%% Example US Census

%load USCensus
x=[1790:10:2000];
y=[3.93,5.31,7.24,9.64,12.87,17.07,23.19,31.44,39.82,50.16,62.95,75.99,91.97,105.71,122.78,131.67,151.33,179.32,203.21,226.5,249.63,281.42];
figure(1),
  plot(x,y,'o')
  axis([1780,x(end),0,y(end)+10])
  xlabel('Decades of Census Data')
  ylabel('US Population in Millions')
  
%% Compute k and y(0) using first two data points.
% Time in years t = 0 corresponds to 1790.
t = 0:10:210;
% Model
y0 = y(1);
k = log(y(2)/y(1))/10;
Pmodel = y0*exp(k*t);
% Compare
tt = t + 1790;
figure(2)
  plot(tt,y,'o',tt,Pmodel)
  axis([1790 2000 0 300])
  xlabel('year')
  ylabel('population')
  legend('Census Data','Model Prediction')
  
%% Compute k and y(0) using least squares. 
p0 = [y(1),0.001];
p=nlinfit(t,y,@(p,x) p(1)*exp(p(2)*x),p0);
exp_fit = p(1)*exp(p(2)*t);
figure(3)
  plot(x,y,'o',x,exp_fit)
  axis([1780,x(end),0,y(end)+10])
  xlabel('Decades of Census Data')
  ylabel('US Population in Millions')
  legend('Census Data','Exponential Fit')