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OptAlgo.m
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OptAlgo.m
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classdef OptAlgo < handle
% =============================================================================
% This is the class of the functions of OPTimization ALGOrithms (OptAlgo).
%
% =============================================================================
properties (Constant)
swarm = 20; % population size
sample = 200; % loop count for evolution (at least 100)
dataPoint = 1000; % amount of data observation
prior = []; % prior information (default empty)
logScale = true; % log-scale for parameter domain (default true)
FUNC = @simulatedMovingBed; % the objective function
end
% Lower level algorithms, in charge of continuous decision variables optimization
% -----------------------------------------------------------------------------
% DE
methods (Static = true, Access = 'public')
function [xValue, yValue] = Differential_Evolution(opt)
% -----------------------------------------------------------------------------
% Differential Evolution algorithm (DE)
%
% DE optimizes a problem by maintaining a population of candidate solutions
% and creating new candidate solutions by combing existing ones according
% to its mathematical formula, and then keeping whichever candidate solution
% has the best score or fitness on the optimization problem at hand
%
% Parameter:
% - FUNC. The objective function from your field
%
% Returns:
% - xValue. The optimal parameter set
% - yValue. The value of objective function with optimal parameter set
% -----------------------------------------------------------------------------
startTime = clock;
if nargin < 1
error('OptAlgo.DE: There are no enough inputs \n');
end
% Get the optimizer options
opt = OptAlgo.getOptionsDE(opt);
% Initialization of the population
[Population, OptPopul] = OptAlgo.initChainDE(opt);
%----------------------------------------------------------------------------------------
% The main loop
% ---------------------------------------------------------------------------------------
for i = 1:opt.nsamples
% The evolution of each generation
[Population, OptPopul] = OptAlgo.evolutionDE(Population, OptPopul, opt);
% Abstract best information so far from the population and display it
yValue = OptPopul(opt.Nchain+1, opt.Nparams+1);
xValue = OptAlgo.pTransfer('exp', OptPopul(1, 1:opt.Nparams));
Value = [xValue, yValue];
save('chainDE.dat', 'Value', '-ascii', '-tabs', '-append');
fprintf('Iter = %5d ---------------- Minimum: %10.3g ---------------- \n', i, yValue);
fprintf('%10.3g | ', xValue); fprintf('\n');
% The convergence criterion: when the standard deviation is smaller
% than the specified tolerence
if mod(i, 5) == 0
delta = std(Population(:, 1:opt.Nparams)) ./ mean(Population(:, 1:opt.Nparams));
if all(abs(delta) < opt.criterion) || i == opt.nsamples
maxIter = i;
break
end
end
end % for i = 1:opt.nsamples
%-------------------------------------------------------------------------------
% Post-process
%-------------------------------------------------------------------------------
% Gather some useful information and store them in the debug mode
result.optTime = etime(clock, startTime)/3600; % in hours
result.convergence = delta;
result.correlation = corrcoef(Population(:, 1:opt.Nparams));
result.Iteration = maxIter;
result.xValue = xValue;
result.yValue = yValue;
fprintf('\n****************************************************** \n');
save(sprintf('result_%2d.mat',fix(rand*100)),'result');
fprintf('Time %5.3g hours elapsed after %5d iterations \n', result.optTime, result.Iteration);
fprintf('Objective value: %10.3g \n', yValue);
fprintf('Optimal set: ');
fprintf(' %g |', xValue);
fprintf('\nThe statistical information of DE is stored as result.mat \n');
fprintf('****************************************************** \n');
end % Differential_Evolution
function opt = getOptionsDE(obj)
% -----------------------------------------------------------------------------
% The parameters for the Optimizer
%
% Return:
% - opt.
