In this chapter, I propose a robust sparse modelling method (RS-SBL) to generate brain behaviour predictive models as well as predictive maps from whole brain fMRI im- ages. With limited sample size, the state-of-the-art compressive sensing technique, Sparse
(a) Active Map
(b) Predictive Map
Figure 5.7: An Example of ‘Individual’ Predictive Map vs. Active Map for Relation vs. Rest Task.
Bayesian Learning, is used as the key technique of RS-SBL to determine sparse relevant voxels during predictive model construction. That is feature selection and modelling are integrated into one single step. The dependency of the choice of significant value in tra- ditional feature selection methods can be therefore removed. My work is the first attempt to integrate random subspace method with SBL. By randomly sampling small subsets of features in voxel space, the spatial correlation and feature-to-sample ratio in each sub- space are largely reduced so that multiple robust classifiers can be constructed. Therefore, aggregating the multiple subspace classifiers returns a strong classifier.
(a) Active Map
(b) Predictive Map
Figure 5.8: An Example of ‘Individual’ Predictive Map vs. Active Map for Relation vs. Match Task.
analysing the fMRI dataset provided by HCP. Among those methods, SVM-Lin and SBC- Lin involve feature space transformation, so all voxels are used for classification. Hence, relevant voxels were difficult to be selected. In addition, as all voxels were used for pre- diction, the predictive models tended to overfit the training data. SBC, on the other hand, directly detected relevant voxels, whereas the resultant predictive map was extremely sparse and its predictive performance was poor. My method, benefiting from the imple- mentation of random subspace, was able to provide meaningful predictive maps with the strongest predictive power. The setting of parameters of my model was controlled by the sparsity of the relevant voxels, which is unknown in real application. I here used the GLM analysis result to estimate the sparsity, this might not result in a selection of the global
optimal parameter settings. However, my method still showed good performance, and better results are expected with global optimal parameters.
My constructed predictive maps highlight the consistent activated brain regions detected by the GLM method, while my results are overall sparse. My method provides the most meaningful predictive maps maintaining the strongest predictive power. However, these sparse solutions may contain noise voxels and discard some relevant ones. In order to enable an accurate interpretation of brain activity from my constructed predictive map, post-processing steps are required. For instance, use a smooth operator (e.g. Gaussian filter) to smooth the detected sparse regions. This can remove the isolated voxels which have high probability to be noise and distribute the activated regions across wide areas so that across subjects analysis can be applied.
In this work, I only concern about classification problem. My method can also be adapted to solve the regression problem by replacing SBC with SBR. Moreover, with the applica- tion of the extended SBC method [PDG10], multi-task classification can be implemented. My method has the potential to be further improved by adding some ensemble methods, such as bagging method. The bagging method can be applied to each subspace, where multiple predictors from a subspace are generated with bootstrapped samples.
Balancing Stability and Predictive
Performance of fMRI Models
6.1
Introduction
The early MVPA studies focus on how well brain states can be predicted. Recently, more and more studies consider its function of voxel selection so that neural activity responding to the brain states can be interpreted, which is an important factor for neuroscientific discovery. Linear sparse modelling has been a popular technique in MVPA studies, as it can be used for implementing predictive model as well as selecting relevant voxels from input voxels using the sparse model parameter. A model is considered to be robust for interpreting neural activity if the selected voxels are all relevant to the specific brain state. The conventional linear sparse modelling methods [YSY+08, CCR+09, RSAM10, VGT12] select the voxels by considering their predictive powers; the selected ones are those that provide the most accurate prediction. However, because the number of samples is always considered to be underdetermined compared to the number of input voxels, if only take the predictive performance into consideration, the selected voxels are specific
to particular dataset and irrelevant voxels may be wrongly selected. In addition, most existing linear sparse modelling methods manage to find the sparsest number of voxels used for prediction. Because of the correlations existing among relevant voxels, these methods can only detect a subset of them. This can introduce a large number of false negative selections while the predictive is almost unaffected. In consequence, with such wrongly selected voxels, even if the predictive performance can be guaranteed, the neural activity is misunderstood. For this reason, an urgent problem for MVPA is to select voxels that can interpret real neural activity as well as provide accurate predictions.
Biomarker discovery which aims to select biomarkers to differentiate diseases from nor- mal states uses similar analysis methods as fMRI MVPA analysis. Both biomarker dis- covery and fMRI MVPA face the same problem as: 1) the training datasets usually have high feature-to-sample ratio; 2) the selected predictors are expected to be predictive and meaningful, where the predictors are always sparse compared to the high dimensional features; 3) correlations existing among predictors. In order to control the robustness of predictors, researchers (e.g. [AHdP+10, ZRS08]) introduced the concept of stability to biomarker discovery techniques. They denoted that a feature is considered to be stable if it is consistently selected when using a selection method with different sample sets. They demonstrate that if the stability is higher, the selected predictors are more robust to noise so that the probability of selecting noisy predictors is highly reduced. However, if an arbitrary fixed set of predictors is chosen, the stability is perfect but the predictive perfor- mance is very poor. This is because some real predictors which are unstable is discarded if only taking the stability into consideration. To deal with this, Kirk et al. [KWB+13] in- vestigated strategies to balance the robustness and predictive performance of biomarkers by optimising both stability and predictive performance simultaneously.
To improve the accuracy of relevant voxel selection of MVPA methods, in this chapter, I introduce the concept of stability to the fMRI MVPA analysis which has not been consid- ered in this field before. I explore the advantages of bringing stability into voxel selection
and propose a novel multivariate voxel selection method which selects the relevant voxels by considering their stability and predictive power simultaneously. The method is im- plemented by wrapping a proposed selection strategy around my novel sparse modelling method, Random Subspace Sparse Bayesian Learning (RS-SBL) (detailed in Chapter 5). My method aims to select voxels that can accurately discriminate different brain states as well as enable precision interpretation of brain activities. By using my selection strat- egy, which combines stability and prediction accuracy assessments, the probabilities of RS-SBL of both selecting irrelevant voxels and unselecting relevant voxels are highly reduced, whereas only small reduction of predictive accuracy is made.
This chapter is organised as follows. In Section 6.2, I first explain the voxel select function of RS-SBL and then describe my proposed method; experimental results of testing my method on both simulation and real datasets are detailed in Section 6.3. In the final section, I make a conclusion of this work.