arXiv subjects

Anuj Srivastava

Publications and source records attributed to Anuj Srivastava.

57 records · Page 4Linked to original sources

arXiv2014

Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance

We consider the statistical analysis of trajectories on Riemannian manifolds that are observed under arbitrary temporal evolutions. Past methods rely on cross-sectional analysis, with the given temporal registration, and consequently may lose the mean structure and artificially inflate observed variances. We introduce a quantity that provides both a cost function for temporal registration and a proper distance for comparison of trajectories. This distance is used to define statistical summaries, such as sample means and covariances, of synchronized trajectories and "Gaussian-type" models to capture their variability at discrete times. It is invariant to identical time-warpings (or temporal reparameterizations) of trajectories. This is based on a novel mathematical representation of trajectories, termed transported square-root vector field (TSRVF), and the $\mathbb{L}^2$ norm on the space of TSRVFs. We illustrate this framework using three representative manifolds---$\mathbb{S}^2$, $\mathrm {SE}(2)$ and shape space of planar contours---involving both simulated and real data. In particular, we demonstrate: (1) improvements in mean structures and significant reductions in cross-sectional variances using real data sets, (2) statistical modeling for capturing variability in aligned trajectories, and (3) evaluating random trajectories under these models. Experimental results concern bird migration, hurricane tracking and video surveillance.

stat.AP↗

arXiv2012

Generative Models for Functional Data using Phase and Amplitude Separation

Constructing generative models for functional observations is an important task in statistical functional analysis. In general, functional data contains both phase (or x or horizontal) and amplitude (or y or vertical) variability. Tradi- tional methods often ignore the phase variability and focus solely on the amplitude variation, using cross-sectional techniques such as fPCA for dimensional reduction and data modeling. Ignoring phase variability leads to a loss of structure in the data and inefficiency in data models. This paper presents an approach that relies on separating the phase (x-axis) and amplitude (y-axis), then modeling these components using joint distributions. This separation, in turn, is performed using a technique called elastic shape analysis of curves that involves a new mathematical representation of functional data. Then, using individual fPCAs, one each for phase and amplitude components, while respecting the nonlinear geometry of the phase representation space; impose joint probability models on principal coefficients of these components. These ideas are demonstrated using random sampling, for models estimated from simulated and real datasets, and show their superiority over models that ignore phase-amplitude separation. Furthermore, the generative models are applied to classification of functional data and achieve high performance in applications involv- ing SONAR signals of underwater objects, handwritten signatures, and periodic body movements recorded by smart phones.

stat.CO↗

arXiv2011

Registration of Functional Data Using Fisher-Rao Metric

We introduce a novel geometric framework for separating the phase and the amplitude variability in functional data of the type frequently studied in growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to derive a proper distance on the quotient space of functions modulo the time-warping group. A convenient square-root velocity function (SRVF) representation transforms the Fisher-Rao metric into the standard $\ltwo$ metric, simplifying the computations. This distance is then used to define a Karcher mean template and warp the individual functions to align them with the Karcher mean template. The strength of this framework is demonstrated by deriving a consistent estimator of a signal observed under random warping, scaling, and vertical translation. These ideas are demonstrated using both simulated and real data from different application domains: the Berkeley growth study, handwritten signature curves, neuroscience spike trains, and gene expression signals. The proposed method is empirically shown to be be superior in performance to several recently published methods for functional alignment.

math.ST↗