% + Nchain. The number of the candidates (particles)
% + Nparams. The number of the optimized parameters
% + bounds. The boundary limitation of the parameters
% + nsamples. The maximum number of algorithm's iteration
% + crossProb. The cross-over probability in DE's formula
% + weight. The weight coefficients in DE's formula
% + strategy. There are 1,2,3,4,5,6 strategies in DE algorithm to
% deal with the cross-over and mutation. As for detais, we
% will refer your the original paper of Storn and Price
% -----------------------------------------------------------------------------
opt = [];
opt.Nchain = OptAlgo.swarm;
opt.Nparams = length(fieldnames(obj.params));
opt.bounds = OptAlgo.pTransfer('log', obj.paramBound);
opt.nsamples = OptAlgo.sample;
opt.criterion = 0.01;
% Check out dimension of the parameter set, and the boundary limiatation
[row, col] = size(opt.Nparams);
if row > 1 || col > 1
error('OptAlgo.getOptionsDE: The initialized dimension of the set of parameters might be wrong \n');
end
[row, col] = size(opt.bounds);
if row ~= opt.Nparams || col ~= 2
error('OptAlgo.getOptionsDE: Please check your setup of the parameter range \n');
end
if mod(opt.nsamples, 5) ~= 0
error('OptAlgo.getOptionsDE: Please set the maximum iteration be divisible to 5 \n');
end
% Those are the specific parameters for the DE algorithm
opt.crossProb = 0.5;
opt.weight = 0.3;
opt.strategy = 5;
end % getOptionsDE
function [Population, OptPopul] = initChainDE(opt)
% -----------------------------------------------------------------------------
% The initilization of the population
%
% Parameter:
% - opt.
% + Nchain. The number of the candidates (particles)
% + Nparams. The number of the optimized parameters
% + bounds. The boundary limitation of the parameters
% + nsamples. The maximum number of algorithm's iteration
% + crossProb. The cross-over probability in DE's formula
% + weight. The weight coefficients in DE's formula
% + strategy. There are 1,2,3,4,5,6 strategies in DE algorithm to
% deal with the cross-over and mutation. As for detais, we
% will refer your the original paper of Storn and Price
%
% Return:
% - Population. The population of the particles, correspondingly the objective value
% - OptPopul. The best fit found among the population.
% -----------------------------------------------------------------------------
if nargin < 1
error('OptAlgo.initChainDE: There are no enough input arguments \n');
end
% Initialization of the parameters
Population = rand(opt.Nchain, opt.Nparams+1);
% Use vectorization to speed up. Keep the parameters in the domain
Population(:, 1:opt.Nparams) = repmat(opt.bounds(:,1)', opt.Nchain, 1) + ...
Population(:, 1:opt.Nparams) .* repmat( (opt.bounds(:,2) - opt.bounds(:,1))', opt.Nchain, 1 );
% Simulation of the sampled points
Population(:, opt.Nparams+1) = arrayfun( @(idx) feval(OptAlgo.FUNC, ...
OptAlgo.pTransfer('exp', Population(idx, 1:opt.Nparams))), 1:opt.Nchain );
% if the parallel toolbox is available
% value = zeros(1, opt.Nchain);
% parameter = Population(1:opt.Nchain, 1:opt.Nparams);
% parfor i = 1:opt.Nchain
% value(i)= feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', parameter(i,:)) )
% end
% Population(:,opt.Nparams+1) = value';
% The statistics of the population
[minValue, row] = min(Population(:, opt.Nparams+1));
% Preallocation and allocation
OptPopul = zeros(opt.Nchain+1, opt.Nparams+1);
OptPopul(opt.Nchain+1, opt.Nparams+1) = minValue;
OptPopul(1:opt.Nchain, 1:opt.Nparams) = repmat( Population(row, 1:opt.Nparams), opt.Nchain, 1 );
end % initChainDE
function [Population, OptPopul] = evolutionDE(Population, OptPopul, opt)
% -----------------------------------------------------------------------------
% The evolution of population
%
% Parameters:
% - Population. The population of the particles, correspondingly the objective value
% - OptPopul. The best fit found among the population.
% - opt. Please see the comments of the function, initChainDE
%
% Return:
% - Population. The population of the particles, correspondingly the objective value
% - OptPopul. The best fit found among the population.
