/* Parameter Estimation determining the growth rate r and the carrying capacity.*/
/* Initializations */
dp[1]:0; /*next time population - current time population*/
n :9 ; /*Number of Population data to be considered. Typically taken to be half of the entire given population data.*/
dt: 10 ; /*Time step of the data provided*/
K[1]: 0;/*Carrying capacity computed. It is of length n-2.*/
r1[1]: 0; 
r2[1]:0 ;
/*Arrays containing the rates r1 and r2 corresponding to K[i] and computed based on P[i] and P[i+1] respectively. */
b: 0;
c1: 0;
c2 : 0;
/*Some temp variables introduced.*/
/******************************************/
data: read_matrix("us_population.dat");
disp(transpose(data)[3][1]);
p[0]:transpose(data)[3][1];
 for i:1 thru n-1 step 1 do
   (p[i]:transpose(data)[3][i+1],
   dp[i]: p[i] - p[i-1]
   ) ;
  
for i:1 thru n-2 step 1 do
   (   
   b : (p[i]*dp[i+1] -p[i+1]*dp[i]) ,
   K[i] : (dp[i+1]*p[i]*p[i]  - dp[i]*p[i+1]*p[i+1])/b   ,
   c1 : dt*( p[i] - p[i]*p[i]/K[i]),
   r1[i] : dp[i]/c1,
   c2 : dt*( p[i+1] - p[i+1]*p[i+1]/K[i]),
   r2[i] : dp[i+1]/c2
   );
 
for i:1 thru n-2 step 1 do
(print("time: ", i , " a: ",dp[i+1]*p[i]*p[i]  - dp[i]*p[i+1]*p[i+1] , "b: ", p[i]*dp[i+1]-p[i+1]*dp[i], "K: a/b: ", 
float(K[i])," r values: " ,r1[i]," = ",r2[i]  ),

print("@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@")
);
/*Plotting the solutions using the parameters obtained above.*/
K : 239.2539884  ;
p0:3.929;
r:0.0311;

population: create_list([1790 + i*10,K*p0/(p0 +(K-p0)*exp(-r*(i*10)))],i,0,15);
/*data: read_matrix("us_population.dat");*/


plot2d(  [[discrete, transpose(data)[1], transpose(data)[3]],[discrete,population]],
[style,points, lines],[point_type,diamond],[color,red , blue]
,[xlabel,"Time"]
,[ylabel,"Population Growth"],[legend, false]
,[title, "Curve Fitting K=239.2539; r=0.0311 "],
[legend, "Discrete Population Data" ,"Fitted Curve"],
[gnuplot_preamble,"set key bottom"]
);