Question

I would like to create a function that finds the parameters p and q of Bass diffusion model, given the data of two time periods.

The model (equation) is the following:

n(T) = p*m + (q-p)*n(T-1) + q/m*n(T-1)^2

where

n(T) = number of addoptions occuring in period T
n(T-1) = number of cumulative adoptions that occured before T
p = coefficient of innovation
q = coefficient of imitation
m = number of eventual adopters

for example if m = 3.000.000
and the data for the years below is the following:

2000: n(T) = 820, n(T-1) = 0
2005: n(T) = 25000, n(T-1) = 18000

then the following equation system has to be solved (in order to determine the values of p and q):

p*m + (q-p)*0 + q/3.000.000 * 0^2 == 820
p*m + (q-p)*18000 + q/3.000.000 * 18000^2 == 25000

By following Matlab documentation I tried to create a function Bass:

function F = Bass(m, p, q, cummulativeAdoptersBefore)

F = [p*m + (q-p)*cummulativeAdoptersBefore(1) + q/m*cummulativeAdoptersBefore(1).^2;
    p*m + (q-p)*cummulativeAdoptersBefore(2) + q/m*cummulativeAdoptersBefore(2).^2];


end

Which should be used in fsolve(@Bass,x0,options) but in this case m, p, q, cummulativeAdoptersBefore(1), and cummulativeAdoptersBefore(2) should be given in x0 and all variables would be considered as unknown instead of just the latter two.

Does anyone know how to solve the system of equations such as above?

Thank you!

Answer 1

fsolve() seeks to minimize the function you supply as argument. Thus, you have to change your equations to

p*m + (q-p)*0     + q/3.000.000 * 0^2     - 820   == 0
p*m + (q-p)*18000 + q/3.000.000 * 18000^2 - 25000 == 0

and in Matlab syntax

function F = Bass(m, p, q, cumulativeAdoptersBefore, cumulativeAdoptersAfter)

    F = [p*m + (q-p)*cumulativeAdoptersBefore(1) ...
             + q/m  *cumulativeAdoptersBefore(1).^2 
             - cumulativeAdoptersAfter(1);
         p*m + (q-p)*cumulativeAdoptersBefore(2) ...
             + q/m  *cumulativeAdoptersBefore(2).^2 
             - cumulativeAdoptersAfter(2)];
end

Note: There is a typo in your Bass function (multiplication instead of sum).

Now you have a function, which takes more parameters than there are unkowns.
One option is to create an anonymous function, which only takes the unknowns as arguments and to fix the other parameters via a closure.
To fit the unkowns p and q, you could use something like

cumulativeAdoptersBefore = [0, 1800];
cumulativeAdoptersAfter = [820, 25000];
m = 3e6;
x = [0, 0]; %# Probably, this is no good starting guess.
xopt = fsolve(@(x) Bass(m, x(1), x(2), cumulativeAdoptersBefore, cumulativeAdoptersAfter), x0);

So fsolve() sees a function taking only a single argument (a vector with two elements) and it also returns a vector value.