Browsing by Author "Carroll, Raymond J."

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    Numerical Methods for Modeling Atomistic Trajectories with Diffusion SDEs
    (2008-08) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
    The stochastic dynamics of small scale systems are often not known fromﾠa prioriﾠphysical considerations. We present data-driven numerical methods which can be used to approximate the nonlinear stochastic dynamics associated with time series of system observables. Given a single time series coming from a simulation or experiment, our approach uses maximum likelihood type estimates to obtain a sequence of local stochastic differential equations. The local models coming from one times series are then patched together using a penalized spline procedure. We provide an effcient algorithm for achieving this which utilizes estimates of the local parameter covariance. We also use goodness-of-fit tests to quantitatively determine when an overdamped Langevin approxi- mation can be used to describe the data. For situations where the overdamped approximation fails, we show that other diffusive models can still be used to approximate the dynamics. In addition, we also briefly discuss how variation observed in different curves, calibrated from different time series, can provide information about "hidden" conformational degrees of freedom not explicitly included in the model and how clustering these curves can help one in learning about the effective underlying free energy surface governing the dynamics of the atomistic system. The methods presented are applied to simulations modeling forced time-dependent transport of potassium transport through a gramicidinﾠAﾠchannel, but have applicability to other forced (and unforced) systems.
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    P-Splines Using Derivative Information
    (2009-04) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
    Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biochemical systems. However efficiently extracting and representing useful physical information from these time series measurements can be challenging. We demonstrate how Penalized splines (P-Splines) can be useful in summarizing complex single-molecule time series data using quantities estimated from the observed data. A design matrix that simultaneously uses noisy function and derivative scatterplot information to refine function estimates using P-spline techniques is introduced. The approach is called the PuDI (P-Splines using Derivative Information) method. We show how Generalized Least Squares fits seamlessly into the PuDI method; several applications demonstrating how inclusion of uncertainty information improves the PuDI function estimates are presented. The PuDI design matrix can be used to assist scatterplot smoothing applications where both unbiased function and derivative estimates are available.
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    PSQR: A Stable and Efficient Penalized Spline Algorithm
    (2009-05) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
    We introduce an algorithm for reliably computing quantities associated with several types of semiparametric mixed models in situations where the condition number on the random effects matrix is large. The algorithm is numerically stable and efficient. It was designed to process penalized spline (P-spline) models without making unnecessary numerical approximations. The algorithm, PSQR (P-splines via QR), is formulated in terms of QR decompositions. PSQR can treat both exactly rank deficient and ill-conditioned matrices. The latter situation often arises in large scale mixed models and/or when a P-spline is estimated using a basis with poor numerical properties, e.g. a truncated power function (TPF) basis. We provide concrete examples where unnecessary numerical approximations introduce both subtle and dramatic errors that would likely go undetected, thus demonstrating the importance of using this reliable numerical algorithm. Simulation results studying a univariate function and a longitudinal data set are used to demonstrate the algorithm. Extensions and the utility of the method in more general semiparametric regression applications are briefly discussed. MATLAB scripts demonstrating implementation are provided in the Supplemental Materials.