Bayesian Registration of Functions with a Gaussian Process Prior

Bayesian Registration of Functions with a Gaussian Process Prior

RELATED PROBLEMS Functions Curves Images Surfaces

FUNCTIONAL DATA Electrocardiogram Signals (Heart Disease) Gait Pressure Measurements (Parkinson s) Gene Expression Data Respiration Strain During 4D-CT Scan Signature Acceleration Curves (Fraud Detection) Hand-grip Strength (Arthritis) Spike Trains Growth Curves

ALIGNMENT PROBLEM DESCRIPTION

MAIN GOALS The function alignment problem has been referred to by many names including registration and warping. Note: I will use these terms interchangeably throughout this talk. Main Goals: 1. Function alignment: principled approach to separate the amplitude and phase variabilities of functional datasets for subsequent statistical analysis. 2. Alignment uncertainty: framework that allows computation of confidence values associated with alignment estimates. 3. Multiple plausible alignments: approach that provides a recipe for identifying multiple good registrations. 4. Statistical analysis of warping functions: tools for computing statistics on the space of warping functions including the mean, covariance, PCA, cluster analysis, etc.

RELATED WORK Bayesian alignment of functions and curves: B-splines: Telesca and Inoue, 2008 (JASA) Multiresolution approach or warplets : Claeskens et al., 2010 (JRSSB) SRVF with Dirichlet prior: Cheng et al., 2016 (Bayesian Analysis) Bayesian deformable models: Allassonniere et al., 2010 (Bernoulli) Computational anatomy: Ma et al., 2008 (NeuroImage); Tang et al., 2013 (PLoS ONE) Other works: Srivastava and Jermyn, 2009 (PAMI); Zhang and Fletcher, 2016 (Algorithmic Advances in Riemannian Geometry and Applications); Wassermann et al., 2014 (WBIR); Simpson et al., 2015 (Medical Image Analysis); etc.

PAIRWISE ALIGNMENT PROBLEM Observation space : set of absolutely continuous functions on [0,1]. Parameter space of warping functions: acts on the function space by composition:. PQRST complex f 1 and f 2 γ f 1, f 2 γ

PAIRWISE ALIGNMENT PROBLEM Most common optimization approach (Dynamic Time Warping): Known issues: (1) lack of symmetry, (2) pinching effect: (one can pinch the entire function f 2 to get arbitrarily close to f 1 in norm), (3) not a proper metric on the quotient space : cannot be used for statistical analysis. Another approach based on the norm: Adding a regularization penalty on γ eliminates the pinching effect, but the other drawbacks remain. The choice of the regularization parameter λ is not obvious in general cases.

SOLUTION BASED ON SRSF Square-root slope function (SRSF):. Inverse-mapping is well defined and easy to compute. Space of SRSFs:. Action of warping group on SRSF space:. Important property:. Equivalence class of SRSF under warping:. Distance can be efficiently computed using Dynamic Programming (DP): This approach overcomes all previously mentioned issues. Separate issue: this is a purely optimization-based approach, which does not allow the user to easily assess alignment uncertainty or to discover multiple plausible alignments.

OUTLINE OF PROPOSED APPROACH Bayesian Model + Riemannian Geometry + Markov Chain Monte Carlo (1) Bayesian Alignment Model: Allows comprehensive exploration of the variable space. Provides credible intervals of warping function estimates. Allows discovering multiple registration solutions through multimodal posteriors. (2) Riemannian Geometry of Warping Group: Enables efficient Riemannian computation on the space of warping functions: tools for computing statistics and implementation of a k-means clustering algorithm to discover multiple modes in the posterior. Allows a geometric prior distribution on the space of warpings. (3) Markov Chain Monte Carlo: Allows sampling from the posterior distribution. We use a specialized preconditioned Crank Nicolson algorithm for functional parameters.

