Improved algorithm for the optimal quantization of single- and multivariate random functions

The Functional Quantization (FQ) method was developed for the approximation of random processes with optimally constructed finite sets of deterministic functions (quanta) and associated probability masses. The quanta and the corresponding probability masses are collectively called a “quantizer”. A method called “Functional Quantization by Infinite-Dimensional Centroidal Voronoi Tessellation” (FQ-IDCVT) was developed for the numerical implementation of FQ. Compared to other methodologies, the FQ-IDCVT algorithm has the advantages of versatility, simplicity, and ease of application. The FQ-IDCVT algorithm was proved to be able to capture the overall stochastic characteristics of random processes. However, for certain time instants of the processes, the FQ-IDCVT algorithm may fail in approximating well the stochastic properties, which results in significant local errors. For instance, it may significantly underestimate the variances and overestimate the correlation coefficients for some time instants. In addition, the FQ concept was developed for the quantization of single-variate processes. A formal description of its application to multivariate processes is lacking in the literature. Therefore, this paper has two objectives: to extend the FQ concept to multivariate processes and to enhance the FQ-IDCVT algorithm so that its accuracy is more uniform across the various time instants for single-variate processes and applicable to multivariate processes. Along this line, the limitations of the FQ-IDCVT algorithm are explained, and a novel improved FQ-IDCVT algorithm is proposed. Four criteria are introduced to quantify the algorithms' global and local quantization performance. For illustration and validation purposes, four numerical examples of single-variate and multivariate, stationary and non-stationary processes with Gaussian and non-Gaussian marginal distributions were quantized. The results of the numerical examples demonstrate the performance and superiority of the proposed algorithm compared to the original FQ-IDCVT algorithm.