A Comparison with Some Other Methods

In this section, we compare the LMDM to other methods used to discover connectivity. Some methods are described in Section 2.3.

1_{http://www.r-project.org/}

2_{http://www.open-std.org/jtc1/sc22/wg14/} 3_{http://www.cs.york.ac.uk/aig/sw/gobnilp/}

The number of Sample Size (T) nodes (n) 100 200 600 1200 3 0.21 0.44 1.37 2.88 4 0.56 1.10 3.29 6.69 6 3.42 6.82 20.67 42.18 10 98.96 199.61 603.66 1099.72 11 167.99 325.50 1001.87 1982.08

Table 3.2: The time in minutes to find the scores for di↵erent numbers of nodes and sample

sizes, and the discount factor was chosen in the range from 0.5 to 1.0 with increments of 0.01.

The Bayesian Network and The Dynamic Bayesian Network

The LMDM is a dynamic version of the Gaussian BN, where, unlike the latter, the LMDM allows the strength of connectivity to change over time. By explicitly modelling drift in the directed connection parameters, the LMDM can discriminate between models whose graphs are Markov equivalent. As a result two models, indistinguishable as BN models, be- come distinct when generalised into LMDMs, see examples of this in Chapter 4. The directed edges of the LMDM can be tentatively associated with a potential causal directionality, as

argued in Queen and Albers (2009), and hint at the e↵ective connectivity rather than the

functional connectivity. The dynamic version of the BN is the DBN. This uses a Vector Autoregression (VAR) type time series sequentially rather than the state space employed in the LMDM, which can also be used to represent certain Granger causal hypotheses.

The Dynamic Granger Causality

Possibly the closest family of competitive models are the dynamic Granger causal

models (Havlicek et al., 2010). However, in general, the scores of these models are not

factorable and so are much slower to search over. Moreover, comparing this with the simplest form of the LMDM we note that regression relationship are lagged whereas in the LMDM they are contemporaneous. Also, the DGC can be seen as the multivariate DLM (MVDLM) where

the covariates are the past values of dependent variables. Queenet al. (2008) compared the

LMDM with the MVDLM. They concluded that the forecast performance of the former was better, considering two model selection measures: the mean squared error (MSE) and the mean absolute deviation (MAD). Note that the LMDM is fitted based only on the individual

observational variances whilst in the MVDLM it is necessary to estimate the covariance matrix of time series variables and so the latter model is more complex.

Other dynamic models

In the previous neuroimaging literature, a sliding time window has been used to

estimate the dynamic correlation among brain regions (Chang and Glover, 2010; Allenet al.,

2012; Leonardi et al., 2013). Some methods also investigate the change points in network

structures (e.g. Cribbenet al., 2012, considered sparse undirected graphs whilst Zhanget al.,

2013, considered the global structure, consisting of a global chain and V dependences among three networks). In contrast to LMDM, these methods study functional connectivity, and also use sophisticated but much more complex statistical computational algorithms rather than conditional conjugate analyses to perform inference.

The other methods, such as LDS, BDS and DCM that we reviewed in Section 2.3, are

more sophisticated but also far too complicated to e↵ectively score quickly enough over a large

model space. Consequently these are not good candidates for use in the initial exploratory

search we have in mind here. An important di↵erence between these methods and LMDM is

that while the dynamic of connectivity is directly estimated in LMDM, most other models

consider connectivity as static or estimate only the di↵erent strengths of connectivity when

modelling a di↵erent experimental situation. We show in our analyses below that in practice

these strengths seem to drift in time.

Of course some authors discuss the possibility of a connectivity for each time point,

ut=t, including another dynamic system for the connectivity,i.e. At=At 1+wat, where

wat ⇠ N(0,Wa) and recall that At is the e↵ective connectivity matrix in the LDS/BDS

model (Bhattacharya et al., 2006; Smith, J.F. et al., 2011). But the inferential techniques

needed for these models are considerably more complicated than for LMDM (e.g. using the

Gibbs sampler scheme), and there are some extra assumptions they need to make,e.g. a fixed

varianceWa, which are not assumed in LMDM. Due the modeled drift in the connectivity,

the LMDM is also able to detect change points in a straightforward manner and so to verify what changes in the interaction among brain regions over time. Such change points are beginning to be studied within this application (see Section 3.5.3). Moreover, unlike other models, the observational variances from the LMDM can be allowed to vary over time (see

Section 3.5.3).

Bhattacharyaet al. (2006) proposed a similar model for LDS, but with the assumption

that the observational errors are temporally dependent. This simpler class can also be implemented as a DLM (and so as an LMDM), as shown by West and Harrison (1997, Chapter 9). Another modification in the model assumption was suggested by Bhattacharya and

Maitra (2011). These authors proposed the autoregressive model for e↵ective connectivity

as At = ⇢At 1+wat, where ⇢ is a parameter to be estimated. So implicitly here we are

using a random walk model for e↵ective connectivity,i.e. Gt=Ip. It would be possible to

useGt=⇢Ipand a similar procedure provided by Petriset al. (2009, Chapter 4) to estimate

the matrixGwithin our method. However, this again gives rise to further complexities and

certain computational issues.

The Time Varying Undirected Graph

Two main di↵erences between the TVUG and the LMDM are firstly the former es-

timates the functional connectivity with undirected edges whilst the LMDM estimates the

e↵ective connectivity. Secondly, although the TVUG estimates the dynamic pattern of the

connectivity, in contrast to the LMDM, it assumes that the time series variables are inde- pendent over time, and they follow the multivariate Gaussian distribution.

Other issues

One big advantage of the LMDM is that the estimation process is straightforward and closed conditional on setting the values of the discount factors. More explicitly its pre-

dictive distributions are products of Student t distributions in which the hyperparameters

are found through well-known Kalman Filter algorithm. Thus, it is easy to find the posterior

distribution of e↵ective connectivity and so to test hypotheses about the parameters. More-

over, from the forecast distribution, the Bayes factor allows us to select a model directly. In contrast, other models need to use approximate inferential methods such as an Expectation

maximization algorithm (EM) or Variational Bayes (VB) that are still quite difficult to im-

plement for these classes and add other problems to the model selection process. They also often use a bootstrap analysis to verify if the e↵ective connectivity is significant dramatically

consuming: a particular problem is that here we are selecting from a large set of alternative hypotheses. So the LMDM provides a very promising fast and informative explanatory data analysis of the nature of the dynamic network.

Another important advantage of the LMDM over most of its competitors is that it comes with a customized suite of diagnostic methods. Therefore, as in all modelling processes, a crucial step is to lead a diagnostic study and so to verify whether the results are valid. In the next section, we propose some diagnostic measures which are especially pertinent to model checking in this particular application. Violations detected through these diagnostics

enable us to embellish the class to accommodate di↵erent features. For example, we are able

to include time-dependent error variances, change points, interaction terms in the regression and so on within the models to better reflect the underlying model and refine the analysis. In Chapter 4, we demonstrate the usefulness of some of these methods to detect any deviations from the simplest LMDM and default and how to improve the model.