Predicting Environmental Patterns Through Mathematical Modeling

Abstract

Earlier studies have applied matrix decomposition methods such as Singular Value Decomposition (SVD) and Non-Negative Matrix Factorization (NMF) in climate-agriculture research. These works showed that SVD can reveal interpretable patterns when climate variables display strong, structured signals. NMF’s non-negative constraints make it less suitable for representing signed patterns. However, the prediction quality of these approaches has varied widely across regions, variables, and crops, highlighting the need to test their performance in different contexts.
The main research question in this study is: To what extent do SVD-based and NMF-based representations, in combination with linear prediction models and rank approximations, capture seasonal and interannual variability in temperature, precipitation, and their hydrothermal interactions in the Netherlands? Both SVD and NMF are applied to the following environment variable data matrices: the temperature matrix, the precipitation matrix, their respective seasonally integrated matrices, and a hydrothermal interaction matrix defined as the integrated product of temperature and precipitation. The seasonal and interannual patterns described by the basis vectors obtained by these decompositions are used to gain insight into the structure of each data matrix. In this thesis, the first components is used to predict seasonal and interannual variations in temperatoral, precipitation, and hydrothermal interaction patterns with a linear prediction model and a rank-approximation. The assessment of these predictions is done by means of the Normalized Mean Squared Error.
The result was that SVD can obtain broad seasonal trends, but these are not always logical in the context of climate behavior. NMF, by contrast, generated positive and interpretable components. The first NMF component consistently represented the dominant seasonal variation, while higher-indexed components resembled residual fluctuations. Overall, NMF provided a clearer and more trustworthy representation of precipitation and hydrothermal interaction patterns, whereas SVD often produced structures that were harder to interpret.
For the prediction of interannual variations, the rank-1 approximation obtained through NMF was generally the most effective. For temperature, NMF rank-1 approximation yielded lower peaks and smaller NMSE values, showing robustness even during extreme years such as 2018 and 2020, and similar improvements were observed for the hydrothermal interaction term. Linear prediction was a simple baseline but proved ineffective. For precipitation, however, prediction remained challenging: both SVD and linear prediction produced high NMSE values, and NMF rank-1 performed similarly to linear prediction, with only temporary improvements in certain years. This highlights that while NMF is the more reliable framework overall, its predictive advantage is not uniform across all variables. ii