In high dimensional regression, it is important to estimate the central and central mean subspaces, to which the projections of the predictors preserve sufficient information about the response and the mean response, respectively. Using the Fourier transform, we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively. Under the normality assumption of the predictors, explicit estimates of the central and central mean subspaces are derived. Bootstrap procedures are used for determining dimensionality and choosing tuning parameters. Simulation results and an application to a real data are reported. Our methods demonstrate competitive performance compared to SIR, SAVE and other existing methods. The approach proposed in the paper provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future.

Matlab programs are available upon request from the authors.

Skull morphology is fundamental to evolution and the biological adaptation of species to their environments. With aquaculture fish species, head size is also important for economic reasons because it has a direct impact on fillet yield. However, little is known about the underlying genetic basis of head size. Catfish is the primary aquaculture species in the United States. In this study, we performed a genome-wide association study using the catfish 250K SNP array with backcross hybrid catfish to map the QTL for head size (head length, head width, and head depth). One significantly associated region on linkage group (LG) 7 was identified for head length. In addition, LGs 7, 9, and 16 contain suggestively associated regions for head length. For head width, significantly associated regions were found on LG9, and additional suggestively associated regions were identified on LGs 5 and 7. No region was found associated with head depth. Head size genetic loci were mapped in catfish to genomic regions with candidate genes involved in bone development. Comparative analysis indicated that homologs of several candidate genes are also involved in skull morphology in various other species ranging from amphibian to mammalian species, suggesting possible evolutionary conservation of those genes in the control of skull morphologies.

No abstract