The R-squared and adjusted R-squared values shown in Table \ref{table:results_high_dimensional_model} would suggest that a reasonable fit has been achieved with this model, with a good level of accuracy, whilst the F-ratio of 5.60 with degrees of freedom 7.78 and 11.22 respectively implies that the relationship established has a p-value somewhere between 0.0041 and 0.0060. As such this result appears to be significant at the 1% level.

Benchmarking functional regression model

However, to ensure that this is the most appropriate fit to the data presented, the high-dimensional model initially developed was subsequently benchmarked against a low-dimensional model (i.e. when the beta basis system for each regression coefficient is made of a small number of B-splines), as well as a constant and a monomial based model. The corresponding \(\beta_i\) coefficients from the benchmarking analysis for the low-dimensional model are presented in Fig. \ref{934895} to Fig. \ref{232642}, whilst the 'goodness-of-fit' measures for all the alternative functional linear regression models are compiled in Table \ref{table:results_benchmarking}: