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gpcca.m
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function [ Pc, chi, A, wk, iopt ] = gpcca(P, sd, kmin, kmax, wk, iopt)
% This file is part of GPCCA.
%
% Copyright (c) 2020, 2018, 2017 Bernhard Reuter
%
% If you use this code or parts of it, cite the following reference:
%
% Reuter, B., Weber, M., Fackeldey, K., Röblitz, S., & Garcia, M. E. (2018). Generalized
% Markov State Modeling Method for Nonequilibrium Biomolecular Dynamics: Exemplified on
% Amyloid β Conformational Dynamics Driven by an Oscillating Electric Field. Journal of
% Chemical Theory and Computation, 14(7), 3579–3594. https://doi.org/10.1021/acs.jctc.8b00079
%
% GPCCA is free software: you can redistribute it and/or modify
% it under the terms of the GNU Lesser General Public License as published
% by the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% -------------------------------------------------------------------------
% This algorithm generates a fuzzy clustering such that the resulting
% membership functions are as crisp (characteristic) as possible.
%
% [ Pc, chi, A, wk, iopt ] = gpcca( P, sd, kmin, kmax, wk, iopt )
%
% Input:
% P (N,N)-matrix to be clustered (row-stochastic)
% sd "initial distribution"
% kmin minimum number of clusters
% kmax maximum number of clusters
% wk (possibly empty) structure with the following
% fields,
% that are initialized here, if they arent passed.
% wk.b Serves to tell SRSchur_num_t.m how many eigenvalues
% or blocks (and Schurvectors) of the Schurmatrix
% (and the Schurvector matrix) shall be sorted:
% If b < 0 then -b blocks will be sorted,
% if b > 0 then b or b+1 eigenvalues will be sorted,
% depending on the sizes of the blocks
% (ALWAYS set b>=kmax+1 here),
% if b = 0 then the whole Schur form will be sorted.
% WARNING: The number of sorted eigenvalues needs to
% be larger than the maximal number of clusters!
% WARNING: If you choose b < 0 you should REALLY
% know what you are doing!
% wk.display If 1, iterative output of the optimization progress
% shown. CAUTION: this slows the process
% significantly down.
% if 0, then not.
% wk.id Identification string for naming of the output
% files.
% wk.init If =1 use A=inv(EVS(index,:)) as starting guess,
% if =0 read A from file
% with identifier interactively passed from the
% command window.
% wk.maxiter Maximum number of optimization iterations:
% typically 2000-100000 for 'nelder-mead',
% typically ~500 for 'levenberg-marquardt',
% typically 100-1000 for 'gauss-newton'.
% -1 means off (internal maximum number of iterations
% of the algorithm is used).
% -2 means to use 50*(k-1)^2 iterations (only in case
% 'nelder-mead').
% wk.parallel Selection of parallel execution of the first
% optimization loop by choosing wk.parallel=1,
% else if wk.parallel=0 the program will be executed
% serially (Standard mode).
% wk.schur Calculate Schurvectors (schur=1) or use existing
% from file (schur=0)
% with identifier interactively passed from the
% command window.
% wk.solver Solver for unconstrained optimization problem.
% One of the following can be chosen:
% 'melder-mead',
% 'levenberg-marquardt' (not advised),
% 'gauss-newton'.
% wk.tolfun Optimization termination tolerance on the function
% value, a positive scalar:
% typically 1e-4 for 'nelder-mead',
% typically 1e-8 for 'levenberg-marquardt'.
% wk.tolx Optimization termination tolerance on x, a positive
% scalar:
% typically 1e-4 for 'nelder-mead',
% typically 1e-10 for 'levenberg-marquardt'.
% wk.xscale Gauss-Newton optimization factor for the scaling
% vector xscale = xscale * ones(size(xguess)).
% wk.xtol Gauss-Newton optimization relative (!) tolerance in
% x (real tolerance related to xscale*xtol).
% iopt (possibly empty) structure with the following
% optional fields,
% that are only necessary, if one wants to perform a
% optional final optimization.
% If one passes at least iopt.solver, the other
% fields are initialized here.
% iopt.display If 1, iterative optimization output is shown;
% if 0, then not.
% iopt.init If =1 use A=inv(EVS(index,:)) as starting guess,
% if =0 read A from file
% with identifier interactively passed from the
% command window,
% if =2 use the the optimized A matrices from the
% first optimization loop as input for the optional
% final optimization.
