BDgraph (version 2.60)

bdgraph.ts: Search algorithm in time series graphical models

Description

This function is for Bayesian model determination in time series graphical models, based on birth-death MCMC method.

Usage

bdgraph.ts( data, Nlength = NULL, n, iter = 1000, burnin = iter / 2, 
            g.prior = 0.5, df.prior = rep( 3, Nlength ), g.start = "empty", 
            save = FALSE, print = 500, cores = NULL )

Arguments

data

The aggregate periodogram \(P_k\), which is arranged as a large \(p x (Nlength*p)\) matrix \([P_1, P_2, ... ,P_Nlength]\).

Nlength

The length of the time series.

n

The number of observations.

iter

The number of iteration for the sampling algorithm.

burnin

The number of burn-in iteration for the sampling algorithm.

g.prior

For determining the prior distribution of each edge in the graph. There are two options: a single value between \(0\) and \(1\) (e.g. \(0.5\) as a noninformative prior) or an (\(p \times p\)) matrix with elements between \(0\) and \(1\).

df.prior

The degree of freedom for complex G-Wishart distribution, \(CW_G(b,D)\), which is a prior distribution of the precision matrix in each frequency.

g.start

Corresponds to a starting point of the graph. It could be "empty" (default) and "full". Option "empty" means the initial graph is an empty graph and "full" means a full graph. It also could be an object with S3 class "bdgraph"; with this option we could run the sampling algorithm from the last objects of previous run (see examples).

save

Logical: if FALSE (default), the adjacency matrices are NOT saved. If TRUE, the adjacency matrices after burn-in are saved.

print

Value to see the number of iteration for the MCMC algorithm.

cores

The number of cores to use for parallel execution. The case cores="all" means all CPU cores to use for parallel execution.

Value

An object with S3 class "bdgraph" is returned:

p_links

An upper triangular matrix which corresponds the estimated posterior probabilities of all possible links.

K_hat

The posterior estimation of the precision matrix.

For the case "save = TRUE" is returned:

sample_graphs

A vector of strings which includes the adjacency matrices of visited graphs after burn-in.

graph_weights

A vector which includes the waiting times of visited graphs after burn-in.

all_graphs

A vector which includes the identity of the adjacency matrices for all iterations after burn-in. It is needed for monitoring the convergence of the BD-MCMC algorithm.

all_weights

A vector which includes the waiting times for all iterations after burn-in. It is needed for monitoring the convergence of the BD-MCMC algorithm.

status

An integer to indicate the iteration where the algorithm exits, since if the sum of all rates is 0 at some iteration, the graph at this iteration is regarded as the real graph. It is 0 if the algorithm doesn't exit.

References

Tank, A., Foti, N., and Fox, E. (2015) Bayesian Structure Learning for Stationary Time Series, arXiv preprint arXiv:1505.03131

Mohammadi, R. and Wit, E. C. (2019). BDgraph: An R Package for Bayesian Structure Learning in Graphical Models, Journal of Statistical Software, 89(3):1-30

Mohammadi, A. and Wit, E. C. (2015). Bayesian Structure Learning in Sparse Gaussian Graphical Models, Bayesian Analysis, 10(1):109-138

Mohammadi, A. et al (2017). Bayesian modelling of Dupuytren disease by using Gaussian copula graphical models, Journal of the Royal Statistical Society: Series C, 66(3):629-645

Letac, G., Massam, H. and Mohammadi, R. (2018). The Ratio of Normalizing Constants for Bayesian Graphical Gaussian Model Selection, arXiv preprint arXiv:1706.04416v2

Dobra, A. and Mohammadi, R. (2018). Loglinear Model Selection and Human Mobility, Annals of Applied Statistics, 12(2):815-845

See Also

bdgraph, bdgraph.mpl, bdgraph.sim, summary.bdgraph, compare

Examples

# NOT RUN {
# Generating time series data
Nlength = 100; N = 150; p = 6; b = 3; threshold = 1e-8;

I = diag( p )
A = 0.5 * matrix( rbinom( p * p, 1, 0.2 ), p, p )

A[ lower.tri( A ) ] = 0
diag( A )           = 0.5

G = matrix( 0, p, p )
K = matrix( 0, p, p * Nlength )

lambda  = seq( 0, Nlength - 1, 1 ) * 2 * base::pi / Nlength
K0      = matrix( 0, p, p * Nlength )
K_times = matrix( 1, p, p )

for( k in 1 : Nlength )
{ # Compute K0
    K0[ , ( k * p - p + 1 ) : ( k * p ) ] = I + t( A ) %*% A + 
        complex( 1, cos( -lambda[ k ] ), sin( -lambda[ k ] ) ) * A +
        complex( 1, cos( lambda[ k ] ), sin( lambda[ k ] ) ) * t( A )
    
    K_times = K_times * ( K0[ , ( k * p - p + 1 ) : ( k * p ) ] != 0 )
    diag( K[ , ( k * p - p + 1 ) : ( k * p ) ] ) = 1
}

G0         = K_times
diag( G0 ) = 0

D = K
# d is the Fourier coefficients of X
d = array( 0, c( p, Nlength, N ) )
x = array( 0, c( p, Nlength, N ) )

for( n in 1 : N )
{ # Generate X
    e = matrix( rnorm( p * Nlength ), p, Nlength )
    
    x[ , 1, n ] = e[ , 1 ]
    for( t in 2 : Nlength )
        x[ , t, n ] = A %*% x[ , t - 1, n ] + e[ , t ]
}

P = 0 * D
for( n in 1 : N )
{ # Compute Pk
    X = x[ , , n ]
    
    for( i in 1 : p )
        d[ i, , n ] = fft( X[ i, ] )
    
    for( i in 1 : Nlength )
        P[,(i*p-p+1):(i*p)] = P[,(i*p-p+1):(i*p)]+d[,i,n] %*% t(Conj(d[,i,n]))
}

bdgraph.obj = bdgraph.ts( P, Nlength, N, iter = 1000 )

summary( bdgraph.obj )

compare( G0, bdgraph.obj, vis = TRUE )
# }