Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: Theory and computing

[EN] We study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preservedThis work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo Regional (FEDER UE) Grant PID2020-115270GB-I00. The authors express their deepest thanks and respect to the reviewers for their valuable commentsCortés, J.; López-Navarro, E.; Romero, J.; Roselló, M. (2021). Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: Theory and computing. European Physical Journal Plus. 136(7):1-23. https://doi.org/10.1140/epjp/s13360-021-01672-wS1231367W.L. Oberkampf, S.M. De Land, B.M. Rutherford, K.V. Diegert, K.F. Alvin, Error and uncertainty in modeling and simulation. Reliab. Eng. Syst. Saf. 75, 333–357 (2002)T. Soong, Random Differential Equations in Science and Engineering, vol. 103 (Academic Press, New York, 1973)Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations, Ser. Stochastic Modelling and Applied Probability, vol. 23. Springer, Berlin Heidelberg (1992)J.L. Bogdanoff, J.E. Goldberg, M. Bernard, Response of a simple structure to a random earthquake-type disturbance. Bull. Seismol. Soc. Am. 51, 293–310 (1961)L. Su, G. Ahmadi, Earthquake response of linear continuous structures by the method of evolutionary spectra. Eng. Struct. 10, 47–56 (1988)X. Jin, Y. Tian, Y. Wang, Z. Huang, Explicit expression of stationary response probability density for nonlinear stochastic systems. Acta Mech. 232, 2101–2114 (2021)D. Lobo, T. Ritto, D. Castello, E. Cataldo, Dynamics of a Duffing oscillator with the stiffness modeled as a stochastic process. Int. J. Non-Linear Mech. 116, 273–280 (2019)Y. Lin, G. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications (McGraw-Hill, Cambridge, 1995)C. To, Nonlinear Random Vibration: Analytical Techniques and Applications (Swets & Zeitlinger, New York, 2000)M. Kaminski, The Stochastic Perturbation Method for Computational Mechanics (Wiley, New York, 2013)J.J. Stoker, Nonlinear Vibrations (Wiley (Interscience), New York, 1950)N. McLachlan, Laplace Transforms and Their Applications to Differential Equations, vol. 103 (Dover Publ. INc., New York, 2014)R.F. Steidel, An Introduction to Mechanical Vibrations (Wiley, New York, 1989)G. Casella, R. Berger, Statistical Inference (Cengage Learning, New Delhi, 2007)H.V. Storch, F.W. Zwiers, Statistical Analysis in Climate Research (Cambridge University Press, Cambridge, 2001)J.V. Michalowicz, J.M. Nichols, F. Bucholtz, Handbook of Differential Entropy (CRC Press, Boca Raton, 2018)H. Banks, H. Shuhua, W. Clayton Thompson, Modelling and Inverse Problems in the Presence of Uncertainty (Ser. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, 2001)Garg, V.K., Wang, Y.-C.: 1 - signal types, properties, and processes. In: Chen, W.-K. (ed.) The Electrical Engineering Handboo

Assessment of stochastic and deterministic models of 6304 quasar lightcurves from SDSS Stripe 82

The optical light curves of many quasars show variations of tenths of a magnitude or more on time scales of months to years. This variation often cannot be described well by a simple deterministic model. We perform a Bayesian comparison of over 20 deterministic and stochastic models on 6304 QSO light curves in SDSS Stripe 82. We include the damped random walk (or Ornstein-Uhlenbeck [OU] process), a particular type of stochastic model which recent studies have focused on. Further models we consider are single and double sinusoids, multiple OU processes, higher order continuous autoregressive processes, and composite models. We find that only 29 out of 6304 QSO lightcurves are described significantly better by a deterministic model than a stochastic one. The OU process is an adequate description of the vast majority of cases (6023). Indeed, the OU process is the best single model for 3462 light curves, with the composite OU process/sinusoid model being the best in 1706 cases. The latter model is the dominant one for brighter/bluer QSOs. Furthermore, a non-negligible fraction of QSO lightcurves show evidence that not only the mean is stochastic but the variance is stochastic, too. Our results confirm earlier work that QSO light curves can be described with a stochastic model, but place this on a firmer footing, and further show that the OU process is preferred over several other stochastic and deterministic models. Of course, there may well exist yet better (deterministic or stochastic) models which have not been considered here.Comment: accepted by AA, 12 pages, 11 figures, 4 table

The effective bandwidth problem revisited

The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are governed by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies a place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of a system with finite buffers, the values $N^{(k)}$ characterize buffer capacities. The paper discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents in particular buffers models. Our approach to this problem is new, and the presentation of our results is simple and clear for real applications.Comment: 29 pages, 11pt, Final version, that will be published as is in Stochastic Model

Gaussian process hyper-parameter estimation using parallel asymptotically independent Markov sampling

Gaussian process emulators of computationally expensive computer codes provide fast statistical approximations to model physical processes. The training of these surrogates depends on the set of design points chosen to run the simulator. Due to computational cost, such training set is bound to be limited and quantifying the resulting uncertainty in the hyper-parameters of the emulator by uni-modal distributions is likely to induce bias. In order to quantify this uncertainty, this paper proposes a computationally efficient sampler based on an extension of Asymptotically Independent Markov Sampling, a recently developed algorithm for Bayesian inference. Structural uncertainty of the emulator is obtained as a by-product of the Bayesian treatment of the hyper-parameters. Additionally, the user can choose to perform stochastic optimisation to sample from a neighbourhood of the Maximum a Posteriori estimate, even in the presence of multimodality. Model uncertainty is also acknowledged through numerical stabilisation measures by including a nugget term in the formulation of the probability model. The efficiency of the proposed sampler is illustrated in examples where multi-modal distributions are encountered. For the purpose of reproducibility, further development, and use in other applications the code used to generate the examples is freely available for download at https://github.com/agarbuno/paims_codesComment: Computational Statistics \& Data Analysis, Volume 103, November 201

Probabilistic Constraint Logic Programming

This paper addresses two central problems for probabilistic processing models: parameter estimation from incomplete data and efficient retrieval of most probable analyses. These questions have been answered satisfactorily only for probabilistic regular and context-free models. We address these problems for a more expressive probabilistic constraint logic programming model. We present a log-linear probability model for probabilistic constraint logic programming. On top of this model we define an algorithm to estimate the parameters and to select the properties of log-linear models from incomplete data. This algorithm is an extension of the improved iterative scaling algorithm of Della-Pietra, Della-Pietra, and Lafferty (1995). Our algorithm applies to log-linear models in general and is accompanied with suitable approximation methods when applied to large data spaces. Furthermore, we present an approach for searching for most probable analyses of the probabilistic constraint logic programming model. This method can be applied to the ambiguity resolution problem in natural language processing applications.Comment: 35 pages, uses sfbart.cl