Statistical inference for perturbed multiscale dynamical systems

Gailus, Siragan; Spiliopoulos, Konstantinos

doi:10.1016/j.spa.2016.06.013

Cited by 17 publications

(20 citation statements)

References 25 publications

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“…Meanwhile, it is shown also in [24] that direct application of the principle of maximum likelihood with discretely-sampled data via Euler-Maruyama approximation produces consistent estimates only if the data is first appropriately subsampled. Most closely related to the present work are [13,31], wherein the authors prove consistency and asymptotic normality of the continuous-data MLE for special cases of (1).…”

Section: Introductionmentioning

confidence: 76%

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Discrete-Time Statistical Inference for Multiscale Diffusions

Gailus¹,

Spiliopoulos²

2018

Multiscale Model. Simul.

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We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE), the limiting variance having been identified explicitly; we furthermore establish consistency and asymptotic normality of a simplified minimum constrast estimator (SMCE), which is however not in general efficient. These results are then extended to the case of high-frequency observations under a condition restricting the rate at which the number of observations may grow vis-à-vis the separation of scales. Numerical simulations illustrate the theoretical results.

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Section: Introductionmentioning

confidence: 76%

“…Given Theorem 1 and Lemma 6, the proof of Lemma 7 is similar to that of Lemma 10 in [13]; we omit the details.…”

Section: Discussionmentioning

confidence: 99%

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Discrete-Time Statistical Inference for Multiscale Diffusions

Gailus¹,

Spiliopoulos²

2018

Multiscale Model. Simul.

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“…where ν 0,Xt t (dy) is a probability measure supported on finitely many global minima of U (X t , ·) (see Theorem 1.3). If an additional assumption on the behaviour of U (x, ·) at its global minima is made then we show that ν 0,Xt t (dy) is time independent and given by a determinantal formula arising from In related works, Spiliopoulos in [Spi13,Spi14], Morse and Spiliopolous in [MS17], and Gailus and Spiliopoulous in [GS17] considered a class of coupled diffusions with multiple time scales in the full dependence setting. Contained therein, after suitable relabelling of the parameters and appropriate choice of coefficients, are results that will apply to (1)-(2) for specific b, ∇ y U and with s(ε) = ε α− 1 2 .…”

Section: Introductionmentioning

confidence: 88%

Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

et al. 2019

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We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described bywhere Bt, Wt are independent Brownian motions on R d and R m respectively, b :. We impose regularity assumptions on b, U and let 0 < α < 1. When s(ε) goes to zero slower than a prescribed rate as ε → 0, we characterize all weak limit points of X ε , as ε → 0, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of U (x, ·) at its global minima we characterize all limit points as Filippov solutions to the differential equation.Laplace's principle. Consequently, for this class of U , we show that every limit point is a generalized Filippov solution to (3) driven by a vector field (see Theorem 1.6). In Section 1.1 we state the model, assumptions made, and the two results precisely and in Section 1.2 we discuss examples of U that satisfy the required assumptions.The factor 1 ε in the drift term in (2), intuitively suggests that the Y ε process is the "fast moving" process as ε → 0 and that the "slow moving" process X ε will see an averaging of Y in this limit. The study of averaging principle in various dynamical systems dates back to the work of Khasminskii and others, summarized in, e.g., Freidlin and Wentzell [FW12], Kabanov and Pergamenshchikov [KP03]. The dynamical systems considered there involve a "slow process" X ε as a solution to an ordinary differential equation (i.e. (1) with no B t term) coupled with the fast process Y ε given by a stochastic differential equation with absence of small noise (i.e. (2) with s(ε) = 1). In this setting, under further assumptions on b, U , the averaging principle leading to characterization of limit points, normal deviations, and large deviations from the averaging principle are detailed in [FW12, Chapter 7]. The ground work for this lies in understanding the long-term behavior of solutions to (2) (for fixed ε > 0), and is laid out in [FW12, Chapters 4-6]. We shall rely on this foundation in prescribing assumptions for U in our main results.Large deviations and generalizations to "full dependence" systems were considered in the works of Veretennikov in [Ver13,Ver94,Ver99]. Motivated by questions from homogenization, [Ver00] considered the fast process (2) with s(ε) = 1 but with presence of small noise for the slow process (i.e. (1) with α = 1 2 ) and established a large deviation principle (LDP) for X ε as ε → 0. One can characterize the limit points of X ε as ε → 0 as the set where the rate function is equal to zero (see [Ver00, Remark 3]). In [Lip96], Liptser considered the joint distribution of the slow process and of the empirical process associated with the fast variable in the one-dimensional setting and derived an LDP. This was recently generalized to multidimensional and full dependence systems by Puhalskii in [Puh16]. The diffusions driving the slow and the fast processes in [Puh16] do not have to be uncorrelated.

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“…For simple models in molecular dynamics, the effect of model misspecification was studied in a series of papers [7,8,16,17,26,28,29] under the assumption of scale sep-aration. In particular, for Brownian particles moving in two-scale potentials it was shown that, when fitting data from the full dynamics to the homogenized equation, the maximum likelihood estimator (MLE) is asymptotically biased [29,Theorem 3.4].…”

Section: Literature Reviewmentioning

confidence: 99%

Drift Estimation of Multiscale Diffusions Based on Filtered Data

et al. 2021

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We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

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Statistical inference for perturbed multiscale dynamical systems

Cited by 17 publications

References 25 publications

Discrete-Time Statistical Inference for Multiscale Diffusions

Discrete-Time Statistical Inference for Multiscale Diffusions

Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

Drift Estimation of Multiscale Diffusions Based on Filtered Data

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