I am kind of new to analyzing circular data and I have a quite complex problem.
I'm trying to correlate 2 multidimensional vectors (N observations x 2 dimensions) where one dimension is circular (i.e. angles in radians).
I have found the corr2 function for 2D correlations which seems to work fine (http://www.mathworks.com/help/images/ref/corr2.html), I am however uncertain if I can use it on circular data (there isn't much information on it in the Matlab Documentation).
I found documentation on the Origin software which states that the correlations can be circular (http://www.originlab.com/doc/Origin-Help/2D-Correlation) but I have no idea if it this generalizes to Malab.
I tried two approaches:
1- I input the linear and circular data as is:
r = corr2([Theta_X Amplitude_X], [Theta_Y Amplitude_Y]);
which yields very good results but I am unsure if they are valid.
2- I "linearized" the data by converting the polar coordinates into Cartesian (pol2cart):
r = corr2([x_X y_X], [x_Y y_Y]);
which yields noisier results.
My question is thus: can I use the corr2 function on circular data? If not, can anyone point me in the right direction?
Related
I have the following problem. I have a N x N real matrix called Z(x; t), where x and t might be vectors in general. I have N_s observations (x_k, Z_k), k=1,..., N_s and I'd like to find the vector of parameters t that better approximates the data in the least square sense, which means I want t that minimizes
S(t) = \sum_{k=1}^{N_s} \sum_{i=1}^{N} \sum_{j=1}^N (Z_{k, i j} - Z(x_k; t))^2
This is in general a non-linear fitting of a matrix function. I'm only finding examples in which one has to fit scalar functions which are not immediately generalizable to a matrix function (nor a vector function). I tried using the scipy.optimize.leastsq function, the package symfit and lmfit, but still I don't manage to find a solution. Eventually, I'm ending up writing my own code...any help is appreciated!
You can do curve-fitting with multi-dimensional data. As far as I am aware, none of the low-level algorithms explicitly support multidimensional data, but they do minimize a one-dimensional array in the least-squares sense. And the fitting methods do not really care about the "independent variable(s)" x except in that they help you calculate the array to be minimized - perhaps to calculate a model function to match to y data.
That is to say: if you can write a function that would take the parameter values and calculate the matrix to be minimized, just flatten that 2-d (on n-d) array to one dimension. The fit will not mind.
I have a 3d matrix of EEG data containing (time x electrode x trial). I would like to collapse the data across trials to obtain average trial values for each electrode and time point.
Can someone please walk be through how to do this in MATLAB?
mean will do it.
meanData = mean(data, 3)
In general, the MATLAB documentation is quite good. Googling what you want done and adding "MATLAB" to your query will produce pretty good results. If you know what function you are looking for, you can type help <function name> in your MATLAB interpreter and it will show you the docs.
Here is the given system I want to plot and obtain the vector field and the angles they make with the x axis. I want to find the index of a closed curve.
I know how to do this theoretically by choosing convenient points and see how the vector looks like at that point. Also I can always use
to compute the angles. However I am having trouble trying to code it. Please don't mark me down if the question is unclear. I am asking it the way I understand it. I am new to matlab. Can someone point me in the right direction please?
This is a pretty hard challenge for someone new to matlab, I would recommend taking on some smaller challenges first to get you used to matlab's conventions.
That said, Matlab is all about numerical solutions so, unless you want to go down the symbolic maths route (and in that case I would probably opt for Mathematica instead), your first task is to decide on the limits and granularity of your simulated space, then define them so you can apply your system of equations to it.
There are lots of ways of doing this - some more efficient - but for ease of understanding I propose this:
Define the axes individually first
xpts = -10:0.1:10;
ypts = -10:0.1:10;
tpts = 0:0.01:10;
The a:b:c syntax gives you the lower limit (a), the upper limit (c) and the spacing (b), so you'll get 201 points for the x. You could use the linspace notation if that suits you better, look it up by typing doc linspace into the matlab console.
Now you can create a grid of your coordinate points. You actually end up with three 3d matrices, one holding the x-coords of your space and the others holding the y and t. They look redundant, but it's worth it because you can use matrix operations on them.
[XX, YY, TT] = meshgrid(xpts, ypts, tpts);
From here on you can perform whatever operations you like on those matrices. So to compute x^2.y you could do
x2y = XX.^2 .* YY;
remembering that you'll get a 3d matrix out of it and all the slices in the third dimension (corresponding to t) will be the same.
Some notes
Matlab has a good builtin help system. You can type 'help functionname' to get a quick reminder in the console or 'doc functionname' to open the help browser for details and examples. They really are very good, they'll help enormously.
I used XX and YY because that's just my preference, but I avoid single-letter variable names as a general rule. You don't have to.
Matrix multiplication is the default so if you try to do XX*YY you won't get the answer you expect! To do element-wise multiplication use the .* operator instead. This will do a11 = b11*c11, a12 = b12*c12, ...
To raise each element of the matrix to a given power use .^rather than ^ for similar reasons. Likewise division.
You have to make sure your matrices are the correct size for your operations. To do elementwise operations on matrices they have to be the same size. To do matrix operations they have to follow the matrix rules on sizing, as will the output. You will find the size() function handy for debugging.
