396 research outputs found
A Multiresolution Stochastic Process Model for Predicting Basketball Possession Outcomes
Basketball games evolve continuously in space and time as players constantly
interact with their teammates, the opposing team, and the ball. However,
current analyses of basketball outcomes rely on discretized summaries of the
game that reduce such interactions to tallies of points, assists, and similar
events. In this paper, we propose a framework for using optical player tracking
data to estimate, in real time, the expected number of points obtained by the
end of a possession. This quantity, called \textit{expected possession value}
(EPV), derives from a stochastic process model for the evolution of a
basketball possession; we model this process at multiple levels of resolution,
differentiating between continuous, infinitesimal movements of players, and
discrete events such as shot attempts and turnovers. Transition kernels are
estimated using hierarchical spatiotemporal models that share information
across players while remaining computationally tractable on very large data
sets. In addition to estimating EPV, these models reveal novel insights on
players' decision-making tendencies as a function of their spatial strategy.Comment: 31 pages, 9 figure
Fast Temporal Wavelet Graph Neural Networks
Spatio-temporal signals forecasting plays an important role in numerous
domains, especially in neuroscience and transportation. The task is challenging
due to the highly intricate spatial structure, as well as the non-linear
temporal dynamics of the network. To facilitate reliable and timely forecast
for the human brain and traffic networks, we propose the Fast Temporal Wavelet
Graph Neural Networks (FTWGNN) that is both time- and memory-efficient for
learning tasks on timeseries data with the underlying graph structure, thanks
to the theories of multiresolution analysis and wavelet theory on discrete
spaces. We employ Multiresolution Matrix Factorization (MMF) (Kondor et al.,
2014) to factorize the highly dense graph structure and compute the
corresponding sparse wavelet basis that allows us to construct fast wavelet
convolution as the backbone of our novel architecture. Experimental results on
real-world PEMS-BAY, METR-LA traffic datasets and AJILE12 ECoG dataset show
that FTWGNN is competitive with the state-of-the-arts while maintaining a low
computational footprint. Our PyTorch implementation is publicly available at
https://github.com/HySonLab/TWGNNComment: arXiv admin note: text overlap with arXiv:2111.0194
A construction of multiwavelets
AbstractA class of r-regular multiwavelets, depending on the smoothness of the multiwavelet functions, is introduced with the appropriate notation and definitions. Oscillation properties of orthonormal systems are obtained in Lemma 1 and Corollary 1 without assuming any vanishing moments for the scaling functions, and in Theorem 1 the existence of r-regular multiwavelets in L2(Rn) is established. In Theorem 2, a particular r-regular multiresolution analysis for multiwavelets is obtained from an r-regular multiresolution analysis for uniwavelets. In Theorem 3, an r-regular multiresolution analysis of split-type multiwavelets, which are perhaps the simplest multiwavelets, is easily obtained by using an r-regular multiresolution analysis for uniwavelets and a (2n − 1)-fold regular multiresolution analysis for uniwavelets. For some split-type multiwavelets, the support or width of the wavelets is shorter than the support or width of the scaling functions without loss of regularity nor of vanishing moments. Examples of split-type multiwavelets in L2(R) are constructed and illustrated by means of figures. Symmetry and antisymmetry are preserved in the case of infinite support
Wavelets on the Interval and Fast Wavelet Transforms
International audienceWe discuss several constructions of orthonormal wavelet bases on the interval, and we introduce a new construction that avoids some of the disadvantages of earlier constructions
Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties
The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets. A large part is devoted to a number of extensions to the newly proposed Stochastic Partial Differential Equation (SPDE) approach for representing Gaussian fields using Gaussian Markov Random Fields (GMRFs). The method is based on that Gaussian Matérn field can be viewed as solutions to a certain SPDE, and is useful for large spatial problems where traditional methods are too computationally intensive to use. A variation of the method using wavelet basis functions is proposed and using a simulation-based study, the wavelet approximations are compared with two of the most popular methods for efficient approximations of Gaussian fields. A new class of spatial models, including the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions, is also constructed using nested SPDEs. The SPDE method is extended to this model class and it is shown that all desirable properties are preserved, such as computational efficiency, applicability to data on general smooth manifolds, and simple non-stationary extensions. Finally, the SPDE method is extended to a larger class of non-Gaussian random fields with Matérn covariance functions, including certain Laplace Moving Average (LMA) models. In particular it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model. A method for estimating spatially dependent temporal trends is also developed. The method is based on using a space-varying regression model, accounting for spatial dependency in the data, and it is used to analyze temporal trends in vegetation data from the African Sahel in order to find regions that have experienced significant changes in the vegetation cover over the studied time period. The problem of estimating such regions is investigated further in the final part of the thesis where a method for estimating excursion sets, and the related problem of finding uncertainty regions for contour curves, for latent Gaussian fields is proposed. The method is based on using a parametric family for the excursion sets in combination with Integrated Nested Laplace Approximations (INLA) and an importance sampling-based algorithm for estimating joint probabilities
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