Spacetime Poster V2 opacity...for eigenfunctions/values and which are orthonormal in an RKHS....
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for eigenfunctions/values and which are orthonormal in an RKHS. Artificial Problem - We study values for and, as is increasingly far from 0. This parameter acts similarly to a normalized length scale. It, along with , controls the long-range behavior. Weather Modeling - We study the GEFS weather data in the western US over a 1 week period and vary two parameters jointly, & and & . We form our time-specific Mercer kernel with the closed form Idea - Choose and to design a nonstationary, time-specific kernel. Michael McCourt, Gregory Fasshauer, David Kozak A Nonstationary Designer Space-Time Kernel Gaussian Random Fields Kernels could be designed to have wider support further from the origin. Kernels could be designed to have the initial data strongly influence later locations. Kernels could be designed to quickly forget what happened at the initial time. Mercer Kernels We use the orthonormal Laguerre polynomials to define suitable eigenfunctions to create a kernel defined only for positive time. where is the modified Bessel function of the first kind. Free Parameters Range: Range: Range: Parametrizing for Space-Time Problems Tillman Gneiting, Marc Genton, et al. Geostatistical Space-Time Models, Stationarity, Separability, and Full Symmetry. Monographs on Statistics and Applied Probability, 107:151--175, 2006. Thomas M. Hamill, Gary T. Bates, et al. NOAA’s Second-Generation Global Medium-Range Ensemble Reforecast Dataset. Bulletin of the American Meterological Society, 94(10):1553–1565, 2013. Gaussian random fields are a powerful and popular tool for modeling spatial and space-time phenomena. Gaussian random fields are defined by two prior beliefs: • A mean function describing the overall trend/expected behavior, and • A covariance kernel describing the covariance which should occur between field values at two points in the domain. Combining these prior beliefs with observed data gives a posterior model with which to make predictions at unobserved locations/times. Design freedom - Predictions are greatly affected by the choice of priors. Designer Covariance Kernels Different covariance kernels impose different prior beliefs; as such, using a "good" covariance kernel can produce a "good" model. Question - How can covariance kernels be chosen effectively? Often, standard covariance kernels are considered, with free parameters chosen through strategies such as maximum likelihood estimation. Examples - Squared Exponential, Matérn, Inverse Multiquadric To improve on this modeling strategy, we may design kernels for specific circumstances or applications to have certain properties. Periodic - Certain frequencies can be more naturally preferred. Divergence free - Gradients may satisfy differential equations. Nonstationarity - Different regions may behave differently. Boundary/limit conditions - Known behaviors can be directly enforced. Smoothness - Better model data from nonsmooth sources. A Designer Kernel for Modeling in Time Shortcoming - Kernels designed for spatial statistics lack the sense of orientation which naturally exists when modeling time. Opportunity - Can we design a kernel which recognizes an initial time, and behaves differently at different times? Mercer's theorem says all positive definite kernels take the form This parameter is the "shift" associated with the Laguerre polynomials. It impacts the behavior of the kernels near the origin. This parameter determines the weight function which defines the sense of orthonormality. Opportunity - Because these kernels have a natural orientation in time, these parameters may be more interpretable in space-time settings. Cross-validation - Can we use a validation RMSE in time (training on data up to and predicting at ) to parametrize these kernels? Conclusion - Cross-validation seems viable, but more investigation is required to effectively leverage the flexibility of this kernel. Spatiotemporal Workshop at NIPS, December 7, 2018, Montréal, QC, USA Gregory Fasshauer, Michael McCourt. Kernel-based Approximation Methods using Matlab. World Scientific Press, 2015.
Transcript of Spacetime Poster V2 opacity...for eigenfunctions/values and which are orthonormal in an RKHS....
for eigenfunctions/values and which are orthonormal in an RKHS.
Artificial Problem - We study values forand, as is increasingly far from 0.
This parameter acts similarly to anormalized length scale. It, along with ,controls the long-range behavior.
Weather Modeling - We study the GEFSweather data in the western US over a 1 weekperiod and vary two parameters jointly, & and & .
We form our time-specific Mercer kernel with the closed form
Idea - Choose and to design a nonstationary, time-specific kernel.
Michael McCourt, Gregory Fasshauer, David Kozak
A Nonstationary Designer Space-Time Kernel
Gaussian Random Fields
Kernels could be designed to have widersupport further from the origin.
Kernels could be designed to have the initial data strongly influence later locations.
Kernels could be designed to quickly forgetwhat happened at the initial time.
Mercer Kernels
We use the orthonormal Laguerre polynomials to define suitableeigenfunctions to create a kernel defined only for positive time.
where is the modified Bessel function of the first kind.
Free ParametersRange:
Range:
Range:
Parametrizing for Space-Time Problems
Tillman Gneiting, Marc Genton, et al. Geostatistical Space-Time Models, Stationarity, Separability, and Full Symmetry. Monographs on Statisticsand Applied Probability, 107:151--175, 2006.Thomas M. Hamill, Gary T. Bates, et al. NOAA’s Second-Generation Global Medium-Range Ensemble Reforecast Dataset. Bulletin of theAmerican Meterological Society, 94(10):1553–1565, 2013.
Gaussian random fields are a powerful and popular tool for modeling spatial and space-time phenomena.
Gaussian random fields are defined by two prior beliefs:
• A mean function describing the overall trend/expected behavior, and
• A covariance kernel describing the covariance which should occur between field values at two points in the domain.
Combining these prior beliefs with observed data gives a posterior model with which to make predictions at unobserved locations/times.
Design freedom - Predictions are greatly a�ected by the choice of priors.
Designer Covariance KernelsDi�erent covariance kernels impose di�erent prior beliefs; as such, using a "good" covariance kernel can produce a "good" model.
Question - How can covariance kernels be chosen e�ectively?
Often, standard covariance kernels are considered, with free parameterschosen through strategies such as maximum likelihood estimation.
Examples - Squared Exponential, Matérn, Inverse Multiquadric
To improve on this modeling strategy, we may design kernels for specificcircumstances or applications to have certain properties.
Periodic - Certain frequencies can be more naturally preferred.Divergence free - Gradients may satisfy di�erential equations.Nonstationarity - Di�erent regions may behave di�erently.Boundary/limit conditions - Known behaviors can be directly enforced.Smoothness - Better model data from nonsmooth sources.
A Designer Kernel for Modeling in TimeShortcoming - Kernels designed for spatial statistics lack the sense of orientation which naturally exists when modeling time.
Opportunity - Can we design a kernel which recognizes an initial time, and behaves di�erently at di�erent times?
Mercer's theorem says all positive definite kernels take the form
This parameter is the "shift" associatedwith the Laguerre polynomials. It impactsthe behavior of the kernels near the origin.
This parameter determines the weightfunction which defines the senseof orthonormality.
Opportunity - Because these kernels have a natural orientation in time, theseparameters may be more interpretable in space-time settings.
Cross-validation - Can we use a validation RMSE in time (training ondata up to and predicting at ) to parametrize these kernels?
Conclusion - Cross-validation seems viable, but more investigation isrequired to e�ectively leverage the flexibility of this kernel.
Spatiotemporal Workshop at NIPS, December 7, 2018, Montréal, QC, USA
Gregory Fasshauer, Michael McCourt. Kernel-based Approximation Methods using Matlab. World Scientific Press, 2015.