% -----------------------------------------------------------------------------
if nargin < 3
error('OptAlgo.evolutionDE: There are no enough input arguments \n');
end
R = opt.Nchain;
C = opt.Nparams;
indexArray = (0:1:R-1);
index = randperm(4);
cr_mutation = rand(R, C) < opt.crossProb;
cr_old = cr_mutation < 0.5;
ShuffRow1 = randperm(R);
idxShuff = rem(indexArray + index(1), R);
ShuffRow2 = ShuffRow1(idxShuff + 1);
idxShuff = rem(indexArray + index(2), R);
ShuffRow3 = ShuffRow2(idxShuff + 1);
idxShuff = rem(indexArray + index(3), R);
ShuffRow4 = ShuffRow3(idxShuff + 1);
idxShuff = rem(indexArray + index(4), R);
ShuffRow5 = ShuffRow4(idxShuff + 1);
PopMutR1 = Population(ShuffRow1, 1:C);
PopMutR2 = Population(ShuffRow2, 1:C);
PopMutR3 = Population(ShuffRow3, 1:C);
PopMutR4 = Population(ShuffRow4, 1:C);
PopMutR5 = Population(ShuffRow5, 1:C);
switch opt.strategy
case 1
% strategy 1
tempPop = PopMutR3 + (PopMutR1 - PopMutR2) * opt.weight;
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
case 2
% strategy 2
tempPop = Population(1:R, 1:C) + opt.weight * (OptPopul(1:R, 1:C) - ...
Population(1:R, 1:C)) + (PopMutR1 - PopMutR2) * opt.weight;
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
case 3
% strategy 3
tempPop = OptPopul(1:R, 1:C) + (PopMutR1 - PopMutR2) .* ((1 -0.9999) * rand(R, C) + opt.weight);
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
case 4
% strategy 4
f1 = (1-opt.weight) * rand(R, 1) + opt.weight;
PopMutR5 = repmat(f1, 1, C);
tempPop = PopMutR3 + (PopMutR1 - PopMutR2) .* PopMutR5;
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
case 5
% strategy 5
f1 = (1-opt.weight) * rand + opt.weight;
tempPop = PopMutR3 + (PopMutR1 - PopMutR2) * f1;
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
case 6
% strategy 6
if (rand < 0.5)
tempPop = PopMutR3 + (PopMutR1 - PopMutR2) * opt.weight;
else
tempPop = PopMutR3 + 0.5 * (opt.weight + 1.0) * (PopMutR1 + PopMutR2 - 2 * PopMutR3);
end
tempPop = Population(1:R, 1:C) .* cr_old + tempPop .* cr_mutation;
end
% Check the boundary limitation
loBound = repmat(opt.bounds(:,1)', R, 1);
upBound = repmat(opt.bounds(:,2)', R, 1);
tempPop(tempPop < loBound) = loBound(tempPop < loBound);
tempPop(tempPop > upBound) = upBound(tempPop > upBound);
% Simulate the new population and compare their objective function values
tempValue(:, 1) = arrayfun( @(idx) feval( OptAlgo.FUNC, ...
