Functional dimension reduction based on fuzzy partition and transformation

Functional sliced inverse regression (FSIR) is the among most popular methods for the functional dimension reduction. However, FSIR has two evident shortcomings. On the one hand, the number of samples in each slice must not be too small and selecting a suitable S is difficult, particularly for data with small sample size, where S indicates the number of slices. On the other hand, FSIR and its related methods are well-known for their poor performance when the link function is an even (or symmetric) dependency. To solve these two problems, we propose three new types of estimation methods. First, we propose the functional fuzzy inverse regression (FFIR) method based on a fuzzy partition. Compared with FSIR that uses a hard partition, the fuzzy partition uses all samples with different weights to estimate the mean in each slice. Therefore, FFIR exhibits good performance even for data with small sample size. Second, we suggest two transformation approaches, namely, FSIRR and FSIRP, avoiding the symmetric dependency between the response and the predictor. FSIRR eliminates the symmetric dependency by transforming the response variable, while FSIRP overcomes the symmetric dependency by transforming the functional predictor. Third, we propose the FFIRR and FFIRP methods by combining the advantages of FFIR and two transformation methods. FFIRR and FFIRP replace the FSIR method on the transformation data via FFIR. Simulation and real data analysis show that three types of proposed methods exhibit better performance than FSIR in terms of the estimation accuracy and stability.