So it is a mathematical fact that if the determinant of a matrix is equal to zero, then the matrix must be singular (not invertible). Now, the problem I am running into is that when I calculate the determinant of my matrix it is equal to zero, however, when I calculate the inverse it exist. I think it has to do with the way R calculates determinants that the two are not agreeing. Here is the code that I am trying (I wont print the results of solve because the matrix is 100 x 100).
> Rinv = solve(R)
>
> det(R)
[1] 0
>
> #Using a Cholesky Factorization
> L = chol(R)
> Q = t(L)
>
> det(L)*det(Q)
[1] 0
For large matrices the determinant can be too large or too small and overflow the double precision.
The determinant is the product of the eigenvalues: for instance, if they are all .0001, your matrix is invertible, but the determinant is 1e-400, which is too small, and can only be represented as 0.
You can look at the logarithm of the determinant instead,
determinant(R, logarithm=TRUE)
or, directly, the eigenvalues
eigen(R, only.values=TRUE)
Related
I am currently in an online class in genomics, coming in as a wetlab physician, so my statistical knowledge is not the best. Right now we are working on PCA and SVD in R. I got a big matrix:
head(mat)
ALL_GSM330151.CEL ALL_GSM330153.CEL ALL_GSM330154.CEL ALL_GSM330157.CEL ALL_GSM330171.CEL ALL_GSM330174.CEL ALL_GSM330178.CEL ALL_GSM330182.CEL
ENSG00000224137 5.326553 3.512053 3.455480 3.472999 3.639132 3.391880 3.282522 3.682531
ENSG00000153253 6.436815 9.563955 7.186604 2.946697 6.949510 9.095092 3.795587 11.987291
ENSG00000096006 6.943404 8.840839 4.600026 4.735104 4.183136 3.049792 9.736803 3.338362
ENSG00000229807 3.322499 3.263655 3.406379 9.525888 3.595898 9.281170 8.946498 3.473750
ENSG00000138772 7.195113 8.741458 6.109578 5.631912 5.224844 3.260912 8.889246 3.052587
ENSG00000169575 7.853829 10.428492 10.512497 13.041571 10.836815 11.964498 10.786381 11.953912
Those are just the first few columns and rows, it has 60 columns and 1000 rows. Columns are cancer samples, rows are genes
The task is to:
removing the eigenvectors and reconstructing the matrix using SVD, then we need to calculate the reconstruction error as the difference between the original and the reconstructed matrix. HINT: You have to use the svd() function and equalize the eigenvalue to $0$ for the component you want to remove.
I have been all over google, but can't find a way to solve this task, which might be because I don't really get the question itself.
so i performed SVD on my matrix m:
d <- svd(mat)
Which gives me 3 matrices (Eigenassays, Eigenvalues and Eigenvectors), which i can access using d$u and so on.
How do I equalize the eigenvalue and ultimately calculate the error?
https://www.rdocumentation.org/packages/base/versions/3.6.2/topics/svd
the decomposition expresses your matrix mat as a product of 3 matrices
mat = d$u x diag(d$d) x t(d$v)
so first confirm you are able to do the matrix multiplications to get back mat
once you are able to do this, set the last couple of elements of d$d to zero before doing the matrix multiplication
It helps to create a function that handles the singular values.
Here, for instance, is one that zeros out any singular value that is too small compared to the largest singular value:
zap <- function(d, digits = 3) ifelse(d < 10^(-digits) * max(abs(d))), 0, d)
Although mathematically all singular values are guaranteed non-negative, numerical issues with floating point algorithms can--and do--create negative singular values, so I have prophylactically wrapped the singular values in a call to abs.
Apply this function to the diagonal matrix in the SVD of a matrix X and reconstruct the matrix by multiplying the components:
X. <- with(svd(X), u %*% diag(zap(d)) %*% t(v))
There are many ways to assess the reconstruction error. One is the Frobenius norm of the difference,
sqrt(sum((X - X.)^2))
I am creating optimization algorithm where I need to plug initial value of A which is a lower triangular matrix with positive diagonal. My first question is how to derive a random lower triangular matrix with positive diagonal as an initial value matrix in r? And is it good idea to choose random matrix? if not what are the ways to do a better initial guess for this type of matrix?
