Fractional Polynomial regression is a form of regression analysis in which the relationship between the independent variable and the dependent variable is modelled as a 1/nth degree polynomial. Thus, this work is used to propose an extension of Fractional Polynomial Regression (FPR) term Lagged Fractional Polynomial Regression (LFPR) which is an alternative method to traditional techniques of analysing the pattern and degree of relationship between two or more continuous non-linear data. The coefficients of the proposed method were estimate using Maximum Likelihood Estimation method. From the results, the LFPR model indicated that for a unit increase in Evaporation, Humidity and Temperature there will be an increase in the millimeter of rainfall series on yearly basis. The value of coefficient of variation (R2) for the LFPR and FPR were 99% and 77%. While the value of adjusted Coefficient of Variation (R2) for LFPR and FPR were 96% and 75% respectively. Hence, the proposed method outperformed and adequately explained the variation in the dependent variable better than Fractional Polynomial Regression based on the values (R2) and adjusted (R2).

Most frequently used models for modeling and forecasting periodic climatic time series do not have the capability of handling periodic variability that characterizes it. In this paper, the Fourier Autoregressive model with abilities to analyze periodic variability is implemented. From the results, FAR(1), FAR(2) and FAR(2) models were chosen based on Periodic Autocorrelation function (PeACF) and Periodic Partial Autocorrelation function (PePACF). The coefficients of the tentative model were estimated using a Discrete Fourier transform estimation method. FAR(1) models were chosen as the optimal model based on the smallest values of Periodic Akaike (PAIC) and Bayesian Information criteria (PBIC). The residual of the fitted models was diagnosed to be white noise. The in-sample forecast showed a close reflection of the original rainfall series while the out-sample forecast exhibited a continuous periodic forecast from January 2019 to December 2020 with relatively small values of Periodic Root Mean Square Error (PRMSE), Periodic Mean Absolute Error (PMAE) and Periodic Mean Absolute Percentage Error (PMAPE). The comparison of FAR(1) model forecast with AR(3), ARMA(2,1), ARIMA(2,1,1) and SARIMA( 1,1,1)(1,1,1)12 model forecast indicated that FAR(1) outperformed the other models as it exhibited a continuous periodic forecast. The continuous monthly periodic rainfall forecast indicated that there will be rapid climate change in Nigeria in the coming yearly and Nigerian Government needs to put in place plans to curtail its effects.

Minimum Message Length MML87 is an information theoretical criterion for model selection and point estimation. In principle, it is a method of inductive inference, and is used in a wide range of approximations and algorithm to determine the ideal model for any given data. In this study, MML87 model selection criterion was investigated and compared with other notably model selection criteria such as Akaike information criterion (AIC), Bayesian information criterion (BIC), Corrected Akaike information criterion (AICc), and Hannan-Quinn (HQ), using Bootstrap Simulation Technique to simulate autoregressive model of order P. We specified three different counts systems as under inferred, correctly inferred and over inferred. Based on the candidate model explored with autoregressive model and the aggregate true model explored, with the estimated parameters. MML87 performed better than all other model selection criteria through the negative log likelihood function and the mean square prediction error estimated. It is more efficient and correctly inferred.