OptAlgo.pTransfer('exp', tempPop(idx, 1:C)) ), 1:R );
% if the parallel toolbox is available
% parfor i = 1:R
% tempValue(i,1)= feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', tempPop(i,:)) )
% end
Population(tempValue < Population(:, C+1), 1:C) = tempPop(tempValue < Population(:, C+1), 1:C);
Population(tempValue < Population(:, C+1), C+1) = tempValue(tempValue < Population(:, C+1));
% Rank the objective function value to find the minimum
[minValue, minRow] = min(Population(:, C+1));
OptPopul(R+1, C+1) = minValue;
OptPopul(1:R, 1:C) = repmat( Population(minRow, 1:C), R, 1 );
end % evolutionDE
end % DE
% PSO
methods (Static = true, Access = 'public')
function [xValue, yValue] = Particle_Swarm_Optimization(opt)
% -----------------------------------------------------------------------------
% Particle Swarm Optimization algorithm (PSO)
%
% PSO optimizes a problem by having a population of candidates
% (particles), and moving these particles around in the search space
% according to mathematical formula ovet the particle's position and
% velocity. Each particle's movement is influenced by its own local
% best-known position but, is also guided toward the best-known positions
% in the search space, which are updated as better positions are found by
% other particles
%
% Returns:
% - xValue. The optimal parameter set
% - yValue. The value of objective function with optimal parameter set
% -----------------------------------------------------------------------------
startTime = clock;
if nargin < 1
error('OptAlgo.PSO: There are no enough inputs \n');
end
% Get the optimizer options
opt = OptAlgo.getOptionsPSO(opt);
% Initialization of the population
[ParSwarm, OptSwarm, ToplOptSwarm] = OptAlgo.initChainPSO(opt);
%-------------------------------------------------------------------------------
% The main loop
%-------------------------------------------------------------------------------
for i = 1:opt.nsamples
% The evolution of the particles in PSO
[ParSwarm, OptSwarm, ToplOptSwarm] = OptAlgo.evolutionPSO ...
(ParSwarm, OptSwarm, ToplOptSwarm, i, opt);
% Abstract best information so far from the population and display it
yValue = OptSwarm(opt.Nchain+1, opt.Nparams+1);
xValue = OptAlgo.pTransfer('exp', OptSwarm(opt.Nchain+1, 1:opt.Nparams));
Value = [xValue, yValue];
save('chainPSO.dat', 'Value', '-ascii', '-tabs', '-append');
fprintf('Iter = %5d ---------------- Minimum: %10.3g ---------------- \n', i, yValue);
fprintf('%10.3g | ', xValue); fprintf('\n');
% convergence criterion
if mod(i, 5) == 0
delta = std(ParSwarm(:, 1:opt.Nparams)) ./ mean(ParSwarm(:, 1:opt.Nparams));
if all(abs(delta) < opt.criterion) || i == opt.nsamples
maxIter = i;
break
end
end
end % for i = 1:opt.nsamples
%-------------------------------------------------------------------------------
% Post-process
%-------------------------------------------------------------------------------
% Gather some useful information and store them in the debug mode
result.optTime = etime(clock, startTime)/3600; % in hours
result.convergence = delta;
result.correlation = corrcoef(ParSwarm(:, 1:opt.Nparams));
result.Iteration = maxIter;
result.xValue = xValue;
result.yValue = yValue;
fprintf('\n****************************************************** \n');
save(sprintf('result_%2d.mat',fix(rand*100)),'result');
fprintf('Time %5.3g hours elapsed after %5d iterations \n', result.optTime, result.Iteration);
fprintf('Objective value: %10.3g \n', yValue);
fprintf('Optimal set: ');
fprintf(' %g |', xValue);
fprintf('\nThe statistical information of PSO is stored as result.mat \n');
fprintf('****************************************************** \n');
end % Particle_Swarm_Optimization
function opt = getOptionsPSO(obj)
% -----------------------------------------------------------------------------
% The parameters for the Optimizer
%
% Return:
% - opt.
% + Nchain. The number of the candidates (particles)
% + Nparams. The number of the optimized parameters
% + bounds. The boundary limitation of the parameters
% + nsamples. The maximum number of algorithm's iteration
% + wMax; wMin. The boundary of the weight in PSO's formula
% + accelCoeff. The accelarate coefficients in PSO's formula
% + topology. The different schemes for the particles' communication
% * Ring Topology. Under this scheme, only the adjacent
% particles exchange the information. So this is the slowest
% scheme to transfer the best position among particles.
% * Random Topology. Under this scheme, the communication is
% isolated a little bit. This is the intermediate one.
% * without Topology. Under this scheme, the the best position
% around the population is going to transfer immediately to
% the rest particles. This is the default one in the
% literatures. However in this case, it will results in the
% unmature convergence.
