A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period
Nikolaos D. Atreas, Persefoni Polychronidou · Numerical Functional Analysis and Optimization · 2008
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.