onto the projection direction as well. Therefore, they should show

per positions in the density function. The top-left inset of Figure

s where three validation points were. It can be seen that the

n points A and C were mapped to the right cluster of the density

validation point B was mapped to the left cluster of the density.

re correct indeed. Incorporating with the priors, the density

(likelihood function) was converted to the posterior probability

Bayes rule. The Bayes rule will be introduced below in this

The bottom-right inset in Figure 3.2 shows the curves of the

probabilities, in which three data points also stayed well at the

ositions they should go. For instance, the validation points A and

on the right side of the crossover point while the validation point

on the left side of the crossover point. The bottom-left inset shows

modal density of projections.

e formulation of LDA

the mean vectors of two Gaussian distributions are denoted by ࢛ଵ

A mapping vector of a projection direction is denoted by w. The

g formula is used to denote the mapping centres (߬ଵ and ߬ଶ) of

ers of a data set in the mapping space (the ݕො space),

߬ଵൌܟ^௧࢛ଵ

߬ଶൌܟ^௧࢛ଶ

(3.5)

st be noted that ߬ଵ∈࣬ and ߬ଶ∈࣬ are two of many mapping

the mapping space ݕො. These two mapping centres (߬ଵ and ߬ଶ)

ay in the centres of two mapping clusters. The distance between

ped centres in the ݕො space is defined as below,

஻^ൌ߬ଶ^െ߬ଵ^ൌܟ௧࢛ଶ^െܟ௧࢛ଵ^{ൌܟ௧ሺ࢛}ଶ^െ࢛ଵ^ሻ

(3.6)

bove equation therefore defines the relationship between two sets

es (the multi-dimensional centres ࢛ଵ and ࢛ଶ as well as the

n centres in the mapping space ߬ଵ and ߬ଶ). The relationship