Using MATLAB I apply Matching Pursuit to approximate a signal. My problem is that I struggle to visualize the time-frequency representation of the selected atoms. I'm trying to produce a Wigner plot similar to the following image (source).
I have looked into the Wavelet Toolbox, Signal Processing Toolbox as well as the open source Time-Frequency Toolbox, but I'm possibly just using the wrong parameters, since my experience with signal processing is quite limited.
Example
Using this data my goal is to reproduce the plot from above.
% fit the signal using MP
itermax = 50;
signal = load('signal.txt');
dict = wmpdictionary(length(signal));
[signal_fit, r, coeff, iopt, qual, X] = wmpalg('OMP', signal, dict, ...
'itermax', itermax);
% wigner plot of the simulated signal
tfrwv(signal_fit) % wigner-ville function from time-frequency toolbox
% wigner plot of each atom
atoms = full(dict(:, iopt)) % selected atoms
for i = 1:itermax
tfrwv(atoms(:, i))
end
Unfortunately, none of the resulting plots comes close to the target visualization. Note, that in the example I use tfrwv with standard parameters which I tweak with the GUI that it opens.
I'd greatly appreciate your help.
Update
I think I have now understood that one needs to use Gabor atoms to get blobs with shapes resembling stretched gaussians. Unfortunately, there are no Gabor functions in the predefined dicts of the Signal Processing Toolbox. However, this question helped me in implementing the needed dictionaries, such that I get atoms which look quite similar to the example:
Since my plots come close but are not perfect, there are still two questions open:
Can all of the blobs that we see in the first example be modeled by Gabor atoms alone, or do I need another dictionary of functions?
How can I combine the indidividual imagesc plots into a single visualization?
To answer your second question 'How can I combine the indidividual imagesc plots into a single visualization?'
If you have multiple 2d matrices that you want to superimpose and display using imagesc, I would suggest taking the element-wise maximum.
For example, I generate two 31x31 grids with gaussians with different mean and variance.
function F = generate2dGauss(mu, Sigma)
x1 = -3:.2:3; x2 = -3:.2:3;
[X1,X2] = meshgrid(x1,x2);
F = mvnpdf([X1(:) X2(:)],mu,Sigma);
F = reshape(F,length(x2),length(x1));
end
F1 = generate2dGauss([1 1], [.25 .3; .3 1]);
F2 = generate2dGauss([-1 -1], [.1 .1; .1 1]);
I can plot them with subplots as in your example,
figure;
subplot(1,2,1);
title('Atom 1');
imagesc(F1);
subplot(1,2,2);
title('Atom 2');
imagesc(F2);
Or I can plot the per element maximum of the two grids.
figure;
title('Both Atoms');
imagesc(max(F1, F2));
You can also experiment with element-wise means, sums, etc, but based on the example you give, I think maximum will give you the cleanest looking result.
Possible pros and cons of different functions:
Maximum will work best if your atoms always have zero-valued backgrounds and no negative values. If the background is zero-valued, but the atoms also contain negative values, the negative values may be covered up by the background of other atoms. If your atom's overlap, the higher value will of course dominate.
Mean will make your peaks less high, but may be more intuitive where you have overlap between atoms.
Sum will make overlapping areas larger valued.
If you have non-zero backgrounds, you could also try using logical indexing. You would have to make some decisions about what to do in overlapping areas, but it would make it easy to filter out backgrounds.
Q. How can I combine the indidividual imagesc plots into a single visualization?
A. Use subplot to draw multiple plots, find below sample with 2 by 2 plots in a figure. Change your equations in code
x = linspace(-5,5);
y1 = sin(x);
subplot(2,2,1)
plot(x,y1)
title('First subplot')
y2 = sin(2*x);
subplot(2,2,2)
plot(x,y2)
title('Second subplot')
y3 = sin(4*x);
subplot(2,2,3)
plot(x,y3)
title('Third subplot')
y4 = sin(6*x);
subplot(2,2,4)
plot(x,y4)
title('Fourth subplot')
Related
I have a geographically distributed data set with X-coordinate, Y-coordinate and corresponding target value of interest D. That is, my data set consists from three vectors: X, Y, D.
Now what I would like to do, is interpolate and extrapolate the target variable D over a coordinate grid of interest. The griddata-function in Matlab seems to be able to help me in this problem, but it only does interpolation over the convex hull determined by my data set.
What I would like to do, is to also extrapolate the data D to any rectangular coordinate grid of interest like so:
I have tried using functions like interp2 and griddedInterpolant, but these functions seem to require that I provide the known data as monotonic matrices (using e.g. meshgrid). That is, if I have understood correctly, I must provide X,Y,D as 2D-grids. But they are not grids, they are non-monotonic vectors.
So how can I proceed?
You can use griddata with option 'v4' that uses biharmonic spline interpolation.
Unlike the other methods, this interpolation is not based on a triangulation.
