Hi @Mariana_Aguilar,

The difference is in how the data is used and pre-processed as you mention : with a B-Spline or P-Spline, you first use a model that correspond to your raw curves data, and then you can calculate Functional Principal Components based on this model (approximation).

With direct models like direct Functional PCA, you directly perform a Singular Value Decomposition (SVD) on your raw data to extract eigenvalues and eigenfunctions. More info here : Types of Functional Model Fits

To determine what would be a suitable option, you can fit different models and evaluate them based on statistical metrics (like information criteria AICc and BIC), and with the visualizations of the Diagnostic plots to evaluate model's adequacy/precision. In your case, it seems that applying B-Splines before extracting Functional Principal components with your curves data is a good option, as the residuals from the models are very low and homogeneous, unlike those from Direct Functional PCA :

Diagnostic plots for B-Splines :

Diagnostic plots for Direct FPCA:

Hope this answer will help you,

Victor GUILLER

Scientific Expertise Engineer

L'Oréal - Data & Analytics