RIEMANNIAN GEOMETRY OF WARPINGS Riemannian metric on (, Fisher-Rao metric): This results in a non-trivial Riemannian geometry of. Simplification: Definition: Define the mapping. Then, given an element, define a new representation using the square-root of its derivative as. We refer to ψ as the square-root density (SRD). Main Result: simplifies to the positive orthant of the Hilbert sphere and the Fisher-Rao metric simplifies to the metric. The Riemannian geometry of the Hilbert sphere is well known providing efficient tools for statistical analysis on the warping group.

GEOMETRIC TOOLS Identity mapping, i.e. no warping:. denotes the inner product. Tangent space:. Geodesic distance:. Exponential and inverse-exponential maps:

STATISTICS OF WARPING FUNCTIONS Given: a sample of warping functions and their corresponding SRDs. Karcher Mean:. Geometric Median (Fletcher et al., 2009):. Both quantities found using a standard gradient descent algorithm. K-means clustering: utilize the Karcher mean and Fisher-Rao distance on the warping group.

PAIRWISE ALIGNMENT MODEL Function discretization:. Pairwise alignment model: Prior truncated to Ψ. C g : pre-specified covariance operator. a and b: pre-specified constants. Sampling from posterior via Metropolis within Gibbs: Update sequentially from full conditionals of g (Metropolis step using pcn) and (direct draw).

Z-MIXTURE pcn MCMC ALGORITHM We use the Z-mixture preconditioned Crank Nicolson (pcn) to update the function g (Cotter et al., 2013): 1. Propose. 2. Accept g with probability. (prior) via Karhunen-Loeve expansion: 1. Assume the eigenpairs of C g are (λ i2, b i (t); i 1). We use a Fourier basis. 2. Sample and let.

Z-MIXTURE pcn MCMC ALGORITHM β is a tuning parameter: 1. Large (small) values imply large (small) jumps in the parameter space. 2. We use a mixture to improve MCMC convergence. Advantages of mixture pcn MCMC algorithm: 1. The acceptance ratio is independent of the dimension of the discretized g. 2. The tuning parameter β is random; flexible to control size of jump.

MULTIPLE ALIGNMENT MODEL Multiple alignment model: Sampling from posterior by updating sequentially: g 1,,g C, q* (Metropolis steps using pcn) and (direct draw).

EXAMPLES: PAIRWISE SIMULATIONS Data: Three simulation settings: (1) randomly generate γ using the first pair of Fourier basis (fourier1), (2) randomly generate γ using the first 20 pairs of Fourier basis (fourier20), (3) generate γ by smoothing a random step function (random) Each simulation setting is repeated ten times. Evaluation: Fisher-Rao distance between posterior mean and true warping.

EXAMPLES: PAIRWISE SIMULATIONS

EXAMPLES: MULTIPLE ALIGNMENT Data: right knee flexion functions for 12 individuals

EXAMPLES: MULTIPLE ALIGNMENT Original Data Aligned Data Cross-sectional Averages Estimated Warps Simulated Gait Data: Pelvis Right Roll Spike Trains Pinch Force

SUMMARY AND FUTURE WORK Contributions: 1. We proposed a novel Bayesian model for pairwise and multiple alignment of functional data. 2. We use the Riemannian geometry of the space of warping functions for: defining a Gaussian prior; efficient sampling from the posterior distribution; computation of statistics of warping functions. 3. The proposed model is very good at exploring the entire variable space. In some cases, it performs better than the optimization-based DP algorithm. Future Work: 1. Theoretical properties of the model. 2. Sensitivity analysis. 3. Alternative priors and posterior sampling strategies for warping functions. 4. Extension to higher-dimensional curves, images and surfaces.

CBMS CONFERENCE CBMS Conference: Elastic Functional and Shape Data Analysis (EFSDA) Where: The Ohio State University, Columbus, OH When: July 16-20, 2018 Primary Lecturer: Prof. Anuj Srivastava Organizers: Sebastian Kurtek (Statistics, OSU), Facundo Memoli (Mathematics, OSU), Yusu Wang (Computer Science and Engineering, OSU), Tingting Zhang (Statistics, University of Virginia), Hongtu Zhu (Biostatistics, MD Anderson Cancer Center)

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