The Experts below are selected from a list of 75 Experts worldwide ranked by ideXlab platform
Anuj Srivastava - One of the best experts on this subject based on the ideXlab platform.
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rate invariant analysis of covariance trajectories
Journal of Mathematical Imaging and Vision, 2018Co-Authors: Zhengwu Zhang, Eric Klassen, Anuj SrivastavaAbstract:Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The execution rates of actions, implying arbitrary parameterizations of trajectories, complicate their analysis. To handle this challenge, we represent covariance trajectories using transported square-Root Vector fields, constructed by parallel translating scaled-velocity Vectors of trajectories to their starting points. The space of such representations forms a Vector bundle on the SPDM manifold. Using a natural Riemannian metric on this Vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the space of this Vector bundle. This metric is invariant to the action of the re-parameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis.
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action recognition using rate invariant analysis of skeletal shape trajectories
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2016Co-Authors: Boulbaba Ben Amor, Anuj SrivastavaAbstract:We study the problem of classifying actions of human subjects using depth movies generated by Kinect or other depth sensors. Representing human body as dynamical skeletons, we study the evolution of their (skeletons’) shapes as trajectories on Kendall’s shape manifold. The action data is typically corrupted by large variability in execution rates within and across subjects and, thus, causing major problems in statistical analyses. To address that issue, we adopt a recently-developed framework of Su et al. [1] , [2] to this problem domain. Here, the variable execution rates correspond to re-parameterizations of trajectories, and one uses a parameterization-invariant metric for aligning, comparing, averaging, and modeling trajectories. This is based on a combination of transported square-Root Vector fields (TSRVFs) of trajectories and the standard Euclidean norm, that allows computational efficiency. We develop a comprehensive suite of computational tools for this application domain: smoothing and denoising skeleton trajectories using median filtering, up- and down-sampling actions in time domain, simultaneous temporal-registration of multiple actions, and extracting invertible Euclidean representations of actions. Due to invertibility these Euclidean representations allow both discriminative and generative models for statistical analysis. For instance, they can be used in a SVM-based classification of original actions, as demonstrated here using MSR Action-3D, MSR Daily Activity and 3D Action Pairs datasets. Using only the skeletal information, we achieve state-of-the-art classification results on these datasets.
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video based action recognition using rate invariant analysis of covariance trajectories
arXiv: Computer Vision and Pattern Recognition, 2015Co-Authors: Zhengwu Zhang, Eric Klassen, Anuj SrivastavaAbstract:Statistical classification of actions in videos is mostly performed by extracting relevant features, particularly covariance features, from image frames and studying time series associated with temporal evolutions of these features. A natural mathematical representation of activity videos is in form of parameterized trajectories on the covariance manifold, i.e. the set of symmetric, positive-definite matrices (SPDMs). The variable execution-rates of actions implies variable parameterizations of the resulting trajectories, and complicates their classification. Since action classes are invariant to execution rates, one requires rate-invariant metrics for comparing trajectories. A recent paper represented trajectories using their transported square-Root Vector fields (TSRVFs), defined by parallel translating scaled-velocity Vectors of trajectories to a reference tangent space on the manifold. To avoid arbitrariness of selecting the reference and to reduce distortion introduced during this mapping, we develop a purely intrinsic approach where SPDM trajectories are represented by redefining their TSRVFs at the starting points of the trajectories, and analyzed as elements of a Vector bundle on the manifold. Using a natural Riemannain metric on Vector bundles of SPDMs, we compute geodesic paths and geodesic distances between trajectories in the quotient space of this Vector bundle, with respect to the reparameterization group. This makes the resulting comparison of trajectories invariant to their re-parameterization. We demonstrate this framework on two applications involving video classification: visual speech recognition or lip-reading and hand-gesture recognition. In both cases we achieve results either comparable to or better than the current literature.
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rate invariant analysis of trajectories on riemannian manifolds with application in visual speech recognition
Computer Vision and Pattern Recognition, 2014Co-Authors: Jingyong Su, Anuj Srivastava, Fillipe D M De Souza, Sudeep SarkarAbstract:In statistical analysis of video sequences for speech recognition, and more generally activity recognition, it is natural to treat temporal evolutions of features as trajectories on Riemannian manifolds. However, different evolution patterns result in arbitrary parameterizations of these trajectories. We investigate a recent framework from statistics literature that handles this nuisance variability using a cost function/distance for temporal registration and statistical summarization & modeling of trajectories. It is based on a mathematical representation of trajectories, termed transported square-Root Vector field (TSRVF), and the L2 norm on the space of TSRVFs. We apply this framework to the problem of speech recognition using both audio and visual components. In each case, we extract features, form trajectories on corresponding manifolds, and compute parametrization-invariant distances using TSRVFs for speech classification. On the OuluVS database the classification performance under metric increases significantly, by nearly 100% under both modalities and for all choices of features. We obtained speaker-dependent classification rate of 70% and 96% for visual and audio components, respectively.