% -----------------------------------------------------------------------------
opt = [];
opt.Nchain = OptAlgo.swarm;
opt.Nparams = length(fieldnames(obj.params));
opt.bounds = OptAlgo.pTransfer('log', obj.paramBound);
opt.nsamples = OptAlgo.sample;
opt.criterion = 0.01;
% Check out the dimension of the set of parameters, and the boundary limitation
[row, col] = size(opt.Nparams);
if row > 1 || col > 1
error('OptAlgo.getOptionsPSO: The initialized dimension of the parameter set might be wrong \n');
end
[row, col] = size(opt.bounds);
if row ~= opt.Nparams || col ~= 2
error('OptAlgo.getOptionsPSO: Please check your setup of the range of parameters \n');
end
if mod(opt.nsamples, 5) ~= 0
error('OptAlgo.getOptionsPSO: Please let the maximum interation be divisible to 5 \n');
end
% Those are the specific parameters for the PSO algorithm
opt.wMax = 0.9;
opt.wMin = 0.4;
opt.accelCoeff = [0.6, 0.4];
opt.topology = 'Random'; % Null; Ring
opt.randCount = int32(opt.Nchain * 0.6);
end % getOptionsPSO
function [ParSwarm, OptSwarm, ToplOptSwarm] = initChainPSO(opt)
% -----------------------------------------------------------------------------
% The initilization of the population
%
% Parameter:
% - opt.
% + Nchain. The number of the candidates (particles)
% + Nparams. The number of the optimized parameters
% + bounds. The boundary limitation of the parameters
% + nsamples. The maximum number of algorithm's iteration
% + wMax; wMin. The boundary of the weight in PSO's formula
% + accelCoeff. The accelarate coefficients in PSO's formula
% + topology. The different schemes for the particles' communication
% * Ring. Under this scheme, only the adjacent
% particles exchange the information. So this is the slowest
% scheme to transfer the best position among particles.
% * Random. Under this scheme, the communication is
% isolated a little bit. This is the intermediate one.
% * Null. Under this scheme, the the best position
% around the population is going to transfer immediately to
% the rest particles. This is the default one in the
% literatures. However in this case, it will results in the
% unmature convergence.
%
% Return:
% - ParSwarm. The swarm of the particles, correspondingly the objective value
% - OptSwarm. The local optima that each particle ever encountered
% - ToplOptSwarm. The global optima that is shared by the rest particles.
% -----------------------------------------------------------------------------
if nargout < 3
error('OptAlgo.initChainPSO: There are not enough output \n');
end
% Initilization of the parameters, and velocity of parameters
ParSwarm = rand(opt.Nchain, 2*opt.Nparams+1);
% Use vectorization to speed up. Keep the parameters in the domain
ParSwarm(:, 1:opt.Nparams) = repmat(opt.bounds(:,1)', opt.Nchain, 1) + ...
ParSwarm(:, 1:opt.Nparams) .* repmat( (opt.bounds(:,2) - opt.bounds(:,1))', opt.Nchain, 1 );
% Simulation of the sampled points
ParSwarm(:, 2*opt.Nparams+1) = arrayfun( @(idx) feval( OptAlgo.FUNC, ...
OptAlgo.pTransfer('exp', ParSwarm(idx, 1:opt.Nparams)) ), 1:opt.Nchain );
% The statistics of the population
OptSwarm = zeros(opt.Nchain+1, opt.Nparams+1);
[minValue, minRow] = min(ParSwarm(:, 2*opt.Nparams+1));
OptSwarm(1:opt.Nchain, 1:opt.Nparams) = ParSwarm(1:opt.Nchain, 1:opt.Nparams);
OptSwarm(1:opt.Nchain, opt.Nparams+1) = ParSwarm(1:opt.Nchain, 2*opt.Nparams+1);
OptSwarm(opt.Nchain+1, 1:opt.Nparams) = ParSwarm(minRow, 1:opt.Nparams);
OptSwarm(opt.Nchain+1, opt.Nparams+1) = minValue;
ToplOptSwarm = OptSwarm(1:opt.Nchain, 1:opt.Nparams);
end % initChainPSO
function [ParSwarm, OptSwarm, ToplOptSwarm] = evolutionPSO(ParSwarm, OptSwarm, ToplOptSwarm, iter, opt)
% -----------------------------------------------------------------------------
% The evolution of particles, according to the local optima and the global optima.
