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113 nips-2006-Learning Structural Equation Models for fMRI

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Author: Enrico Simonotto, Heather Whalley, Stephen Lawrie, Lawrence Murray, David Mcgonigle, Amos J. Storkey

Abstract: Structural equation models can be seen as an extension of Gaussian belief networks to cyclic graphs, and we show they can be understood generatively as the model for the joint distribution of long term average equilibrium activity of Gaussian dynamic belief networks. Most use of structural equation models in fMRI involves postulating a particular structure and comparing learnt parameters across different groups. In this paper it is argued that there are situations where priors about structure are not ﬁrm or exhaustive, and given sufﬁcient data, it is worth investigating learning network structure as part of the approach to connectivity analysis. First we demonstrate structure learning on a toy problem. We then show that for particular fMRI data the simple models usually assumed are not supported. We show that is is possible to learn sensible structural equation models that can provide modelling beneﬁts, but that are not necessarily going to be the same as a true causal model, and suggest the combination of prior models and learning or the use of temporal information from dynamic models may provide more beneﬁts than learning structural equations alone. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Most use of structural equation models in fMRI involves postulating a particular structure and comparing learnt parameters across different groups. [sent-3, score-0.687]

2 In this paper it is argued that there are situations where priors about structure are not ﬁrm or exhaustive, and given sufﬁcient data, it is worth investigating learning network structure as part of the approach to connectivity analysis. [sent-4, score-0.68]

3 1 Introduction Structural equation modelling (SEM) is a technique widely used in the behavioural sciences. [sent-8, score-0.213]

4 It has also appeared as a standard approach for analysis of what has become known as effective connectivity in the functional magnetic resonance imaging (fMRI) literature and is still in common use despite the increasing interest in dynamical methods such as dynamic causal models [6]. [sent-9, score-0.999]

5 Simply put, effective connectivity analysis involves looking at the possible causal inﬂuences between brain regions given measurements of the activity of those regions. [sent-10, score-0.904]

6 Structural equation models are a Gaussian modelling tool, and are similar to Gaussian belief networks. [sent-11, score-0.367]

7 In fact Gaussian belief networks can be seen as a subset of valid structural equation models. [sent-12, score-0.569]

8 However structural equation models do not have the same acyclicity constraints as belief networks. [sent-13, score-0.692]

9 It should be noted that the graphical form used in this paper is at odds with traditional SEM representations, and consistent with that used for belief networks, as those will be more familiar to the expected audience. [sent-14, score-0.142]

10 Within the fMRI context, the use of structural equation generally takes the following form. [sent-15, score-0.465]

11 First certain regions of interests (commonly called seeds) are chosen according to some understanding of what brain regions might be of interest or of importance. [sent-16, score-0.285]

12 Then neurobiological knowledge is used to propose a connectivity model. [sent-17, score-0.466]

13 This connectivity model states what regions are connected to what other regions, and the direction of the connectivity. [sent-18, score-0.593]

14 This connectivity model is used to deﬁne a structural equation model. [sent-19, score-0.931]

15 In this paper we consider what can be done when it is hard to specify connectivity a priori, and ask how much we can achieve by learning network structures from the fMRI data itself. [sent-21, score-0.582]

16 The novel developments of this paper include the examination of various generative representations for structural equation models which allow straightforward comparisons with belief networks and other models such as dynamic causal models. [sent-22, score-1.045]

17 We implement Bayesian Information Criterion approximations to the evidence and use this in a Metropolis-Hastings sampling scheme for learning structural equation models. [sent-23, score-0.465]

18 These models are then applied to toy data, and to fMRI data, which allows the examination of the types of assumptions typically made. [sent-24, score-0.206]

19 1 Related Work: Structural Equation Models Structural equation models and path analysis have a long history. [sent-26, score-0.269]

20 Linear Gaussian structural equation models can be split into the case of path analysis [20], where the all the variables are directly measurable and structural equation models with latent variables [1], where latent variable models are allowed. [sent-29, score-1.475]

