restreigen

restreigen computes eigenvalues restriction (without Dykstra algorithm)

Syntax

  • out=restreigen(eigenvalues, niini, restr)example
  • out=restreigen(eigenvalues, niini, restr, tol)example
  • out=restreigen(eigenvalues, niini, restr, tol, userepmat)example

Description

restreigen restricts the eigenvalues according to the constraint specified in scalar restr. This function is called in every concentration step of function tclust and can also be used inside function MixSim to generate groups with a prespecified level of overlapping

example

out =restreigen(eigenvalues, niini, restr) Example using all default options.

example

out =restreigen(eigenvalues, niini, restr, tol) Second example of eigenvalue restriction.

example

out =restreigen(eigenvalues, niini, restr, tol, userepmat) Compare speed.

Examples

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  • Example using all default options.
  • Suppose v=3 and k=4 so the matrix containing the eigenvalues is 3-by-4 First column of matrix eigenvalues contains the eigenvalues of the first group Second column of matrix eigenvalues contains the eigenvalues of the second group Thrid column of matrix eigenvalues contains the eigenvalues of the third group Fourth column of matrix eigenvalues contains the eigenvalues of the fourth group

    rng(10,'twister')
    eigenvalues=abs(10*randn(3,4));
    % niini is the vector containing the sizes of the 4 groups
    niini=[30;40;20;10];
    out=restreigen(eigenvalues,niini,1.1)
    disp('Input matrix of unrestricted eigenvalues')
    disp(eigenvalues)
    disp('Output matrix of restricted eigenvalues')
    disp(out)
    disp('Ratio between largest and smallest unrestricted eigenvalues')
    disp(max(max(eigenvalues))/min(min(eigenvalues)))
    disp('Ratio between largest and smallest restricted eigenvalues')
    disp(max(max(out))/min(min(out)))

  • Second example of eigenvalue restriction.
  • eigenvalues=abs(randn(3,4));
    eigenvalues(:,3)=0;
    niini=[30;40;20;10];
    restreigen(eigenvalues,niini,1.1)
    eigenvalues(:,3)=1;
    restreigen(eigenvalues,niini,1.1)

  • Compare speed.
  • We compare the speed of restreigneasy with that of restreigen. We use userepmat=2 if the current MATLAB version if >=R2017a or userepmat =1 if MATLAB version is >=R2013a but <R2017a else we use userepmat =0

    v=10;
    k=8;
    tol=1e-8;
    if verLessThanFS(9.2)== false
    % If MATLAB version is at least 2017a
    userepmat=2;
    elseif verLessThanFS(8.1) == false
    % if MATLAB version is at least R2013b  
    userepmat=1;
    else
    userepmat=0;
    end
    oldroutinetime=0;
    newroutinetime=0;
    rng(1)
    for j=1:10000
    eigenvalues=100*abs(randn(v,k));
    % niini is the vector containing the sizes of the 4 groups
    niini=randi(100,[k,1]);
    tic;
    outold=restreigeneasy(eigenvalues,niini,1.1);
    % Uncomment the line below if you want 
    % outold=restreigen(eigenvalues,niini,1.1,tol,1);
    oldroutinetime=oldroutinetime+toc;
    tic;
    outnew=restreigen(eigenvalues,niini,1.1,tol,userepmat);
    newroutinetime=newroutinetime+toc;
    if max(max(abs(outold-outnew)))>1e-5
    error('The two routines are different')
    end
    end
    disp(['Computing time of restreigeneasy: ' num2str(oldroutinetime)])
    disp(['Computing time of restreigen: ' num2str(newroutinetime)])

    Input Arguments

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    eigenvalues — Eigenvalues. Matrix.

    v x k matrix containing the eigenvalues of the covariance matrices of the k groups.

    v is the number of variables of the dataset which has to be clustered.

    Data Types: single| double

    niini — Cluster size. Vector.

    k x 1 vector containing the size of the k clusters

    Data Types: single| double

    restr — Restriction factor. Scalar.

    Scalar containing the restr parameter in tclust program.

    More in detail, parameter restr defines the cluster's shape restrictions, which are applied on all clusters during each iteration.

    Setting restr to 1, yields the strongest restriction, forcing all eigenvalues/determinants to be equal and so the method looks for similarly scattered (respectively spherical) clusters.

    Data Types: single| double

    Optional Arguments

    tol — tolerance. Scalar defining the tolerance of the procedure.

    The default value is 1e-8

    Example: 'tol',[1e-18]

    Data Types: double

    userepmat — use repmat, bsxfun or implicit expansion. Scalar.

    If userepmat is equal to 1, function repmat is used instead of bsxfun inside the procedure. Remark: repmat is built in from MATLAB 2013b so it is faster to use repmat if the current version of MATLAB is >2013a.

    If userepmat is 2, implicit expansion is used instead of bsxfun. Note that implicit expansion has been introduced only in 2017a therefore it will not work with previous releases.

    Example: 'userepmat',1

    Data Types: double

    Output Arguments

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    out —Restricted eigenvalues. Matrix

    v-by-k matrix containing restricted eigenvalues.

    The ratio between two possible elements in matrix out is not greater than restr

    References

    Fritz H., Garcia-Escudero, L.A. and Mayo-Iscar, A. (2013), A fast algorithm for robust constrained clustering, "Computational Satistics and Data Analysis", Vol. 61, pp. 124-136.

    [Available at http://www.eio.uva.es/infor/personas/tclust_algorithm.pdf]

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