In my current analysis, I am trying to multiply a matrix (flm), of dimension nxm, with the inverse of a matrix nxmxp, and then use this result to multiply it by the inverse of the matrix (flm).
I was trying using the following code:
flm = repmat(Data.fm.flm(chan,:),[1 1 morder]); %chan -> is a vector 1by3
A = (flm(:,:,:)/A_inv(:,:,:))/flm(:,:,:);
However. due to the problem of dimensions, I am getting the following error message:
Error using ==> mrdivide
Inputs must be 2-D, or at least one
input must be scalar.
To compute elementwise RDIVIDE, use
RDIVIDE (./) instead.
I have no idea on how to proceed without using a for loop, so anyone as any suggestion?
I think you are looking for a way to conveniently multiply matrices when one is of higher dimensionality than the other. In that case you can use bxsfun to automatically 'expand' the smaller matrix.
x = rand(3,4);
y = rand(3,4,5);
bsxfun(#times,x,y)
It is quite simple, and very efficient.
Make sure to check out doc bsxfun for more examples.
Related
I have a linear system A*x=b. Here x is the unknown value, so I have to solve for x. A is a sparse matrix whose diagonal and off-diagonal elements have some non-zero values. The other elements is zero or close to zero.
Using MATLAB, I have two options
x = inv(A) * b
x = A \ b
But both give me NaNs in the result. I know this will happen because A is a sparse matrix. So, I tried pinv() which is pseudo-inverse. This time I got some results.
Is it OK to use pseudo-inverse in a situation like this, where inverse has failed to produce a result? What type of result does pseudo inverse produce? Is it reliable or it is a form of error?
I am using Matlab R2011a. I have a matrix A which is of size 27*3355432. This matrix has a lot of zeros. I need to compute the comand eig(A'*A) but I can't even do the A'*A. I have tried using the sparse matrix B = sparse(A)and then computing B'*B but I get the error:
??? Error using ==> mtimes
Both logical inputs must be scalar.
To compute elementwise TIMES, use TIMES (.*) instead
Truth is I am not a Matlab expert. Is there a way to produce such a database?
Here is my testing function:
function diff = svdtester()
y = rand(500,20);
[U,S,V] = svd(y);
%{
y = sprand(500,20,.1);
[U,S,V] = svds(y);
%}
diff_mat = y - U*S*V';
diff = mean(abs(diff_mat(:)));
end
There are two very similar parts: one finds the SVD of a random matrix, the other finds the SVD of a random sparse matrix. Regardless of which one you choose to comment (right now the second one is commented-out), we compute the difference between the original matrix and the product of its SVD components and return that average absolute difference.
When using rand/svd, the typical return (mean error) value is around 8.8e-16, basically zero. When using sprand/svds, the typical return values is around 0.07, which is fairly terrible considering the sparse matrix is 90% 0's to start with.
Am I misunderstanding how SVD should work for sparse matrices, or is something wrong with these functions?
Yes, the behavior of svds is a little bit different from svd. According to MATLAB's documentation:
[U,S,V] = svds(A,...) returns three output arguments, and if A is m-by-n:
U is m-by-k with orthonormal columns
S is k-by-k diagonal
V is n-by-k with orthonormal columns
U*S*V' is the closest rank k approximation to A
In fact, usually k will be somethings about 6, so you will get rather "rude" approximation. To get more exact approximation specify k to be min(size(y)):
[U, S, V] = svds(y, min(size(y)))
and you will get error of the same order of magnitude as in case of svd.
P.S. Also, MATLAB's documentations says:
Note svds is best used to find a few singular values of a large, sparse matrix. To find all the singular values of such a matrix, svd(full(A)) will usually perform better than svds(A,min(size(A))).
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MATLAB is running out of memory but it should not be
I want to perform PCA analysis on a huge data set of points. To be more specific, I have size(dataPoints) = [329150 132] where 328150 is the number of data points and 132 are the number of features.
I want to extract the eigenvectors and their corresponding eigenvalues so that I can perform PCA reconstruction.
However, when I am using the princomp function (i.e. [eigenVectors projectedData eigenValues] = princomp(dataPoints); I obtain the following error :
>> [eigenVectors projectedData eigenValues] = princomp(pointsData);
Error using svd
Out of memory. Type HELP MEMORY for your options.
Error in princomp (line 86)
[U,sigma,coeff] = svd(x0,econFlag); % put in 1/sqrt(n-1) later
However, if I am using a smaller data set, I have no problem.
How can I perform PCA on my whole dataset in Matlab? Have someone encountered this problem?
Edit:
I have modified the princomp function and tried to use svds instead of svd, but however, I am obtaining pretty much the same error. I have dropped the error bellow :
Error using horzcat
Out of memory. Type HELP MEMORY for your options.
Error in svds (line 65)
B = [sparse(m,m) A; A' sparse(n,n)];
Error in princomp (line 86)
[U,sigma,coeff] = svds(x0,econFlag); % put in 1/sqrt(n-1) later
Solution based on Eigen Decomposition
You can first compute PCA on X'X as #david said. Specifically, see the script below:
sz = [329150 132];
X = rand(sz);
[V D] = eig(X.' * X);
Actually, V holds the right singular vectors, and it holds the principal vectors if you put your data vectors in rows. The eigenvalues, D, are the variances among each direction. The singular vectors, which are the standard deviations, are computed as the square root of the variances:
S = sqrt(D);
Then, the left singular vectors, U, are computed using the formula X = USV'. Note that U refers to the principal components if your data vectors are in columns.
U = X*V*S^(-1);
Let us reconstruct the original data matrix and see the L2 reconstruction error:
X2 = U*S*V';
L2ReconstructionError = norm(X(:)-X2(:))
It is almost zero:
L2ReconstructionError =
6.5143e-012
If your data vectors are in columns and you want to convert your data into eigenspace coefficients, you should do U.'*X.
This code snippet takes around 3 seconds in my moderate 64-bit desktop.
Solution based on Randomized PCA
Alternatively, you can use a faster approximate method which is based on randomized PCA. Please see my answer in Cross Validated. You can directly compute fsvd and get U and V instead of using eig.
You may employ randomized PCA if the data size is too big. But, I think the previous way is sufficient for the size you gave.
My guess is that you have a huge data set. You don't need all of the svd coefficients. In this case, use svds instead of svd :
Taken directly from Matlab help:
s = svds(A,k) computes the k largest singular values and associated singular vectors of matrix A.
From your question, I understand that you don't call svd directly. But you might as well take a look at princomp (It is editable!) and alter the line that calls it.
You probably needed to calculate an n by n matrix in your computation somehow that is to say:
329150 * 329150 * 8btyes ~ 866GB`
of space which explains why you're getting a memory error. There seems to be an efficient way to calculate pca using princomp(X, 'econ') which I suggest you give it a try.
More on this in stackoverflow and mathworks..
Manually compute X'X (132x132) and svd on it. Or find NIPALS script.
I have a sparse matrix S.
I perform the following operation
D1 = diag(sum(S,2)), basically forming a diagonal matrix.
Now I need to perform (D1)^(-0.5), but I get an error
"Error using mpower, use full(x)^full(y)"
Converting to full will defeat the purpose of using a sparse matrix.
Any advice will be very helpful.
Raising a diagonal matrix to a power can be done simply by doing the operation on the diagonal elements elementwise... so:
D1_diagonal_elements = sum(S,2);
your_result = diag(D1_diagonal_elements .^ (-0.5));