I can imagine there are much better solutions based on your specific applications, but we can set the diagonal to half-Normal (i.e. |e| where e ~ N(0,1)) and set the lower-triangular off-diagonal elements to standard Normal values. ...
n <- 10
M <- diag(abs(rnorm(n)))
M[lower.tri(M, diag = FALSE)] <- rnorm(n*(n-1)/2)
Rao QE is a weighted Euclidian distance matrix. I have the vectors for the elements of the d_ijs in a data table dt, one column per element (say there are x of them). p is the final column. nrow = S. The double sums are for the lower left (or upper right since it is symmetric) elements of the distance matrix.
If I only needed an unweighted distance matrix I could simply do dist() over the x columns. How do I weight the d_ijs by the product of p_i and p_j?
And example data set is at https://github.com/GeraldCNelson/nutmod/blob/master/RaoD_example.csv with the ps in the column called foodQ.ratio.
You still start with dist for the raw Euclidean distance matrix. Let it be D. As you will read from R - How to get row & column subscripts of matched elements from a distance matrix, a "dist" object is not a real matrix, but a 1D array. So first do D <- as.matrix(D) or D <- dist2mat(D) to convert it to a complete matrix before the following.
Now, let p be the vector of weights, the Rao's QE is just a quadratic form q'Dq / 2:
c(crossprod(p, D %*% p)) / 2
Note, I am not doing everything in the most efficient way. I have performed a symmetric matrix-vector multiplication D %*% p using the full D rather than just its lower triangular part. However, R does not have a routine doing triangular matrix-vector multiplication. So I compute the full version than divide 2.
This doubles computation amount that is necessary; also, making D a full matrix doubles memory costs. But if your problem is small to medium size this is absolutely fine. For large problem, if you are R and C wizard, call BLAS routine dtrmv or even dtpmv for the triangular matrix-vector computation.
Update
I just found this simple paper: Rao's quadratic entropy as a measure of functional diversity based on multiple traits for definition and use of Rao's EQ. It mentions that we can replace Euclidean distance with Mahalanobis distance. In case we want to do this, use my code in Mahalanobis distance of each pair of observations for fast computation of Mahalanobis distance matrix.
I am trying to generate a positive definite matrix (A'*A) of dimensions 8x8.
where A is 1x8.
I tried it for many randomly generated matrix A but not able to generate it.
octave-3.6.1.exe:166> A= (rand(1,8)+rand(1,8)*1i);
octave-3.6.1.exe:167> chol(A'*A);
error: chol: input matrix must be positive definite
Can anyone please tell me what is going wrong here. Thanks for the help in advance.
It's not possible to do that, since no matrix of that form is positive definite.
Claim: Given a 1xn (real, n>1) matrix A, the symmetric matrix M = A'A is not positive definite:
Proof: By definition, M is positive definite iff x'Mx > 0 for all non zero x. That is, iff x'A'Ax = (Ax)'Ax = (Ax)^2 = (A_1 x_1 + ... + A_n x_n) > 0 for all non zero x.
Since the real values A_i are linearly dependent, there exists x_i, not all zero, such that A_1 x_1 + ... + A_n x_n = 0. We found a non zero vector x such that x'Mx = 0, so M is not positive definite.
A different proof, that can be applied directly to the complex case is this: Let A be an 1xn (complex, n>1) matrix. Positive definiteness implies invertibility, so M = A*A must have full rank to be positive definite. It clearly has rank 1, so it's not invertible and thus not positive definite.
Here is how I routinelly create SPD matrix
1) Create a random Symetric Matrix
2) Make sure that all the diagonal values are greater than the sum of any row or column they appear in.
