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Abstract: Constructing generative models for functional observations is an important task in statistical functional analysis. In general, functional data contains both phase (or or horizontal) and amplitude (or or vertical) variability. Traditional 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 (-axis) and amplitude (-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, it imposes joint probability models on principal coefficients of these components while respecting the nonlinear geometry of the phase representation space. 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 involving SONAR signals of underwater objects, handwritten signatures, and periodic body movements recorded by smart phones. [Copyright &y& Elsevier]

In this paper we present a method for clustering and classification of acoustic color data based on statistical analysis of functions using square-root velocity functions (SVRF). The convenience of the SVRF is that it transforms the Fisher-Rao metric into the standard L2metric. As a result, a formal distance can be calculated using geodesic paths. Moreover, this method allows optimal deformations between acoustic color data to be computed for any two targets allowing for robustness to measurement error. Using the SVRF formulation statistical models can then be constructed using principal component analysis to model the functional variation of acoustic color data. Empirical results demonstrate the utility of functional data analysis for improving performance results in pattern recognition using acoustic color data.

Functional variables are often used as predictors in regression problems. A commonly used parametric approach, called scalar-on-function regression , uses the L 2 inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase–amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. Here we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices. [ABSTRACT FROM AUTHOR]

The problem of estimating a target-layer trajectory, modeled by a straight line, in 2D point clouds that contain target locations and overwhelming clutter is studied. These point clouds are generated by an image-based pre-processing tool, termed ATR, operating on SONAR image data that results in: (1) point locations and (2) an ATR score: a measure of the “target-likeness” for each point. The model of choice assumes that the observed point cloud is a superposition of two spatial processes: (1) a 1D Poisson process along the target-layer line, corrupted by 2D Gaussian noise, denoting target locations and (2) a 2D Poisson process denoting clutter. It is further assumed that the target-likeness measure follows known probability distributions for both target locations and clutter. The line is parameterized by distance from the origin and the angle with respect to a horizontal axis, and the likelihood of these parameters for observed data is derived. Using a maximum-likelihood approach, a gradient-based estimate for line parameters and other nuisance parameters is developed. A formal procedure that tests for the presence of a target-layer trajectory in the point cloud data is additionally developed. The success of this method in both simulated and real datasets collected by NSWC PCD is demonstrated. [ABSTRACT FROM AUTHOR]