In QDA, we need to estimate Σ𝑘 for each class 𝑘∈{1,…,𝐾} rather than assuming Σ𝑘=Σ as in LDA. The discriminant function of LDA is quadratic in 𝑥 in the image below.

Since QDA estimates a covariance matrix for each class, it has a greater number of effective parameters than LDA. We can derive the number of parameters in the following way.

• We need 𝐾 class priors 𝜋𝑘. Since ∑𝐾𝑖=1𝜋𝑘=1, we do not need a parameter for one of the priors. Thus, there are 𝐾−1 free parameters for the priors.
• Since there are 𝐾 centroids, 𝜇𝑘, with 𝑝 entries each, there are 𝐾𝑝 parameters relating to the means.
• From the covariance matrix, Σ𝑘, we only need to consider the diagonal and the upper right triangle. This region of the covariance matrix has 𝑝(𝑝+1)2 elements. Since 𝐾 such matrices need to be estimated, there are 𝐾𝑝(𝑝+1)2 parameters relating to the covariance matrices.

Thus, the effective number of QDA parameters is 𝐾−1+𝐾𝑝+𝐾𝑝(𝑝+1)2.