B\'ezier interpolation improves the inference of dynamical models from data

Bézier interpolation improves the inference of dynamical models from data

Kai Shimagaki and John P. Barton

Phys. Rev. E 107, 024116 – Published 13 February 2023

Abstract

Many dynamical systems, from quantum many-body systems to evolving populations to financial markets, are described by stochastic processes. Parameters characterizing such processes can often be inferred using information integrated over stochastic paths. However, estimating time-integrated quantities from real data with limited time resolution is challenging. Here, we propose a framework for accurately estimating time-integrated quantities using Bézier interpolation. We applied our approach to two dynamical inference problems: Determining fitness parameters for evolving populations and inferring forces driving Ornstein-Uhlenbeck processes. We found that Bézier interpolation reduces the estimation bias for both dynamical inference problems. This improvement was especially noticeable for data sets with limited time resolution. Our method could be broadly applied to improve accuracy for other dynamical inference problems using finitely sampled data.

Received 7 October 2022
Accepted 23 January 2023

DOI:https://doi.org/10.1103/PhysRevE.107.024116

Physics Subject Headings (PhySH)

Evolutionary & population dynamics Genomics Population genetics Stochastic inference

Genomes Viruses

Fokker–Planck equation Path-integral methods

Interdisciplinary PhysicsPhysics of Living Systems

Authors & Affiliations

Kai Shimagaki ¹ and John P. Barton ^1,2,*

¹Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
²Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, Pennsylvania 15213, USA

^*Corresponding author: jpbarton@pitt.edu

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Vol. 107, Iss. 2 — February 2023

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Figure 1
Bézier interpolation generates smooth curves. Cubic Bézier curves smoothly interpolate between discretely sampled frequency trajectories generated from a Wright-Fisher model. Simulation parameters. $L = 50$ sites, population size $N = 10^{3}$ , mutation rate $μ = 10^{- 3}$ , with simulations over $T = 300$ generations. Data points are sampled every 50 generations and interpolated using cubic Bézier and linear interpolation.
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Figure 2
Bézier interpolation reduces bias in estimated selection coefficients. (a) Wright-Fisher simulation with selection and mutation. Each trajectory drawn as a solid line is true complete data, and filled circles are a subset of the complete data, which is observed every $Δ t = 75$ generations and used for selection coefficient prediction. (b) Selection coefficients for the frequency trajectories in (a) were estimated by MPL with Bézier and linear interpolation. MPL with Bézier interpolation greatly reduces estimation bias for inferred selection coefficients when the time interval between sampled observations is large. Simulation parameters. $L = 50$ sites with 10 beneficial, 10 deleterious, and 30 neutral mutations with selection coefficients of $s = 0.03$ , $s = - 0.03$ , and $s = 0$ , respectively. Other parameters of the WF models are the same as in Fig. 1.
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Figure 3
MPL with Bézier interpolation improves prediction precision and reduces estimation bias. (a),(b) When the observation time interval is longer ( $Δ t = 75$ ), the PPV curve for Bézier interpolation is universally higher than the curve for linear interpolation for both deleterious and beneficial cases. (c) The selection coefficient distributions estimated by MPL with linear interpolation visibly shrank toward zero and were biased, while distributions estimated by MPL with Bézeir interpolation did not considerably shrink and have the mean values near the true selection values.
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Figure 4
Bézier method suppresses interpolation error, especially for off-diagonal pairwise covariances. (a) Sampling time interval dependence for interpolation errors $E (Δ t)$ for diagonal covariances and (b) for off-diagonal pairwise covariances. We simulated WF dynamics using the model described in Fig. 2 and generated data sets that evolved to the 300th generation for each trial. For example, when $Δ t = 100$ , results only use data from $t = 0, 100, 200, and 300$ . (c) The autocorrelation of off-diagonal covariance elements decays faster than diagonal ones. To simplify the analysis, we evaluated the autocorrelation function from generation $t = 50$ . The diagonal autocorrelation shows nonmonotonic decay after long times due to mutant frequencies that approach the frequency boundaries (i.e., 0 and 1).
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Figure 5
Bézier interpolation can improve the inference accuracy of parameters in the OU process. (a) Comparison between true and inferred OU parameters using piecewise constant, linear, and Bézier interpolation. Linear regression slope values are included in each panel. Inferred interaction parameters using Bézier interpolation correspond most closely with the true parameters. (b) Dependence of the slope between true and inferred parameters on the time sampling interval $Δ t = 1$ , shown separately for the (b) diagonal and (c) off-diagonal interaction parameters of the $J$ matrix. In both diagonal and off diagonal, the slope values decrease more gradually with increasing $Δ t$ for Bézier interpolation than for linear interpolation.
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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Bézier interpolation improves the inference of dynamical models from data

Kai Shimagaki and John P. Barton

Phys. Rev. E 107, 024116 – Published 13 February 2023

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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Bézier interpolation improves the inference of dynamical models from data

Kai Shimagaki and John P. Barton

Phys. Rev. E 107, 024116 – Published 13 February 2023

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