I have a square matrix of size C by C and I want to build a diagonal block matrix with it repeated N=2C(1+C) times. The problem is that the value of C can change, and so I don't know if I could use blkdiag, as I don't know the number of parameters I should enter because the size of the matrix is a variable that the user chooses. How could I do this in MATLAB?
you can use
M = kron(eye(N),A);
where A is the CxC matrix that repeats N times....
Because this will scale rather quickly , for sparse implementation use:
M = kron(speye(N),A);
Related
Considering a discrete dynamical system where x[0]=rand() denotes the initial condition of the system.
I have generated an m by n matrix by the following step -- generate m vectors with m different initial conditions each with dimension N (N indicates the number of samples or elements). This matrix is called R. Using R how do I create a Toeplitz matrix, T? T
Mathematically,
R = [ x_0[0], ....,x_0[n-1];
..., ,.....;
x_m[0],.....,x_m[n-1]]
The toeplitz matrix T =
x[n-1], x[n-2],....,x[0];
x[0], x[n-1],....,x[1];
: : :
x[m-2],x[m-3]....,x[m-1]
I tried working with toeplitz(R) but the dimension changes. The dimension should no change, as seen mathematically.
According to the paper provided (Toeplitz structured chaotic sensing matrix for compressive sensing by Yu et al.) there are two Chaotic Sensing Matrices involved. Let's explore them separately.
The Chaotic Sensing Matrix (Section A)
It is clearly stated that to create such matrix you have to build m independent signals (sequences) with m different initials conditions (in range ]0;1[) and then concatenate such signals per rows (that is, one signal = one row). Each of these signals must have length N. This actually is your matrix R, which is correctly evaluated as it is. Although I'd like to suggest a code improvement: instead of building a column and then transpose the matrix you can directly build such matrix per rows:
R=zeros(m,N);
R(:,1)=rand(m,1); %build the first column with m initial conditions
Please note: by running randn() you select values with Gaussian (Normal) distribution, such values might not be in range ]0;1[ as stated in the paper (right below equation 9). As instead by using rand() you take uniformly distributed values in such range.
After that, you can build every row separately according to the for-loop:
for i=1:m
for j=2:N %skip first column
R(i,j)=4*R(i,j-1)*(1-R(i,j-1));
R(i,j)=R(i,j)-0.5;
end
end
The Toeplitz Chaotic Sensing Matrix (Section B)
It is clearly stated at the beginning of Section B that to build the Toeplitz matrix you should consider a single sequence x with a given, single, initial condition. So let's build such sequence:
x=rand();
for j=2:N %skip first element
x(j)=4*x(j-1)*(1-x(j-1));
x(j)=x(j)-0.5;
end
Now, to build the matrix you can consider:
how do the first row looks like? Well, it looks like the sequence itself, but flipped (i.e. instead of going from 0 to n-1, it goes from n-1 to 0)
how do the first column looks like? It is the last item from x concatenated with the elements in range 0 to m-2
Let's then build the first row (r) and the first column (c):
r=fliplr(x);
c=[x(end) x(1:m-1)];
Please note: in Matlab the indices start from 1, not from 0 (so instead of going from 0 to m-2, we go from 1 to m-1). Also end means the last element from a given array.
Now by looking at the help for the toeplitz() function, it is clearly stated that you can build a non-squared Toeplitz matrix by specifying the first row and the first column. Therefore, finally, you can build such matrix as:
T=toeplitz(c,r);
Such matrix will indeed have dimensions m*N, as reported in the paper.
Even though the Authors call both of them \Phi, they actually are two separate matrices.
They do not take the Toeplitz of the Beta-Like Matrix (Toeplitz matrix is not a function or operator of some kind), neither do they transform the Beta-Like Matrix into a Toeplitz-matrix.
You have the Beta-Like Matrix (i.e. the Chaotic Sensing Matrix) at first, and then the Toeplitz-structured Chaotic Sensing Matrix: such structure is typical for Toeplitz matrices, that is a diagonal-constant structure (all elements along a diagonal have the same value).
In Matlab, I have created a matrix A with size (244x2014723)
and a matrix B with size (244x1)
I was able to calculate the correlation matrix using corr(A,B) which yielded in a matrix of size 2014723x1. So, every column of matrix A correlates with matrix B and gives one row value in the matrix of size 2014723x1.
My question is when I ask for a covariance matrix using cov(A,B), I get an error saying A and B should be of same sizes. Why do I get this error? How is the method to find corr(A,B) any different from cov(A,B)?
The answer is pretty clear if you read the documentation:
cov:
If A and B are matrices of observations, cov(A,B) treats A and B as vectors and is equivalent to cov(A(:),B(:)). A and B must have equal size.
corr
corr(X,Y) returns a p1-by-p2 matrix containing the pairwise correlation coefficient between each pair of columns in the n-by-p1 and n-by-p2 matrices X and Y.
The difference between corr(X,Y) and the MATLAB® function corrcoef(X,Y) is that corrcoef(X,Y) returns a matrix of correlation coefficients for the two column vectors X and Y. If X and Y are not column vectors, corrcoef(X,Y) converts them to column vectors.
