Theory of fractional covariance matrix and its applications in PCA and 2D-PCA

• We extend fractional covariance matrix (FCM) to fractional order forms by the given new definition.
• We propose two new techniques of dimensionality reduction by using FCM to PCA and 2D-PCA.
• Two new techniques are superior to the standard PCA and 2D-PCA if choosing different order between 0 and 1, which expands the transition recognition ranges of PCA and 2D-PCA.

In this paper, according to the definition and applications of fractional moments, we give new definitions of the fractional variance and fractional covariance. Furthermore, we give the definition of fractional covariance matrix. Based on fractional covariance matrix, principal component analysis (PCA) and two-dimensional principal component analysis (2D-PCA), we propose two new techniques, called fractional principal component analysis (FPCA) and two-dimensional fractional principal component analysis (2D-FPCA), which extends PCA and 2D-PCA to fractional order form, and extends the transition recognition ranges of PCA and 2D-PCA. To evaluate the performances of FPCA and 2D-FPCA, a series of experiments are performed on two face image databases: ORL and Yale. Experiments show that two new techniques are superior to the standard PCA and 2D-PCA if choosing different order between 0 and 1.