% gradient descent quadratic bowl test
clc %clear console and return cursor to home
format long

%actual function
tm = linspace(0,10,100);
N0 = 10;
R0 = 0.75;
Nd_smth = N0*exp(R0*tm);
M = length(Nd_smth);

%generate vals from random distr with mean = mn and stdv = stdv
rng('default'); %fix seed for random # generator to matlab startup seed
mn = 0; %mean
stdv = 1000;
r_pb = mn + stdv.*randn(1,M); 
Nd = r_pb+Nd_smth; %data
plot(tm,Nd,'bx')

N = @(N0,R0,t) N0.*exp(R0*t); 
F = @(N0,R0,t) 0.5*sum( (N(N0,R0,t)-Nd).^2 );

F_N0 = @(N0,R0,t) sum( (N(N0,R0,t)-Nd).*exp(R0*t) );
F_R0 = @(N0,R0,t) sum( (N(N0,R0,t)-Nd).*(N0*t.*exp(R0*t)) );

% initial guess
N0 = 100; R0 = 0.01; 
lambda = 1000;
delta = 1e-9; %timestep or learning rate
N_max = 600000; Tol = 1e-3;
plot(tm,Nd,'bx',tm,N(N0,R0,tm),'r-')
pause; Out_max = 10;

for i = 1:N_max
    N0_old = N0;
    N0 = N0 - delta*(F_N0(N0,R0,tm));
    R0 = R0 - (1/lambda)*delta*(F_R0(N0_old,R0,tm));

    %plot curve every 5000 iterations
    if rem(i,5000) == 0
        sprintf("N0=%f, R0=%f, i=%f",N0,R0,i)
        pause(0.1)
        plot(tm,Nd,'bx',tm,N(N0,R0,tm),'r-')
    end
end
disp(N0)
disp(R0)
plot(tm,Nd,'bx',tm,N(N0,R0,tm),'r-')