82,627 research outputs found
Efficient state-space inference of periodic latent force models
Latent force models (LFM) are principled approaches to incorporating solutions to differen-tial equations within non-parametric inference methods. Unfortunately, the developmentand application of LFMs can be inhibited by their computational cost, especially whenclosed-form solutions for the LFM are unavailable, as is the case in many real world prob-lems where these latent forces exhibit periodic behaviour. Given this, we develop a newsparse representation of LFMs which considerably improves their computational efficiency,as well as broadening their applicability, in a principled way, to domains with periodic ornear periodic latent forces. Our approach uses a linear basis model to approximate onegenerative model for each periodic force. We assume that the latent forces are generatedfrom Gaussian process priors and develop a linear basis model which fully expresses thesepriors. We apply our approach to model the thermal dynamics of domestic buildings andshow that it is effective at predicting day-ahead temperatures within the homes. We alsoapply our approach within queueing theory in which quasi-periodic arrival rates are mod-elled as latent forces. In both cases, we demonstrate that our approach can be implemented efficiently using state-space methods which encode the linear dynamic systems via LFMs.Further, we show that state estimates obtained using periodic latent force models can re-duce the root mean squared error to 17% of that from non-periodic models and 27% of thenearest rival approach which is the resonator model (S ̈arkk ̈a et al., 2012; Hartikainen et al.,2012.
Efficient State-Space Inference of Periodic Latent Force Models
Latent force models (LFM) are principled approaches to incorporating
solutions to differential equations within non-parametric inference methods.
Unfortunately, the development and application of LFMs can be inhibited by
their computational cost, especially when closed-form solutions for the LFM are
unavailable, as is the case in many real world problems where these latent
forces exhibit periodic behaviour. Given this, we develop a new sparse
representation of LFMs which considerably improves their computational
efficiency, as well as broadening their applicability, in a principled way, to
domains with periodic or near periodic latent forces. Our approach uses a
linear basis model to approximate one generative model for each periodic force.
We assume that the latent forces are generated from Gaussian process priors and
develop a linear basis model which fully expresses these priors. We apply our
approach to model the thermal dynamics of domestic buildings and show that it
is effective at predicting day-ahead temperatures within the homes. We also
apply our approach within queueing theory in which quasi-periodic arrival rates
are modelled as latent forces. In both cases, we demonstrate that our approach
can be implemented efficiently using state-space methods which encode the
linear dynamic systems via LFMs. Further, we show that state estimates obtained
using periodic latent force models can reduce the root mean squared error to
17% of that from non-periodic models and 27% of the nearest rival approach
which is the resonator model.Comment: 61 pages, 13 figures, accepted for publication in JMLR. Updates from
earlier version occur throughout article in response to JMLR review
Non-stationary self-similar Gaussian processes as scaling limits of power law shot noise processes and generalizations of fractional Brownian motion
We study shot noise processes with Poisson arrivals and non-stationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: 1) the conditional variance functions of the noises have a power law and 2) the conditional noise distributions are piecewise. In both cases, the limit processes are self-similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self-similar Gaussian processes with non-stationary increments, via the time-domain integral representation, which are natural generalizations of fractional Brownian motions.Published versio
Analyzing long-term correlated stochastic processes by means of recurrence networks: Potentials and pitfalls
Long-range correlated processes are ubiquitous, ranging from climate
variables to financial time series. One paradigmatic example for such processes
is fractional Brownian motion (fBm). In this work, we highlight the potentials
and conceptual as well as practical limitations when applying the recently
proposed recurrence network (RN) approach to fBm and related stochastic
processes. In particular, we demonstrate that the results of a previous
application of RN analysis to fBm (Liu \textit{et al.,} Phys. Rev. E
\textbf{89}, 032814 (2014)) are mainly due to an inappropriate treatment
disregarding the intrinsic non-stationarity of such processes. Complementarily,
we analyze some RN properties of the closely related stationary fractional
Gaussian noise (fGn) processes and find that the resulting network properties
are well-defined and behave as one would expect from basic conceptual
considerations. Our results demonstrate that RN analysis can indeed provide
meaningful results for stationary stochastic processes, given a proper
selection of its intrinsic methodological parameters, whereas it is prone to
fail to uniquely retrieve RN properties for non-stationary stochastic processes
like fBm.Comment: 8 pages, 6 figure
Efficient simulation of Brown-Resnick processes based on variance reduction of Gaussian processes
Brown-Resnick processes are max-stable processes that are associated to
Gaussian processes. Their simulation is often based on the corresponding
spectral representation which is not unique. We study to what extent simulation
accuracy and efficiency can be improved by minimizing the maximal variance of
the underlying Gaussian process. Such a minimization is a difficult
mathematical problem that also depends on the geometry of the simulation
domain. We extend Matheron's (1974) seminal contribution in two aspects: (i)
making his description of a minimal maximal variance explicit for convex
variograms on symmetric domains and (ii) proving that the same strategy reduces
the maximal variance also for a huge class of non-convex variograms
representable through a Bernstein function. A simulation study confirms that
our non-costly modification can lead to substantial improvements among Gaussian
representations. We also compare it with three other established algorithms.Comment: 19 pages, 3 figures, 4 tables; To appear with the Applied Probability
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