A parsimonious, computationally efficient machine learning method for spatial regression

Aug 5, 2026

A parsimonious, physically inspired approach called the modified planar rotator method (MPRS) is proposed for spatial and temporal regression. The method is non-parametric and models spatial or temporal dependence through short-range, distance-dependent interactions while avoiding assumptions about the underlying probability distribution. Learning relies on a fully autonomous algorithm using equilibrium conditional Monte Carlo simulations, supports scattered observations and arbitrary spatial dimensions, and shows strong gap-filling performance for rough, non-Gaussian data. Tests on synthetic and real datasets in 1–3D demonstrate competitive accuracy without parameter tuning versus kriging and inverse distance weighting.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages42

Reading Time1 h 46 min

AccessFree to download

Introduction

Spatial prediction and its applications

Existing interpolation methods and limitations

Computationally efficient alternatives

Short-range interaction modeling with random fields

Springer Nature 2021 LATEX template arXiv :2309 . 16448v1 [ stat .ML] 28 Sep 2023 A parsimonious, computationally e􀀎cient machine learning method for spatial regression Milan ukovi􀀔c1* and Dionissios T. Hristopulos2 1* Department of Theoretical Physics and Astrophysics, Institute 􀀔 of Physics, Faculty of Science, Pavol Jozef Saf􀀓arik University, Park Angelinum 9, Ko􀀔sice, 041 54, Slovak Republic. 2 Department of Electrical and Computer Engineering, Technical University of Crete, Akrotiri Campus, Chania, 73100, Crete, Greece. *Corresponding author(s). E-mail(s): [milan.zukovic@upjs.sk](milan.zukovic@upjs.sk) ; Contributing authors: [dchristopoulos@tuc.gr](dchristopoulos@tuc.gr) ; Abstract We introduce the modi􀀌ed planar rotator method (MPRS), a physically inspired machine learning method for spatial/temporal regression. MPRS is a non-parametric model which incorporates spatial or temporal correlations via short-range, distance-dependent “interactions”without assuming a speci􀀌c form for the underlying probability distribution. Predictions are obtained by means of a fully autonomous learning algorithm which employs equilibrium conditional Monte Carlo simulations. MPRS is able to handle scattered data and arbitrary spatial dimensions. We report tests on various synthetic and real-word data in one, two and three dimensions which demonstrate that the MPRS prediction performance (without parameter tuning) is competitive with standard interpolation methods such as ordinary kriging and inverse distance weighting. In particular, MPRS is a particularly e􀀋ective gap-􀀌lling method for rough and non-Gaussian data (e.g., daily precipitation Springer Nature 2021 LATEX template 2 Computationally e􀀎cient spatial regression time series) . MPRS shows superior computational e􀀎ciency and scalability for large samples. Massive data sets involving millions of nodes can be processed in a few seconds on a standard personal computer. Keywords: machine learning, interpolation, time series, scattered data, non-Gaussian model, precipitation, autonomous algorithm 1 Introduction The spatial prediction (interpolation) problem arises in various 􀀌elds of science and engineering that study spatially distributed variables. In the case of scattered data, 􀀌lling gaps facilitates understanding of the spatial features, visualization of the observed process, and it is also necessary to obtain fully populated grids of spatially dependent parameters used in partial di􀀋erential equations. Spatial prediction is highly relevant to many disciplines, such as environmental mapping, risk assessment (Christakos, 2012) and environmental health studies (Christakos and Hristopulos, 2013), subsurface hydrology (Kitanidis, 1997; Rubin, 2003), mining (Goovaerts, 1997), and oil reserves estimation (Hohn, 1988; Hamzehpour and Sahimi, 2006) . In addition, remote sensing images often include gaps with missing data (e.g. , clouds, snow, heavy precipitation, ground vegetation coverage, etc.) that need to be 􀀌lled (Rossi et al, 1994) . Spatial prediction is also useful in image analysis (Winkler, 2003; Gui and Wei, 2004) and signal processing (Unser and Blu, 2005; Ramani and Unser, 2006) including medical applications (Parrott et al, 1993; Cao and Worsley, 2001) . Spatial interpolation methods in the literature include simple deterministic approaches, such as inverse distance weighting (Shepard, 1968) and minimum curvature (Sandwell, 1987), as well as the widely-used family of kriging Springer Nature 2021 LATEX template Computationally e􀀎cient spatial regression 3 estimators (Cressie, 1990) . The latter are stochastic methods, with their popularity being due to favorable statistical properties (optimality, linearity, and unbiasedness under ideal conditions) . Thus, kriging usually outperforms other interpolation methods in prediction accuracy. However, the computational complexity of kriging increases cubically with the sample size and thus beco

FAQs

What is the modified planar rotator method (MPRS) used for?

MPRS is a machine learning method for spatial and temporal regression, particularly for spatial prediction (interpolation) when data are scattered or have gaps.

How does MPRS incorporate spatial or temporal correlation?

MPRS uses short-range, distance-dependent “interactions” to capture correlations without assuming a specific form for the underlying probability distribution.

Why is MPRS computationally efficient compared with kriging?

MPRS is designed to avoid the cubic growth in computational complexity that makes kriging impractical for large samples, and it can scale efficiently to massive datasets.

Aug 5, 2026

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