[Numpy-discussion] USING NUMPY FOR AN ENROLLMENT PROJECTION PROBLEM

Ajith Prasad (OFN) ofnap at nus.edu.sg
Sat Mar 25 05:00:03 CST 2006


I posted this problem - arising from my work at the local university -
some years back in comp.lang.python and received a solution that did not
require the use of matrix algebra. The solution was:

    def enrollment(year, intake, survivalrate): 
        return sum([intake[year-i]*rate 
                    for i, rate in enumerate(survivalrate)]) 


I would welcome help in formulating a solution using NumPy. Thanks in
advance for any suggestion. 


The Problem: 


The enrolment E(n) of an institution at the beginning of year n is the 
sum of the intake for year n, I(n), and the survivors from the intakes 
of previous r years. Thus, if s(1) is the 1-year intake survival rate, 
s(2) is the 2-year survival rate, etc, we have: 


E(n)= I(n)+I(n-1)*s(1)+ I(n-2)*s(2)+...+I(n-r)*s(r) 
E(n+1)= I(n+1)+I(n)*s(1)+I(n-1)*s(2)+... +I(n-r-1)*s(r) 
. 
. 
. 
E(n+k)= I(n+k)+I(n+k-1)*s(1)+I(n+k-2)*s(2)+...+I(n+k-r)*s(r) 


Given: 
(a) the actual intakes for the current and previous r years, I(n), 
I(n-1),I(n-2),..,I(n-r), and the planned intakes for the next n+k 
years: I(n+1), I(n+2),..., I(n+k), we have the intake vector I = 
(I(n-r), I(n-r-1),...,I(n),I(n+1),..., I(n+k)); and 
(b) the survival rate vector, s = (1,s(1), s(2),...,s(r)) 
Find: 
The  k*1 enrolment projection column vector, E = 
(E(n+1),E(n+2),...,E(n+k)) in terms of a k*(r+1) matrix P (derived from
I) and the (r+1)*1 column vector, s. 


I = P*s 


Is there a compact Python representation of the relevant matrix P 
where: 


P = [I(n+1) I(n) I(n-1).. . I(n-r) 
     I(n+2) I(n+1) I(n)...  I(n-r-1) 
     . 
     . 
     I(n+k) I(n+k-1) I(n+k-2)... I(n+k-r)] 






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