21 Furthermore structural equation models can also be characterised by the inclusion of exogenous inﬂuences. [sent-31, score-0.55]

22 Structural equation models have been analysed and understood in Bayesian terms before. [sent-32, score-0.215]

23 They form a part of the causal modelling framework of Pearl [11], and have been discussed within that context, as well as a number of others [11, 4, 13, 10]. [sent-33, score-0.225]

24 Approaches to learning structural equation models have not played a signiﬁcant part in fMRI methods. [sent-34, score-0.55]

25 For dynamic causal models (rather than structural equation models) the issue of model comparison was dealt with in [12], but large scale structure learning was not considered. [sent-37, score-0.825]

26 In fMRI literature, SEMs have generally been used to model ‘effective connectivity’, or rather modelling the causal relationships between different brain regions. [sent-38, score-0.334]

27 In fact it seems SEMs have been the most widely used model for connectivity analyses in neuroimaging. [sent-41, score-0.466]

28 In all of the studies cited above the underlying structure was presumed known or presumed to be one of a small number of possibilities. [sent-42, score-0.207]

29 The presumption in much fMRI connectivity analysis is that we can obtain models for activity dependence from neuro-anatomical sources. [sent-45, score-0.606]

30 The problem with this is that it fails to account for the fact that connectivity analysis is usually done with a limited number of regions. [sent-46, score-0.466]

31 Furthermore, just because regions are physically connected does not mean there is any actual functional inﬂuence in a particular context. [sent-49, score-0.206]

32 Hence it has to be accepted that neuro-anatomically derived connectivity is a ﬁrst guess at best. [sent-50, score-0.466]

33 It is not the purpose of this paper to propose that anatomical connectivity be ignored, but instead it asks what happens if we go to the other extreme: can we say something about connectivity from the data? [sent-51, score-1.04]

34 In reality anatomical connectivity models are needed, and can be used to provide good priors for the connections and even for the relative connection strengths. [sent-52, score-0.93]

35 Statistically there are huge equivalences in structural equation models that will not be determined by the data alone. [sent-53, score-0.592]

36 3 Understanding Structural Equation Models In this section two generative views of structural equation modelling are presented. [sent-54, score-0.582]

37 The idea behind structural equation modelling is that it represents causal dependence between different variables. [sent-55, score-0.69]

38 The fact that cyclic structures are allowed in structural equation models could be seen as an implicit assumption of some underlying dynamic which the structural equation model is an equilibrium rep- resentation of. [sent-56, score-1.284]

39 Indeed that is commonly how effective connectivity models are interpreted in an fMRI context. [sent-57, score-0.595]

40 Two linear models, both of which produce a structural equation model prior, are presented here. [sent-58, score-0.465]

41 1 The Traditional Model The standard SEM view is that the core SEM structure is a covariance produced by the solution to a set of linear equations x = Ax + ω with Gaussian term ω. [sent-61, score-0.132]

42 This does not have any direct generative elucidation, but can instead be thought of as relating to a deterministic dynamical system subject to uncertain ﬁxed input. [sent-62, score-0.14]

43 Suppose we have a dynamical system xt+1 = Axt + ω, subject to some input ω, where we presume the system input is unknown and Gaussian distributed. [sent-63, score-0.169]

44 To generate from the model, we sample ω, run the dynamical system to its ﬁxed point, and use that ﬁxed point as a sample of x. [sent-64, score-0.145]

45 This ﬁxed point is given by x = (I − A)−1 ω which produces the standard SEM covariance structure for x. [sent-65, score-0.132]

46 2 Average Activity Of A Gaussian Dynamic Bayesian Network An alternative and potentially appealing view is that the the SEM represents the distribution of the long term activity of the nodes in a Gaussian dynamic Bayesian network (Kalman ﬁlter). [sent-69, score-0.174]

