Developmental stage annotation of Drosophila gene expression pattern images via an entire solution path for LDA

Gene expression in a developing embryo occurs in particular cells (spatial patterns) in a time-specific manner (temporal patterns), which leads to the differentiation of cell fates. Images of a Drosophila melanogaster embryo at a given developmental stage, showing a particular gene expression pattern revealed by a gene-specific probe, can be compared for spatial overlaps. The comparison is fundamentally important to formulating and testing gene interaction hypotheses. Expression pattern comparison is most biologically meaningful when images from a similar time point (developmental stage) are compared. In this paper, we present LdaPath, a novel formulation of Linear Discriminant Analysis (LDA) for automatic developmental stage range classification. It employs multivariate linear regression with the L₁-norm penalty controlled by a regularization parameter for feature extraction and visualization. LdaPath computes an entire solution path for all values of regularization parameter with essentially the same computational cost as fitting one LDA model. Thus, it facilitates efficient model selection. It is based on the equivalence relationship between LDA and the least squares method for multiclass classifications. This equivalence relationship is established under a mild condition, which we show empirically to hold for many high-dimensional datasets, such as expression pattern images. Our experiments on a collection of 2705 expression pattern images show the effectiveness of the proposed algorithm. Results also show that the LDA model resulting from LdaPath is sparse, and irrelevant features may be removed. Thus, LdaPath provides a general framework for simultaneous feature selection and feature extraction.