V M Kurbanov - One of the best experts on this subject based on the ideXlab platform.
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on the riesz inequality and the basis property of systems of Root Vector functions of a discontinuous dirac operator
Differential Equations, 2019Co-Authors: V M Kurbanov, L Z BuksaevaAbstract:We consider a discontinuous Dirac operator on the interval (0, 2π). It is assumed that its coefficient (potential) is a complex-valued matrix function integrable on (0, 2π). Criteria are established for the Riesz and unconditional basis properties of the system of Root Vector functions in L 2 2 (0, 2π). A theorem about the equivalent basis property in L 2 (0, 2π), 1 ∞, is proved.
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on bessel property and unconditional basicity of the systems of Root Vector functions of a dirac type operator
Azerbaijan Journal of Mathematics, 2017Co-Authors: V M Kurbanov, Elchin J Ibadov, Gunel R HajiyevaAbstract:In this paper, we consider Dirac type one-dimensional operator $Dy=B\frac{dy}{dx} +P\left(x\right)y$, $y\left(x\right)=\left(y_{1} \left(x\right),y_{2} \left(x\right)\right)^{T} $, where $B=\left(\begin{array}{l} {0\, \, \, b_{1} } \\ {b_{2} \, \, 0} \end{array}\right)\, ;\, \, b_{1} >0,\, \, b_{2} <0$, $P\left(x\right)=\left(\begin{array}{l} {\, \, \, p_{1} \left(x\right)\, \, \, \, \, 0} \\ {\, \, \, \, \, \, 0\, \, \, \, \, \, p_{2} \left(x\right)} \end{array}\right),$ $p_{1} \left(x\right)$ and $p_{2} \left(x\right)$ are complex-valued functions defined on arbitrary finite interval $G\left(a,b\right)$ of a real straightline, and establish the criterion of Bessel property and unconditional basicity of the system of Root Vector-functions of this operator.
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componentwise uniform equiconvergence of expansions in Root Vector functions of the dirac operator with the trigonometric expansion
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in Root Vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved.
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two sided estimates for Root Vector functions of the dirac operator
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We consider the one-dimensional Dirac operator. We derive a shift formula for its Root Vector functions and prove anti-a priori and two-sided estimates for various Lp-norms of these functions.
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riesz inequality for systems of Root Vector functions of the dirac operator
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We establish a criterion for the Riesz property of systems of Root Vector functions of the one-dimensional Dirac operator.
Heino Bohn Nielsen - One of the best experts on this subject based on the ideXlab platform.
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unit Root Vector autoregression with volatility induced stationarity
Journal of Empirical Finance, 2014Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates are found to have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic. Moreover, they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that T-convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations illustrate the asymptotic theory.
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unit Root Vector autoregression with volatility induced stationarity
Research Papers in Economics, 2012Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that vT-convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.
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unit Root Vector autoregression with volatility induced stationarity
CREATES Research Papers, 2012Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that square-Root T convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.
A I Ismailova - One of the best experts on this subject based on the ideXlab platform.
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componentwise uniform equiconvergence of expansions in Root Vector functions of the dirac operator with the trigonometric expansion
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in Root Vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved.
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two sided estimates for Root Vector functions of the dirac operator
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We consider the one-dimensional Dirac operator. We derive a shift formula for its Root Vector functions and prove anti-a priori and two-sided estimates for various Lp-norms of these functions.
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riesz inequality for systems of Root Vector functions of the dirac operator
Differential Equations, 2012Co-Authors: V M Kurbanov, A I IsmailovaAbstract:We establish a criterion for the Riesz property of systems of Root Vector functions of the one-dimensional Dirac operator.
Anders Rahbek - One of the best experts on this subject based on the ideXlab platform.
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unit Root Vector autoregression with volatility induced stationarity
Journal of Empirical Finance, 2014Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates are found to have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic. Moreover, they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that T-convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations illustrate the asymptotic theory.
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unit Root Vector autoregression with volatility induced stationarity
Research Papers in Economics, 2012Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that vT-convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.
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unit Root Vector autoregression with volatility induced stationarity
CREATES Research Papers, 2012Co-Authors: Anders Rahbek, Heino Bohn NielsenAbstract:We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-Roots, while at the same time maintain stationarity despite such unit-Roots. Specifically, the model bridges Vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit Roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit Root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit Roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that square-Root T convergence to Gaussian distributions apply despite unit Roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.
