Bayesian Spatial Point Process Modeling of Neuroimaging Data
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1 Bayesian Spatial Point Process Modeling of Neuroimaging Data Timothy D. Johnson Department of Biostatistics University of Michigan Johnson (University of Michigan) June 14, / 17
2 Outline 1 Acknowledgements 2 Background/Motivation 3 Bayesian Spatial Point Process Models 4 MS Data Analysis Johnson (University of Michigan) June 14, / 17
3 Acknowledgements Jian Kang, Department of Biostatistics and Bioinformatics, Emory University Thomas E. Nichols, Department of Statistics, University of Warwick Ernst Wilhelm Radü, Department of Neurology, University Hospital Basel Tor D. Wager, Department of Psychology & Neuroscience, University of Colorado, Boulder Lisa Feldman Barrett, Department of Psychology, Northeastern University US NIH/NINDS grant: 1 R01 NS A1 Johnson (University of Michigan) June 14, / 17
4 Background/Motivation Multiple Sclerosis autoimmune disease affecting the central nervous system affects women more than men commonly diagnosed between 20 and 40 years of age. caused by damage to the myelin sheath surrounding axons slows nerve impulse damage caused by inflammation cause is unknown, theories include viral infection genetics environmental factors Johnson (University of Michigan) June 14, / 17
5 Background/Motivation Multiple Sclerosis symptoms symptoms vary due to location and severity of attack muscle symptoms bowel/bladder problems eye problems brain/nerve problems speech problems, etc episodes can last from days to months episodes alternate with periods of reduced or no symptoms no known cure Johnson (University of Michigan) June 14, / 17
6 Background/Motivation Multiple Sclerosis Subtypes CIS clinically isolated syndrome first neurological episode PRL progressive relapsing gradual progression, short phases of remission PRP primary progressive gradual progression, frequent phases of exacerbation RLRM relapsed-remitting alternating phases of relapse followed by phases of remission most common form ( 80% of cases) SCP secondary chronic progressive progressive decline between acute attacks, no definite periods of remission Johnson (University of Michigan) June 14, / 17
7 MS imaging Background/Motivation Data are lesion s centers-of-mass T1 weighted MRI Johnson (University of Michigan) June 14, / 17
8 The Data Background/Motivation Johnson (University of Michigan) June 14, / 17
9 Bayesian Spatial Point Process Models Cox Processes Point process (PP) A point process is a probabilistic structure used to model a random number of points in random locations Johnson (University of Michigan) June 14, / 17
10 Cox Processes Bayesian Spatial Point Process Models Point process (PP) A point process is a probabilistic structure used to model a random number of points in random locations Poisson point process A PP, Y, is a Poisson PP (on brain B) with intensity function λ if 1 the number of points, N(B), has a Poisson distribution with mean B λ(y)dy 2 given the number of points, N(B), the locations of the points are i.i.d. with density λ(y)/ B λ(y)dy (normalized intensity) Johnson (University of Michigan) June 14, / 17
11 Cox Processes Bayesian Spatial Point Process Models Point process (PP) A point process is a probabilistic structure used to model a random number of points in random locations Poisson point process A PP, Y, is a Poisson PP (on brain B) with intensity function λ if 1 the number of points, N(B), has a Poisson distribution with mean B λ(y)dy 2 given the number of points, N(B), the locations of the points are i.i.d. with density λ(y)/ B λ(y)dy (normalized intensity) Cox process Let λ be a random function. If conditional on λ, Y is a Poisson PP with intensity λ, then Y is called a Cox process driven by λ. Johnson (University of Michigan) June 14, / 17
12 Bayesian Spatial Point Process Models Overdispersed Data A Cox process has larger variance than a Poisson process The data warrant this extra variability Mean Var # Subjects CIS PRL PRP RLRM SCP Hence a Cox process is warranted Johnson (University of Michigan) June 14, / 17
13 Bayesian Spatial Point Process Models What is an intensity function? Not the density of the point pattern Integrates to the expected number of points (for a Poisson PP) Integrating over any subregion results in the expected number of points in that subregion Probabilistically, determines the # of points and their locations If we condition on the observed # of points, the normalized intensity is the density function of the locations (Binomial process) Johnson (University of Michigan) June 14, / 17
14 Bayesian Spatial Point Process Models Poisson/gamma random field model A (Bayesian) nonparametric representation of the intensity function Intensity function is the convolution of a Gaussian kernel and a Gamma random field A gamma random field can be represented as an infinite sum of its jump heights ν m at it jump locations θ m G(A) = ν m I(θ m A) GRF(U(A), β) m=1 U(A) is the uniform distribution β is the inverse scale Johnson (University of Michigan) June 14, / 17
15 Bayesian Spatial Point Process Models Gamma field GRF(U(dx), β) U(dx) uniform distribution, β inverse scale Jump locations θ m i.i.d. uniformly over the brain Arrival times ζ m follow a unit rate Poisson process Jump heights ν m = E 1 1 { B ζ m}/β, where E 1 (t) = t e u u 1 du Jump Heights Intensity function Johnson (University of Michigan) June 14, / 17
16 Bayesian Spatial Point Process Models Naïve Bayesian Classifier Use the entire T1-black whole data The point process models only use lesion s centers-of-mass Should give advantage to the Naïve Bayesian classifier Feature vector consists of 0 s, and 1 s if voxels within lesion boundary, score it 1 Each element in vector treated independently Binomial model at each vector for each subtype continuity corrected A priori, assume each subtype is equally probable Johnson (University of Michigan) June 14, / 17
17 MS Data Analysis Posterior Mean Intensity (Poisson Gamma random field model) Johnson (University of Michigan) June 14, / 17
18 MS Data Analysis Prediction Results Naïve Bayes Table: LOOCV classification results CIS PRL PRP RLRM SCP CIS PRL PRP RLRM SCP Overall correct classification: Average correct classification: Johnson (University of Michigan) June 14, / 17
19 MS Data Analysis Prediction Results Poisson/gamma random field model Table: LOOCV classification results CIS PRL PRP RLRM SCP CIS PRL PRP RLRM SCP Overall correct classification: Average correct classification: Johnson (University of Michigan) June 14, / 17
20 MS Data Analysis Questions? Johnson (University of Michigan) June 14, / 17
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