Consistency and relative efficiency of subspace methods

Deistler, Manfred; Peternell, K.; Scherrer, Wolfgang

doi:10.1016/0005-1098(95)00089-6

Cited by 105 publications

(22 citation statements)

References 8 publications

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“…From that point of view, all we can know (from second-order statistics) is the spectrum, and so if, naturally, we want a unique model, the only model we can obtain is the causal stable minimum phase model: the ISS model. The standard approach to SS model fitting is the so-called state-space subspace method (Deistler, Peternell, & Scherer, 1995; Bauer, 2005), and indeed it delivers an ISS model. The alternative approach of fitting a VARMA model (Hannan & Deistler, 1988; Lutkepohl, 1993) is equivalent to getting an ISS model.…”

Section: State Spacementioning

confidence: 99%

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State-Space Analysis of Granger-Geweke Causality Measures with Application to fMRI

Solo

2016

Neural Computation

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The recent interest in the dynamics of networks and the advent, across a range of applications, of measuring modalities that operate on different temporal scales have put the spotlight on some significant gaps in the theory of multivariate time series. Fundamental to the description of network dynamics is the direction of interaction between nodes, accompanied by a measure of the strength of such interactions. Granger causality and its associated frequency domain strength measures (GEMs) (due to Geweke) provide a framework for the formulation and analysis of these issues. In pursuing this setup, three significant unresolved issues emerge. First, computing GEMs involves computing submodels of vector time series models, for which reliable methods do not exist. Second, the impact of filtering on GEMs has never been definitively established. Third, the impact of downsampling on GEMs has never been established. In this work, using state-space methods, we resolve all these issues and illustrate the results with some simulations. Our analysis is motivated by some problems in (fMRI) brain imaging, to which we apply it, but it is of general applicability.

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Section: State Spacementioning

confidence: 99%

“…Suppose we fit an SS model to data z t , t = 1, …, T using, for example, so-called state-space subspace methods (Deistler et al, 1995; Bauer, 2005) or VARMA methods in, for example, Lutkepohl (1993). Let

{\hat{F}}_{Y \to X}

{\hat{F}}_{X \to Y}

{\hat{F}}_{Y . X}

{\hat{F}}_{X o Y}

be the corresponding GEM estimators.…”

Section: Granger Causalitymentioning

confidence: 99%

State-Space Analysis of Granger-Geweke Causality Measures with Application to fMRI

Solo

2016

Neural Computation

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“…Emails: {atsiamis,pappasg}@seas.upenn.edu ρ(A) < 1) and is limited to asymptotic results. In [7,8] it is shown that the identification error can decrease as fast as O 1/ √ N up to logarithmic factors, as the number of output data N grows to infinity. While asymptotic results have been established, a finite sample analysis of subspace algorithms remains an open problem [2].…”

mentioning

confidence: 99%

Finite Sample Analysis of Stochastic System Identification

Tsiamis

Pappas

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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In this paper, we analyze the finite sample complexity of stochastic system identification using modern tools from machine learning and statistics. An unknown discrete-time linear system evolves over time under Gaussian noise without external inputs. The objective is to recover the system parameters as well as the Kalman filter gain, given a single trajectory of output measurements over a finite horizon of length N . Based on a subspace identification algorithm and a finite number of N output samples, we provide non-asymptotic high-probability upper bounds for the system parameter estimation errors. Our analysis uses recent results from random matrix theory, self-normalized martingales and SVD robustness, in order to show that with high probability the estimation errors decrease with a rate of 1/ √ N . Our non-asymptotic bounds not only agree with classical asymptotic results, but are also valid even when the system is marginally stable. * The authors are with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104. Emails: {atsiamis,pappasg}@seas.upenn.edu ρ(A) < 1) and is limited to asymptotic results. In [7,8] it is shown that the identification error can decrease as fast as O 1/ √ N up to logarithmic factors, as the number of output data N grows to infinity. While asymptotic results have been established, a finite sample analysis of subspace algorithms remains an open problem [2]. Another open question is whether the asymptotic stability condition ρ(A) < 1 can be relaxed to marginal stability ρ(A) ≤ 1.From a machine learning perspective, finite sample analysis has been a standard tool for comparing algorithms in the non-asymptotic regime. Early work in finite sample analysis of system identification can be found in [13][14][15]. A series of recent papers [16][17][18] studied the finite sample properties of system identification from a single trajectory, when the system state is fully observed (C = I). Finite sample results for partially observed systems (C = I), which is a more challenging problem, appeared recently in [19][20][21]. These papers provide a non-asymptotic convergence rate of 1/ √ N for the recovery of matrices A, B, C, D up to a similarity transformation. The results rely on the assumption that that the system can be driven by external inputs, i.e. B, D = 0. In [19], the analysis of the classical Ho-Kalman realization algorithm was explored. In [20], it was shown that with a prefiltering step, consistency can be achieved even for marginally stable systems where ρ (A) ≤ 1. In [21], identification of a system of unknown order is considered. Finite sample properties of system identification algorithms have also been used to design controllers [22]. The dual problem of Kalman filtering has not been studied yet in this context.In this paper, we perform the first finite sample analysis of system (1) in the case B, D = 0, when we have no inputs, also known as stochastic system identification (SSI) [2]. This problem is more challenging than the case B, D...

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“…i 0 the estimatesN i are consistent, where i 0 ¼ intðdq bic Þ which is the integer closer to the product of d and the optimal lag length for an autoregressive (AR) approximation of z t , chosen by using the Schwarz (1978) criterion over 0 q ( log T) a for some constant 0 < a < 1. Specifically, d > 1 is a sufficient condition in the stationary case (see Deistler et al, 1995), whereas d > 2 is required in the integrated case (see Bauer, 2005b). However, in finite samples the estimatesN i differ for different i, resulting in distinct forecasts.…”

Section: Forecasting By Exploiting Different Values Of Imentioning

confidence: 99%