% iopt.maxiter Maximum number of optimization iterations:
% typically 2000-100000 for 'nelder-mead',
% typically ~500 for 'levenberg-marquardt',
% typically 100-1000 for 'gauss-newton'.
% -1 means off (internal maximum number of iterations
% of the algorithm is used).
% -2 means to use 50*(k-1)^2 iterations (only in case
% 'nelder-mead').
% iopt.parallel Selection of parallel execution of the optional
% optimization loop by choosing iopt.parallel=1,
% else if iopt.parallel=0 the program will be
% executed serially (Standard mode).
% iopt.solver Solver for unconstrained optimization problem.
% One of the following can be chosen:
% 'melder-mead',
% 'levenberg-marquardt' (not advised),
% 'gauss-newton'.
% iopt.tolfun Optimization termination tolerance on the function
% value, a positive scalar:
% typically 1e-4 for 'nelder-mead',
% typically 1e-8 for 'levenberg-marquardt'.
% iopt.tolx Optimization termination tolerance on x, a positive
% scalar:
% typically 1e-4 for 'nelder-mead',
% typically 1e-10 for 'levenberg-marquardt'.
% iopt.xscale Gauss-Newton optimization factor for the scaling
% vector xscale = xscale * ones(size(xguess)).
% iopt.xtol Gauss-Newton optimization relative (!) tolerance in
% x (real tolerance related to xscale*xtol).
%
% Output:
% Pc Coarse grained stochastic transition matrix as
% returned from the last optimization.
% chi (N,k)-matrix with membership values.
% A (k,k)-tranformation matrix A, chi = evs * A
% wk See above. All values are given back. Might vary
% from input values, if those were unfeasible. If no
% input was passed, the here initialized values
% are given back.
% iopt See above. Values are only given back, if at least
% iopt.solver was passed as input.
% Might vary from input values, if those were
% unfeasible. If only iopt.solver was passed, the
% here initialized values are given back.
% Cite:
%
% [1] P.Deuflhard, M.Weber: Robust Perron Cluster Analysis in Conformation
% Dynamics,
% Lin. Alg. App. 2005, 398c, 161-184.
% [2] S. Roeblitz and M. Weber: Fuzzy Spectral Clustering by PCCA+.
% In: Classification and Clustering: Models, Software and Applications,
% WIAS Report No. 26, Berlin 2009.
% http://www.wias-berlin.de/publications/wias-publ/run.jsp?template=...
% abstract&type=Report&year=2009&number=26
%
% Written by by Bernhard Reuter, Theoretical Physics II,
% University of Kassel, 2017,
% based on algorithms and code from Susanna Roeblitz and Marcus Weber,
% Zuse Institute Berlin, 2012
%--------------------------------------------------------------------------
% densify P and sd, if sparse
P = full(P) ;
sd = full(sd) ;
% make sure P, sd, kmin, kmax arent empty
assert(~isempty(P),'gpcca:EmptyInput','Variable P is empty') ;
assert(~isempty(sd),'gpcca:EmptyInput','Variable sd is empty') ;
assert(~isempty(kmin),'gpcca:EmptyInput','Variable kmin is empty') ;
assert(~isempty(kmax),'gpcca:EmptyInput','Variable kmax is empty') ;
% make sure kmin, kmax are integer values and kmin<=kmax
assert(mod(kmin,1) == 0, 'gpcca:k_InputError', ...
'kmin is not an integer value') ;
assert(mod(kmax,1) == 0, 'gpcca:k_InputError', ...
'kmax is not an integer value') ;
assert(kmax >= kmin, 'gpcca:k_InputError', 'kmax !>= kmin') ;
%--------------------------------------------------------------------------
class_t1 = class(P) ;
class_t2 = class(sd) ;
if (strcmpi(class_t1,class_t2))
class_t = class_t1 ;
else
error('gpcca:P_sd_DataTypeError', ...
['class(P) is not equal class(sd)! This will lead to ', ...
'numeric precision errors!'])
end
function r = num_t(expression)
if (nargin > 0)
if(strcmpi(class_t,'mp')), r = mp(expression) ;
else
if isnumeric(expression)
r = expression ;
else
r = eval(expression) ;
end ;
end ;
else
r = class_t ;
end ;
end % num_t
% -----------------------------------------------------------------------
% Global variable 'figureVisible' defines, if figures are visible.
