To improve computational efficiency, we took a second look
at a non-iterative algorithm for estimating the vector of
proportions.
The problem is reformulated as follows. Notice that only c-1 of the c fractions in the proportion vector are independent. The remaining fraction is completely determined by knowledge of the others. Therefore, we formulate the problem in terms of the first c-1 proportions alone.
Denoting
We obtain the set of equations
This is a set of n equations in c-1 variables, and
is an unconstrained problem. Therefore, we may use the
standard least-squares approach
Minimize (over all 
)
The solution to this minimization problem is
is given by
This method resulted in good estimates on the synthetic data, as well as the AVHRR data set. These results are discussed in the next session.