Other option is using scatteredInterpolant. You can choose to use any of 'linear' or 'nearest' methods for extrapolation
I found out one way using scatteredInterpolant:
xy = -2.5 + 5*gallery('uniformdata',[200 2],0);
x = xy(:,1);
y = xy(:,2);
v = x.*exp(-x.^2-y.^2);
F1 = scatteredInterpolant(x,y,v, 'natural');
[xq,yq] = ndgrid(-5:.1:5) % Make the grid
vq1 = F1(xq,yq); % Evaluate function values at grid of interest
surf(xq,yq,vq1)
hold on
plot3(x,y,v, 'ro', 'MarkerFaceColor', 'red')
xlabel('X')
ylabel('Y')
zlabel('V')
title('Interpolation and exrapolation based on scattered data')
The problem is, you can do extrapolation with only three methods: 'linear', 'nearest', 'natural'.
i have this formula to find fourier series in matlab
f(n)= (f(t),exp(jnt))
and the inner product is: =(1\2*pi)integral((between pi and
-pi)(f1*f2'*dt))
now i want to find fourier coefficients in matlab for this vector(f(t)=t)
where t is a vector that it's lenght is 1000.
i need to find the 2k+1 fourier coefficients by approximate amount when k=2 , which means n=(-2,-1,0,1,2) and then Compare it to the Analytical Calculation.
this is what i did so far:
clc
t = linspace(-pi,pi,1000);
f=t;
plot(t,f); hold all;
dt=2*pi/1000;
cnPlusVal=0;
cnMinusVal=0;
FourierS1=0;
FourierS2=0;
k=2;
for l = 1:k
cnPlusVal=cnPlusVal+f.*exp(-i*l*t)*(dt/2*pi) ;
cnMinusVal=cnMinusVal+f.*exp(i*l*t)*(dt/2*pi);
FourierS1=FourierS1+cnPlusVal.*(exp(i*l*t));
FourierS2=FourierS2+cnMinusVal.*(exp(i*-l*t));
end
now in order to Compare it to the Analytical Calculation i need to plot the forier series .. any help of how to do this in the same graph for f ?
You have two problems to deal with here:
your first plot is on a completely different scale when compared to the output series;
you cannot infer a good axis scope using the limits of the series, because they contain complex numbers.
Here is the workaround I propose you:
figure();
plot(t,FourierS1);
x_lim = get(gca(),'XLim');
y_lim = get(gca(),'YLim');
hold on;
plot(t,f);
set(gca(),'XLim',x_lim,'YLim',y_lim);
hold off;
Basically:
you plot the Fourier serie;
you retain the current x-axis and y-axis limits of the plot;
you plot f over the current plot using the hold function properly;
you revert the plot limits to the previous scope.
Here is the output:
I have various plots (with hold on) as show in the following figure:
I would like to know how to find equations of these six curves in Matlab. Thanks.
I found interactive fitting tool in Matlab simple and helpful, though somewhat limited in scope:
The graph above seems to be linear interpolation. Given vectors X and Y of data, where X contains the arguments and Y the function points, you could do
f = interp1(X, Y, x)
to get the linearly interpolated value f(x). For example if the data is
X = [0 1 2 3 4 5];
Y = [0 1 4 9 16 25];
then
y = interp1(X, Y, 1.5)
should give you a very rough approximation to 1.5^2. interp1 will match the graph exactly, but you might be interested in fancier curve-fitting operations, like spline approximations etc.
Does rxns stand for reactions? In that case, your curves are most likely exponential. An exponential function has the form: y = a*exp(b * x) . In your case, y is the width of mixing zone, and x is the time in years. Now, all you need to do is run exponential regression in Matlab to find the optimal values of parameters a and b, and you'll have your equations.
The advice, though there might be better answer, from me is: try to see the rate of increase in the curve. For example, cubic is more representative than quadratic if the rate of increase seems fast and find the polynomial and compute the deviation error. For irregular curves, you might try spline fitting. I guess there is also a toolbox in matlab for spline fitting.
There is a way to extract information with the current figure handle (gcf) from you graph.
For example, you can get the series that were plotted in a graph:
% Some figure is created and data are plotted on it
figure;
hold on;
A = [ 1 2 3 4 5 7] % Dummy data
B = A.*A % Some other dummy data
plot(A,B);
plot(A.*3,B-1);
% Those three lines of code will get you series that were plotted on your graph
lh=findall(gcf,'type','line'); % Extract the plotted line from the figure handle
xp=get(lh,'xdata'); % Extract the Xs
yp=get(lh,'ydata'); % Extract the Ys
There must be other informations that you can get from the "findall(gcf,...)" methods.
I 'm having a problem with creating a joint density function from data. What I have is queue sizes from a stock as two vectors saved as:
X = [askQueueSize bidQueueSize];
I then use the hist3-function to create a 3D histogram. This is what I get:
http://dl.dropbox.com/u/709705/hist-plot.png
What I want is to have the Z-axis normalized so that it goes from [0 1].
How do I do that? Or do someone have a great joint density matlab function on stock?
This is similar (How to draw probability density function in MatLab?) but in 2D.
What I want is 3D with x:ask queue, y:bid queue, z:probability.
Would greatly appreciate if someone could help me with this, because I've hit a wall over here.
