Proper ARMA Modeling and Forecasting in the Generalized Segre’s Quaternions Domain

Abstract

The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts of properness are suggested and statistical tests to check if a GSQ random vector is proper or not are proposed. Further, a method to determine in which specific hypercomplex algebra is most likely to achieve, if possible, the properness properties is given. Next, both the linear estimation and prediction problems are studied in the GSQ domain. Finally, ARMA modeling and forecasting for proper GSQ time series are tackled. Experimental results show the superiority of the proposed approach over its counterpart in the Hamilton quaternion domain.

Keywords

• ARMA models
• <i>E</i>_<i>α</i>₀-properness
• generalized Segre’s quaternions
• <i>H</i>_<i>α</i>₀-properness
• prediction
• proper signal processing