I want to use adjacency matrix in MATLAB to solve a maximum flow problem formulated in Linear Programming. I want to try with adjacency matrix because using incidence matrix gives huge matrices due to the number of edges.
Related
I have a task where I need to train a machine learning model to predict a set of outputs from multiple inputs. My inputs are 1000 iterations of a set of 3x 1 vectors, a set of 3x3 covariance matrices and a set of scalars, while my output is just a set of scalars. I cannot use regression learner app because these inputs need to have the same dimensions, any idea on how to unify them?
One possible way to solve this is to flatten the covariance matrix into a vector. Once you did that, you can construct a 1000xN matrix where 1000 refers to the number of samples in your dataset and N is the number of features. For example if your features consist of a 3x1 vector, a 3x3 covariance matrix and lets say 5 other scalars, N could be 3+3*3+5=17. You then use this matrix to train an arbitrary model such as a linear regressor or more advanced models like a tree or the like.
When training machine learning models it is important to understand your data and exploit its structure to help the learning algorithms. For example we could use the fact that a covariance matrix is symmetric and positive semi-definite and thus lives in a closed convex cone. Symmetry of the matrix implies that it lives in a subspace of the set of all 3x3 matrices. In fact the dimension of the space of 3x3 symmetric matrices is only 6. You can use that knowledge to reduce redundancy in your data.
I have a correlation matrix for N random variables. Each of them is uniformly distributed within [0,1]. I am trying to simulate these random variables, how can I do that? Note N > 2. I was trying to using Cholesky Decomposition and below is my steps:
get the lower triangle of the correlation matrix (L=N*N)
independently sample 10000 times for each of the N uniformly distributed random variables (S=N*10000)
multiply the two: L*S, and this gives me correlated samples but the range of them is not within [0,1] anymore.
How can I solve the problem?
I know that if I only have 2 random variables I can do something like:
1*x1+sqrt(1-tho^2)*y1
to get my correlated sample y. But if you have more than two variables correlated, not sure what should I do.
You can get approximate solutions by generating correlated normals using the Cholesky factorization, then converting them to U(0,1)'s using the normal CDF. The solution is approximate because the normals have the desired correlation, but converting to uniforms is a non-linear transformation and only linear xforms preserve correlation.
There's a transformation available which will give exact solutions if the transformed Var/Cov matrix is positive semidefinite, but that's not always the case. See the abstract at https://www.tandfonline.com/doi/abs/10.1080/03610919908813578.
I am simulating a physical system, where I need to calculate the eigenvalues and vectors of a very large (~10000x10000) matrix.
So far I have used the in-built Eig algorithm in MATLAB but it is very slow for large matrices. Is there other algorithms in MATLAB that would do a better job or can I somehow improve the performance of Eig? Specifically it turns out that I only need the first ~100 eigenvectors of the matrix starting from the smallest numerical eigenvalue. Is there a way to get the algorithm to calculate only the first N eigenvectors and eigenvalues to save computation time? Of course this would only work if the eigenvectors come out sorted but they seem to do so, because of the symmetry of the Matrix I am using.
Your matrix has mostly zeros, so you should make it a sparse matrix. You'll then be able to use EIGS to calculate a smaller number of eigenvalues and eigenvectors.
http://www.mathworks.com/help/matlab/ref/eigs.html
i am doing Principal Component Analysis,and want help to know if can represent
summation from i to m (X(i)*X(i)^T) in terms of data matrix..direct multiplication of two matrices.
Can this be done..or need i use a for loop and do it.
Currently i have tried
sum=zeros(n,n);
for i=1:m
sum=sum+ X(i,:)*(X(i,:)^T);
end
My goal is to find the principal eigen values of the resulting matrix.
Thanks in advance
Say the shape of the data matrix X is (Dim, Num), you can just compute sum of all sample correlations with:
S = X*X'
For implementing PCA, also don't forget to divide the matrix by the amount of samples.
Sigma = (1/N)X*X'
If your data has zero mean, this is also the covariance matrix.
I am working on a project that involves the computation of the eigenvectors of a very large sparse matrix.
To be more specific I have a Matrix that is the laplacian of a big graph and I am interested in finding the eigenvector associated to the second smallest eigenvalue.
Of course Matlab takes ages to compute the eigenvectors, even because it computes all of them.
Any suggestions?
Thank you very much
Andrea
Have you tried this usage of eigs:
[v,c]=eigs(A,2,'sm');
for example:
A = delsq(numgrid('C',256));
[v,c]=eigs(A,2,'sm');
generates a ~50K x 50K sparse matrix and find its 2 smallerst eigenvalues and eigenvectors in about 1 second in my old laptop...