Convex aggregative modelling of infinite memory nonlinear systems

Wachel, Paweł

doi:10.1080/00207179.2016.1143977

Cited by 12 publications

(20 citation statements)

References 16 publications

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“…for some constants C,  > 0 and for all = 1, 2, … , where X [ ] n−i = X n−i · 1 (i < ) are system inputs reset after th sample; cf definition 2.1 in the work of Wachel. 6 Using this observation, the bound in (18) can be derived from the proof of a more general result in theorem 3.1 in the aforementioned work. 6 Remark 2.…”

Section: Theorem 1 Let Assumptions 1 To 4 Hold and Additionally Lementioning

confidence: 96%

“…Proof Due to Assumptions 2 and 3 (that the nonlinearity is bounded and the linear system is stable), the system possesses the exponential fading memory property , that is, it satisfies the inequality

(||, true \sum_{i = 0}^{\infty} λ_{i} m false(X_{n - i} false) - true \sum_{i = 0}^{\infty} λ_{i} m ((), X_{n - i}^{false[τ false]})) \leq 2 \sup_{x \in false[- 1, 1 false]} false| m false(x false) false| (||, true \sum_{i = τ}^{\infty} λ_{i}) \leq C e^{- scriptC τ},

for some constants

C, scriptC > 0

and for all τ =1,2,…, where

X_{n - i}^{false[τ false]} = X_{n - i} \cdot bold1 ((), i < τ)

are system inputs reset after τ th sample; cf definition 2.1 in the work of Wachel . Using this observation, the bound in can be derived from the proof of a more general result in theorem 3.1 in the aforementioned work …”

Section: Algorithmsmentioning

confidence: 99%

“…The aggregative algorithms were proposed inter alia by Juditsky and Nemirovski and, recently, adapted to nonlinear system modeling in the works of Wachel and Śliwiński and Wachel. () In principle, they reveal a resemblance to the popular in signal and image processing area algorithms based on l 1 ‐regularization (with the least absolute shrinkage and selection operator method being the most prominent example; see, for example, the works of James et al and Gao et al); however, the sparse representation of the identified object is not required. In addition, without the l 1 ‐constraint, this kind of algorithms reduces to the classical least square method, which has been already studied in the context of estimating a regression function by orthogonal series, in both fixed and random design 11(chapter 18)…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we apply the aggregative algorithm proposed in the work of Wachel and Śliwiński to recover a nonlinearity in Hammerstein systems and compare it with the standard orthogonal one. For brevity of presentation, we only consider the instances based on Fourier and Legendre orthogonal expansions.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinearity recovery by standard and aggregative orthogonal series algorithms

Śliwiński

Wachel

Lagosz

2018

Appl Stoch Models Bus & Ind

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In this paper, the problem of nonlinearity recovery in Hammerstein systems is considered. Two algorithms are presented: the first is a standard orthogonal series algorithm, whereas the other, ie, the aggregative one, exploits the convex programming approach. The finite sample size properties of both approaches are examined, compared, and illustrated in a numerical experiment. The aggregative algorithm performs better when the number of measurements is comparable to the number of parameters; however, it also imposes additional smoothness restrictions on the recovered nonlinearities.

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Section: Theorem 1 Let Assumptions 1 To 4 Hold and Additionally Lementioning

confidence: 96%

(||, true \sum_{i = 0}^{\infty} λ_{i} m false(X_{n - i} false) - true \sum_{i = 0}^{\infty} λ_{i} m ((), X_{n - i}^{false[τ false]})) \leq 2 \sup_{x \in false[- 1, 1 false]} false| m false(x false) false| (||, true \sum_{i = τ}^{\infty} λ_{i}) \leq C e^{- scriptC τ},

for some constants

C, scriptC > 0

and for all τ =1,2,…, where

X_{n - i}^{false[τ false]} = X_{n - i} \cdot bold1 ((), i < τ)

Section: Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinearity recovery by standard and aggregative orthogonal series algorithms

Śliwiński

Wachel

Lagosz

2018

Appl Stoch Models Bus & Ind

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“…Next, since it is interesting to evaluate ("score") and compare the presented algorithms in a formal manner, we introduce the model assessment algorithm based on the aggregative modelling technique, proposed for memoryless models in [19] and extended to finite-memory nonlinear one in [20]. …”

Section: The Proposed Approachmentioning

confidence: 99%

Recognition and Modeling of Atypical Children Behavior

Postawka

Śliwiński

2015

Artificial Intelligence and Soft Computing

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Abstract. According to reports from medical community the number of autistic children's birth is more and more alarming. Early diagnosis and regular rehabilitation are crucial. The problem with verbal and emotional communication is very common. In a form of short survey, a few similar issues and their solutions have been examined in terms of input data type, feature selection, pattern recognition and formal mathematical modeling. Then we propose a system for autistic children rehabilitation, surveillance and emotions translation. These new solutions have been compared with those reported in the literature. The preliminary experiments provide rather satisfactory results.

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Kernel-based identification of Wiener–Hammerstein system

Mzyk

Wachel

2017

Automatica

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Convex aggregative modelling of infinite memory nonlinear systems

Cited by 12 publications

References 16 publications

Nonlinearity recovery by standard and aggregative orthogonal series algorithms

Nonlinearity recovery by standard and aggregative orthogonal series algorithms

Recognition and Modeling of Atypical Children Behavior

Kernel-based identification of Wiener–Hammerstein system

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