Co-Clustering with Outlier Detection for Functional Data

• Joint Research Centre, European Commission, Ispra, Italy

• KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium

Keywords: Co-clustering – Functional data – Multivariate – Trimming

Introduction

Co-clustering is a powerful technique for simultaneously grouping the rows and columns of a data matrix to uncover hidden patterns and structures. Unlike traditional clustering methods that treat rows and columns independently, co-clustering capitalizes on the interactions between these dimensions, offering significant advantages in applications such as microarray data analysis, text mining, and recommendation systems.
We explore the Latent Block Model (LBM), a versatile approach to co-clustering that models data through an underlying unobserved block structure, effectively managing various data types including continuous, binary, count and functional data (Slimen et al. [2018], Bouveyron et al. [2022]). The inherent adaptability of LBM makes it a valuable tool for data analysis and exploration. However, real-world data often deviate from theoretical assumptions, containing anomalies or outliers that can bias model estimation (García-Escudero et al. [2008]). These anomalies, while problematic, may also carry crucial information (Rousseeuw et al. [2019], Torti et al. [2021], Portela and Olsen [2023]).

Contribution

To address the challenges posed by outliers, we propose a cell-wise trimmed version of the Latent Block Model for multivariate normal data (Fibbi et al. [2024]). Our approach incorporates a trimming step in each iteration to discard anomalous data points, enhancing the robustness of parameter estimation. This process involves the introduction of a mask to identify outlier cells, which are determined using the Mahalanobis distance relative to the estimated distribution of their respective clusters.
The proposed method employs a modified Stochastic EM (SEM) algorithm, specifically the SEM-Gibbs algorithm, which is resilient to local minima and effectively handles missing data. In the M-step, the parameter estimation is done robustly.
Further, we tackle model selection challenges inherent in co-clustering, proposing a heuristic approach that combines a trimmed version of the ICL criterion for determining the number of clusters and the G-statistic for optimizing the trimming level (Lomet [2012]). This dual-step process ensures robust cluster retrieval and outlier management.

Study case

Our simulation studies, based on synthetic data, confirm the method’s accuracy in recovering partitions and parameters, as well as its efficacy in outlier detection. We generated the synthetic data by adapting the MixSim framework [Melnykov et al., 2012, 2024] to the co-clustering needs, with some adaptations of functions MixSim.m and simdataset.m in the FSDA toolbox for MATLAB (Riani et al. [2012, 2015]). A statistically principled extension of MixSim to co-clustering is however an open research problem.
The final contribution of this work is the application of the new framework to a set of functional data related to the energy market (Bernardi and Sangalli [2022]). The data are processed using functional principal component analysis to reduce them to three parameters that capture the main behaviours of the curve. Our aim is to determine if this analysis can reveal anomalous variations in energy market pricing. The original datasets are publicly available in the Italian energy service provider (GSE, “Gestore Servizi Energetici”) website.