47 This deﬁnes a Markov chain, and is the evolution equation of a Gaussian dynamic Bayesian network. [sent-74, score-0.209]

48 This interpretation says that if we have some latent system running as a Gaussian dynamic Bayesian network, but our measuring equipment is only capable of capturing longer term averages of the network activity then our measurements are distributed according to an SEM. [sent-80, score-0.258]

49 By formulating the generative framework we see it is important to restrict the form of connectivity model in this way. [sent-83, score-0.5]

50 (2) where Kω is the covariance of ω, Kσ the covariance of σ etc. [sent-87, score-0.156]

51 The next simplest case would involve presuming once again that there are no inputs but that in fact the observations are stochastic functions of the latent variables. [sent-92, score-0.147]

52 In this section we provide prior distributions for the parameters of the structural equation model. [sent-106, score-0.503]

53 For the purposes of this paper we take Aij = 0, presume we have no particular a priori bias towards positive or negative connections and a uniform prior over structures. [sent-108, score-0.274]

54 An independent prior over connections seems reasonable as two separate connections between different brain regions would have no a priori reason to be related. [sent-109, score-0.573]

55 We can calculate all the relevant derivatives for the SEM straightforwardly, and adapt the parameters to maximize the posterior of the structural equation model. [sent-117, score-0.567]

56 This will enable us to sample from an approximate posterior distribution of structures to ﬁnd a sample which best represents the data. [sent-121, score-0.272]

57 6 Sampling From SEMs In order to represent the posterior distribution over network structures, we resort to a sampling approach. [sent-122, score-0.142]

58 Because there are no acyclicity constraints, MCMC proposals are simpler than the comparable situation for belief networks in that no acyclicity checking needs to be done for the proposals. [sent-123, score-0.25]

59 We choose highly sparse swap matrices, but to reduce the possibility of transitioning randomly about the larger graphs, without ever considering smaller networks we introduce a bias towards removing connections rather than adding connections in generating the swap matrix. [sent-125, score-0.401]

60 22 0     This connectivity matrix is represented graphically in Figure 11. [sent-151, score-0.466]

61 15 of the cases, (c) the highest posterior structure from the sample (d) a random sample. [sent-165, score-0.203]

62 The graphs for the maximum posterior sample and a random sample are shown in Figure 1. [sent-166, score-0.196]

63 We can conclude from this that we can gain some information from learning SEM structures, but as with learning any graphical models there are many symmetries and equivalences, so it is vital not to infer too much from the learnt structures. [sent-169, score-0.168]

64 We used the auditory paced ﬁnger-tapping task; brieﬂy, a single subject tapped his right index ﬁnger, paced by an auditory tone (1. [sent-172, score-0.288]

65 5mm resolution) was acquired to facilitate anatomical localisation of the functional data. [sent-183, score-0.242]

66 After removal of the ﬁrst two volumes to account for T1 saturation effects, cerebral volumes were realigned to correct for both within- and between-session subject motion). [sent-190, score-0.24]

67 For comparison with previous extant work, the most signiﬁcant voxel in each cluster was chosen as a seed, giving 13 seeds representing 13 separate anatomical regions. [sent-195, score-0.286]

68 When it was obvious that a given cluster encompassed more than one distinct anatomical region, seeds were also selected for other regions covered by the cluster. [sent-196, score-0.323]

69 A high resolution structural scan was acquired using a 3D T1-weighted sequence (TI = 600 ms). [sent-201, score-0.39]

70 For an effective connectivity analysis, a number of brain regions (seeds) were chosen on the basis of the results of a functional connectivity study [19] and taking regard of areas which may be of particular clinical interest. [sent-212, score-1.252]

71 Hence we are interested in learning a 28 by 28 connectivity matrix. [sent-214, score-0.466]

72 The stability of the log posterior along with estimations of cross-correlation against lag were used as heuristics to determine convergence prior to obtaining 10000 sample points. [sent-217, score-0.187]

