IDENTIFICATION OF OPTIMAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODEL FOR METEOROLOGICAL DATA

Abstract

Autoregressive Integrated Moving Average (ARIMA) model of various orders was presented, with a view to identifying the optimal model. In this study, ARIMA (p, d, q) model of various orders was presented in both causal and inverted forms, behavioural patterns of both causality and invertibility parameters were investigated, parameters of the models were estimated using Ordinary Least Square approach and optimal model from a class of models was identified using Normalized Bayesian Information Criterion (BIC) as model selection criterion. It was deduced that behaviour of causality parameter, ѱᵢ and invertibility parameter,π, depend on positive and negative values of auto regressive parameter ɸ and moving average parameter θ respectively. ѱᵢ is skewed to right and sinusoidal for positive and negative values of ɸ respectively. ѱᵢ of ARMA (1, q) model increases as the value of q increases for positive values of ɸ. Also, ѱ increases as ɸ increases as shown in Figure 2.4 for positive values of θ. Similarly, πᵢ, of ARIMA (p, d, 1) model for various integer values of d are sinusoidal for positive values of θ irrespective of ɸₙ's arrangement, [πᵢ] increases as d increases and the lower the integer value of d, the faster [πᵢ] converges to zero. For negative integer values of θ , πᵢ, of ARlMA (p, d, 1) model oscillates as the value of d increases with [πᵢ]d₌ ₖ < [πᵢ]d₌ ₖ ₊ ₂ for all even integer value k > 0 provided i ≥ 3 and I[πᵢ]d₌ ₖ > [πᵢ]d₌ ₖ ₊ ₂ for all odd integer value k > 0. Similarly, πᵢ, of ARIMA (p, d, I) model was found to be sinusoidal for p = 1,2,3 and 4 and d = 0 and 1 with[πᵢ] ᵨ ₌ ₄ > [πᵢ] ᵨ ₌ ₃ > [πᵢ] ᵨ ₌ ₂ > [πᵢ] ᵨ ₌ ₁ for all positive values of θ. For all negative values of θ, [πᵢ] ᵨ ₌ ₖ < [πᵢ] ᵨ ₌ ₖ ₊ ₂ for odd integer value k > 0 and [πᵢ] ᵨ ₌ ₖ < [πᵢ] ᵨ ₌ ₖ ₊ ₂ for even integer value k > 0. ARIMA (p, d, q) model was formulated for daily maximum meteorological temperature data of Ondo, Nigeria (07°06'27" N, 04°50'19" E) and Zaira, Nigeria (11°04'N, 07°42'E), daily rain amount of Calabar, Nigeria (07°06'27"N, 08°20'E) and Ondo, Nigeria (07°06'27"N, 04°50'19"E) and daily cloud of Warri, Nigeria (05°30'N, 05°41'E) and Benin City, Nigeria (07° 06'27" N, 05° 31' E) from January 1995 to December 2005. The choice of ARIMA model of orders p and q was intended to retain any persistence in natural process. To determine the performance of models, Normalized BIC was adopted. ARIMA (1, 1, 1) model was adequate-for modelling daily maximum temperature of On do, Nigeria and Zaira, Nigeria, parameters of the model was estimated and redundant variable was removed for each of the cities. Similarly for modelling daily rain amount of Calabar, Nigeria and Ondo, Nigeria, and daily cloud of Warri, Nigeria and Benin City, Nigeria, ARIMA (1, 0, 1) model was found adequate and parameters of the model were estimated.