Since the observables at particular time instants in a temporal sequence exhibit dependencies, they are not independent samples. Thus, it is not plausible to apply i.i.d. assumption-based dimensionality reduction methods to sequence data. This paper presents a novel supervised dimensionality reduction approach for sequence data, called Linear Sequence Discriminant Analysis (LSDA). It learns a linear discriminative projection of the feature vectors in sequences to a lower-dimensional subspace by maximizing the separability of the sequence classes such that the entire sequences are holistically discriminated. The sequence class separability is constructed based on the sequence statistics, and the use of different statistics produces different LSDA methods. This paper presents and compares two novel LSDA methods, namely M-LSDA and D-LSDA. M-LSDA extracts model-based statistics by exploiting the dynamical structure of the sequence classes, and D-LSDA extracts the distance-based statistics by computing the pairwise similarity of samples from the same sequence class. Extensive experiments on several different tasks have demonstrated the effectiveness and the general applicability of the proposed methods.