In matlab it is easy to generate a normally distributed random vector with a mean and a standard deviation. From the help randn:
Generate values from a normal distribution with mean 1 and standard
deviation 2.
r = 1 + 2.*randn(100,1);
Now I have a covariance matrix C and I want to generate N(0,C).
But how could I do this?
From the randn help:
Generate values from a bivariate normal distribution with specified mean
vector and covariance matrix.
mu = [1 2];
Sigma = [1 .5; .5 2]; R = chol(Sigma);
z = repmat(mu,100,1) + randn(100,2)*R;
But I don't know exactly what they are doing here.
This is somewhat a math question, not a programming question. But I'm a big fan of writing great code that requires both solid math and programming knowledge, so I'll write this for posterity.
You need to take the Cholesky decomposition (or any decomposition/square root of a matrix) to generate correlated random variables from independent ones. This is because if X is a multivariate normal with mean m and covariance D, then Y = AX is a multivariate normal with mean Am and covariance matrix ADA' where A' is the transpose. If D is the identity matrix, then the covariance matrix is just AA' which you want to be equal to the covariance matrix C you are trying to generate.
The Cholesky decomposition computes such a matrix A and is the most efficient way to do it.
For more information, see: http://web.as.uky.edu/statistics/users/viele/sta601s03/multnorm.pdf
You can use the following built-in matlab function to do your job
mvnrnd(mu,SIGMA)
Related
If X is a multivariate t random variable with mean=[1,2,3,4,5] and a covariance matrix C, how to simulate points in matlab? I try mvtrnd in matlab, but clearly the sample mean does not give mean close to [1,2,3,4,5]. Also, when I test three simple examples, say X1 with mean 0 and C1=[1,0.3;0.3,1], X2 with mean 0 and C2=[0.5,0.15;0.15,0.5] and X3 with mean 0 and C3=[0.4,0.12;0.12,0.4] and use mvtrnd(C1,3,1000000), mvtrnd(C2,3,1000000) amd mvtrnd(C2,3,1000000) respectively, I find the sample points in each case give nearly the correlation matrix [1,0.3;0.3,1] but the sample covariance computed all give near [3,1;1,3]. Why and how to fix it?
The Mean
The t distribution has a zero mean unless you shift it. In the documentation for mvtrnd:
the distribution of t is that of a vector having a multivariate normal
distribution with mean 0, variance 1, and covariance matrix C, divided
by an independent chi-square random value having df degrees of
freedom.
Indeed, mean(X) will approach [0 0] for X = mvtrnd(C,df,n); as n gets larger.
The Correlation
Matching the correlation is straightforward as it addresses a part of the relationship between the two dimensions of X.
% MATLAB 2018b
df = 5; % degrees of freedom
C = [0.44 0.25; 0.25 0.44]; % covariance matrix
numSamples = 1000;
R = corrcov(C); % Convert covariance to correlation matrix
X = mvtrnd(R,df,numSamples); % X ~ multivariate t distribution
You can compare how well you matched the correlation matrix R using corrcoef or corr().
corrcoef(X) % Alternatively, use corr(X)
The Covariance
Matching the covariance is another matter. Admittedly, calling cov(X) will reveal that this is lacking. Recall that the diagonal of the covariance is the variance for the two components of X. My intuition is that we fixed the degrees of freedom df, so there is no way to match the desired variance (& covariance).
A useful function is corrcov which converts a covariance matrix into a correlation matrix.
Notice that this is unnecessary as the documentation for mvtrnd indicates
C must be a square, symmetric and positive definite matrix. If its
diagonal elements are not all 1 (that is, if C is a covariance matrix
rather than a correlation matrix), mvtrnd rescales C to transform it
to a correlation matrix before generating the random numbers.
I want to make similar graphs to this given on the picture:
I am using Fisher Iris data and employ PCA to reduce dimensionality.
this is code:
load fisheriris
[pc,score,latent,tsquare,explained,mu] = princomp(meas);
I guess the eigenvalues are given in Latent, that shows me only four features and is about reduced data.
My question is how to show all eigenvalues of original matrix, which is not quadratic (150x4)? Please help! Thank you very much in advance!
The short (and useless) answer is that the [V, D] eig(_) function gives you the eigenvectors and the eigenvalues. However, I'm afraid I have bad news for you. Eigenvalues and eigenvectors only exist for square matrices, so there are no eigenvectors for your 150x4 matrix.
All is not lost. PCA actually uses the eigenvalues of the covariance matrix, not of the original matrix, and the covariance matrix is always square. That is, if you have a matrix A, the covariance matrix is AAT.
The covariance matrix is not only square, it is symmetric. This is good, because the singular values of a matrix are related to the eigenvalues of it's covariance matrix. Check the following Matlab code:
A = [10 20 35; 5 7 9]; % A rectangular matrix
X = A*A'; % The covariance matrix of A
[V, D] = eig(X); % Get the eigenvectors and eigenvalues of the covariance matrix
[U,S,W] = svd(A); % Get the singular values of the original matrix
V is a matrix containing the eigenvectors, and D contains the eigenvalues. Now, the relationship:
SST ~ D
U ~ V
I use '~' to indicate that while they are "equal", the sign and order may vary. There is no "correct" order or sign for the eigenvectors, so either is valid. Unfortunately, though, you will only have four features (unless your array is meant to be the other way around).
