MACI, 3(2011), 343-346
L.R. Castro, M.C. Maciel, S.M. Castro (Eds.)
A NN-BASED AUTOREGRESSIVE MODEL THAT CONSIDERS THE ENERGY ASSOCIATED OF TIME SERIES FOR FORECASTING C. RODRÍGUEZ RIVERO, J. PUCHETA, J. BAUMGARTNER, M. HERRERA, C. SALAS AND V. SAUCHELLI Departments of Electrical and Electronic Engineering, Mathematics Research Laboratory Applied to Control (LIMAC), Faculty of Exact, Physical and Natural Sciences - National University of Córdoba, Velez Sarsfield Av. 1611 (University Campus) Córdoba, Argentina,
[email protected], www.mathlabappliedcontrol.blogspot.com. Departments of Electrical Engineering, Faculty of Technologies and Applied Sciences, National University of Catamarca, Catamarca, Argentina.
Abstract: In this work a neural network (NN) based autoregressive model for time series forecasting that takes into account the energy associated with the series is presented. The criterion for fitting comprises to yield time series values from forecasted time series area. These values are approximated by the NN to generate a primitive calculated as an area by the predictor filter. The NN output will tend to approximate the current value available from the series which has the same Hurst Parameter as the real time series. The approach is tested over a time series obtained from samples of the Mackey-Glass delay differential equations (MG) and serve to be applied for meteorological variables measurements such as soil moisture series, daily rainfall and monthly cumulative rainfall time series forecasting. Keywords: Neural networks, time series forecast, area, Primitive, Hurst’s parameter, Mackey-Glass.
1. INTRODUCTION Natural phenomena prediction is a challenging topic, useful for control problems from agricultural activities and decision-making that helps the producer to decide. There are several approaches based on NN that face the rainfall forecast problem for energy demand purposes, for water availability and seedling growth by taking an ensemble of measurement points [1]. Here, the proposed approach is based on the classical NAR filter using time-lagged feed-forward neural networks, where the forecast data used from the MG benchmark equation is simulated by a Monte Carlo [2] approach. The number of filter parameters is put function of the roughness of the time series, in such a way that the error between the smoothness of the time series data and the forecasted data modifies the number of the filter parameters.
2. OVERVIEW OF THE MG EQUATION AND FBM Samples of MG equation are used to model natural phenomena and have been implemented by different authors to perform comparisons between different techniques for forecasting and regression models [3]. Here we propose an algorithm to predict values of time series taken from the solution of the MG equation. The MG equation is explained by the time delay differential equation defined as •
y (t ) =
α y (t − τ ) − β y (t ), 1 + y c (t − τ )
(1)
where α, β, and c are parameters and τ is the delay time. According as τ increases, the solution turns from periodic to chaotic. Equation (1) is solved by a standard fourth order Runge-Kutta integration step. By
343
setting the parameter β ranging between 0.1 and 0.9 the stochastic dependence of the deterministic time series obtained varies according to its roughness. The performance of the proposed method is tested with the SMAPE index and it is compared with a traditional NN based predictor. Due to the random process, it is proposed to use the Hurst’s parameter H in the learning process to modify on-line the number of patterns, of iterations and filter inputs. This H gives information about the roughness of a signal, and also to determine its stochastic dependence. The definition of the Hurst's parameter is defined by Mandelbrot through its stochastic representation B H (t ) =
+
⎛ ⎜ 1 ⎞ ⎜⎝ ⎛ Γ ⎜Η + ⎟ 2⎠ ⎝ 1
∫ (t − s ) t
H−
1 2
0
⎛ ∫ ⎜⎝ (t − s ) 0
−∞
H−
1 2
− (− s )
H−
1 2
⎞ ⎟dB (s ) ⎠
(2)
⎞ dB (s )⎟ ⎠
where, Γ(·) represents the Gamma function
(3)
∞
Γ (α ) = ∫ x α −1e − x dx 0
and 0
