Functional linear regression is an important topic in functional data analysis. The model has been extended to exponential families, in particular to binary response but to the best of our knowledge, there is no study and application of multinomial responses. Thanks to sparsity of wavelet coefficients, we use the wavelet basis as a dimension reduction for functional data and combine it with L1 penalized regression in order to estimate the unknown parameters in the multinomial functional regression model. Our study is inspired by and applied to acceleration signals from a scientific experiment on horse lameness. Our aim is to develop a method that uses acceleration signals from trooting horses to detect the lameness and identify the lame limb. We propose to use a multinomial functional regression which apart from classification enables us to investigate which parts of the signals have a significant effect in the classification procedure.