I am wondering how to draw samples in matlab, where I have precision matrix and mean as the input argument.
I know mvnrnd is a typical way to do so, but it requires the covariance matrix (i.e inverse of precision)) as the argument.
I only have precision matrix, and due to the computational issue, I can't invert my precision matrix, since it will take too long (my dimension is about 2000*2000)
Good question. Note that you can generate samples from a multivariant normal distribution using samples from the standard normal distribution by way of the procedure described in the relevant Wikipedia article.
Basically, this boils down to evaluating A*z + mu where z is a vector of independent random variables sampled from the standard normal distribution, mu is a vector of means, and A*A' = Sigma is the covariance matrix. Since you have the inverse of the latter quantity, i.e. inv(Sigma), you can probably do a Cholesky decomposition (see chol) to determine the inverse of A. You then need to evaluate A * z. If you only know inv(A) this can still be done without performing a matrix inverse by instead solving a linear system (e.g. via the backslash operator).
The Cholesky decomposition might still be problematic for you, but I hope this helps.
If you want to sample from N(μ,Q-1) and only Q is available, you can take the Cholesky factorization of Q, L, such that LLT=Q. Next take the inverse of LT, L-T, and sample Z from a standard normal distribution N(0, I).
Considering that L-T is an upper triangular dxd matrix and Z is a d-dimensional column vector,
μ + L-TZ will be distributed as N(μ, Q-1).
If you wish to avoid taking the inverse of L, you can instead solve the triangular system of equations LTv=Z by back substitution. μ+v will then be distributed as N(μ, Q-1).
Some illustrative matlab code:
% make a 2x2 covariance matrix and a mean vector
covm = [3 0.4*(sqrt(3*7)); 0.4*(sqrt(3*7)) 7];
mu = [100; 2];
% Get the precision matrix
Q = inv(covm);
%take the Cholesky decomposition of Q (chol in matlab already returns the upper triangular factor)
L = chol(Q);
%draw 2000 samples from a standard bivariate normal distribution
Z = normrnd(0,1, [2, 2000]);
%solve the system and add the mean
X = repmat(mu, 1, 2000)+L\Z;
%check the result
mean(X')
var(X')
corrcoef(X')
% compare to the sampling from the covariance matrix
Y=mvnrnd(mu,covm, 2000)';
mean(Y')
var(Y')
corrcoef(Y')
scatter(X(1,:), X(2,:),'b')
hold on
scatter(Y(1,:), Y(2,:), 'r')
For more efficiency, I guess you can search for some package that efficiently solves triangular systems.
I'm probably being a little dense but I'm not very mathsy and can't seem to understand the covariance element of creating multivariate data.
I'm after two columns of random data (representing two correlated variables).
I think I am right in needing to use the mvnrnd function and I understand that 'mu' must be a column of my mean vectors. As I need 4 distinct classes within my data these are going to be (1, 1) (-1 1) (1 -1) and (-1 -1). I assume I will have to do the function 4x with a different column of mean vectors each time and then combine them to get my full data set.
I don't understand what I should put for SIGMA - Matlab help tells me that it must be 'a d-by-d symmetric positive semi-definite matrix, or a d-by-d-by-n array' i.e. a covariance matrix. I don't understand how I create a covariance matrix for numbers that I am yet to generate.
Any advice would be greatly appreciated!
Assuming that I understood your case properly, I would go this way:
data = [normrnd(0,1,5000,1),normrnd(0,1,5000,1)]; %% your starting data series
MU = mean(data,1);
SIGMA = cov(data);
Now, it should be possible to feed mvnrnd with MU and SIGMA:
r = mvnrnd(MU,SIGMA,5000);
plot(r(:,1),r(:,2),'+') %% in case you wanna plot the results
I hope this helps.
I think your aim is to generate the simulated multivariate gaussian distributed data. For example, I use
k = 6; % feature dimension
mu = rand(1,k);
sigma = 10*eye(k,k);
unit matrix by 10 times is a symmetric positive semi-definite matrix. And the gaussian distribution will be more round than other type of sigma.
then you can use it as the above example of mvnrnd function and see the plot.
I compute the multinomial Gaussian density for some huge number of times in a project where I update the covariance matrix by rank-1. Instead of computing the covariance from scratch, I used the cholupdate function to add a new sample to the covariance and remove a new sample to the covariance. By this way, the update is told to be in $O(n^2)$ as opposed to $O(n^3)$ Cholesky factorization of the covariance matrix.
persistent R
if (initialize) % or isempty(R)
% compute covariance V
R = chol(V);
else
R = cholupdate(R,xAdded);
detVar = prod(diag(R))^2;
Rt = R';
coeff = 1/sqrt((2*pi)^dimension*detVar);
y = Rt\x;
logp = log(coeff) - 1/2 * norm(y)^2;
Actually the code is quite complicated but I simplified it here. I wonder if there is a faster way to compute the inverse (the Rt\x part in the code) of an upper triangular matrix in MATLAB. Do you have any ideas to do it more efficiently in MATLAB.
Note that computing the determinant is also faster this way. So the new method will also not bad for the computation of the determinant.
The mldivide function is smart enough to check for triangular matrices, in which case it uses a forward/backward substitution method to efficiently solve the linear system:
AX=B <--> X=inv(A)*B <--> X=A\B
(compute x1, substitute it in second equation and compute x2, substitute in third ...)