% Even if figures are invisible (figureVisible=false), the figures
% are still generated and printed to files.
global figureVisible
if figureVisible == false
% all subsequent figures invisible
set(0,'DefaultFigureVisible','off');
end
% -----------------------------------------------------------------------
% initialize the workspace
[wk, fileid] = initialize_workspace(wk)
assert(~(wk.init==0 & wk.parallel==1), ...
'gpcca:IncompatibleOptionsError', ...
['You choose parallel execution of the first optimization loop ' ...
'combined with initialization of A from file. These two ' ...
'options are mutually exclusive!']) ;
% make sure that one isnt messing with wk.b
if (wk.b > 0)
assert(wk.b > kmax, 'gpcca:k_InputError', ['wk.b <= kmax: ', ...
'The number of sorted eigenvalues needs to be larger ', ...
'than the maximal number of clusters!']) ;
else
disp(['You chose wk.b < 0: You should REALLY know what you ', ...
'are doing!'])
end
% initialize recording to diary
name = strcat(fileid,'-','diary','.txt') ;
diary(name) ;
% save the stochastic matrix P and the initial distribution sd
disp (' ')
Switch = inputT(['If you want to save the P matrix (in ASCII ' ...
'format), type 1, else type 0: '], 'Switch') ;
disp (' ')
if isempty(Switch) || (Switch~=0 && Switch~=1)
error('invalid input')
elseif Switch==0
disp ('P wont be saved')
else
name=strcat(fileid,'-','P','.txt') ;
save_t(name,P,'-ascii')
end
disp (' ')
Switch = inputT(['If you want to save the sd vector (in ASCII ' ...
'format), type 1, else type 0: '], 'Switch') ;
disp (' ')
if isempty(Switch) || (Switch~=0 && Switch~=1)
error('invalid input')
elseif Switch==0
disp ('sd wont be saved')
else
name=strcat(fileid,'-','sd','.txt') ;
save_t(name,sd,'-ascii')
end
% plot the stochastic matrix P
disp (' ')
Switch = inputT(['If you want to plot the sd P matrix, ' ...
'type 1, else type 0 (dont do this for large matrices!): '], 'Switch') ;
disp (' ')
if isempty(Switch) || (Switch~=0 && Switch~=1)
error('invalid input')
elseif Switch==0
disp ('P wont be plotted')
else
name=strcat(fileid,'-','P-plot') ;
plotmatrix(double(P),name) ;
end
% -----------------------------------------------------------------------
% If you want to use gpcca on eigen- instead of Schur-vectors, replace the
% calculation and sd-orthonormalization of the Schur-vectors by the
% calculation and sd-orthonormalization of the eigenvectors. The
% sd-orthonormalization is imperative for the correct function of the
% following implementation of GPCCA!
if (wk.schur == 1)
[ X, RR ] = do_schur( P, sd, fileid, wk.b ) ;
else
inputname = inputT(['Enter the name of the ordered Schurvectors ',...
'to use for GPCCA (IN QUOTES): '],'inputname') ;
X = load_t(inputname,'-ascii',num_t) ;
dummy = (X'*diag(sd)*X - num_t(eye(size(X,2))) > (num_t('10000')*eps(num_t))) ;
assert(~any(dummy(:)),'gpcca:X_OrthogonalityError', ...
'Schurvectors appear to not be orthogonal')
clearvars dummy
dummy = ( abs(X(:,1) - 1) < ( numeric_t('1000') * eps(numeric_t) ) ) ;
assert(all(dummy(:)), 'gpcca:FirstColumnError', ...
'X(:,1) isnt equal 1!')
inputname = inputT(['Enter the name of the ordered Schurmatrix ',...
'matching to X to use for GPCCA (IN QUOTES): '],'inputname') ;
RR = load_t(inputname,'-ascii','double') ;
% check RR for correctness
assert(size(RR,1)==size(RR,2),'gpcca:MatrixShapeError1', ...
'RR Matrix is not quadratic but %d x %d',size(RR,1),size(RR,2))
% check if the number of rows of X matches with the shape of RR
assert( size(X,1)==size(RR,1), 'gpcca:MatrixShapeError2', ...
['The number of rows size(X,1)=%d of X doesnt match with the ', ...
