Parameter Estimation and Variable Selection for Big Systems of Linear Ordinary Differential Equations: A Matrix-Based Approach

ABSTRACT

Ordinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a “Big System” with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector-based two-stage method that are currently available. We applied this new method to two real datasets—a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters—to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice. Supplementary materials for this article are available online.

Keywords:

• Complex system
• Eigenvalue updating algorithm
• High dimension
• Matrix-based variable selection
• Ordinary differential equation
• Separable least squares

Supplementary Materials

Supplementary Text. This file includes the following supporting material: (a) algorithms for ST-SLS and ST-SLS-VS, (b) definitions of two norms used in this study, (c) technical details of the optimization procedures used in this study, (d) generalizations of the proposed methods for heterogeneous linear ODE systems, (e) the closed-form gradient of the solutions curves with respect to eigenvalues, and (f) detailed proofs of theorems.

Supplementary Table S1. The gene regulatory network reconstructed from the time-course yeast microarray data.

Supplementary Table S2. The overall stock interaction network reconstructed from the Standard & Poor stock market data.

Supplementary File S3. Per-sector stock interaction sub-networks reconstructed from the Standard & Poor stock market data.

Acknowledgment

LW and XQ contributed equally to this article. The authors thank Dr. Hongqi Xue for his thoughtful discussions and helpful comments on the earlier version of this article, and Dr. Michelle Carey for her suggestion of the stock market data.

Additional information

Funding

This work is supported in part by NIH RO1 AI087135, Respiratory Pathogens Research Center (NIAID contract number HHSN272201200005C), the University of Rochester CTSA award number UL1 TR002001 from the National Center for Advancing Translational Sciences of the National Institutes of Health, the University of Rochester Center for AIDS Research (NIH 5 P30 AI078498-08), and National Natural Science Foundation of China Grants 11526096 and 11601185.