I want to compute the trace of an n-fold exterior product of a matrix in Matlab. For the definition of the exterior product, please refer to here and so I am interested in computing the Taylor coefficients in the expansion of the determinant. Is this already implemented in Matlab or do I have to do anything new?
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I will try to explain my doubt in the better way possible:
I'm trying to solve the reaction-diffusion equation with PDE Toolbox (Matlab), the syntax to get the solution (u) is:
parabolic - Solve parabolic PDE problem
This MATLAB function produces the solution to the FEM formulation of the scalar PDE problem:
u1 = parabolic(u0,tlist,b,p,e,t,c,a,f,d)
c,a,f,d are the coefficients of the parabolic equation:
d(∂u/∂t)−∇⋅(c∇u)+au=f,
But I want this:
u=parabolic(u0,tlist,b,p,e,t,c,#coeffunction,f,d);
u0 is the initial solution, tlist is a vector array of times where I record the solution, b corresponds to boundary conditions, (p,e,t) are the elements of the mesh, and #coeffunction is a function handle which corresponds to the coefficient "a".
My question is: Anyone knows if is it possible to specify "a" as a variable coefficient which have a dependency with tlist?
Although not the PDE toolbox, the FEATool Matlab FEM toolbox allows entering and solving PDEs with nonlinear and time dependent coefficients. Equation and boundary expressions can be defined as constants or string expressions (including time, dependent variables, derivatives, space dimension coordinates, and even external user defined Matlab functions). For example coefficients can be defined as
fea.coef = { 'coef1' [] [] 42 ;
'coef2' [] [] '2*u-ux^2+sin(2*pi*t)' ;
'coef3' [] [] 'my_fun(t)' };
where now coef1-coef3 can be used freely in the equations and boundary conditions. This answer shows one heat transfer example illustrating this approach.
In your case with a reaction-diffusion equation you could possibly just use the Matlab GUI as convection-diffusion-reaction PDE equations are pre-defined as enter
u*t
as the reaction source term for the corresponding R coefficient. And if you then prefer working with m-script files you can just export the GUI model as a FEA model .m text file.
When solving the log likelihood expression for autoregressive models, I cam across the variance covariance matrix Tau given under slide 9 Parameter estimation of time series tutorial. Now, in order to use
fminsearch
to maximize the likelihood function expression, I need to express the likelihood function where the variance covariance matrix arises. Can somebody please show with an example how I can implement (determinant of Gamma)^-1/2 ? Any other example apart from autoregressive model will also do.
How about sqrt(det(Gamma)) for the sqrt-determinant and inv(Gamma) for inverse?
But if you do not want to implement it yourself you can look at yulewalkerarestimator
UPD: For estimation of autocovariance matrix use xcov
also, this topic is a bit more explained here
I have a system of dynamic equations that ultimately can be written in the well-known "spring-mass-damper" form:
[M]{q''}+[C]{q'}+[K]{q}={0}
[M], [C], [K]: n-by-n Coefficient Matrices
{q}: n-by-1 Vector of the Degrees of Freedom
(the ' mark represents a time derivative)
I want to find the eigenvalues and eigenvectors of this system. Obviously due to the term [C]{q'}, the standard MATLAB function eig() will not be useful.
Does anyone know of a simple MATLAB routine to determine the eigenvalues, eigenvectors of this system? The system is homogeneous so an efficient eigenvalue analysis should be very feasible, but I'm struggling a bit.
Obviously I can use brute force and a symbolic computing software to find the gigantic characteristic polynomial. But this seems inefficient for me, especially because I'm looping this through the other parts of the code to determine frequencies as a function of other varied parameters.
I need to create a diagonal matrix containing the Fourier coefficients of the Gaussian wavelet function, but I'm unsure of what to do.
Currently I'm using this function to generate the Haar Wavelet matrix
http://www.mathworks.co.uk/matlabcentral/fileexchange/33625-haar-wavelet-transformation-matrix-implementation/content/ConstructHaarWaveletTransformationMatrix.m
and taking the rows at dyadic scales (2,4,8,16) as the transform:
M= 256
H = ConstructHaarWaveletTransformationMatrix(M);
fi = conj(dftmtx(M))/M;
H = fi*H;
H = H(4,:);
H = diag(H);
etc
How do I repeat this for Gaussian wavelets? Is there a built in Matlab function which will do this for me?
For reference I'm implementing the algorithm in section 4 of this paper:
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04218361
I maybe would not being answering the question, but i will try to help you advance.
As far as i know, the Matlab Wavelet Toolbox only deal with wavelet operations and coefficients, increase or decrease resolution levels, and similar operations, but do not exposes the internal matrices serving to doing the transformations from signals and coefficients.
Hence i fear the answer to this question is no. Some time ago, i did this for some of the Hart Class wavelet, and i actually build the matrix from the scratch, and then i compared the coefficients obtained with the Built-in Matlab Wavelet Toolbox, hence ensuring your matrices are good enough for your algorithm. In my case, recursive parameter estimation for time varying models.
For the function ConstructHaarWaveletTransformationMatrix it is really simple to create the matrix, because the Hart Class could be really simple expressed as Kronecker products.