'shape (%d,%d) of RR!'],...
size(X,1), size(RR,1), size(RR,2) )
end
% -----------------------------------------------------------------------
[ badblocks, badblockrows ] = find_twoblocks( double(X), double(RR), fileid ) ;
clearvars RR
disp(' ')
disp('The following cluster numbers will be excluded in all further analysis ')
disp('since using them would lead to splitting of 2x2-blocks of eigenvalues!')
disp(badblockrows)
% -----------------------------------------------------------------------
EVS = X ;
% convert to double for futher calculations (but not EVS!)
sd = double(sd) ;
%pi = double(pi) ;
P = double(P) ;
% -----------------------------------------------------------------------
disp (' ')
disp ('Decide, if you want to apply the minChi-Criterion.')
minChi_switch = inputT(['If you want to use minChi, type 1, ' ...
'else type 0: '], 'minChi_switch') ;
disp (' ')
if isempty(minChi_switch) || (minChi_switch~=0 && minChi_switch~=1)
error('invalid input')
elseif minChi_switch==0
disp ('minChi-Criterion wont be used')
else % start of minChi procedure
[ ~ ] = use_minChi( EVS, kmin, kmax, fileid ) ;
end % end of minChi procedure
% -----------------------------------------------------------------------
disp(' ')
kmin = inputT(['Your choice for the number of clusters to start ' ...
'the optimization step: '],'kmin') ;
kmax = inputT(['Your choice for the number of clusters to end the ' ...
'optimization step: '],'kmax') ;
disp(' ')
% make sure kmin and kmax are defined
assert(~isempty(kmin),'gpcca:EmptyKeyboardInput', ...
'Variable kmin is empty') ;
assert(~isempty(kmax),'gpcca:EmptyKeyboardInput', ...
'Variable kmax is empty') ;
% make sure kmin, kmax are integer values and kmin<=kmax
assert(mod(kmin,1) == 0, 'gpcca:KeyboardInputError', ...
'kmin is not an integer value') ;
assert(mod(kmax,1) == 0, 'gpcca:KeyboardInputError', ...
'kmax is not an integer value') ;
assert(kmax >= kmin, 'gpcca:KeyboardInputError', 'kmax !>= kmin') ;
% decide for "best" number of clusters (other criteria are possible):
% the optimal value for trace(S) can be at most kt
% therefore crispness=trace(S)/kt<1, but should be as large as
% possible
disp (' ')
disp ('Optimize crispness vs. number of clusters')
disp ('=============================================')
disp (' ')
assert(isa(EVS,num_t),'gpcca:EVS_DataTypeError', ...
'Variable is type %s not %s',class(EVS),num_t)
% initialize A
A_cell = cell([1,kmax]) ;
for kt = kmin:kmax
if badblocks(kt) == false
evs = EVS(:,1:kt) ;
A_cell{kt} = initialize_A(evs, wk.init) ;
end
end
[ Pc, A_cell, chi, val_vec, opt_vec ] = optimization_loop( P, sd, ...
double(EVS), A_cell, kmin, kmax, wk, fileid ) ;
A = A_cell{kmax} ;
% -----------------------------------------------------------------------
% save the 'val'-vector and the crispness-vector
name=strcat(fileid,'-','opt_vec','-','n=',int2str(kmin),'-', ...
int2str(kmax),'.txt') ;
save_t(name,opt_vec,'-ascii')
name=strcat(fileid,'-','val_vec','-','n=',int2str(kmin),'-', ...
int2str(kmax),'.txt') ;
save_t(name,val_vec,'-ascii')
% plot the crispness over the number of clusters
fig5 = figure ;
plot(kmin:kmax,opt_vec(:,2),'-x')
xlabel('number of clusters')
ylabel('crispness')
name=strcat(fileid,'-','crispness-figure','-','n=',int2str(kmin), ...
'-',int2str(kmax)) ;
h = gcf ; % Current figure handle
set(h,'PaperPositionMode','manual') ;
set(h,'PaperUnits','inches') ;
set(h,'PaperPosition',[0 0 3.33 2.63]) ;
set(h,'PaperUnits','inches') ;
set(h,'PaperSize',[3.33 2.63]) ;
savefig(fig5,strcat(name,'.fig'),'compact')
print(fig5,strcat(name,'.pdf'),'-dpdf')
print(fig5,strcat(name,'.png'),'-dpng','-r600')
% -----------------------------------------------------------------------
% initialize optional parameters for optional final optimization
[iopt, fileid, final_opt] = initialize_optional(iopt, wk.id)
% test, if final optimization shall be performed
if final_opt == true
assert(~(iopt.init==0 & iopt.parallel==1), ...
