1,444 research outputs found
Inferring the Residual Waiting Time for Binary Stationary Time Series
summary:For a binary stationary time series define to be the number of consecutive ones up to the first zero encountered after time , and consider the problem of estimating the conditional distribution and conditional expectation of after one has observed the first outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state the stopping times along which we estimate have density one
Inferring the residual waiting time for binary stationary time series
summary:For a binary stationary time series define to be the number of consecutive ones up to the first zero encountered after time , and consider the problem of estimating the conditional distribution and conditional expectation of after one has observed the first outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state the stopping times along which we estimate have density one
On universal estimates for binary renewal processes
A binary renewal process is a stochastic process taking values in
where the lengths of the runs of 1's between successive zeros are
independent. After observing one would like to predict the
future behavior, and the problem of universal estimators is to do so without
any prior knowledge of the distribution. We prove a variety of results of this
type, including universal estimates for the expected time to renewal as well as
estimates for the conditional distribution of the time to renewal. Some of our
results require a moment condition on the time to renewal and we show by an
explicit construction how some moment condition is necessary.Comment: Published in at http://dx.doi.org/10.1214/07-AAP512 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Intermittent estimation for finite alphabet finitarily Markovian processes with exponential tails
summary:We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first outputs of a stationary process where the random variables may take finitely many possible values. Our schemes are universal in the class of finitarily Markovian processes that have an exponential rate for the tail of the look back time distribution. In addition explicit rates are given. A necessary restriction is that the scheme proposes an estimate only at certain stopping times, but these have density one so that one rarely fails to give an estimate
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