A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period

Abstract

We introduce a class of sparse matrices U _m (A _{p 1} ) of order m by m, where m is a composite natural number, p ₁ is a divisor of m, and A _{p 1} is a set of nonzero real numbers of length p ₁. The construction of U _m (A _{p 1} ) is achieved by iteration, involving repetitive dilation operations and block-matrix operations. We prove that the matrices U _m (A _{p 1} ) are invertible and we compute the inverse matrix (U _m (A _{p 1} ))⁻¹ explicitly. We prove that each row of the inverse matrix (U _m (A _{p 1} ))⁻¹ has only two nonzero entries with alternative signs, located at specific positions, related to the divisors of m. We use the structural properties of the matrix (U _m (A _{p 1} ))⁻¹ in order to build a nonlinear estimator for prediction of nearly periodic time series of length m with fixed period.

Keywords:

• Prediction
• Sparse matrices
• Time series

AMS Subject Classification:

• 65F50
• 65F30
• 15A09
• 60G25