% Modified version of the MATLAB template file FitNumDifSR.m that was
% modified for the solution of Example 3 of Supplemental Unit S4 of "A
% First Course on Kinetics and Reaction Engineering."
%
function S4_Example_3(p_guess)
    % Known quantities and constants (in consistent units)
    A = 10.0;
    
    % Set variables
    x = [2	0.2	0.2
    2	0.6	0.6
    2	0.2	1
    2	1	0.2
    1	0.2	0.2
    1	0.6	0.6
    1	0.2	1
    1	1	0.2
    0.67	0.2	0.2
    0.67	0.6	0.6
    0.67	0.2	1
    0.67	1	0.2];
    
    % Experimental response variables
    y_hat = [0.208
    0.491
    0.911
    0.091
    0.112
    0.333
    0.694
    0.060
    0.083
    0.274
    0.545
    0.042];
    
	% Function that evaluates the ODEs
    function dudv = odeqns(v,u)
        % Current parameter values are available in column vector p
        % Current set variable values are available in column vector x_set
        term = A*p(1)*u(1)*u(2)^2/(1.0 + p(2)*u(1) + p(3)*u(2));
        dudv = [
            -term
			-term
            2*term
        ];
    end % of internal function odeqns
    
    % Function that solves the model equations and calculates the model-
    % predicted responses
    function y = nlmodel(p_current,x)
        % Declare y
        y = zeros(n_data,1);
        % Make the current parameters available to the model equations
        p = p_current;
        
        for i = 1:n_data
            % Make the set variables available to the model equations
            for j = 1:n_set
                x_set(j) = x(i,j);
            end
            
            % Initial and final values
            v0 = 0;
            u0 = [
                x_set(2)
                x_set(3)
                0.0
            ];
            vf = x_set(1);
            
            % Solve the set of ordinary differential model equations
            [v,uu] = ode45(@odeqns,[v0 vf],u0);
            last_value = length(v);
            u = uu(last_value,:);
            
            % calculate the response, y, by evaluating 
            % f(u(1),...,u(n),x_set(1),...,x_set(n_set))
            y(i) = (x_set(2)-u(1))/x_set(2);	
        end
    end % of internal function nlmodel

    % All the necessary internal functions are now defined

    % get the number of data points and set variables
    n_data = size(x,1);
    n_set = size(x,2);
    
    % create global variables to make parameters and set variables
    % available to the internal function that evaluates the model equations
    p = zeros(length(p_guess),1);
    x_set = zeros(size(x,2),1);

    % Perform the regression analysis and report the results
    [pf,resid,J,sigma,mse] = nlinfit(x,y_hat,@nlmodel,p_guess);
    confint = nlparci(pf,resid,'covar',sigma);
    pf_u = (confint(:,2) - confint(:,1))/2.0;
    
    % Calculate the model-predicted responses
    y = zeros(n_data,1);
    y = nlmodel(pf,x);
    
    % Display the correlation coefficient, parameters and uncertainties
    format shortE;
    r_squared = 1.0 - (sum(resid.^2))/(sum((y_hat-mean(y_hat)).^2))
    display('Best Values for the Parameters:')
    pf
    display('95% Confidence Intervals for the Parameters:')
    pf_u

    % Generate the graphical output
    n_set = size(x,2);
    if n_set == 1
        % Model plot
        % Use many data points for the model so the curve is smooth
        xx = linspace(min(x),max(x))';
        yy = nlmodel(pf,xx);
        figure;
        plot(x,y,'r',x,y_hat,'o')
        xlabel('X')
        ylabel('Y')
        title('Model Plot')
        legend('Model','Data')
    else
        % Parity plot
        figure;
        plot(y_hat,y_hat,'r',y_hat,y,'o')
        xlabel('Measured Response')
        ylabel('Predicted Response')
        title('Parity Plot')

        % Residuals plots for each set variable
        for n = 1:n_set
            figure;
            plot(x(:,n),zeros(length(y),1),'r',x(:,n),resid,'o')
            xlabel(['x',num2str(n)])
            ylabel('Residual')
            title(['Residuals Plot vs. x',num2str(n)])
        end
    end

end % of S4_Example_3.m