[Paper Review] Accelerated least squares estimation for systems of ordinary differential equations

Itai Dattner, Shota Gugushvili|arXiv (Cornell University)|Mar 27, 2015

Control Systems and Identification24 references4 citations

TL;DR

This paper proposes the Accelerated Least Squares (ACCEL) method for efficient parameter estimation in systems of ordinary differential equations with noisy observations. Starting from a $√n$-consistent initial estimator, it applies a single Newton-Raphson step to achieve asymptotically equivalent performance to full least squares with minimal computational cost, enabling fast point and interval estimates.

ABSTRACT

We study the problem of parameter estimation for a system of ordinary differential equations based on noisy observations on the solution of the system. A classical estimation approach to this problem is the least squares method. Owing to a highly nonlinear character of the least squares criterion function and the need to employ repetitive numerical integration of the system, the latter method becomes computationally intense for most realistic systems. We propose the accelerated (ACCEL) least squares method, which starts from a preliminary \sqrt{n}-consistent estimator of the parameter of interest and next through a Newton-Raphson type step turns it into an asymptotically equivalent estimator to the least squares estimator. Additional computational burden of this step is negligible. We demonstrate excellent practical performance of the ACCEL least squares approach via simulations and real data examples. The method enables the researcher to obtain both point and interval estimates.

Motivation & Objective

• Address the high computational cost of traditional least squares estimation in systems of ODEs with noisy data.
• Overcome the challenge of nonlinear, non-convex optimization in ODE parameter estimation due to repeated numerical integration.
• Develop a computationally efficient alternative to full least squares that maintains asymptotic efficiency.
• Enable practical point and interval estimation for realistic ODE systems with limited computational resources.

Proposed method

• Begin with a preliminary $√n$-consistent estimator of the ODE parameters obtained via a computationally light method.
• Apply a single Newton-Raphson update step to the least squares criterion function using the initial estimator as a starting point.
• Leverage the local quadratic approximation of the least squares criterion around the initial estimator to minimize additional integration cost.
• Ensure the final estimator is asymptotically equivalent to the full least squares estimator with negligible extra computation.
• Use numerical integration (e.g., Runge-Kutta) only once per data point in the initial step, and avoid repeated integration in the refinement phase.
• Derive the Newton-Raphson update using the gradient and Hessian of the least squares objective, exploiting smoothness of the ODE solution map.

Experimental results

Research questions

• RQ1 Can a computationally efficient alternative to full least squares estimation be developed for ODE systems with noisy observations?
• RQ2 Does a single Newton-Raphson step from a $√n$-consistent initial estimator yield asymptotically equivalent results to full least squares?
• RQ3 How does the ACCEL method compare in computational efficiency and estimation accuracy to standard least squares and other estimators?
• RQ4 Can the ACCEL method reliably produce both point and interval estimates in practical ODE parameter estimation problems?

Key findings

• The ACCEL method achieves asymptotic equivalence to the full least squares estimator with only a single Newton-Raphson update step.
• The method reduces computational burden significantly by avoiding repeated numerical integration required in standard least squares.
• Simulations and real data examples demonstrate strong practical performance, with estimation accuracy comparable to full least squares.
• The approach enables reliable point and interval estimation, making it suitable for real-world applications with limited computational resources.
• The computational cost of the refinement step is negligible compared to the initial estimation phase.
• The method maintains statistical efficiency while drastically improving runtime, especially for complex or high-dimensional ODE systems.