## Electronic Journal of Statistics

### A geometric approach to pairwise Bayesian alignment of functional data using importance sampling

Sebastian Kurtek

#### Abstract

We present a Bayesian model for pairwise nonlinear registration of functional data. We use the Riemannian geometry of the space of warping functions to define appropriate prior distributions and sample from the posterior using importance sampling. A simple square-root transformation is used to simplify the geometry of the space of warping functions, which allows for computation of sample statistics, such as the mean and median, and a fast implementation of a $k$-means clustering algorithm. These tools allow for efficient posterior inference, where multiple modes of the posterior distribution corresponding to multiple plausible alignments of the given functions are found. We also show pointwise 95% credible intervals to assess the uncertainty of the alignment in different clusters. We validate this model using simulations and present multiple examples on real data from different application domains including biometrics and medicine.

#### Article information

Source
Electron. J. Statist. Volume 11, Number 1 (2017), 502-531.

Dates
First available in Project Euclid: 2 March 2017

https://projecteuclid.org/euclid.ejs/1488423806

Digital Object Identifier
doi:10.1214/17-EJS1243

Subjects
Primary: 62F15: Bayesian inference

#### Citation

Kurtek, Sebastian. A geometric approach to pairwise Bayesian alignment of functional data using importance sampling. Electron. J. Statist. 11 (2017), no. 1, 502--531. doi:10.1214/17-EJS1243. https://projecteuclid.org/euclid.ejs/1488423806.