%
% Parameters:
% - ParSwarm. The swarm of the particles, correspondingly the objective value
% - OptSwarm. The local optima that each particle ever encountered
% - ToplOptSwarm. The global optima that is shared by the rest particles.
% - iter. The iteration number in the main loop
% - opt. Please see the comments of the function, initChainPSO
%
% Return:
% - ParSwarm. The swarm of the particles, correspondingly the objective value
% - OptSwarm. The local optima that each particle ever encountered
% - ToplOptSwarm. The global optima that is shared by the rest particles.
% -----------------------------------------------------------------------------
if nargin < 5
error('OptAlgo.evolutionPSO: There are no enough input arguments \n')
end
if nargout ~= 3
error('OptAlgo.evolutionPSO: There are no enough output arguments \n')
end
R = opt.Nchain;
C = opt.Nparams;
% Different strategies of the construction of the weight
weight = opt.wMax - iter * ((opt.wMax - opt.wMin) / opt.nsamples);
% weight = 0.7;
% weight = (opt.wMax - opt.wMin) * (iter / opt.nsamples)^2 ...
% + (opt.wMin - opt.wMax) * (2 * iter / opt.nsamples) + opt.wMax;
% weight = opt.wMin * (opt.wMax / opt.wMin)^(1 / (1 + 10 * iter / opt.nsamples));
% The difference of current position and the best position the particle has encountered
LocalOptDiff = OptSwarm(1:R, 1:C) - ParSwarm(1:R, 1:C);
% The difference of current position and the best position the swarm has encountered
if strcmp(opt.topology, 'Null')
GlobalOptDiff = OptSwarm(R+1, 1:C) - ParSwarm(:, 1:C);
elseif strcmp(opt.topology, 'Random') || strcmp(opt.topology, 'Ring')
GlobalOptDiff = ToplOptSwarm(:, 1:C) - ParSwarm(:, 1:C);
end
% The evolution of the velocity matrix, according to LocalOptDiff and GlobalOptDiff
% TempVelocity = weight .* ParSwarm(:,C+1:2*C) + ...
% opt.accelCoeff(1) * unifrnd(0,1.0) .* LocalOptDiff + ...
% opt.accelCoeff(2) * unifrnd(0,1.0) .* GlobalOptDiff;
ParSwarm(:, C+1:2*C) = weight .* ParSwarm(:, C+1:2*C) + ...
opt.accelCoeff(1) .* LocalOptDiff + opt.accelCoeff(2) .* GlobalOptDiff;
% The evolution of the current positions, according to the Velocity and the step size
stepSize = 1; % stepSize = 0.729;
ParSwarm(:, 1:C) = ParSwarm(:, 1:C) + stepSize .* ParSwarm(:, C+1:2*C);
% Check the boundary limitation
loBound = repmat(opt.bounds(:,1)', R, 1);
upBound = repmat(opt.bounds(:,2)', R, 1);
ParSwarm(ParSwarm(:, 1:C) < loBound) = loBound(ParSwarm(:, 1:C) < loBound);
ParSwarm(ParSwarm(:, 1:C) > upBound) = upBound(ParSwarm(:, 1:C) > upBound);
% Simulation of the sampled points
ParSwarm(:, 2*C+1) = arrayfun( @(idx) feval( OptAlgo.FUNC, ...