'gpcca:IncompatibleOptionsError', ...
['You choose parallel execution of the optional optimization loop ' ...
'combined with initialization of A from file. These two ' ...
'options are mutually exclusive!']) ;
% change the diary
name = strcat(fileid,'-','diary','.txt') ;
diary(name) ;
% -------------------------------------------------------------------
% optional final optimization for beforehand determined optimal
% cluster number
disp(' ')
koptmin = inputT(['Your choice for the number of clusters to ' ...
'start the final optimization step: '],'koptmin') ;
koptmax = inputT(['Your choice for the number of clusters to end ' ...
'the final optimization step: '],'koptmax') ;
disp (' ')
% make sure koptmin and koptmax are defined
assert(~isempty(koptmin),'gpcca:EmptyKeyboardInput', ...
'Variable koptmin is empty') ;
assert(~isempty(koptmax),'gpcca:EmptyKeyboardInput', ...
'Variable koptmax is empty') ;
% make sure koptmin, koptmax are integer values and
% koptmin<=koptmax
assert(mod(koptmin,1) == 0, 'gpcca:KeyboardInputError', ...
'koptmin is not an integer value') ;
assert(mod(koptmax,1) == 0, 'gpcca:KeyboardInputError', ...
'koptmax is not an integer value') ;
assert(koptmax >= koptmin, 'gpcca:KeyboardInputError', ...
'koptmax !>= koptmin') ;
disp (' ')
disp ('Perform final optimization')
disp ('==========================')
disp (' ')
assert(isa(EVS,num_t),'gpcca:EVS_DataTypeError', ...
'Variable is type %s not %s', class(EVS), num_t)
% initialize A_cell, if iopt.init~=2, else use A_cell from the
% previous optimization
if ( iopt.init == 0 || iopt.init == 1 )
A_cell = cell([1,koptmax]) ;
for kt = koptmin:koptmax
if badblocks(kt) == false
evs = EVS(:,1:kt) ;
A_cell{kt} = initialize_A(evs, wk.init) ;
end
end
elseif iopt.init == 2
assert( koptmin >= kmin, 'gpcca:KeyboardInputError', ...
['You choosed to use the optimized A matrices from the first '...
'optimization loop as input for the final optimization but '...
'koptmin is smaller than kmin!'] ) ;
assert( koptmax <= kmax, 'gpcca:KeyboardInputError', ...
['You choosed to use the optimized A matrices from the first '...
'optimization loop as input for the final optimization but '...
'koptmax is bigger than kmax!'] ) ;
else
error('gpcca:IoptInitError',['Couldnt '...
'initialize A for the final optimization since iopt.init ' ...
'was neither =0, =1 nor =2!'])
end
[ Pc, A_cell, chi, val_vec, opt_vec ] = optimization_loop( P, sd,...
double(EVS), A_cell, koptmin, koptmax, iopt, fileid ) ;
A = A_cell{koptmax} ;
% -------------------------------------------------------------------
% plot the crispness over the number of clusters, if more than
% one final optimization was performed
if koptmax > koptmin
fig6 = figure ;
plot(koptmin:koptmax,opt_vec(:,2),'-x')
xlabel('number of clusters')
ylabel('crispness')
name=strcat(fileid,'-','crispness-figure','-','n=', ...
int2str(koptmin),'-',int2str(koptmax)) ;
h = gcf ; % Current figure handle
set(h,'PaperPositionMode','manual') ;
set(h,'PaperUnits','inches') ;
set(h,'PaperPosition',[0 0 3.33 2.63]) ;
set(h,'PaperUnits','inches') ;
set(h,'PaperSize',[3.33 2.63]) ;
savefig(fig6,strcat(name,'.fig'),'compact')
print(fig6,strcat(name,'.pdf'),'-dpdf')
print(fig6,strcat(name,'.png'),'-dpng','-r600')
% save the 'val'-vector and the crispness-vector
name=strcat(fileid,'-','opt_vec','-','n=',int2str(koptmin),'-', ...
int2str(koptmax),'.txt') ;
save_t(name,opt_vec,'-ascii')
name=strcat(fileid,'-','val_vec','-','n=',int2str(koptmin),'-', ...
int2str(koptmax),'.txt') ;
save_t(name,val_vec,'-ascii')
end
% -------------------------------------------------------------------
end
% -----------------------------------------------------------------------
set(0,'DefaultFigureVisible','on'); % all subsequent figures visible
diary OFF ;
end