jones MC, Pewsey A (2009). “Sinh-arcsinh distributions.” Biometrika, 96(4), 761–780. ISSN 0006-3444.

Examples

## Not run: 
# Load the data for this example and set up the model object
data(ma2)
model <- newModel(fnSimVec = ma2_sim_vec, fnSum = ma2_sum, simArgs = ma2$sim_args,
                  theta0 = ma2$start, fnLogPrior = ma2_prior)
thetaExact <- c(0.6, 0.2)

# reduce the number of iterations M if desired for all methods below
# Method 1: standard BSL
resultMa2BSL <- bsl(y = ma2$data, n = 500, M = 300000, model = model, covRandWalk = ma2$cov,
                    method = "BSL", verbose = 1L)
show(resultMa2BSL)
summary(resultMa2BSL)
plot(resultMa2BSL, thetaTrue = thetaExact, thin = 20)

# Method 2: unbiased BSL
resultMa2uBSL <- bsl(y = ma2$data, n = 500, M = 300000, model = model, covRandWalk=ma2$cov,
                     method = "uBSL", verbose = 1L)
show(resultMa2uBSL)
summary(resultMa2uBSL)
plot(resultMa2uBSL, thetaTrue = thetaExact, thin = 20)

# Method 3: BSLasso (BSL with glasso shrinkage estimation)
# tune the penalty parameter fisrt
ssy <- ma2_sum(ma2$data)
lambdaAll <- list(exp(seq(-5.5,-1.5,length.out=20)))
set.seed(100)
penaltyGlasso <- selectPenalty(ssy = ssy, n = 300, lambdaAll, theta = thetaExact,
                        M = 100, sigma = 1.5, model = model, method = "BSL", shrinkage = "glasso")
penaltyGlasso
plot(penaltyGlasso)

resultMa2BSLasso <- bsl(y = ma2$data, n = 300, M = 250000, model = model, covRandWalk=ma2$cov,
                        method = "BSL", shrinkage = "glasso", penalty = 0.027, verbose = 1L)
show(resultMa2BSLasso)
summary(resultMa2BSLasso)
plot(resultMa2BSLasso, thetaTrue = thetaExact, thin = 20)

# Method 4: BSL with Warton's shrinkage and Whitening
# estimate the Whtieing matrix and tune the penalty parameter first
W <- estimateWhiteningMatrix(20000, model, method = "PCA", thetaPoint = ma2$start)
gammaAll <- list(seq(0.3, 0.8, 0.02))
set.seed(100)
penaltyWarton <- selectPenalty(ssy = ssy, n = 300, gammaAll, theta = thetaExact,
                        M = 100, sigma = 1.2, model = model, method = "BSL", shrinkage = "Warton",
                        whitening = W)
penaltyWarton
plot(penaltyWarton, logscale = FALSE)

resultMa2Whitening <- bsl(y = ma2$data, n = 300, M = 250000, model = model, covRandWalk=ma2$cov,
                        method = "BSL", shrinkage = "Warton", whitening = W,
                        penalty = 0.52, verbose = 1L)
show(resultMa2Whitening)
summary(resultMa2Whitening)
plot(resultMa2Whitening, thetaTrue = thetaExact, thin = 20)

# Method 5: semiBSL, the summary statistics function is different from previous methods
model2 <- newModel(fnSimVec = ma2_sim_vec, fnSum = ma2_sum, simArgs = ma2$sim_args,
                  sumArgs = list(epsilon = 2), theta0 = ma2$start, fnLogPrior = ma2_prior)
sim <- simulation(model, n = 1e4, theta = ma2$start, seed = 1) # run a short simulation
plot(density(sim$ssx[, 1])) # the first marginal summary statistic is right-skewed
resultMa2SemiBSL <- bsl(y = ma2$data, n = 500, M = 200000, model = model2, covRandWalk=ma2$cov,
                        method = "semiBSL", verbose = 1L)
show(resultMa2SemiBSL)
summary(resultMa2SemiBSL)
plot(resultMa2SemiBSL, thetaTrue = thetaExact, thin = 20)

# Method 6: BSL with consideration of model misspecification (mean adjustment)
resultMa2Mean <- bsl(y = ma2$data, n = 500, M = 200000, model = model, covRandWalk=ma2$cov,
                        method = "BSLmisspec", misspecType = "mean", verbose = 1L)
show(resultMa2Mean)
summary(resultMa2Mean)
plot(resultMa2Mean, thetaTrue = thetaExact, thin = 20)

# Method 7: BSL with consideration of model misspecification (variance inflation)
resultMa2Variance <- bsl(y = ma2$data, n = 500, M = 200000, model = model, covRandWalk=ma2$cov,
                     method = "BSLmisspec", misspecType = "variance", verbose = 1L)
show(resultMa2Variance)
summary(resultMa2Variance)
plot(resultMa2Variance, thetaTrue = thetaExact, thin = 20)

# Plotting the results together for comparison
# plot using the R default plot function
oldpar <- par()
par(mar = c(5, 4, 1, 2), oma = c(0, 1, 2, 0))
combinePlotsBSL(list(resultMa2BSL, resultMa2uBSL, resultMa2BSLasso, resultMa2SemiBSL), which = 1,
                thetaTrue = thetaExact, thin = 20, label = c("bsl", "uBSL", "bslasso", "semiBSL"),
                col = c("black", "red", "blue", "green"), lty = 1:4, lwd = 1)
mtext("Approximate Univariate Posteriors", outer = TRUE, cex = 1.5)

# plot using the ggplot2 package
combinePlotsBSL(list(resultMa2BSL, resultMa2uBSL, resultMa2BSLasso, resultMa2SemiBSL), which = 2,
    thetaTrue = thetaExact, thin = 20, label = c("bsl", "ubsl", "bslasso", "semiBSL"),
    options.color = list(values=c("black", "red", "blue", "green")),
    options.linetype = list(values = 1:4), options.size = list(values = rep(1, 4)),
    options.theme = list(plot.margin = grid::unit(rep(0.03,4), "npc"),
        axis.title = ggplot2::element_text(size=12), axis.text = ggplot2::element_text(size = 8),
        legend.text = ggplot2::element_text(size = 12)))
par(mar = oldpar$mar, oma = oldpar$oma)

## End(Not run)


[Package BSL version 3.2.4 Index]