A Trigonometric Fractional Polynomial Autoregressive Time Series Model for Complex Structure Datasets

Abstract

Time series data exhibiting complex structures such as persistence, nonlinear trends, seasonal, cyclical and irregular variations are observed across many fields. Classical linear time series models including Autoregressive Moving Average (ARMA), Autoregressive Integrated Moving Average (ARIMA), and Seasonal Autoregressive Integrated Moving Average (SARIMA) have been widely applied to such data. However, their effectiveness becomes limited when multiple forms of variation coexist. In particular, datasets that display long-run growth or decay alongside smooth cyclical movements are often inadequately represented by stochastic and fixed seasonal models. This instigated the development of flexible hybrid modelling frameworks capable of capturing the temporal dynamics. This study developed a Fractional Polynomial Trigonometric Autoregressive (TFPAR) time series model that integrates autoregressive dependence, fractional polynomial trend and trigonometric regression terms within a unified framework. The autoregressive component captures temporal persistence; fractional polynomial structure accounts for nonlinear long-run growth or decay, while the trigonometric terms model smooth seasonal and cyclical behaviour using sinusoidal functions. The development of the model was done based on the Box–Jenkins methodology comprising identification, estimation, diagnostic checking, validation and forecasting. Exploratory data analysis was used for model identification, autocorrelation and partial autocorrelation functions and spectral analysis were also used to detect dominant frequencies. Parameter estimation was carried out using Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE) techniques. Adequacy and performance of the model were evaluated using residual diagnostics, coefficients of determination (R² and adjusted R²), information criteria (AIC and BIC), and forecast accuracy measures (Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE)). The exploratory analysis of Nigerian monthly temperature and carbon (IV) oxide (CO₂) emission series from the result revealed strong persistence, evident nonlinear trends and pronounced low-frequency cyclical behaviour. Autocorrelation functions decay slowly indicating long memory while periodogram analysis identifies dominant deterministic frequencies beyond fixed seasonality. Estimation results confirm the statistical significance of the autoregressive, fractional polynomial and trigonometric components. The fitted models for temperature and CO₂ emission series revealed a R² values of 0.94 and 0.96, with corresponding adjusted R² values of 0.93 and 0.95 respectively, indicating substantial explanatory power and good model fit. Diagnostic checks further supported model adequacy, with Durbin–Watson statistics of 2.01 for temperature and 1.98 for CO₂ emissions, suggesting no residual serial correlation. Residual autocorrelation plots showed no significant remaining structure and residual distributions approximated normality. The developed model demonstrated clear superiority over competing models, recording the lowest information criteria values. For temperature, AIC = −412.6 and BIC = −389.4, while for CO₂ emissions, AIC = −438.1 and BIC = −415.9, compared with higher values under other specifications. Forecast evaluation further confirmed improved performance, with temperature forecasts yielding RMSE = 0.42, MAE = 0.31, and MAPE = 2.9%, and CO₂ emission forecasts producing RMSE = 1.87, MAE = 1.42, and MAPE = 3.4%. The study concluded that integrating fractional polynomial trends and trigonometric cycles within an autoregressive framework provides a reliable and effective alternative for modelling and forecasting time series data characterised by persistence, nonlinear evolution and smooth cyclical behaviour due to its superior explanatory and predictive performance, and therefore recommends the adoption of the TFPAR model for analysing and forecasting such complex time series.