The Gaussian Wavelet case as i fear should be done from the scratch too...
THe steps i suggest would be;
Although MATLAB dont include explicitely the matrices, you can use the Matlab built-in functions to recover the Gaussian Wavelets, and thus compose the matrix for your algorithm.
Build every column of the matrix with every Gaussian Wavelet, for every resolution levels you are requiring (the dyadic scales). Use the Matlab Wavelets toolbox for recover the shapes.
After this, compare the coefficients obtained by you, with the coefficients of the toolbox. This way you will correct the order of the Matrix row.
Numerically, being fj the signal projection over Vj (the PHI signals space, scaling functions) at resolution level j, and gj the signal projection over Wj (the PSI signals space, mother functions) at resolution level j, we can write:
f=fj0+sum_{j0}^{j1-1}{gj}
Hence, both fj0 and gj will induce two matrices, lets call them PHIj and PSIj matrices:
f=PHIj0*cj0+sum_{j0}^{j1-1}{PSIj*dj}
The PHIj columns contain the scaled and shifted scaling wavelet signal (one, for j0 only) for the approximation projection (the Vj0 space), and the PSIj columns contain the scaled and shifted mother wavelet signals (several, from j0 to j1-1) for the detail projection (onto the Wj0 to Wj1-1 spaces).
Hence, the Matrix you need is:
PHI=[PHIj0 PSIj0... PSIj1]
Thus you can express you original signal as:
f=PHI*C
where C is a vector of approximation and detail coefficients, for the levels:
C=[cj0' dj0'...dj1']'
The first part, for addressing the PHI build can be achieved by writing:
function PHI=MakePhi(l,str,Jmin,Jmax)
% [PHI]=MakePhi(l,str,Jmin,Jmax)
%
% Build full PHI Wavelet Matrix for obtaining wavelet coefficients
% (extract)
%FILTER
[LO_R,HI_R] = wfilters(str,'r');
lf=length(LO_R);
%PHI BUILD
PHI=[];
laux=l([end-Jmax end-Jmax:end]);
PHI=[PHI MakeWMatrix('a',str,laux)];
for j=Jmax:-1:Jmin
laux=l([end-j end-j:end]);
PHI=[PHI MakeWMatrix('d',str,laux)];
end
the wfilters is a MATLAB built in function, giving the required signal for the approximation and or detail wavelet signals.
The MakeWMatrix function is:
function M=MakeWMatrix(typestr,str,laux)
% M=MakeWMatrix(typestr,str,laux)
%
% Build Wavelet Matrix for obtaining wavelet coefficients
% for a single level vector.
% (extract)
[LO_R,HI_R] = wfilters(str,'r');
if typestr=='a'
F_R=LO_R';
else
F_R=HI_R';
end
la=length(laux);
lin=laux(2); lout=laux(3);
M=MakeCMatrix(F_R,lin,lout);
for i=3:la-1
lin=laux(i); lout=laux(i+1);
Mi=MakeCMatrix(LO_R',lin,lout);
M=Mi*M;
end
and finally the MakeCMatrix is:
function [M]=MakeCMatrix(F_R,lin,lout)
% Convolucion Matrix
% (extract)
lf=length(F_R);
M=[];
for i=1:lin
M(:,i)=[zeros(2*(i-1),1) ;F_R ;zeros(2*(lin-i),1)];
end
M=[zeros(1,lin); M ;zeros(1,lin)];
[ltot,lin]=size(M);
lmin=floor((ltot-lout)/2)+1;
lmax=floor((ltot-lout)/2)+lout;
M=M(lmin:lmax,:);
This last matrix should include some interpolation routine for having better general results in each case.
I expect this solve part of your problem.....
Hyp
I run
Y_testing_obtained = classify(X_testing, X_training, Y_training);
and the error I get is
Error using ==> classify at 246
The pooled covariance matrix of TRAINING must be positive definite.
X_training is 1550 x 5 matrix. Can you please tell me what this error means, i.e. why is it appearing, and how to work around it?
Thanks
Explanation: When you run the function classify without specifying the type of discriminant function (as you did), Matlab uses Linear Discriminant Analysis (LDA). Without going into too much details on LDA, the algorithms needs to calculate the covariance matrix of X_testing in order to solve an optimisation problem, and this matrix has to be positive definite (see Wikipedia: Positive-definite matrix). The underlying assumption is that your data is represented by a multivariate probability distribution, which always has a positive definite covariance matrix unless one or more variables are exact linear combinations of the others.
To solve your problem: It is possible that one of your variables is a linear combination of the others. You can try selecting a sensible subset of your variables, or perform Principal Component Analysis (PCA) on the training data and then classify using the first few principal components. Or, you could specify the type of discriminant function and choose one of the two naive Bayes classifiers, for example:
Y_testing_obtained = classify(X_testing, X_training, Y_training, 'diaglinear');
As a side note, you also need to have more observations (rows) than variables (columns), but in your case this is not the problem as you seem to have 1550 observations and 5 variables.
Finally, you can also have a look at the answers posted to a similar question on the Matlab forum.
Try regularizing the data using cvshrink function in Matlab