73 [15] for a Schizophrenia studies), we found that samples from the posterior of this model were in fact so highly connected that displaying them would infeasible. [sent-220, score-0.179]

74 For D2 a connectivity of 350 of the 752 total possible connections was typical. [sent-221, score-0.595]

75 We can generalise the path analysis model by making the region activities latent variables, and allow the measurement variables to be noisy versions of those regions. [sent-224, score-0.205]

76 For dataset D1, we sample posterior structures given the training data with T = 100. [sent-228, score-0.225]

77 In addition an a priori connectivity (a) (b) Figure 2: Structure for (a) the hand speciﬁed model (b) the highest posterior sample. [sent-230, score-0.612]

78 (a) (b) (c) Figure 3: Graphical structure for (a) the highest posterior structure from the sample (b) random sample. [sent-231, score-0.257]

79 structure is proposed for the regions in the study, taking into account the task involved. [sent-235, score-0.142]

80 This was obtained by using knowledge of neuroanatomical connectivity drawn from studies using tract-tracing in non-human primates. [sent-236, score-0.503]

81 It was produced independent of the connectivity analysis and without knowledge of its results, but taking into account the seed locations and their corresponding activities. [sent-237, score-0.524]

82 Both these models perform better than other random models with equivalent numbers of connections. [sent-249, score-0.17]

83 In reality learnt models are going to be used in new situations and situations with less data. [sent-250, score-0.209]

84 By estimating the model parameters on 100 data points, instead of 2000, we ﬁnd that the learnt model performs very slightly better than the hand speciﬁed model (log odds ratio of 63 on a 574 point test set), and both perform better than the full covariance (log odds of 292). [sent-252, score-0.307]

85 Maximum posterior samples and a random sample are illustrated in Figure 3. [sent-255, score-0.187]

86 Even so this is signiﬁcantly greater than the idealised connectivity structures typically used in most studies. [sent-257, score-0.542]

87 One further approach is to assume a fully connected structure, but where the connectivity is in two categories. [sent-258, score-0.505]

88 We have priors on connectivity with the same values of Tij as before for the strong connections and much larger values for the weaker connections. [sent-259, score-0.661]

89 Following this procedure we ﬁnd that models of the form of 3c are typical samples from the posterior where only the larger connections are shown. [sent-261, score-0.354]

90 Again connections such as those between the Cuneus/Precuneus and the Superior Frontal Gyrus, the Thalamic connections, and some of the cross-hemispheric connections are amongst those that would be expected. [sent-262, score-0.258]

91 This approach is related to recent work on the use of sparse priors for effective connectivity [18]. [sent-263, score-0.576]

92 9 Future Directions This work demonstrates that if we learn structural equation models from data, we ﬁnd there is little evidence for the simple forms of path analysis model which is in common use in the fMRI literature. [sent-264, score-0.604]

93 We suggest that learning connectivity can be a reasonable complement to current procedures where prior speciﬁcation is hard. [sent-265, score-0.54]

94 It should be expected that combining learnt structures with prior anatomical models will help in the speciﬁcation of more accurate connectivity assumptions, as it will reduce the number of equivalence and focus on more reasonable structural forms. [sent-268, score-1.227]

95 Furthermore future comparisons can be made using a sample of reasonable models instead of a single a priori chosen model. [sent-269, score-0.212]

96 We would also expect that the major gains in learning models with come from the focus on dynamical networks which do not suffer from speciﬁcity problems. [sent-270, score-0.171]

97 The preedictive value of changes in effective connectivity for human learning. [sent-283, score-0.51]

98 Attentional modulation of effective connectivity from V2 to V5/MT in humans. [sent-303, score-0.51]

99 Structural equation modelling and its application to network analysis in functional brain imaging. [sent-331, score-0.441]

100 Altered effective connectivity during working memory performance in schizophrenia: a study with fMRI and structural equation modeling. [sent-375, score-0.975]

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