I couldn't see a simple way of doing this. You can get the histogram counts back from hist3 using
[N C] = hist3(X);
and the idea would be to normalise them with:
N = N / sum(N(:));
but I can't find a nice way to plot them back to a histogram afterwards (You can use bar3(N), but I think the axes labels will need to be set manually).
The solution I ended up with involves modifying the code of hist3. If you have access to this (edit hist3) then this may work for you, but I'm not really sure what the legal situation is (you need a licence for the statistics toolbox, if you copy hist3 and modify it yourself, this is probably not legal).
Anyway, I found the place where the data is being prepared for a surf plot. There are 3 matrices corresponding to x, y, and z. Just before the contents of the z matrix were calculated (line 256), I inserted:
n = n / sum(n(:));
which normalises the count matrix.
Finally once the histogram is plotted, you can set the axis limits with:
xlim([0, 1]);
if necessary.
With help from a guy at mathworks forum, this is the great solution I ended up with:
(data_x and data_y are values, which you want to calculate at hist3)
x = min_x:step:max_x; % axis x, which you want to see
y = min_y:step:max_y; % axis y, which you want to see
[X,Y] = meshgrid(x,y); *%important for "surf" - makes defined grid*
pdf = hist3([data_x , data_y],{x y}); %standard hist3 (calculated for yours axis)
pdf_normalize = (pdf'./length(data_x)); %normalization means devide it by length of
%data_x (or data_y)
figure()
surf(X,Y,pdf_normalize) % plot distribution
This gave me the joint density plot in 3D. Which can be checked by calculating the integral over the surface with:
integralOverDensityPlot = sum(trapz(pdf_normalize));
When the variable step goes to zero the variable integralOverDensityPlot goes to 1.0
Hope this help someone!
There is a fast way how to do this with hist3 function:
[bins centers] = hist3(X); % X should be matrix with two columns
c_1 = centers{1};
c_2 = centers{2};
pdf = bins / (sum(sum(bins))*(c_1(2)-c_1(1)) * (c_2(2)-c_2(1)));
If you "integrate" this you will get 1.
sum(sum(pdf * (c_1(2)-c_1(1)) * (c_2(2)-c_2(1))))
I have a Matlab figure I want to use in a paper. This figure contains multiple cdfplots.
Now the problem is that I cannot use the markers because the become very dense in the plot.
If i want to make the samples sparse I have to drop some samples from the cdfplot which will result in a different cdfplot line.
How can I add enough markers while maintaining the actual line?
One method is to get XData/YData properties from your curves follow solution (1) from #ephsmith and set it back. Here is an example for one curve.
y = evrnd(0,3,100,1); %# random data
%# original data
subplot(1,2,1)
h = cdfplot(y);
set(h,'Marker','*','MarkerSize',8,'MarkerEdgeColor','r','LineStyle','none')
%# reduced data
subplot(1,2,2)
h = cdfplot(y);
set(h,'Marker','*','MarkerSize',8,'MarkerEdgeColor','r','LineStyle','none')
xdata = get(h,'XData');
ydata = get(h,'YData');
set(h,'XData',xdata(1:5:end));
set(h,'YData',ydata(1:5:end));
Another method is to calculate empirical CDF separately using ECDF function, then reduce the results before plotting with PLOT.
y = evrnd(0,3,100,1); %# random data
[f, x] = ecdf(y);
%# original data
subplot(1,2,1)
plot(x,f,'*')
%# reduced data
subplot(1,2,2)
plot(x(1:5:end),f(1:5:end),'r*')
Result
I know this is potentially unnecessary given MATLAB's built-in functions (in the Statistics Toolbox anyway) but it may be of use to other viewers who do not have access to the toolbox.
The empirical CMF (CDF) is essentially the cumulative sum of the empirical PMF. The latter is attainable in MATLAB via the hist function. In order to get a nice approximation to the empirical PMF, the number of bins must be selected appropriately. In the following example, I assume that 64 bins is good enough for your data.
%# compute a histogram with 64 bins for the data points stored in y
[f,x]=hist(y,64);
%# convert the frequency points in f to proportions
f = f./sum(f);
%# compute the cumulative sum of the empirical PMF
cmf = cumsum(f);
Now you can choose how many points you'd like to plot by using the reduced data example given by yuk.
n=20 ; % number of total data markers in the curve graph
M_n = round(linspace(1,numel(y),n)) ; % indices of markers
% plot the whole line, and markers for selected data points
plot(x,y,'b-',y(M_n),y(M_n),'rs')
verry simple.....
try reducing the marker size.
x = rand(10000,1);
y = x + rand(10000,1);
plot(x,y,'b.','markersize',1);
For publishing purposes I tend to use the plot tools on the figure window. This allow you to tweak all of the plot parameters and immediately see the result.
If the problem is that you have too many data points, you can:
1). Plot using every nth sample of the data. Experiment to find an n that results in the look you want.
2). I typically fit curves to my data and add a few sparsely placed markers to plots of the fits to differentiate the curves.
Honestly, for publishing purposes I have always found that choosing different 'LineStyle' or 'LineWidth' properties for the lines gives much cleaner results than using different markers. This would also be a lot easier than trying to downsample your data, and for plots made with CDFPLOT I find that markers simply occlude the stairstep nature of the lines.