The Estimating Parameter and Number of Knots for Nonparametric Regression Methods in Modelling Time Series Data

Abstract

This research aims to explore and compare several nonparametric regression techniques, including smoothing splines, natural cubic splines, B-splines, and penalized spline methods. The focus is on estimating parameters and determining the optimal number of knots to forecast cyclic and nonlinear patterns, applying these methods to simulated and real-world datasets, such as Thailand’s coal import data. Cross-validation techniques are used to control and specify the number of knots, ensuring the curve fits the data points accurately. The study applies nonparametric regression to forecast time series data with cyclic patterns and nonlinear forms in the dependent variable, treating the independent variable as sequential data. Simulated data featuring cyclical patterns resembling economic cycles and nonlinear data with complex equations to capture variable interactions are used for experimentation. These simulations include variations in standard deviations and sample sizes. The evaluation criterion for the simulated data is the minimum average mean square error (MSE), which indicates the most efficient parameter estimation. For the real data, monthly coal import data from Thailand is used to estimate the parameters of the nonparametric regression model, with the MSE as the evaluation metric. The performance of these techniques is also assessed in forecasting future values, where the mean absolute percentage error (MAPE) is calculated. Among the methods, the natural cubic spline consistently yields the lowest average mean square error across all standard deviations and sample sizes in the simulated data. While the natural cubic spline excels in parameter estimation, B-splines show strong performance in forecasting future values.