The functional parameter object used in the beta basis system is then redefined based on the refined smoothing parameter identified in order to ensure that the functional linear regression analysis converges on a model that has the best chances of performing well out-of-sample.
The functional linear regression analysis can now be run with the identified smoothing parameters and scalar response variables to identify the \(\beta_i\) coefficients and the associated variance that enables determination of the 95% confidence bounds (see sections 9.4.3 and 9.4.4 of \cite{Ramsay_2009} respectively). Fig. \ref{820059} to Fig. \ref{942889} show the resulting \(\beta_i\) coefficients obtained for the number of non-corporates and the number of cited references by priority year, when considering the emergence phase of development and using a high-dimensional regression fit (i.e. when the beta basis system for each regression coefficient is made of a large number of B-splines):