% Modified version of the MATLAB template file FitNumAlgSR.m that was
% modified for the solution of Example 2 of Supplemental Unit S4 of "A
% First Course on Kinetics and Reaction Engineering."
%
function S4_Example_2(p_guess)
    % Known quantities and constants
    A = 10.0;
    
    % Enter the set variables
    x = [5	0.2	0.2
    5	0.6	0.6
    5	0.2	1
    5	1	0.2
    10	0.2	0.2
    10	0.6	0.6
    10	0.2	1
    10	1	0.2
    15	0.2	0.2
    15	0.6	0.6
    15	0.2	1
    15	1	0.2];

    % Enter the experimentally measured responses
    y_hat = [0.200
    0.422
    0.234
    0.128
    0.119
    0.312
    0.191
    0.092
    0.090
    0.239
    0.170
    0.073];
    
    % Function that evaluates the model equations
	function g = nlaeqns(u)
        % Current parameter values are available in column vector p
        % Current set variable values are available in column vector x_set
        term2 = A*p(1)*u(1)*u(2)^2/(1.0 + p(2)*u(1) + p(3)*u(2));
		g = [
			x_set(1)*x_set(2) - term2 - x_set(1)*u(1);
            x_set(1)*x_set(3) - term2 - x_set(1)*u(2);
            u(3) - 2*(x_set(2) - u(1));
		];
    end % of internal function nlaeqns

    % Function that calculates the responses using the model solution
    function y = nlmodel(p_current,x)
        % Make the current parameters available to the model equations
        p = p_current;
        y = zeros(n_data,1);
        
        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
            
            % Guesses
            % The measured response is available here as y_hat(i)
            u_guess = [
                0.5
                0.5
                0.5
            ];
        
            % Solve the set of nonlinear modle equations
            u = fsolve(@nlaeqns, u_guess);
            
            % Once you are sure fsolve is obtaining converged solutions,
            % you can suppress the "equations solved" messages by
            % commenting out the line above this comment and uncommenting
            % the next two lines.
            % options = optimoptions('fsolve','Display','off');
            % u = fsolve(@nlaeqns, u_guess, options);
            
            % Calculate the response
            y(i) = u(3)/(u(1) + u(2) + u(3));
        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
    [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 and the correlation
    % coefficient
    y = zeros(n_data,1);
    y = nlmodel(pf,x);

    % 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
    
    % 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
end % of S4_Example_2.m