This thesis mainly works on the parametric graphical modelling of multivariate time series.
The idea of graphical model is that each missing edge in the graph corresponds to a zero
partial coherence between a pair of component processes. A vector autoregressive process
(VAR) together with its associated partial correlation graph defines a graphical interaction
(GI) model. The current estimation methodologies are few and lacking of details when
fitting GI models. Given a realization of the VAR process, we seek to determine its graph
via the GI model; we proceed by assuming each possible graph and a range of possible
autoregressive orders, carrying out the estimation, and then using model-selection criteria
AIC and/or BIC to select amongst the graphs and orders.
We firstly consider a purely time domain approach by maximizing the conditional maximum
likelihood function with zero constraints; this non-convex problem is made convex
by a ‘relaxation’ step, and solved via convex optimization. The solution is exact with high
probability (and would be always exact if a certain covariance matrix was block-Toeplitz).
Alternatively we look at an iterative algorithm switching between time and frequency domains.
It updates the spectral estimates using equations that incorporate information from
the graph, and then solving the multivariate Yule-Walker equations to estimate the VAR
process parameters. We show that both methods work very well on simulated data from GI
models.
The methods are then applied on real EEG data recorded from Schizophrenia patients,
who suffer from abnormalities of brain connectivity. Though the pretreatment has been
carried out to remove improper information, the raw methods do not provide any interpretive
results. Some essential modification is made in the iterative algorithm by spectral
up-weighting which solves the instability problem of spectral inversion efficiently. Equivalently
in convex optimization method, adding noise seems also to work but interpretation of
eigenvalues (small/large) is less clear. Both methods essentially delivered the same results
via GI models; encouragingly the results are consistent from a completely different method
based on nonparametric/multiple hypothesis testing.