Effect of Data Transformation on Residue

Recently, Aguilar-Ruiz [2005] considers a data matrix containing both scaling and shifting factors and shows that the mean squared residue [Cheng and Church, 2000], called RESIDUE(II) in this paper, is useful to discover shifting patterns, but not appropriate to find scaling patterns. This finding draws our attention on the weakness of RESIDUE(II) measure and the need of new approaches to discover both scaling and shifting patterns in the considered matrix. To resolve the weakness of RESIDUE(II) in finding scaling patterns, we propose a simple remedy that still uses the same residue measure. The main idea is to remove hidden scaling factors in the considered data matrix by taking a specific data transformation. We investigate various data transformations including no transformation, double centering, mean centering, standard deviation normalization, and Z-score transformation. Further, we apply these data transformations to row/column dimension of data matrix models with different global/local scaling and global/local shifting factors. First, we characterize the properties of the data transformations on different data matrix models, including six Euclidean co-clustering schemes in Bregman co-clustering algorithms [Banerjee et al., 2007] and other existing data models in the literature. In particular, we formally analyze the effect of each data transformation on the two residues [Cho et al., 2004], here called RESIDUE(I) and RESIDUE(II), respectively. Then, we apply all the data transformations to publicly available human cancer gene expression datasets and empirically validate the analysis results by using the minimum sum squared residue co-clustering (MSSRCC) algorithms [Cho et al., 2004]. In conclusion, through column standard deviation normalization or column Z-score transformation, we are able to overcome the shortcoming of RESIDUE(II) in finding scaling patterns and discover both scaling and shifting patterns.