arXiv subjects

Martin Lysy

Publications and source records attributed to Martin Lysy.

22 records · Page 2Linked to original sources

arXiv2015

Model comparison and assessment for single particle tracking in biological fluids

State-of-the-art techniques in passive particle-tracking microscopy provide high-resolution path trajectories of diverse foreign particles in biological fluids. For particles on the order of 1 micron diameter, these paths are generally inconsistent with simple Brownian motion. Yet, despite an abundance of data confirming these findings and their wide-ranging scientific implications, stochastic modeling of the complex particle motion has received comparatively little attention. Even among posited models, there is virtually no literature on likelihood-based inference, model comparisons, and other quantitative assessments. In this article, we develop a rigorous and computationally efficient Bayesian methodology to address this gap. We analyze two of the most prevalent candidate models for 30 second paths of 1 micron diameter tracer particles in human lung mucus: fractional Brownian motion (fBM) and a Generalized Langevin Equation (GLE) consistent with viscoelastic theory. Our model comparisons distinctly favor GLE over fBM, with the former describing the data remarkably well up to the timescales for which we have reliable information.

stat.AP↗

arXiv2014

Functional derivatives applied to error propagation of uncertainties in topography to large-aperture scintillometer-derived heat fluxes

Scintillometer measurements allow for estimations of the refractive index structure parameter $C_n^2$ over large areas in the atmospheric surface layer. Turbulent fluxes of heat and momentum are inferred through coupled sets of equations derived from the Monin--Obukhov similarity hypothesis. One-dimensional sensitivity functions have been produced that relate the sensitivity of heat fluxes to uncertainties in single values of beam height over flat terrain. However, real field sites include variable topography. We develop here, using functional derivatives, the first analysis of the sensitivity of scintillometer-derived sensible heat fluxes to uncertainties in spatially distributed topographic measurements. Sensitivity is shown to be concentrated in areas near the center of the beam path and where the underlying topography is closest to the beam height. Relative uncertainty contributions to the sensible heat flux from uncertainties in topography can reach 20\,\% of the heat flux in some cases. Uncertainty may be greatly reduced by focusing accurate topographic measurements in these specific areas. A~new two-dimensional variable terrain sensitivity function is developed for quantitative error analysis. This function is compared with the previous one-dimensional sensitivity function for the same measurement strategy over flat terrain. Additionally, a~new method of solution to the set of coupled equations is produced that eliminates computational error.

physics.ao-ph↗

arXiv2013

Statistical Inference for Stochastic Differential Equations with Memory

In this paper we construct a framework for doing statistical inference for discretely observed stochastic differential equations (SDEs) where the driving noise has 'memory'. Classical SDE models for inference assume the driving noise to be Brownian motion, or "white noise", thus implying a Markov assumption. We focus on the case when the driving noise is a fractional Brownian motion, which is a common continuous-time modeling device for capturing long-range memory. Since the likelihood is intractable, we proceed via data augmentation, adapting a familiar discretization and missing data approach developed for the white noise case. In addition to the other SDE parameters, we take the Hurst index to be unknown and estimate it from the data. Posterior sampling is performed via a Hybrid Monte Carlo algorithm on both the parameters and the missing data simultaneously so as to improve mixing. We point out that, due to the long-range correlations of the driving noise, careful discretization of the underlying SDE is necessary for valid inference. Our approach can be adapted to other types of rough-path driving processes such as Gaussian "colored" noise. The methodology is used to estimate the evolution of the memory parameter in US short-term interest rates.

stat.ME↗

arXiv2012

Shrinkage Estimation in Multilevel Normal Models

This review traces the evolution of theory that started when Charles Stein in 1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197--206, Univ. California Press] showed that using each separate sample mean from $k\ge3$ Normal populations to estimate its own population mean $μ_i$ can be improved upon uniformly for every possible $μ=(μ_1,...,μ_k)'$. The dominating estimators, referred to here as being "Model-I minimax," can be found by shrinking the sample means toward any constant vector. Admissible minimax shrinkage estimators were derived by Stein and others as posterior means based on a random effects model, "Model-II" here, wherein the $μ_i$ values have their own distributions. Section 2 centers on Figure 2, which organizes a wide class of priors on the unknown Level-II hyperparameters that have been proved to yield admissible Model-I minimax shrinkage estimators in the "equal variance case." Putting a flat prior on the Level-II variance is unique in this class for its scale-invariance and for its conjugacy, and it induces Stein's harmonic prior (SHP) on $μ_i$.

stat.ME↗