A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines
Abstract
An unbiased stochastic estimator of tr(I–A), where A is the influence matrix associated with the calculation of Laplacian smoothing splines, is described. The estimator is similar to one recently developed by Girard but satisfies a minimum variance criterion and does not require the simulation of a standard normal variable. It uses instead simulations of the discrete random variable which takes the values 1, -1 each with probability 1/2. Bounds on the variance of the estimator, similar to those established by Girard, are obtained using elementary methods. The estimator can be used to approximately minimize generalised cross validation (GCV) when using discretized iterative methods for fitting Laplacian smoothing splines to very large data sets. Simulated examples show that the estimated trace values, using either the estimator presented here or the estimator of Girard, perform almost as well as the exact values when applied to the minimization of GCV for n as small as a few hundred, where n is the number of data points.
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The one-dimensional integral in Equation 38 can be computed with numerical ODE solvers and the Skilling-Hutchinson trace estimator.
“One can compute the one-dimensional integral above with numerical ODE solvers and the Skilling-Hutchinson trace estimator [97,217].”
From Diffusion Models: A Comprehensive Survey of Methods and Applications (2024) · cites this paper as [97]
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Diffusion Models: A Comprehensive Survey of Methods and Applications
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., & Yang, M.-H. (2024) · ACM Computing Surveys · doi:10.1145/3626235 · Cited by 1,704 · see citing works
Efficient variance components analysis across millions of genomes
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- Authors
- Hutchinson, M. F.
- Affiliations
- Australian National University (AU)
- Corresponding author
- Hutchinson, M.F.
- ORCID
- Hutchinson, M.F.
- Venue
- Communications in Statistics - Simulation and Computation
- Publisher
- Informa UK Limited
- Published
- 1989-01
- Volume
- 18
- Issue
- 3
- Pages
- 1059-1076
- ISSN
- 0361-0918 (ISSN-L)
- 1532-4141
- OpenAlex
- W4244155866
- Cited by
- 381
- References
- 11
- Citations by year
- 2026: 60 · 2025: 45 · 2024: 38 · 2023: 30 · 2022: 30 · 2021: 27 · 2020: 16 · 2019: 12 · 2018: 8 · 2017: 7 · 2016: 11 · 2015: 10 · 2014: 8 · 2013: 9 · 2012: 15
- FWCI
- 1.67
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Communications in Statistics - Simulation and Computation (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Applied Mathematics › Statistical and numerical algorithms
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- influence matrix, trace estimation, variance bounds, large data sets
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