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P-splines using derivative information. (English) Zbl 1204.62154

Summary: Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biomolecular systems. We demonstrate how a collection of penalized-splines (p-splines) can be useful in quantitatively summarizing such data. In this work, functions estimated using p-splines are associated with stochastic differential equations (SDEs). It is shown how quantities estimated in a single SDE summarize fast-scale phenomena, whereas variation between curves associated with different SDEs partially reflects noise induced by motion evolving on a slower time scale. P-splines assist in “semiparametrically” estimating nonlinear SDEs in situations where a time-dependent external force is applied to a single-molecule system. The p-splines introduced simultaneously use function and derivative scatterplot information to refine curve estimates. We refer to the approach as the PuDI (P-splines using Derivative Information) method. It is shown how generalized least squares ideas fit seamlessly into the PuDI method. Applications demonstrating how utilizing uncertainty information/approximations along with generalized least squares techniques improve PuDI fits are presented. Although the primary application here is in estimating nonlinear SDEs, the PuDI method is applicable to situations where both unbiased function and derivative estimates are available.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P35 Applications of statistics to physics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62M99 Inference from stochastic processes
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
92C05 Biophysics
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