The Description of the Model

Consider the column vector Y0

t = (Yt(1), . . . , Yt(n)) which denotes the data fromn regions

at time t. Denote their observed values designated respectively by y0_t = (yt(1), . . . , yt(n)). Let the time series until time t for region r = 1, . . . , n be Yt_(r)0 _{= (Y1(r), . . . , Yt(r)) and} the time series for possible parents of region r at time t be Xt(r)0 = {Yt(1), . . . , Yt(r

1)} for r = 2, . . . , n. Note that the n regions in a DAG can always be ordered to ensure

that P a(r) _✓ Xt(r), where P a(r) is the parent set of Yt(r). The MDM is defined by n observation equations, a system equation and initial information (Queen and Smith, 1993). The observation equations specify the time-varying regression parameters of each region on its parents. The system equation is a multivariate autoregressive model for the evolution

of time-varying regression coefficients, and the initial information is given through a prior

density for regression coefficients. Thus, the linear multiregression dynamic model is specified

in terms of a collection of conditional regression DLMs (West and Harrison, 1997), as follows.

We write theobservation equations as

Yt(r) =Ft(r)0✓t(r) +vt(r), vt(r)⇠N(0, Vt(r));

wherer = 1, . . . , n;t= 1, . . . , T;_N(_·,_·) is a Gaussian distribution;Ft(r) is a known function of P a(r) and is usually defined as Ft(r) = M(r)Y⇤t, where M(r) is pr ⇥(n+ 1) matrix

M(r) is (1,0, . . . ,0) representing the intercept;pr=|P a(r)|+1 counts the number of parents of regionrplus one (for the intercept);Y_t⇤= (1,Y0_t)0. The observational error,vt(r), is taken

to be independent over t, with variance Vt(r). The pr-dimensional time-varying regression

coefficient is ✓0_t(r) = (✓(1)_t (r), . . . ,✓_t(pr)(r)). Generally the parameter ✓(1)_t (r) represents the intercept of the regression of regionr whilst✓(i)_t (r) for i >1 represents the e↵ective connec- tivity strength for the (i 1)thparent of regionr. Concatenating thenregression coefficients as✓0_t= (✓0_t(1), . . . ,✓_t0(n)) gives a vector of lengthp=Pn_r=1pr.

We next write thesystem equation as

✓t=Gt✓t 1+wt, wt⇠N(0,Wt); (3.1)

where Gt = blockdiag{Gt(1), . . . ,Gt(n)}, each Gt(r) being a pr ⇥pr matrix, wt is the innovation for the latent regression coefficients, and Wt = blockdiag{Wt(1), . . . ,Wt(n)},

eachWt(r) being apr⇥pr matrix. The errorwt is assumed independent ofvs for all tand

s; vs= (vs(1), . . . , vs(n)). For most of the development we need only consider Gt(r) =Ipr,

whereIpr is the pr-dimensional identity matrix.

For instance, suppose the graphical structure given by Figure 3.1, then the model equations are written as:

✓t(r) = ✓t 1(r) +wt(r); wt(r)⇠N (0,Wt(r)) ; Yt(1) = ✓(1)t (1) +vt(1); Yt(2) = ✓(1)t (2) +✓ (2) t (2)Yt(1) +vt(2); Yt(3) = ✓(1)t (3) +✓ (2) t (3)Yt(1) +✓t(3)(3)Yt(2) +vt(3); vt(r)⇠N (0, Vt(r)),

for r = 1, . . . ,3, p1 = 1, p2 = 2 and p3 = 3. The e↵ective connectivity strengths of this

example are then✓_t(2)(2), ✓_t(2)(3) and✓_t(3)(3).

Finally, theinitial information is written as

(✓0|y0)⇠N(m0,C0); (3.2)

where✓0|y0expresses the prior knowledge of the regression parameters, before observing any

of the parameters and C0 is the p⇥p variance-covariance matrix. C0 can be defined as blockdiag{C0(1), . . . ,C0(n)}, with eachC0(r) being a pr square matrix.

vt(2)! vt(3)! vt(1)! Time%t" Data" Connec*vity" DAG" θt(1)(1)% θt(1)(2)% θt(1)(3)% θt(2)(3)% θt(3)(3)% θt(2)(2)% Time%t+1" Time%t21" Yt(1)! Yt(2)! Yt(3)!

Figure 3.1: Dependence structure for the MDM considering Region 1 as the parent of Region 2 and

Region 3; and Region 2 as the parent of Region 3. The solid circles represent observed variables,

Yt(r). The dashed circles represent latent variables: blue for observational errors,vt(r); violet for the

intercept of the regression of Regionr, ✓t(1)(r);r = 1,2,3; and orange for the e↵ective connectivity

strength between two regions,✓(2)t (2),✓

(2)

t (3) and✓

(3)

t (3).

There are five important features of this model class discussed in the literature. 1. Although the predictive distributions of each node given its parents are Student t dis-

tributed, because the covariates enter the scale function of these conditionals, the joint distribution can be highly non-Gaussian. Queen and Smith (1993) provided exam- ples of this. This feature is useful for fMRI studies, because models that assume that processes are not jointly Gaussian may be better fitted to fMRI data than ones that assume joint Gaussianity;

2. As the values of variables of a particular node and its parents are observed simultane- ously, to make predictions, it is necessary to know the marginal forecast distribution for

each node in timet, given only the past. This distribution is not generally of a simple

form, but it is not hard to calculate its expectation and covariance matrix. Queen and Smith (1993) demonstrated the mean and covariance matrix of the marginal forecast distribution, considering the corrected linear MDM (CLMDM). The linear MDM as- sumes that the residuals have a Gaussian distribution and the relation between nodes and their parents is linear, where their parents are explanatory variables. In contrast,

the CLMDM uses the residuals of models fitted for parents as regression covariates. The one step ahead mean and covariance matrix of the LMDM were found by Queen

et al. (2008) and are described in Appendix A. Also, Queen et al. (2008) argued the problem that the covariance between root nodes is zero in the LMDM, which sometimes is not expected in a real situation. Therefore they proposed to include in the model a set of variables that explain the correlation between roots as a parent of them; 3. Each LMDM is defined in part by a directed acyclic graph (DAG) whose vertices are

observed fMRI series at a given time. In addition, its directed edges represent the existence of a dependence on those contemporaneous observations that are explicitly included as regressors to the receiving variable. In our context, therefore, these directed edges denote the hypothesis that direct contemporaneous relationships might exist between a variable and its parents. The directionality of the edges can be interpreted as being ‘causal’ in a sense that is carefully argued in Queen and Albers (2009); 4. Dependence relationships between each component and its contemporaneous parents

— as represented by the corresponding regression coefficients — are allowed to drift

with time. Therefore, unlike a static BN, the MDM models dynamic links and so

allows us to discriminate between models that would be Markov equivalent in their

static versions. Queen and Albers (2009) showed this result using real traffic flows

data. We will also discuss this question considering di↵erent sample sizes and dynamic

levels using synthetic data in Chapter 4;

5. The class of MDM can be further modified to include other features that might be nec- essary in a straightforward and convenient manner. For instance, Queen and Albers (2009) showed that a causal relationship could be better identified using the interven-

tion process. In addition, Anacleto Junioret al. (2013a) worked with heteroscedasticity

and measurement errors in the LMDM. Yet Anacleto Junioret al. (2013b) dealt with

cycle problems in the time series using cubic splines in the MDM. Some methods used to check and to embellish the MDM are discussed in Section 3.5.