On the Bilateral Prediction Error Matrix of a Multivariate Stationary Stochastic Process

1979 ◽  
Vol 10 (2) ◽  
pp. 247-252 ◽  
Author(s):  
A. G. Miamee ◽  
H. Salehi
Keyword(s):  
Stochastic Process ◽  
Prediction Error ◽  
Stationary Stochastic Process ◽  
Error Matrix

Spectral characterization of the Wold–Zasuhin decomposition and prediction-error operator

1991 ◽  
Vol 110 (3) ◽  
pp. 559-567 ◽  
Author(s):  
S. C. Power
Keyword(s):  
Stochastic Process ◽  
Prediction Error ◽  
General Relation ◽  
Spectral Characterization ◽  
Structure Theory ◽  
Iterative Approach ◽  
Error Matrix ◽  
Series Representations ◽  
Series Expression

Over thirty years ago Wiener and Masani pointed out in the introduction of their celebrated paper [31] that for a general multivariate stationary stochastic process no relation had been given for the prediction-error matrix in terms of the spectrum of the process. In particular it was unknown how to characterize the rank of the process in spectral terms (cf. Masani[12], p. 369, question 1). Despite explicit progress in this connection with certain regular processes, such as the series representations in [11, 19, 22, 32], or the iterative approach of [28, 29], and despite progress in the structure theory of degenerate processes ([8, 10, 14, 15, 26]), a general relation or series expression has remained elusive.

Stochastic Kriging for Crashworthiness Optimization Accounting for Simulation Noise

2020 ◽  
Author(s):  
Seyed Saeed Ahmadisoleymani ◽  
Samy Missoum
Keyword(s):  
Stochastic Process ◽  
Optimization Algorithm ◽  
Stationary Stochastic Process ◽  
Expected Improvement ◽  
Restraint System ◽  
Stochastic Kriging ◽  
Occupant Restraint System ◽  
Crashworthiness Optimization ◽  
Surrogate Based Optimization ◽  
Available Information

Abstract Vehicle crash simulations are notoriously costly and noisy. When performing crashworthiness optimization, it is therefore important to include available information to quantify the noise in the optimization. For this purpose, a stochastic kriging can be used to account for the uncertainty due to the simulation noise. It is done through the addition of a non-stationary stochastic process to the deterministic kriging formulation. This stochastic kriging, which can also be used to include the effect of random non-controllable parameters, can then be used for surrogate-based optimization. In this work, a stochastic kriging-based optimization algorithm is proposed with an infill criterion referred to as the Augmented Expected Improvement, which, unlike its deterministic counterpart the Expect Improvement, accounts for the presence of irreducible aleatory variance due to noise. One of the key novelty of the proposed algorithm stems from the approximation of the aleatory variance and its update during the optimization. The proposed approach is applied to the optimization of two problems including an analytical function and a crashwor-thiness problem where the components of an occupant restraint system of a vehicle are optimized.

Some comments on spectral representations of non-stationary stochastic processes

1973 ◽  
Vol 10 (04) ◽  
pp. 881-885 ◽  
Author(s):  
H. Tong
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Stationary Stochastic Process ◽  
Sufficient Condition ◽  
Stationary Stochastic Processes ◽  
Spectral Representations ◽  
Oscillatory Processes

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.

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