restreigeneasy

restreigeneasy computes eigenvalues restriction (without Dykstra algorithm)

Syntax

  • out=restreigeneasy(eigenvalues, niini, restr)example
  • out=restreigeneasy(eigenvalues, niini, restr, tol)example

Description

restreigeneasy restricts the eigenvalues according to the constraint specified in scalar restr. This function is a readable and understandble (but sometimes slower) version of routine restreigen.

example

out =restreigeneasy(eigenvalues, niini, restr) Compiting time of three implementations of the eigenvalues restriction.

example

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

Examples

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  • Compiting time of three implementations of the eigenvalues restriction.
  • vmin  = 300; vmax  = 300; vstep = 1; vv = vmin:vstep:vmax;
    kmin  = 4;   kmax  = 16;  kstep = 4; kk = kmin:kstep:kmax;
    cycles = 1;
    t=zeros(floor((vmax-vmin)/vstep),length(kk)); t_easy=t; t2_easy=t;
    ik=0;
    for k = kk
    ik=ik+1;
    iv=0;
    for v=vv
    disp(['p=',num2str(v),' - k=',num2str(k)]);
    iv = iv+1;
    eigenvalues=abs(10*randn(v,k));
    niini=randi(1000,k,1);
    t0=tic;
    for s=1:cycles, out  = restreigen(eigenvalues,niini,1.1,1e-8);end
    t(iv,ik) = toc(t0);
    t1=tic;
    for s=1:cycles, out1 = restreigenmemopt(eigenvalues,niini,1.1,1e-8);end
    t_easy(iv,ik)=toc(t1);
    t2=tic;
    for s=1:cycles, out2 = restreigeneasy(eigenvalues,niini,1.1,1e-8);end
    t2_easy(iv,ik)=toc(t2);
    if sum(sum(out2-out1))>0
    disp('Zob!');
    end
    end
    end
    figure;
    vvv = 1;
    h2=plot(1:ik,t_easy(vvv,:),'-r',1:ik,t2_easy(vvv,:),'-k',1:ik,t(vvv,:),'-b');
    set(h2,'LineWidth',1.5);
    legend('restreigenmemopt' , 'restreigeneasy', 'restreigen','FontSize',20);
    xlabel('Number of groups k','FontSize',18);
    set(gca,'Xtick',1:ik,'XtickLabel',kk,'XTickLabelRotation',45);
    title(['Execution time for v=' num2str(vv(vvv))],'FontSize',20);
    p=300 - k=4
    p=300 - k=8
    p=300 - k=12
    p=300 - k=16
    
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

  • 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=restreigeneasy(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)))

    Related Examples

    expand all

  • 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

    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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