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Pairwise curve synchronization for functional data
 Biometrika
, 2008
"... Increasingly, data collected by scientists in different fields are in the form of trajectories or curves. These curves can often be viewed as realizations of a composite process driven by both amplitude and time variation. We consider the situation where functional variation is dominated by time var ..."
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Cited by 20 (3 self)
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Increasingly, data collected by scientists in different fields are in the form of trajectories or curves. These curves can often be viewed as realizations of a composite process driven by both amplitude and time variation. We consider the situation where functional variation is dominated by time variation, and develop a curvesynchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in a first step. These initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in a second step. A truncated averaging process is used to obtain robust estimation of individual warping functions. The method compares well with other available warping approaches and is illustrated with Berkeley growth data and gene expression data for multiple sclerosis.
kmean alignment for curve clustering
 Comput. Statist. Data Anal
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Signal estimation under random timewarpings and nonlinear signal alignment. In
 Eds.), Proceedings of Advances in Neural Information Processing Systems (NIPS
"... While signal estimation under random amplitudes, phase shifts, and additive noise is studied frequently, the problem of estimating a deterministic signal under random timewarpings has been relatively unexplored. We present a novel framework for estimating the unknown signal that utilizes the actio ..."
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Cited by 8 (2 self)
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While signal estimation under random amplitudes, phase shifts, and additive noise is studied frequently, the problem of estimating a deterministic signal under random timewarpings has been relatively unexplored. We present a novel framework for estimating the unknown signal that utilizes the action of the warping group to form an equivalence relation between signals. First, we derive an estimator for the equivalence class of the unknown signal using the notion of Karcher mean on the quotient space of equivalence classes. This step requires the use of FisherRao Riemannian metric and a squareroot representation of signals to enable computations of distances and means under this metric. Then, we define a notion of the center of a class and show that the center of the estimated class is a consistent estimator of the underlying unknown signal. This estimation algorithm has many applications: (1) registration/alignment of functional data, (2) separation of phase/amplitude components of functional data, (3) joint demodulation and carrier estimation, and (4) sparse modeling of functional data. Here we demonstrate only (1) and (2): Given signals are temporally aligned using nonlinear warpings and, thus, separated into their phase and amplitude components. The proposed method for signal alignment is shown to have state of the art performance using Berkeley growth, handwritten signatures, and neuroscience spike train data. 1
On the definition of phase and amplitude variability in functional data analysis
 Test
"... We introduce a mathematical framework in which a functional data registration problem can be soundly and coherently set. In detail, we show that the introduction of a metric/semimetric and of a group of warping function respect to which the metric/semimetric is invariant is the key to a clear and ..."
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Cited by 4 (1 self)
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We introduce a mathematical framework in which a functional data registration problem can be soundly and coherently set. In detail, we show that the introduction of a metric/semimetric and of a group of warping function respect to which the metric/semimetric is invariant is the key to a clear and not ambiguous definition of phase and amplitude variability. Moreover, an amplitudetototal variability index is proposed. This index turns to be useful in practical situations to measure to what extent amplitude variability affects the data and to compare the effectiveness of different registration methods. 1
A Multiresolution Approach to Time Warping achieved by a Bayesian PriorPosterior Transfer Fitting Strategy
"... multiresolution approach to time warping achieved by a ..."
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Cited by 2 (1 self)
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multiresolution approach to time warping achieved by a
Moments Based Functional Synchronization
, 2005
"... A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. A common “synchronization” approach involves equating “landmarks ” such as peaks or troughs. The landmarks method can work well b ..."
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A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. A common “synchronization” approach involves equating “landmarks ” such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. It may also involve a manual identification of marker events. We develop automated alignment methods based on equating the “moments ” of a given set of curves. These moments do not depend on the identification of markers. For example, the first moment is a measure of the average value of a curve in the x, or time, axis while the second moment measures its spread. We explore both linear and nonlinear synchronization procedures. Finally, we discuss the advantages of utilizing, not only the “amplitude ” information, which measures the general shape of the curves, but also the “warping ” information, which measures the way the curves have been distorted in time. Illustrations are provided on functional analyses involving principal components, clustering, classification and regression.
GRASS GIS: A multipurpose open source GIS
 ENVIRONMENTAL MODELLING & SOFTWARE
, 2012
"... ..."
Predictive interfaces Gazebased interfaces Feature selection Feature representation
, 2014
"... Sketchbased interaction ..."
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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. Traditional methods often ignore the phase variability and focus sol ..."
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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. Traditional methods often ignore the phase variability and focus solely on the amplitude variation, using crosssectional 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 (xaxis) and amplitude (yaxis), 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 phaseamplitude 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.
disease
, 2012
"... Segmentation, alignment and statistical analysis of biosignals with application to ..."
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Segmentation, alignment and statistical analysis of biosignals with application to