Usually for (1) I use random number between 0 and 1. Its then easy to figure out a number to use for each diagonal entries.
Cheers,
How does it actually reduce noise..can you suggest some nice tutorials?
SVD can be understood from a geometric sense for square matrices as a transformation on a vector.
Consider a square n x n matrix M multiplying a vector v to produce an output vector w:
w = M*v
The singular value decomposition M is the product of three matrices M=U*S*V, so w=U*S*V*v. U and V are orthonormal matrices. From a geometric transformation point of view (acting upon a vector by multiplying it), they are combinations of rotations and reflections that do not change the length of the vector they are multiplying. S is a diagonal matrix which represents scaling or squashing with different scaling factors (the diagonal terms) along each of the n axes.
So the effect of left-multiplying a vector v by a matrix M is to rotate/reflect v by M's orthonormal factor V, then scale/squash the result by a diagonal factor S, then rotate/reflect the result by M's orthonormal factor U.
One reason SVD is desirable from a numerical standpoint is that multiplication by orthonormal matrices is an invertible and extremely stable operation (condition number is 1). SVD captures any ill-conditioned-ness in the diagonal scaling matrix S.
One way to use SVD to reduce noise is to do the decomposition, set components that are near zero to be exactly zero, then re-compose.
Here's an online tutorial on SVD.
You might want to take a look at Numerical Recipes.
Singular value decomposition is a method for taking an nxm matrix M and "decomposing" it into three matrices such that M=USV. S is a diagonal square (the only nonzero entries are on the diagonal from top-left to bottom-right) matrix containing the "singular values" of M. U and V are orthogonal, which leads to the geometric understanding of SVD, but that isn't necessary for noise reduction.
With M=USV, we still have the original matrix M with all its noise intact. However, if we only keep the k largest singular values (which is easy, since many SVD algorithms compute a decomposition where the entries of S are sorted in nonincreasing order), then we have an approximation of the original matrix. This works because we assume that the small values are the noise, and that the more significant patterns in the data will be expressed through the vectors associated with larger singular values.
In fact, the resulting approximation is the most accurate rank-k approximation of the original matrix (has the least squared error).
To answer to the tittle question: SVD is a generalization of eigenvalues/eigenvectors to non-square matrices.
Say,
$X \in N \times p$, then the SVD decomposition of X yields X=UDV^T where D is diagonal and U and V are orthogonal matrices.
Now X^TX is a square matrice, and the SVD decomposition of X^TX=VD^2V where V is equivalent to the eigenvectors of X^TX and D^2 contains the eigenvalues of X^TX.
SVD can also be used to greatly ease global (i.e. to all observations simultaneously) fitting of an arbitrary model (expressed in an formula) to data (with respect to two variables and expressed in a matrix).
For example, data matrix A = D * MT where D represents the possible states of a system and M represents its evolution wrt some variable (e.g. time).
By SVD, A(x,y) = U(x) * S * VT(y) and therefore D * MT = U * S * VT
then D = U * S * VT * MT+ where the "+" indicates a pseudoinverse.
One can then take a mathematical model for the evolution and fit it to the columns of V, each of which are a linear combination the components of the model (this is easy, as each column is a 1D curve). This obtains model parameters which generate M? (the ? indicates it is based on fitting).
M * M?+ * V = V? which allows residuals R * S2 = V - V? to be minimized, thus determining D and M.
Pretty cool, eh?
The columns of U and V can also be inspected to glean information about the data; for example each inflection point in the columns of V typically indicates a different component of the model.
Finally, and actually addressing your question, it is import to note that although each successive singular value (element of the diagonal matrix S) with its attendant vectors U and V does have lower signal to noise, the separation of the components of the model in these "less important" vectors is actually more pronounced. In other words, if the data is described by a bunch of state changes that follow a sum of exponentials or whatever, the relative weights of each exponential get closer together in the smaller singular values. In other other words the later singular values have vectors which are less smooth (noisier) but in which the change represented by each component are more distinct.