BSL (version 0.1.1)

ma2: 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. (2018).

The function ma2_sim(theta,options) simulates an MA(2) time series.

The function ma2_sum(x) returns the summary statistics for a given data set. Since the summary statistics are the data, this function simply returns the data.

ma2_prior(theta) evaluates the (unnormalised) prior, which is uniform subject to several restrictions related to invertibility of the time series.

Usage

data(ma2)
ma2_sim(theta, options)
ma2_sum(x)
ma2_prior(theta)

Arguments

theta

A vector of proposed model parameters, \(\theta_1\) and \(\theta_2\)

options

A list of options for simulating data from the model. For this example, the list contains only \(T\), the number of observations.

x

Observed or simulated data in the format of a vector of length \(T\).

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 \(T=50\). Further information about this model and the specific choices of tuning parameters used in BSL and BSLasso can be found in An et al. (2018).

• data: A time series dataset, in the form of a vector of length \(T\)

• sim_options: A list containing \(T=50\)

• start: A vector of suitable initial values of the parameters for MCMC

• cov: Covariance matrix of the multivariate normal random walk, in the form of a \(2 x 2\) matrix

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 \(t=1,\ldots,T\) and \(z_t ~ N(0,1)\) for \(t=-1,0,\ldots,T\). 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.

References

An, Z., South, L. F., Nott, D. J. & Drovandi, C. C. (2018). Accelerating Bayesian synthetic likelihood with the graphical lasso. https://eprints.qut.edu.au/102263/

Examples

# NOT RUN {
# Loading the data for this example
data(ma2)
true_ma2 <- c(0.6,0.2)

# Performing BSL
resultMa2BSL <- bsl(y = ma2$data, n = 500, M = 300000, start = ma2$start, cov_rw = ma2$cov,
                 fn_sim = ma2_sim, fn_sum = ma2_sum, fn_prior = ma2_prior,
                 sim_options = ma2$sim_options, theta_names = c('theta1', 'theta2'))
summary(resultMa2BSL)
plot(resultMa2BSL, true_value = true_ma2, thin = 20)

# Performing tuning for BSLasso 
lambda_all <- list(exp(seq(-3,0.5,length.out=20)), exp(seq(-4,-0.5,length.out=20)), 
                 exp(seq(-5.5,-1.5,length.out=20)), exp(seq(-7,-2,length.out=20)))

sp_ma2 <- selectPenalty(ssy = ma2_sum(ma2$data), n = c(50, 150, 300, 500), lambda_all,
                 theta = true_ma2, M = 100, sigma = 1.5, fn_sim = ma2_sim,
                 fn_sum = ma2_sum, sim_options = ma2$sim_options)
sp_ma2
plot(sp_ma2)

# Performing BSLasso with a fixed penalty
resultMa2BSLasso <- bsl(y = ma2$data, n = 300, M = 250000, start = ma2$start, cov_rw = ma2$cov,
                 fn_sim = ma2_sim, fn_sum = ma2_sum, penalty = 0.027, fn_prior = ma2_prior,
                 sim_options = ma2$sim_options, theta_names = c('theta1', 'theta2'))
summary(resultMa2BSLasso)
plot(resultMa2BSLasso, true_value = true_ma2, thin = 20)

# Plotting the results together for comparison
combinePlotsBSL(resultMa2BSL, resultMa2BSLasso, true_value = true_ma2, thin = 20)
# }
# NOT RUN {
# }