function mod_inverse_prob(q_guess)
% Modified Inverse Least Squares Estimation Problem
% Implementation of the inverse problem which finds the optimal values for the 
%     parameters C and K for the vibrating beam (damped or undamped model) 
%     using the least squares function
% q_guess, vector with guess values for the paramters C and K,
%          where q_guess(1) = C = c/m and q_guess(2) = K = k/m
% J_q_guess, cost function value associated with the guess values for C and
%            K (residual)
% q_optimal, optimal values for C and K
% J_q_optimal, cost function value associated with the optimal values
%              for C and K (residual)
% variance_hat, variance associated with the optimal values for C and K
initial_t = load('trac3.x');            % reads in time data for frequency of 6.15 Hz
initial_a = load('trac3.txt');          % reads in acceleration data for frequency of 6.15 Hz
% find index of a peak acceleration point (corresponds to maximum displacement)
max_val = find(initial_a==max(initial_a));    
t = initial_t(max_val:1024);                 % change indexing of the time and acceleration data                 
a = initial_a(max_val:1024);  

% plot data
figure (1)
plot(t,a,'r--')  % plot of acceleration data versus time
grid on
xlabel('Time')
ylabel('Acceleration')
title('Beam Acceleration Data')

% initial conditions (initial displacement = -(y"(t)/K and initial velocity = 0)
IC = [(-a(1)/q_guess(2));0];    
a = a - mean(a);   % center acceleration data about 0
% Use undamped model if C = 0, otherwise use damped model
if (q_guess(1) == 0)
    % cost associated with guess for K
    J_q_guess = undamped_cost_func(q_guess,t,a,IC) 
    % find the optimal value for K and the associated cost function value
    [q_optimal,J_q_optimal] = fminsearch(@undamped_cost_func,q_guess,[],t,a,IC) 
    % compute the solution to the undamped spring-mass-dashpot system using
    %    optimal K value
    mod_sol = undamped_mod_sol(q_optimal(2),t,IC);   
    % compute the acceleration values from the model with optimal K
    a_mod = -q_optimal(2)*mod_sol(:,1);
else
    % cost associated with the guess values for C and K
    J_q_guess = damped_cost_func(q_guess,t,a,IC) 
    % find the optimal values for C and K and the associated cost function
    %      value
    [q_optimal,J_q_optimal] = fminsearch(@damped_cost_func,q_guess,[],t,a,IC)
    % compute the solution to the damped spring-mass-dashpot system using
    %     the optimal values for C and K
    mod_sol = damped_mod_sol(q_optimal(1),q_optimal(2),t,IC); 
    % compute the acceleration values from the model with optimal C and K
    a_mod = -q_optimal(1)*mod_sol(:,2) - q_optimal(2)*mod_sol(:,1);
end
% variance associated with the optimal C* and K*
variance_hat = (1/(length(a)-length(q_optimal)))*length(a)*J_q_optimal   
r_j = a - a_mod;   % residuals between acceleration data and acceleration from model 
% Plots
figure (2)
plot(t,a_mod,'b-v',t,a,'r-*')  % plot of acceleration data and acceleration from model versus time
legend('model','experiment')
grid on
xlabel('Time')
ylabel('Acceleration')
title('Model versus Experimental Data')
figure (3)
plot(t,r_j,'r+')      % plot of errors versus time 
grid on
xlabel('Time')
ylabel('Error')
title('Error versus time')