%% Example US Census
clear all, close all
%load adis data
x=1981:2004;
y=[260 992 2717 5341 8224 13195 21355 32196 35230 43352 45524 47572 ...
    106618 77474 71073 68808 60270 47915 46143 41795 43158 43578 44963 42514];
figure(1),
  plot(x,y,'o')
  axis([1981,2004,0,max(y)])
  xlabel('Years')
  ylabel('aids cases')
x = x(1:12); y = y(1:12);
figure(2)
  plot(x,y,'o')
  axis([1981,1992,0,max(y)])
  xlabel('year')
  ylabel('aids cases')
   
%% Compute k and y(0) in exponential model using least squares. 
t = 0:1:length(x)-1;
p0 = [y(1),1];
p=nlinfit(t,y,@(p,x) p(1)*exp(p(2)*x),p0);
exp_fit = p(1)*exp(p(2)*t);
figure(2)
  plot(x,y,'o',x,exp_fit)
  axis([1981,1992,0,max(y)])
  xlabel('Years')
  ylabel('aids cases')
  legend(['Data','Exponential Fit'])

  %% Compute k, M, and y(0) in logisitic model using least squares. 
t = 0:1:length(x)-1;
p0 = [y(1),1,max(y)];
p=nlinfit(t,y,@(p,x) p(1)*p(3) ./ (p(1) + (p(3)-p(1))*exp(-p(2)*x)),p0);
logistic_fit = p(1)*p(3) ./ (p(1) + (p(3)-p(1))*exp(-p(2)*t));
figure(3)
  plot(x,y,'o',x,logistic_fit)
  axis([1981,1992,0,max(y)])
  xlabel('Years')
  ylabel('aids cases')
  legend(['Data','Logistic Fit'])