Estimating the concentration of optically active constituents of sea water by Takagi-Sugeno models with quadratic rule consequents

Determining the concentrations of chlorophyll, suspended particulate matter and coloured dissolved organic matter in the sea water is basic to support the monitoring of upwelling phenomena, algae blooms, and changes in the marine ecosystem. Since these concentrations affect the spectral distribution of the solar light back-scattered by the water body, their estimation can be computed by using a set of remotely sensed multispectral measurements of the reflected sunlight. In this paper, the relation between the concentrations of interest and the average subsurface reflectances is modelled by means of a set of second-order Takagi-Sugeno (TS) fuzzy rules. Unlike first-order TS rules, which adopt linear functions as consequent, second-order TS rules exploit quadratic functions, thus improving the modelling capability of the rule in the subspace determined by the antecedent. First, we show how we can build a second-order TS model through a simple transformation, which allows estimating the consequent parameters using standard linear least-squares algorithms, and by adopting one of the most used methods proposed in the literature to generate first-order TS models. Then, we compare first-order and second-order TS models against mean square error and interpretability of rules. We highlight how second-order TS models allow us to achieve better approximation than first-order TS models though maintaining interpretability of the rules. Finally, we show how second-order TS models perform considerably better (the mean square error is lower by two orders of magnitude) than the specific implementations of radial basis function networks and multi-layer perceptron networks used in previous papers for the same application domain.