Exact smoothing for stationary and non-stationary time series

Casals, José Francisco Forniés; Jerez, Miguel; Sotoca, Sonia

doi:10.1016/s0169-2070(99)00030-8

Cited by 18 publications

(11 citation statements)

References 14 publications

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“…Result 2 (variance of smoothed estimates when the system has unit roots). Casals et al (2000) show that for any SS model with unit eigenvalues, corresponding to a time series with unit AR roots, the exact covariance of fi xed-interval smoothed estimates of the states is where P * tN is the smoother covariance derived from a Kalman fi lter with null initial conditions. Application of Result 2 to a non-stationary subsystem requires the initial state covariance given in (D.4), P 1 , to show unbounded uncertainty.…”

Section: Previous Resultsmentioning

confidence: 99%

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Modelling and forecasting time series sampled at different frequencies

Carro

Jerez

López

2008

Journal of Forecasting

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This paper discusses how to specify an observable high-frequency model for a vector of time series sampled at high and low frequencies. To this end we first study how aggregation over time affects both, the dynamic components of a time series and their observability, in a multivariate linear framework. We find that the basic dynamic components remain unchanged but some of them, mainly those related to the seasonal structure, become unobservable. Building on these results, we propose a structured specification method built on the idea that the models relating the variables in high and low sampling frequencies should be mutually consistent. After specifying a consistent and observable high-frequency model, standard state-space techniques provide an adequate framework for estimation, diagnostic checking, data interpolation and forecasting. Our method has three main uses. First, it is useful to disaggregate a vector of lowfrequency time series into high-frequency estimates coherent with both, the sample information and its statistical properties. Second, it may improve forecasting of the low-frequency variables, as the forecasts conditional to high-frequency indicators have in general smaller error variances than those derived from the corresponding low-frequency values. Third, the resulting forecasts can be updated as new high-frequency values become available, thus providing an effective tool to assess the effect of new information over medium term expectations. An example using national accounting data illustrates the practical application of this method.

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Section: Previous Resultsmentioning

confidence: 99%

“…where vai t denotes the unobserved value of VAI in quarter t and ind t is the corresponding adjusted indicator. Finally, by applying a fi xed-interval smoothing to the sample (Casals Jerez and Sotoca, 2000) we obtain the disaggregates and forecasts shown in Figure 2.…”

Section: Steps (3) and (4): Specifi Cation And Estimation Of The Quarmentioning

confidence: 99%

Modelling and forecasting time series sampled at different frequencies

Carro

Jerez

López

2008

Journal of Forecasting

Self Cite

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“…We will therefore do the same as in the first step: (a) use the results in De Jong (1988) to estimate 1 a and its variance, conditional to z 2 , (b) apply the smoother due to Casals, Jerez and Sotoca (2000) to condition it to z 2 , and (c) obtain an estimate of the stationary sub-system state vector and its covariance, so that the diffuse initial conditions will not affect the filtering results, see (4.11)-(4.12).…”

Section: First Step: Decompositionmentioning

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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“…The general framework on which all models in this chapter are cast, is the so called State Space systems, that have experienced a remarkable attention during the last decades, as the extended literature about it reveals [3], [7], [13], [15], [16], [17], [21], [24], [26] and [27].…”

Section: State Space Systemsmentioning

confidence: 99%

“…The former relates the output to the states of the system, while the latter reflects the dynamic behavior of the system by relating the current value of the states to their past values. There are a number of different formulations of these equations, but one fairly general representation is given by equations (1) (see [3] and [21] R Q C H E and , , , , ,  are, respectively, the n  n, n  r, m  n, and m  s, r  r and s  s system matrices, some elements of which are known and others that need to be estimated in some way.…”

Section: State Space Systemsmentioning

confidence: 99%

Digital Filters for Maintenance Management

Márquez¹,

Pedregal²

2011

Digital Filters

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Faults in mechanisms must be detected quickly and reliably in order to avoid important losses. Detection systems should be developed to minimize maintenance costs and are generally based on consistent models, but as simple as possible. Also, the models for detecting faults must adapt to external and internal conditions to the mechanism. The present chapter deals with three particular maintenance algorithms for turnouts in railway infrastructure by means of discrete filters that comply with these general objectives. All of them have the virtue of being developed within a well-known and common framework, namely the State Space with the help of the Kalman Filter (KF) and/or complementary Fixed Interval Smoother (FIS) algorithms. The algorithms are tested on real applications and thorough results are shown. IntroductionFaults in any important mechanisms must be detected quickly and reliably if the information is to be useful. Generally such mechanisms may be modeled as discrete dynamic systems, where data must be processed on line. When feasible, the detection system should use a model as simple as possible for detecting faults quickly by analyzing data in real time. The models for detecting faults must adapt to external and internal conditions to the mechanism, since both of them may affect the system as a whole.The present chapter deals with maintenance systems for turnouts in railway infrastructure by means of discrete filters. Turnouts are assembled from switches and a crossing where the moving parts are often described as the "points" move by the point mechanism. The standard railway point mechanism is a complex electro-mechanical device with many potential failure modes.Several approaches for maintenance of such devices are shown in this chapter and briefly described in this introduction. All of them have the virtue of being developed within a wellknown common framework, namely the State Space (SS) with the help of the Kalman Filter (KF) and/or complementary Fixed Interval Smoother (FIS) algorithms, exposed in general terms in the following section. Based on this common framework, the following subsections in this introduction show the particular applications shown in later sections of the chapter. Filtering with Integrated Random Walks (IRW)One possible way to analyze faults on line is to work with a reference dynamic system for their analysis. If the absolute value of the difference between the actual data and the reference data (i.e. the profile without any fault) is analyzed, the majority of faults may be detected by means of a simplified univariate dynamic system, like the one explored in [9]. The dynamic system and the use of the SS framework and the KF in this study allow increasing the reliability of the model presented that is the basic input to a rule-based decision mechanism. When applied to the linear discrete data filtering problem, the KF is a powerful algorithm, because it supports estimations of past, present and, most importantly, future states. It can therefore be used in predictive m...

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Exact smoothing for stationary and non-stationary time series

Cited by 18 publications

References 14 publications

Modelling and forecasting time series sampled at different frequencies

Modelling and forecasting time series sampled at different frequencies

Minimally conditioned likelihood for a nonstationary state space model

Digital Filters for Maintenance Management

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