Weighted Linear Recurrent Forecasting in Singular Spectrum Analysis

Singular Spectrum Analysis (SSA) is an increasingly popular time series filtering and forecasting technique. Owing to its widespread applications in a variety of fields, there is a growing interest towards improving its forecasting capabilities. As such, this paper takes into consideration the Recurrent forecasting approach in SSA (SSA-R) and presents a new mechanism for improving the accuracy of forecasts attainable via this method. The proposed Recurrent SSA-R approach is referred to as Weighted SSA-R (W:SSA-R), and we propose using a weighting algorithm for weigthing the coefficients of the Linear Recurrent Relation (LRR). The performance of forecasts from the W:SSA-R approach are compared with forecasts from the established SSA-R approach. We exploit real data and various simulated time series for the comparison, so as to provide the reader with more conclusive findings. Our results confirm that the W:SSA-R approach can provide comparatively more accurate forecasts and is indeed a viable solution for improving forecasts by SSA.

On the geometry behind a recurrent relation

We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.

The Hyperspherical Functions of a Derivative

We present the results of theoretical research of the generalized hypersherical function (HS) by generalizing two known functions related to the sphere hypersurface and hypervolume and the recurrent relation between them. By introducing two-dimensional degrees of freedomkandn(and the third, radiusr), we develop the derivative functions for all three arguments and the possibilities of their use. The symbolical evolution, numerical experiment, and graphical presentation of functions are realized using the Mathcad Professional and Mathematica softwares.

On the Recurrent Solution of Renewal-Type Integral Equations

In 1987, Sheldon M. Ross [4] gave an interesting recurrent relation method for approximations of important functions in renewal theory. Using his idea and Feller's technique for inverting Laplace transforms, we obtain a general method for recurrent approximations to the solutions of renewal-type integral equations.

On the summation of certain trigonometric series

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the serieswhere α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), whereWhen κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.