Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions

Ansley, Craig F.; Kohn, Robert

doi:10.1214/aos/1176349739

Cited by 216 publications

(137 citation statements)

References 12 publications

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“…The most representative work in the data transformation approach is Ansley and Kohn (1985), hereafter AK, who proposed a sophisticated data transformation that cancels the nonstationary components of the model. As AK recognize, their approach has two shortcomings: it needs a complex and nonstandard filtering and requires the data transformation to be independent of the parameter values.…”

Section: Introductionmentioning

confidence: 99%

Minimally conditioned likelihood for a nonstationary state space model

Casals

Sotoca

Jerez

2014

Mathematics and Computers in Simulation

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Abstract:Computing the gaussian likelihood for a nonstationary state-space model is a difficult problem which has been tackled by the literature using two main strategies: data transformation and diffuse likelihood. The data transformation approach is cumbersome, as it requires nonstandard filtering. On the other hand, in some nontrivial cases the diffuse likelihood value depends on the scale of the diffuse states, so one can obtain different likelihood values corresponding to different observationally equivalent models. In this paper we discuss the properties of the minimally-conditioned likelihood function, as well as two efficient methods to compute its terms with computational advantages for specific models. Three convenient features of the minimally-conditioned likelihood are: (a) it can be computed with standard Kalman filters, (b) it is scale-free, and (c) its values are coherent with those resulting from differencing, being this the most popular approach to deal with nonstationary data.

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Section: Introductionmentioning

confidence: 99%

Minimally conditioned likelihood for a nonstationary state space model

Casals

Sotoca

Jerez

2014

Mathematics and Computers in Simulation

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“…In an abstract sense, a similar situation occurs in the initialization of a Kalman filter -the forecast covariance matrix generally is not available at the first time step. To deal with incompletely specified initial conditions, Ansley and Kohn (1985) proposed a method that is equivalent to assuming a diffuse prior distribution for the unspecified part of the initial state. A distribution is said to be diffuse if its covariance matrix is arbitrarily large (de Jong, 1991).…”

Section: Introductionmentioning

confidence: 99%

“…The diffuse assumption often corresponds to the limit of complete lack of knowledge in Bayesian analysis, from which the Kalman filter can be derived (Maybeck, 1979). Ansley and Kohn (1985) and de Jong (1991) discuss the extension of the Kalman filter to partially diffuse covariance matrices.…”

Section: Introductionmentioning

confidence: 99%

“…We propose two specific algorithms called the Diffuse Ensemble Kalman Filter (DEnKF) and the Diffuse Ensemble Transform Kalman Filter (DETKF). Our derivation of the DEnFs is essentially independent of Ansley and Kohn (1985) and de Jong (1991), as it is tailored to the special needs of an ensemble Kalman filter. It should be recognized, however, that the derivation of a diffuse filter is subtle.…”

Section: Introductionmentioning

confidence: 99%

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The diffuse ensemble filter

Yang

DelSole

2009

Nonlin. Processes Geophys.

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Abstract.A new class of ensemble filters, called the Diffuse Ensemble Filter (DEnF), is proposed in this paper. The DEnF assumes that the forecast errors orthogonal to the first guess ensemble are uncorrelated with the latter ensemble and have infinite variance. The assumption of infinite variance corresponds to the limit of "complete lack of knowledge" and differs dramatically from the implicit assumption made in most other ensemble filters, which is that the forecast errors orthogonal to the first guess ensemble have vanishing errors. The DEnF is independent of the detailed covariances assumed in the space orthogonal to the ensemble space, and reduces to conventional ensemble square root filters when the number of ensembles exceeds the model dimension. The DEnF is well defined only in data rich regimes and involves the inversion of relatively large matrices, although this barrier might be circumvented by variational methods. Two algorithms for solving the DEnF, namely the Diffuse Ensemble Kalman Filter (DEnKF) and the Diffuse Ensemble Transform Kalman Filter (DETKF), are proposed and found to give comparable results. These filters generally converge to the traditional EnKF and ETKF, respectively, when the ensemble size exceeds the model dimension. Numerical experiments demonstrate that the DEnF eliminates filter collapse, which occurs in ensemble Kalman filters for small ensemble sizes. Also, the use of the DEnF to initialize a conventional square root filter dramatically accelerates the spin-up time for convergence. However, in a perfect model scenario, the DEnF produces larger errors than ensemble square root filters that have covariance localization and inflation. For imperfect forecast models, the DEnF produces smaller errors than the ensemble square root filter with inflation. These experiments suggest that the DEnF has some advantages relative to the ensemble square root filters in the regime of small ensemble size, imperfect model, and copious observations.

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“…Some of the more successful methods for treating the problem of the start-up conditions that have been proposed have arisen within the context of the Kalman filter and the associated smoothing algorithms-see Ansley and Kohn (1985), De Jong (1991), and Durbin and Koopman (2001), for example. The context of the Kalman filter is a wide one; and it seems that the necessary results can be obtained more easily by restricting the context.…”

Section: Implementing the Filtersmentioning

confidence: 99%

Trend Estimation and De-Trending

Pollock

Optimisation, Econometric and Financial Analysis

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Abstract. An account is given of a variety of linear filters which can be used for extracting trends from economic time series and for generating de-trended series. A family of rational square-wave filters is described which enable designated frequency ranges to be selected or rejected. Their use is advocated in preference to other filters which are commonly used in quantitative economic analysis.

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Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions

Cited by 216 publications

References 12 publications

Minimally conditioned likelihood for a nonstationary state space model

Minimally conditioned likelihood for a nonstationary state space model

The diffuse ensemble filter

Trend Estimation and De-Trending

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