Recomposing vector input to algorithm from matrix output

0
votes

I've written some code to implement an algorithm that takes as input a vector q of real numbers, and returns as an output a complex matrix R. The Matlab code below produces a plot showing the input vector q and the output matrix R.

Given only the complex matrix output R, I would like to obtain the input vector q. Can I do this using least-squares optimization? Since there is a recursive running sum in the code (rs_r and rs_i), the calculation for a column of the output matrix is dependent on the calculation of the previous column.

Perhaps a non-linear optimization can be set up to recompose the input vector q from the output matrix R?

Looking at this in another way, I've used an algorithm to compute a matrix R. I want to run the algorithm "in reverse," to get the input vector q from the output matrix R.

If there is no way to recompose the starting values from the output, thereby treating the problem as a "black box," then perhaps the mathematics of the model itself can be used in the optimization? The program evaluates the following equation:

Equations

The Utilde(tau, omega) is the output matrix R. The tau (time) variable comprises the columns of the response matrix R, whereas the omega (frequency) variable comprises the rows of the response matrix R. The integration is performed as a recursive running sum from tau = 0 up to the current tau timestep.

Here are the plots created by the program posted below:

q value inputmatrix output

Here is the full program code:

N = 1001;
q = zeros(N, 1); % here is the input
q(1:200) = 55;
q(201:300) = 120;
q(301:400) = 70;
q(401:600) = 40;
q(601:800) = 100;
q(801:1001) = 70;
dt = 0.0042;
fs = 1 / dt;
wSize = 101;
Glim = 20;
ginv = 0;
R = get_response(N, q, dt, wSize, Glim, ginv); % R is output matrix
rows = wSize; 
cols = N;

figure; plot(q); title('q value input as vector'); 
ylim([0 200]); xlim([0 1001])

figure; imagesc(abs(R)); title('Matrix output of algorithm')
colorbar

Here is the function that performs the calculation:

function response = get_response(N, Q, dt, wSize, Glim, ginv)

fs = 1 / dt; 
Npad = wSize - 1; 
N1 = wSize + Npad;
N2 = floor(N1 / 2 + 1);
f = (fs/2)*linspace(0,1,N2);
omega = 2 * pi .* f';
omegah = 2 * pi * f(end);
sigma2 = exp(-(0.23*Glim + 1.63));

sign = 1;
if(ginv == 1)
    sign = -1;
end

ratio = omega ./ omegah;
rs_r = zeros(N2, 1);  
rs_i = zeros(N2, 1);   
termr = zeros(N2, 1);
termi = zeros(N2, 1);
termr_sub1 = zeros(N2, 1);
termi_sub1 = zeros(N2, 1);
response = zeros(N2, N);

 % cycle over cols of matrix
for ti = 1:N               

    term0 = omega ./ (2 .* Q(ti));
    gamma = 1 / (pi * Q(ti));

    % calculate for the real part
    if(ti == 1)
        Lambda = ones(N2, 1);
        termr_sub1(1) = 0;  
        termr_sub1(2:end) = term0(2:end) .* (ratio(2:end).^-gamma);  
    else
        termr(1) = 0; 
        termr(2:end) = term0(2:end) .* (ratio(2:end).^-gamma); 
        rs_r = rs_r - dt.*(termr + termr_sub1);
        termr_sub1 = termr;
        Beta = exp( -1 .* -0.5 .* rs_r );

        Lambda = (Beta + sigma2) ./ (Beta.^2 + sigma2);  % vector
    end 

    % calculate for the complex part  
    if(ginv == 1)  
        termi(1) = 0;
        termi(2:end) = (ratio(2:end).^(sign .* gamma) - 1) .* omega(2:end);
    else
        termi = (ratio.^(sign .* gamma) - 1) .* omega;
    end
    rs_i = rs_i - dt.*(termi + termi_sub1);
    termi_sub1 = termi;
    integrand = exp( 1i .* -0.5 .* rs_i );

    if(ginv == 1) 
        response(:,ti) = Lambda .* integrand;
    else        
        response(:,ti) = (1 ./ Lambda) .* integrand;
    end  
end % ti loop
2
matlabmathematical-optimizationnumerical-methods

2 Answers

3
votes

No, you cannot do so unless you know the "model" itself for this process. If you intend to treat the process as a complete black box, then it is impossible in general, although in any specific instance, anything can happen.

Even if you DO know the underlying process, then it may still not work, as any least squares estimator is dependent on the starting values, so if you do not have a good guess there, it may converge to the wrong set of parameters.

0
votes

It turns out that by using the mathematics of the model, the input can be estimated. This is not true in general, but for my problem it seems to work. The cumulative integral is eliminated by a partial derivative.

N = 1001;
q = zeros(N, 1);
q(1:200) = 55;
q(201:300) = 120;
q(301:400) = 70;
q(401:600) = 40;
q(601:800) = 100;
q(801:1001) = 70;
dt = 0.0042;
fs = 1 / dt;
wSize = 101;
Glim = 20;
ginv = 0;
R = get_response(N, q, dt, wSize, Glim, ginv);
rows = wSize; 
cols = N;
cut_val = 200;

imagLogR = imag(log(R));

Mderiv = zeros(rows, cols-2);
for k = 1:rows
   val = deriv_3pt(imagLogR(k,:), dt);
   val(val > cut_val) = 0;
   Mderiv(k,:) = val(1:end-1);
end

disp('Running iteration');
q0 = 10;
q1 = 500;
NN = cols - 2;
qout = zeros(NN, 1);
for k = 1:NN
    data = Mderiv(:,k); 
    qout(k) = fminbnd(@(q) curve_fit_to_get_q(q, dt, rows, data),q0,q1);
end

figure; plot(q); title('q value input as vector'); 
ylim([0 200]); xlim([0 1001])

figure;
plot(qout); title('Reconstructed q')
ylim([0 200]); xlim([0 1001])

Here are the supporting functions:

function output = deriv_3pt(x, dt)

% Function to compute dx/dt using the 3pt symmetrical rule
% dt is the timestep

N = length(x);
N0 = N - 1;
output = zeros(N0, 1);
denom = 2 * dt;

for k = 2:N0 
   output(k - 1) = (x(k+1) - x(k-1)) / denom;  
end


function sse = curve_fit_to_get_q(q, dt, rows, data)

fs = 1 / dt;
N2 = rows;
f = (fs/2)*linspace(0,1,N2);  % vector for frequency along cols
omega = 2 * pi .* f';
omegah = 2 * pi * f(end);
ratio = omega ./ omegah;

gamma = 1 / (pi * q);

termi = ((ratio.^(gamma)) - 1) .* omega;

Error_Vector =  termi - data;
sse = sum(Error_Vector.^2);

OriginalEstimated