Particle Swarm Training of a Neural Network for the Lower Upper Bound Estimation of the Prediction Intervals of Time Series

Many time series forecasting applications use ranges rather than point forecasts. Producing forecasts in the form of Prediction Intervals (PIs) is natural, since intervals are an important component of many mathematical models. The LUBE (Lower Upper Bound Estimation) method is aimed at finding ranges based on solving optimization problems taking into account interval width and coverage. Using the Particle Swarm Training of simple neural network, we look for a solution to the optimization problem of the Coverage Width-Based Criterion (CWC), which is the exponential convolution of conflicting criteria PICP (Prediction Interval Coverage Probability) and PINRW (Prediction Interval Normalized Root-mean-square Width). Based on the concept of the Pareto compromise, it is introduced as a Pareto front in the space of specified criteria. The Pareto compromise is constructed as a relationship between conflicting criteria based on the found solution to the optimization problem. The data under consideration are the financial time series of the MOEX closing prices. Our findings reveal that a relatively simple neural network, comprising eight neurons and their corresponding 26 parameters (weights of neuron connections and neuron signal biases), is sufficient to yield reliable PIs for the investigated financial time series. The novelty of our approach lies in the use of a simple network structure (containing fewer than 100 parameters) to construct PIs for a financial time series. Additionally, we offer an experimental construction of the Pareto frontier, formed by the PICP and PINRW criteria.