Hansen, Lars Kai

Madsen, Kristoffer Hougaard

Rasmussen, Peter Mondrup

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Visualization of nonlinear kernel models in neuroimaging by sensitivity mapsRasmussen, Peter Mondrup; Hansen, Lars Kai; Madsen, Kristoffer HougaardPublication date:2010Link back to DTU OrbitCitation (APA):Rasmussen, P. M., Hansen, L. K., & Madsen, K. H. (2010). Visualization of nonlinear kernel models inneuroimaging by sensitivity maps. Abstract from 16th Annual Meeting of the Organization for Human BrainMapping, Barcelona, Spain.Visualization of nonlinear kernel models in neuroimaging bysensitivity mapsP. M. Rasmussen†*, K. H. Madsen‡, T. E. Lund*, L. K. Hansen††DTU Informatics, Technincal University of Denmark. ‡Danish Research Center for Magnetic Resonance, Hvidovre, Denmark.Center of Functionally Integrative Neuroscience, Aarhus, Denmark.AimRecently there has been renewed interest indecoding information about mental states fromdistributed representational patterns in neuro-imaging. It is highly desirable to identify inwhich brain locations the discriminative infor-mation resides. Here we investigate a modelvisualization for nonlinear kernel methods bas-ed on sensitivity analysis.Materials and MethodsData setsSimulation: XOR coupling.fMRI: Finger tapping experiment. 8 subjects.Block design.Two conditions: left and right handfinger tapping.10 rep. of each condition persubject.Classification methods- Support vector machine (SVM).- Logistic regression (LogReg).- Relevance vector machine (RVM).Model evaluationNPAIRS framework. Prediction accuracy andpattern reproducibility metrics.ConclusionNonlinear classification models provide moreflexibility than linear models. In addition to builda model with a good classification accuracya competing goal is to interpret the model interms of the neural representation that drivesthe predictions. The sensitivity map is a straight-forward, yet powerful procedure for generationof meaningful global summary maps of nonlin-ear kernel methods.References[1] U. Kjems, L. K. Hansen, J. Anderson, S. Frutiger, S. Muley, J. Sidtis, D. Rottenberg, and S. C. Strother. Thequantitative evaluation of functional neuroimaging experiments: Mutual information learning curves. Neuroim-age, 15(4):772–786, 2002.[2] Stephen LaConte, Stephen Strother, Vladimir Cherkassky, Jon Anderson, and Xiaoping Hu. Support vectormachines for temporal classification of block design fmri data. Neuroimage, 26(2):317–329, Jun 2005.[3] Michael E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine LearningResearch, 1:211–244, June 2001.Nonlinear classificationFigure 1: Analysis of spatial brain patterns by usinga multivariate approach. Each scan is represented interms of a high dimensional pattern vector (here onlyshown two dimensions denoted by voxel A and voxelB). In a classification scheme the goal is to assign thecorrect label/brain state to a new brain scan based onthe spatial pattern vector. The symbols corresponds totwo hypothetical brain states. In the middle panel thebrain states are separable by a linear decision bound-ary, while a nonlinear decision boundary is required inthe example in the left panel.Figure 2: Simulation of regions that interact through a non-linear relationship. The interrelation between the two regionsis important to the classification task. This configuration isaimed at modeling a set of brain regions in which the interac-tion is modulated by the brain state. There are four spatiallycontiguous regions that can activate denoted by (A,B,C,D).Initially we let regions (A,B,D) be activated by random se-quence taking values ±1. The target signal, coding for aninitial baseline taking values tn = −1, and for an active statet = 1. The region (C) is activated with an XOR-sequencerelative to region (A) and tn, so that Cn = An tn. The tworegions (B,D) are unrelated to the stimulus reference func-tion tn but are included to ensure that the brain state can notbe decoded based on regional measures, such as varianceor non-Gaussianity. Additionally, Gaussian noise was addedto the images to obtain a signal to noise ratio of 0.2. Clas-sification accuracy with an SVM was 90% while the corre-sponding model with a linear kernel achieved 50% accuracy(corresponding to random guessing). Top panel left to right:simulation setup, example of input image, sensitivity map,”true map”.Results - fMRI IFigure 3: Classification of (Left) vs. (RIGHT) finger tap-ping with LogReg, RVM, and the SVM. Scans were rep-resented in terms of the Gaussian kernel. The kernelwidth was set equal to the average input-space dis-tance to the nearest 25 % points across all training ex-amples. Model visualization was performed using thesensitivity map. The SVM showed highest predictionaccuracy (99.1%), an average reproducibility of 0.86,and on average 64 support vectors out of 80 possibleretained in the model. The LogReg model had a pre-diction accuracy (98.9%) and a reproducibility of 0.89.The RVM had a prediction accuracy of 98.0% and a re-producibility of 0.44. Furthermore, the RVM showed anexceptional degree of sparsity compared to the SVMwith only three relevance vectors on average retainedin the model. Models are visualized via reproduciblesensitivity maps. The maps are arbitrary thresholded toshow voxel in the upper 10 percentile.Mathematical ModelingConsider a labeled data set D = {xn, tn}Nn=1,where x is a V dimensional feature vector that,in the context of classification in fMRI, containsa brain scan volume or some derived features,while t is the corresponding target variable.In a probabilistic framework the aim in super-vised learning is to estimate the predictive dis-tribution over t for a new input xp(t|x,D) =∫p(t|x,w)p(w|x,D)dw, (1)where w are model parameters, while p(t|x,w)is the likelihood and p(w|x,D) is the posteriordistribution over the model parameters.Binary classification: p (t|x,w) = σ (tw′ϕ(x)) .Linear model y (x) = w′ϕ(x), where basis func-tions ϕ may be kernels.kx holds the elements k(xn,x), where n is theindex of a specific training example. Examplesare the linear kernel k(xn,x) = x′nx,and the Gaussian kernelk(xn,x) = exp(−||xn − x||2/2σ2).Model visualizationUsual visualization of linear kernel methods in functional neuroimagingy (x) = w′kx = w′Xx (2)where training examples are ordered in the rows of X. w′X is then considered as adiscriminative map. We propose the sensitivity map as a general visualization of kernelmethods in neuroimaging.sj =∫ (∂ log p(t|x,D)∂xj)2p(t,x)dtdx, (3)where the visualization sj resides in the same domain as the input data. In somemodels it is convenient to perform the integrals, in most cases we will approximate byempirical means over the training set D. The sensitivity analysis or sensitivity map is asimple measurement of to what extent the predictive performance of a model dependson a given input.For logistic regressionsj =∫γ(w,x)(∂w′kx∂xj)2p(x)dx (4)where γ(w, x) = (1− σ (w′kx)) σ (w′kx).

Visualization of nonlinear kernel models in neuroimaging by sensitivity maps

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Visualization of nonlinear kernel models in neuroimaging by sensitivity maps

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