Asked by Ahmad Hamad
on 11 Jul 2018

Hello,

I have data that represent 16 cosine shaped curves, but the data is in the form of scattered points (x_i y_i) i= 1,2,3 .... N. please relate to the attached plot. The points are not associated to the functions, further more, I don't have the exact functions but only their models, more specifically, each function follows the form: f_k = A_k cos(phi_0k + omega*x), omega is fixed for all the functions, only the amplitude A_k and the phase shift phi_k are specific to each function. Is there an easy way to associate each data point to one of the 16 curves?

Thanks in advance.

Answer by Anton Semechko
on 11 Jul 2018

Edited by Anton Semechko
on 11 Jul 2018

Accepted Answer

Below is an example where I use brute-force search to find an optimal set of sinusoid parameters that best fit an unorganized dataset; like the one you have. Fitted model parameters can be subsequently used to classify the individual data points. I omitted that latter part, but it wont be too difficult to implement.

SAMPLE SINUSOIDS

RESULTS OF THE SEARCH

function brute_force_sinusiod_fit_demo

% Generate a sample data set composed on N sinusoids with the same angular

% frequency but varying amplitudes and phases

% -------------------------------------------------------------------------

N=16; % # of sinusoids

w=3; % angular frequency

A_o=(1+rand(N,1))/2; % amplitudes in the range [0.5 1]

phi_o=rand(N,1)*pi; % phases in the range [0 pi]

f=@(x,A,phi) A*cos(w*x+phi);

[F,X]=deal(cell(N,1));

for n=1:N

X{n}=sort(pi*rand(1E3,1)); % unevenly spaced samples in time

F{n}=f(X{n},A_o(n),phi_o(n)); % measured signal

end

F=cell2mat(F);

X=cell2mat(X);

% Scramble the order of samples so it becomes difficult to tell which point

% came from what signal

id_prm=randperm(numel(F));

F=F(id_prm);

X=X(id_prm);

figure('color','w')

plot(X,F,'.','MarkerSize',5)

drawnow

% Attempt to recover parameters of the N curves from simulated data

% -------------------------------------------------------------------------

A_rng=[0.4 1.2]; % expected range of amplitudes

phi_rng=[0 pi]; % expected range of phase shifts

% Search grid

phi=linspace(phi_rng(1),phi_rng(2),200);

dphi=phi(2)-phi(1);

A=linspace(A_rng(1),A_rng(2),200);

dA=A(2)-A(1);

[phi,A]=meshgrid(phi,A);

% Search A-phi parameter space

Ng=numel(A);

MAE=median(abs(F))*ones(size(A)); % expected error

for i=1:Ng

fprintf('%u/%u\n',i,Ng)

Fi=f(X,A(i),phi(i)); % output of the model

dF=abs(F-Fi); % absolute residuals

dF(dF>5*dA)=[]; % remove data points that deviate from Fi by more than 5*dA

if isempty(dF), continue; end

dF(dF>3*median(dF))=[];

if isempty(dF), continue; end

MAE(i)=median(dF); % quality of the fit

end

% Extract N best fits

[A_fit,phi_fit]=deal(zeros(N,1));

MAE2=MAE;

for n=1:N

% Absolute minimum

[~,id_min]=min(MAE2(:));

A_fit(n)=A(id_min);

phi_fit(n)=phi(id_min);

% Set neighbourhood around absolute minimum to Inf

D=((A-A_fit(n))/dA).^2 + ((phi-phi_fit(n))/dphi).^2;

MAE2(D<=(2^2))=Inf;

end

% Visualize

% -------------------------------------------------------------------------

figure('color','w')

subplot(2,1,1)

imagesc(phi_rng,A_rng,MAE)

hold on

plot(phi_fit,A_fit,'or','MarkerSize',10,'MarkerFaceColor','none')

set(gca,'YDir','normal','XLim',phi_rng+dphi*[-1 1],'YLim',A_rng+dA*[-1 1],'FontSize',15)

axis equal

xlabel('phase','FontSize',20)

ylabel('amplitude','FontSize',20)

title('Best Fit Model Parameters','FontSize',25)

subplot(2,1,2)

imagesc(phi_rng,A_rng,MAE)

hold on

plot(phi_o,A_o,'or','MarkerSize',10,'MarkerFaceColor','none')

set(gca,'YDir','normal','XLim',phi_rng+dphi*[-1 1],'YLim',A_rng+dA*[-1 1],'FontSize',15)

axis equal

xlabel('phase','FontSize',20)

ylabel('amplitude','FontSize',20)

title('Actual Model Parameters','FontSize',25)

Ahmad Hamad
on 12 Jul 2018

@Anton Semechko,

Thank you for the code, Hough transform seems to be the way to go but the parameter space is huge, keep in mind that in your example you assumed w is given but that is not the case. It's common between all the signals, that is true, but it is also unknown.

Anton Semechko
on 12 Jul 2018

Yeah, that problem is significantly more challenging, but can be tackled using a similar approach.

There are various heuristics you can use along the way to speed-up the search. Assuming sinusoids have different amplitudes, you can fit them in decreasing order of amplitude, and remove data points best explained by the fitted model along the way. Maxim amplitude can be estimated directly from the data without any optimization. Frequency and phase of the signal with maximum amplitude can be optimized using the same general strategy I used in the demo above. Frequency found for this first signal can either be fixed when fitting remaining sinusoids, or you can implement a more robust strategy that allows you to test this solution on some sub-set of data.

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Answer by Matt J
on 11 Jul 2018

dpb
on 11 Jul 2018

Ah...that's not a bad idea at all...

Ahmad Hamad
on 12 Jul 2018

Matt J
on 12 Jul 2018

You could start with coarse sampling, e.g., 10x10x10 and gradually refine.

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