Determine the Signal Curve from Parameters of a Power Curve by Noisy Measurement

Question

I have a class of signals described by function: $$ f(inc,d,t)=inc\cdot t^d $$ where inc and d have a finite set of values like 1, 2, 3, i.e. $$ inc, d\in \left \{1,2,3 \right \} $$ and $$ 0\leq t<1 $$

Example plots:

I need to determine parameters inc and d given a discrete signal of f obscured by additive white Gaussian noise. Preferably an algorithm capable of working at real time.

Some candidates are:

• sole differentiation (this amplifies noise)
• filtering + differentiation (loses information)
• curve fitting (least squares?)

Polynomial fitting seems to be an overkill, because function is known and is monotonic. Isotonic regression seems too general, because it is applicable to all increasing/decreasing functions. Maybe there is a better solution when we have a concrete function and a set of possible parameters.

Answer 1

Since your domain for the parameters is limited (Only 9 options) the best way for White Noise would be going through them and pick then one with the least Mean Squared Error (MSE) which is the parameter to minimize for AWGN.

in MATLAB it will be something like:

function [ paramAlpha, paramBeta ] = EstimateModelParameters( vT, vY )

vParamAlpha = [1, 2, 3];
vParamBeta  = [1, 2, 3];

bestMse = 1e50;

vX = zeros(size(vY, 1), 1, class(vY));

for ii = 1:length(vParamAlpha)
    currParamAlpha = vParamAlpha(ii);
    for jj = 1:length(vParamBeta)
        currParamBeta = vParamBeta(jj);
        vX(:) = currParamAlpha * (vT .^ currParamBeta);
        currMse = mean((vX - vY) .^ 2);
        if(currMse < bestMse)
            bestMse     = currMse;
            paramAlpha  = currParamAlpha;
            paramBeta   = currParamBeta;
        end
    end
end


end

In a simple performance simulation I got the following:

Which means that for STD in reasonable value for this case (Since your maximum value is around ~3) your performance will be great.

The code is easily adaptable to any values of the parameters you'd like.
Though I think such brute force methods are reasonable for up to ~100 combinations.

Another thing you might try is use Affine Model on the log of the values. Then apply some "Rounding" to the values. Though this will change the properties of the noise, in real world it seems to be effective way.

The full code is available on my StackExchange Signal Processing Q70753 GitHub Repository (Look at the SignalProcessing\Q70753 folder).

Answer 2

To me polynomial fitting is not overkill, as it uses the maximal knowledge on the data. A monic polynomial (only one $x^\delta$ term) could be a good approach, especially with robust regression, for maximum noise resistance. If $\mathrm{inc}$ and $d$ are in moderate-size finite subsets, performing all combinations and choosing the best of all (with an appropriate metric) would be a way to go.

You can also perform a logarithmic transformation, impose constraints on coefficients.

Question :

• are all unknowns supposed to be integers?

PS : I may come back later with simulations.