One way you could get the covariances of your vector with each column of you matrix is to use a loop. Another way (might be in-efficient depending on the size) is
C = cov([B,A])
and then look at the first row (or column) or C.
See link
In the more about section, the equation describing how cov is computed for cov(A,B) makes it clear why they need to be the same size. The summation is over only one variable which enumerates the elements of A,B.
Please see the following issue:
P=rand(4,4);
for i=1:size(P,2)
for j=1:size(P,2)
[r,p]=corr(P(:,i),P(:,j))
end
end
Clearly, the loop will cause the number of correlations to be doubled (i.e., corr(P(:,1),P(:,4)) and corr(P(:,4),P(:,1)). Does anyone have a suggestion on how to avoid this? Perhaps not using a loop?
Thanks!
I have four suggestions for you, depending on what exactly you are doing to compute your matrices. I'm assuming the example you gave is a simplified version of what needs to be done.
First Method - Adjusting the inner loop index
One thing you can do is change your j loop index so that it only goes from 1 up to i. This way, you get a lower triangular matrix and just concentrate on the values within the lower triangular half of your matrix. The upper half would essentially be all set to zero. In other words:
for i = 1 : size(P,2)
for j = 1 : i
%// Your code here
end
end
Second Method - Leave it unchanged, but then use unique
You can go ahead and use the same matrix like you did before with the full two for loops, but you can then filter the duplicates by using unique. In other words, you can do this:
[Y,indices] = unique(P);
Y will give you a list of unique values within the matrix P and indices will give you the locations of where these occurred within P. Note that these are column major indices, and so if you wanted to find the row and column locations of where these locations occur, you can do:
[rows,cols] = ind2sub(size(P), indices);
Third Method - Use pdist and squareform
Since you're looking for a solution that requires no loops, take a look at the pdist function. Given a M x N matrix, pdist will find distances between each pair of rows in a matrix. squareform will then transform these distances into a matrix like what you have seen above. In other words, do this:
dists = pdist(P.', 'correlation');
distMatrix = squareform(dists);
Fourth Method - Use the corr method straight out of the box
You can just use corr in the following way:
[rho, pvals] = corr(P);
corr in this case will produce a m x m matrix that contains the correlation coefficient between each pair of columns an n x m matrix stored in P.
Hopefully one of these will work!
this works ?
for i=1:size(P,2)
for j=1:i
Since you are just correlating each column with the other, then why not just use (straight from the documentation)
[Rho,Pval] = corr(P);
I don't have the Statistics Toolbox, but according to http://www.mathworks.com/help/stats/corr.html,
corr(X) returns a p-by-p matrix containing the pairwise linear correlation coefficient between each pair of columns in the n-by-p matrix X.
What's the best way to do the following (in Matlab) if I have two matrices A and B, let'say both of size m-by-n:
C = zeros(m,m);
for t=1:n
C=C+A(:,t)*B(:,t)';
end
This is nothing more than
C = A*B';
where A and B are each m-by-n. I'm not sure that you're going to get more efficient than that unless the matrices have special properties.
One place where you might get a benefit from using bsxfun for matrix multiplication is when the dimensions are sufficiently large (probably 100-by-100 or more) and one matrix is diagonal, e.g.:
A = rand(1e2);
B = diag(rand(1,1e2));
C = bsxfun(#times,A,diag(B).');
This occurs in many matrix transforms – see the code for sqrtm for example (edit sqrtm).
I'm currently working on improving the blending part of the image mosaic sample application on VLfeat's homepage. In this final blending stage I want to combine two non-nan-sparse images, both being the output from two images interpolations using interp2 with nan-flag. Specifically, given two image matrices A and B and blended matrix C all of same dimension M-by-N, I want for each matrix position (i,j) in A and B want to check whehter
both A and B have a defined value in (i,j) so make C(i,j) the average of them or,
either A or B have a defined (~isnan()) value in (i,j) so put that in C(i,j) or,
neither A nor B have a defined value in (i,j) thereby leaving C(i,j) as is
assuming C is initialized to all nan values.
I haven't find a simple nor elegant way of doing this without having to
reshape A and B into vectors
find non-nan vector indexes AI=find(~isnan(A)) and BI=find(~isnan(B))
find intersection II of AI and BI
use II, AI and BI to modify a vector C of same length as A and B as mentioned in three steps above
reshape C back to M-by-N to finally get result wanted
I have tried to express the same steps using matrices and matrix indexes without success. Is this the only way of doing this in MATLAB? It seems kind of cumbersome.
Maybe I'm missing something (I do not understand why you need to reshape matrix to vectors, for example).
Anyway, here is a try:
% Unconditionnaly compute the average into temp D matrix
D=(A+B)/2;
% Restore A and B values where B and A are NaN
D(isnan(A))=B(isnan(A));
D(isnan(B))=A(isnan(B));
% Only modify C wherever the final result is not NaN
C(~isnan(D))=D(~isnan(D));