Plotting vector fields can be done with quiver. To plot the components separately you have more options: surf, contour and others. Look up the help docs and they will link to similar types. The plot family are mainly about lines so they aren't much help for fields without creative use of the markers, colours and alpha.
To plot the curve, or any other contour, you don't have to test the values of a matrix - it won't work well anyway because of the granularity - you can use the contour plot with specific contour values.
Solving systems of dynamic equations is completely possible, but you will be doing a numeric simulation and your results will again be subject to the granularity of your grid. If you have closed form solutions, like your phi expression, they may be easier to work with conceptually but harder to get working in matlab.
This kind of problem is tractable in matlab but it involves some non-basic uses which are pretty hard to follow until you've got your head round Matlab's syntax. I would advise to start with a 2d grid instead
[XX, YY] = meshgrid(xpts, ypts);
and compute some functions of that like x^2.y or x^2 - y^2. Get used to plotting them using quiver or plotting the coordinates separately in intensity maps or surfaces.
Due to the nature of my problem, I want to evaluate the numerical implementations of the Radon transform in Matlab (i.e. different interpolation methods give different numerical values).
while trying to code my own Radon, and compare it to Matlab's output, I found out that my radon projection sizes are different than Matlab's.
So a bit of intuition of how I compute the amount if radon samples needed. Let's do the 2D case.
The idea is that the maximum size would be when the diagonal (in a rectangular shape at least) part is proyected in the radon transform, so diago=sqrt(size(I,1),size(I,2)). As we dont wan nothing out, n_r=ceil(diago). n_r should be the amount of discrete samples of the radon transform should be to ensure no data is left out.
I noticed that Matlab's radon output is always even, which makes sense as you would want a "ray" through the rotation center always. And I noticed that there are 2 zeros in the endpoints of the array in all cases.
So in that case, n_r=ceil(diago)+mod(ceil(diago)+1,2)+2;
However, it seems that I get small discrepancies with Matlab.
A MWE:
% Try: 255,256
pixels=256;
I=phantom('Modified Shepp-Logan',pixels);
rd=radon(I,pi/4);
size(rd,1)
s=size(I);
diagsize=sqrt(sum(s.^2));
n_r=ceil(diagsize)+mod(ceil(diagsize)+1,2)+2
rd=
367
n_r =
365
As Matlab's Radon transform is a function I can not look into, I wonder why could it be this discrepancy.
I took another look at the problem and I believe this is actually the right answer. From the "hidden documentation" of radon.m (type in edit radon.m and scroll to the bottom)
Grandfathered syntax
R = RADON(I,THETA,N) returns a Radon transform with the
projection computed at N points. R has N rows. If you do not
specify N, the number of points the projection is computed at
is:
2*ceil(norm(size(I)-floor((size(I)-1)/2)-1))+3
This number is sufficient to compute the projection at unit
intervals, even along the diagonal.
I did not try to rederive this formula, but I think this is what you're looking for.
This is a fairly specialized question, so I'll offer up an idea without being completely sure it is the answer to your specific question (normally I would pass and let someone else answer, but I'm not sure how many readers of stackoverflow have studied radon). I think what you might be overlooking is the floor function in the documentation for the radon function call. From the doc:
The radial coordinates returned in xp are the values along the x'-axis, which is
oriented at theta degrees counterclockwise from the x-axis. The origin of both
axes is the center pixel of the image, which is defined as
floor((size(I)+1)/2)
For example, in a 20-by-30 image, the center pixel is (10,15).
This gives different behavior for odd- or even-sized problems that you pass in. Hence, in your example ("Try: 255, 256"), you would need a different case for odd versus even, and this might involve (in effect) padding with a row and column of zeros.
I have an unknown scalar fonction defined into a partial space (a pyramid portion), for this function, I have several measurements points into the coordonates mesurePoints, where the mesure mesure is known :
size(mesurePoints) = [n 3]
size(mesure) = n
I also have my space discretized into a clood of equidistant points wich I'll call interpolPoints,
I would like to obtain interpolated values interp_mesure on the points interpolPoints based on my measurements mesure on the points mesurePoints.
I tried to use interp3,
interp_mesure = interp3(...
mesurePoints(:,1),mesurePoints(:,2),mesurePoints(:,2),...
mesure,...
interpolPoints(:,1),interpolPoints(:,2),interpolPoints(:,3));
but I get the error that V (mesure) should be a 3D array, but I am confuse, my data isn't 3D, it is 3D dependant, but it's a scalar data, how can I proceed? Is interpol3 not adapted to my problem?
Edit 1 : Here is a similar problem to illustrate mine : how do you interpolate temperature in a volume if you have some temperature measurements in this volume?
Edit 2 : as no matlab solution have come to mind yet, I use a hand-made interpolation weighted by inverse distance with a power factor, the result is good close to points but as my points are quite scattered, the result is not good in empty areas.
I'm having trouble understanding exactly what is your data, but I get the impression that maybe interp1 or interp2 would be better suited to your needs, as your data isn't organized as a 3-D array