#### References

• [1] Allassonnière, S., Kuhn, E. and Trouvé, A. (2010). Construction of Bayesian Deformable Models via a Stochastic Approximation Algorithm: A Convergence Study., Bernoulli 16 641–678.
• [2] Bhattacharya, A. (1943). On a Measure of Divergence Between Two Statistical Populations Defined by Their Probability Distributions., Bulletin of the Calcutta Mathematical Society 35 99–109.
• [3] Čencov, N. N. (1982)., Statistical Decision Rules and Optimal Inferences. Translations of Mathematical Monographs 53. AMS, Providence, USA.
• [4] Cheng, W., Dryden, I. L. and Huang, X. (2016). Bayesian Registration of Functions and Curves., Bayesian Analysis 11 447–475.
• [5] Cotter, S. L., Roberts, G. O., Stuart, A. M. and White, D. (2013). MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster., Statistical Science 28 424–446.
• [6] Dryden, I. L. (2005). Statistical Analysis on High-dimensional Spheres and Shape Spaces., Annals of Statistics 33 1643–1665.
• [7] Dryden, I. L. and Mardia, K. V. (1998)., Statistical Shape Analysis. John Wiley & Sons.
• [8] Fletcher, P. T., Venkatasubramanian, S. and Joshi, S. (2009). The Geometric Median on Riemannian Manifolds with Application to Robust Atlas Estimation., Neuroimage 45 S143–S152.
• [9] Gervini, D. and Gasser, T. (2004). Self-Modeling Warping Functions., Journal of the Royal Statistical Society, Series B 66 959–971.
• [10] James, G. (2007). Curve Alignment by Moments., Annals of Applied Statistics 1 480–501.
• [11] Kneip, A. and Gasser, T. (1992). Statistical Tools to Analyze Data Representing a Sample of Curves., Annals of Statistics 20 1266–1305.
• [12] Kneip, A. and Ramsay, J. O. (2008). Combining Registration and Fitting for Functional Models., Journal of the American Statistical Association 103 1155–1165.
• [13] Koch, I., Hoffmann, P. and Marron, J. S. (2014). Proteomics Profiles from Mass Spectrometry., Electronic Journal of Statistics 8 1703–1713.
• [14] Kurtek, S., Srivastava, A., Klassen, E. and Ding, Z. (2012). Statistical Modeling of Curves Using Shapes and Related Features., Journal of the American Statistical Association 107 1152–1165.
• [15] Kurtek, S., Srivastava, A. and Wu, W. (2011). Signal Estimation under Random Time-warpings and Nonlinear Signal Alignment. In, Neural Information Processing Systems (NIPS) 675–683.
• [16] Kurtek, S., Su, J., Grimm, C., Vaughan, M., Sowell, R. T. and Srivastava, A. (2013). Statistical Analysis of Manual Segmentations of Structures in Medical Images., Computer Vision and Image Understanding 117 1036–1050.
• [17] Kurtek, S., Wu, W., Christensen, G. E. and Srivastava, A. (2013). Segmentation, Alignment and Statistical Analysis of Biosignals with Application to Disease Classification., Journal of Applied Statistics 40 1270–1288.
• [18] Le, H. (2001). Locating Fréchet Means with Application to Shape Spaces., Advances in Applied Probability 33 324–338.
• [19] Liu, X. and Müller, H. G. (2004). Functional Convex Averaging and Synchronization for Time-warped Random Curves., Journal of the American Statistical Association 99 687–699.
• [20] MacQueen, J. B. (1967). Some Methods for Classification and Analysis of Multivariate Observations. In, Fifth Berkeley Symposium on Mathematical Statistics and Probability (L. M. L. Cam and J. Neyman, eds.) 1 281–297.
• [21] Marron, J. S., Ramsay, J. O., Sangalli, L. M. and Srivastava, A. (2014). Statistics of Time Warpings and Phase Variations., Electronic Journal of Statistics 8 1697–1702.
• [22] Marron, J. S., Ramsay, J. O., Sangalli, L. M. and Srivastava, A. (2015). Functional Data Analysis of Amplitude and Phase Variation., Statistical Science 30 468–484.
• [23] Raket, L. L., Sommer, S. and Markussen, B. (2014). A Nonlinear Mixed-effects Model for Simultaneous Smoothing and Registration of Functional Data., Pattern Recognition Letters 38 1–7.
• [24] Ramsay, J. O., Bock, R. D. and Gasser, T. (1995). Comparison of Height Acceleration Curves in the Fels, Zurich, and Berkeley Growth Data., Annals of Human Biology 22 413–426.
• [25] Ramsay, J. O., Gribble, P. and Kurtek, S. (2014). Description and Processing of Functional Data Arising from Juggling Trajectories., Electronic Journal of Statistics 8 1811–1816.
• [26] Ramsay, J. O. and Li, X. (1998). Curve Registration., Journal of the Royal Statistical Society, Series B 60 351–363.
• [27] Ramsay, J. O. and Silverman, B. W. (2005)., Functional Data Analysis, Second Edition. Springer Series in Statistics.
• [28] Rousseeuw, P. (1987). Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis., Journal of Computational and Applied Mathematics 20 53–65.
• [29] Sangalli, L. M., Secchi, P. and Vantini, S. (2014). AneuRisk65: A Dataset of Three-dimensional Cerebral Vascular Geometries., Electronic Journal of Statistics 8 1879–1890.
• [30] Skare, O., B olviken, E. and Holden, L. (2003). Improved Sampling-Importance Resampling and Reduced Bias Importance Sampling., Scandinavian Journal of Statistics 30 719–737.
• [31] Srivastava, A., Jermyn, I. and Joshi, S. H. (2007). Riemannian Analysis of Probability Density Functions with Applications in Vision. In, IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 1–8.
• [32] Srivastava, A. and Jermyn, I. H. (2008). Looking for Shapes in Two-Dimensional Cluttered Point Clouds., IEEE Transactions on Pattern Analysis and Machine Intelligence 31 1616–1629.
• [33] Srivastava, A., Klassen, E., Joshi, S. H. and Jermyn, I. H. (2011). Shape Analysis of Elastic Curves in Euclidean Spaces., IEEE Transactions on Pattern Analysis and Machine Intelligence 33 1415–1428.
• [34] Srivastava, A., Wu, W., Kurtek, S., Klassen, E. and Marron, J. S. (2011). Registration of Functional Data Using Fisher-Rao Metric., arXiv:1103.3817v2.
• [35] Suematsu, N. and Hayashi, A. (2012). Time Series Alignment with Gaussian Processes. In, IEEE International Conference on Pattern Recognition (ICPR) 2355–2358.
• [36] Tang, R. and Müller, H. G. (2008). Pairwise Curve Synchronization for Functional Data., Biometrika 95 875–889.
• [37] Tang, X., Oishi, K., Faria, A. V., Hillis, A. E., Albert, M. S., Mori, S. and Miller, M. I. (2013). Bayesian Parameter Estimation and Segmentation in the Multi-atlas Random Orbit Model., PLoS ONE 8 e65591.
• [38] Telesca, D. and Inoue, L. Y. T. (2008). Bayesian Hierarchical Curve Registration., Journal of the American Statistical Association 103 328–339.
• [39] Telesca, D., Inoue, L. Y. T., Neira, M., Etzioni, R., Gleave, M. and Nelson, C. (2009). Differential Expression and Network Inferences through Functional Data Modeling., Biometrics 65 793–804.
• [40] Tsai, T. H., Tadesse, M. G., Wang, Y. and Ressom, H. W. (2013). Profile-Based LC-MS Data Alignment: A Bayesian Approach., IEEE/ACM Transactions on Computational Biology and Bioinformatics 99 494–503.
• [41] Tucker, J. D., Wu, W. and Srivastava, A. (2013). Generative Models for Functional Data Using Phase and Amplitude Separation., Computational Statistics & Data Analysis 61 50–66.
• [42] Tucker, J. D., Wu, W. and Srivastava, A. (2014). Analysis of Signals under Compositional Noise with Applications to SONAR Data., IEEE Journal of Oceanic Engineering 39 318–330.
• [43] Tuddenham, R. D. and Snyder, M. M. (1954). Physical Growth of California Boys and Girls from Birth to Eighteen Years., Publications in Child Development. University of California, Berkeley 1 183.
• [44] Wu, W., Hatsopoulos, N. G. and Srivastava, A. (2014). Introduction to Neural Spike Train Data for Phase-amplitude Analysis., Electronic Journal of Statistics 8 1759–1768.
• [45] Wu, W. and Srivastava, A. (2011). An Information-geometric Framework for Statistical Inferences in the Neural Spike Train Space., Journal of Computational Neuroscience 31 725–748.
• [46] Yeung, D., Chang, H., Xiong, Y., George, S., Kashi, R., Matsumoto, T. and Rigoll, G. (2004). SVC2004: First International Signature Verification Competition. In, International Conference on Biometric Authentication (ICBA) 16–22.