OptAlgo.pTransfer('exp', ParSwarm(idx, 1:C)) ), 1:R );
% Update the LocalOpt for each particle
OptSwarm(ParSwarm(:, 2*C+1) < OptSwarm(1:R, C+1), 1:C) = ParSwarm(ParSwarm(:, 2*C+1) < OptSwarm(1:R, C+1), 1:C);
OptSwarm(ParSwarm(:, 2*C+1) < OptSwarm(1:R, C+1), C+1) = ParSwarm(ParSwarm(:, 2*C+1) < OptSwarm(1:R, C+1), 2*C+1);
for row = 1:R
% Update the GlobalOpt around the whole group
if strcmp(opt.topology, 'Random')
for i = 1:opt.randCount
rowtemp = randi(R, 1);
if OptSwarm(row, C+1) > OptSwarm(rowtemp, C+1)
minrow = rowtemp;
else
minrow = row;
end
end
ToplOptSwarm(row,:) = OptSwarm(minrow, 1:C);
elseif strcmp(opt.topology, 'Ring')
if row == 1
ValTemp2 = OptSwarm(R, C+1);
else
ValTemp2 = OptSwarm(row-1, C+1);
end
if row == R
ValTemp3 = OptSwarm(1, C+1);
else
ValTemp3 = OptSwarm(row+1, C+1);
end
[~, mr] = sort([ValTemp2, OptSwarm(row, C+1), ValTemp3]);
if mr(1) == 3
if row == R
minrow = 1;
else
minrow = row+1;
end
elseif mr(1) == 2
minrow = row;
else
if row == 1
minrow = R;
else
minrow = row-1;
end
end
ToplOptSwarm(row,:) = OptSwarm(minrow, 1:C);
end % if strcmp(opt.topology)
end % for row = 1:R
% Statistics
[minValue, minRow] = min(OptSwarm(:, C+1));
OptSwarm(R+1, 1:C) = OptSwarm(minRow, 1:C);
OptSwarm(R+1, C+1) = minValue;
end % evolutionPSO
end % PSO
% MCMC
methods (Static = true, Access = 'public')
function [xValue, yValue] = Markov_Chain_Monte_Carlo(opt)
%------------------------------------------------------------------------------
% Markov Chain Monte Carlo (MCMC) simulation
%
% The central problem is that of determining the posterior probability for
% parameters given the data, p(thelta|y). With uninformative prior distribution
% and normalization constant p(y), the task is reduced to that of maximizing the
% likelihood of data, p(y|thelta).
%
% "Delayed rejection" was implemented in order to increase the acceptance ratio.
%
% If the Jacobian matrix can be obtained, it will be used to generate R
%
% Parameter:
% - FUNC. The objective function from your field
%
% Returns:
% - xValue. The optimal parameter set
% - yValue. The value of objective function with optimal parameter set
%------------------------------------------------------------------------------
startTime = clock;
if nargin < 1
error('OptAlgo.MCMC: There are no enough inputs \n');
end
% Get the MCMC options
opt = OptAlgo.getOptionsMCMC(opt);
% Preallocation
accepted = 0; n0 = 1;
lasti = 0; chaincov = []; chainmean = []; wsum = [];
chain = zeros(opt.nsamples, opt.Nparams+1);
% Inilization of a starting point
[oldpar, SS, Jac] = OptAlgo.initPointMCMC(opt);
% Construct the error standard deviation: sigma square
if SS < 0, sumSquare = exp(SS); else, sumSquare = SS; end
sigmaSqu = sumSquare / (opt.nDataPoint - opt.Nparams);
sigmaSqu_0 = sigmaSqu;
if ~isempty(Jac)
% Construct the R = chol(covMatrix) for candidate generatio
[~, S, V] = svd(Jac);
% Truncated SVD
if opt.Nparams == 1, S = S(1); else, S = diag(S); end
S(S < 1e-3) = 1e-3;
% Fisher information matrix
% covariance matrix = sigmaSqu * inv(Fisher information matrix) = v'*s^{-2}*v
covMatrix = V * diag(1./S.^2)* V' * sigmaSqu;
% Construct the R = chol(covMatrix) for candidate generation
R = chol(covMatrix .* 2.4^2 ./ opt.Nparams);
else
% If the Jacobian matrix cannot be obtained,
% a set of samples is used to generate R
[R, oldpar, SS] = OptAlgo.burnInSamples(opt);
end
%------------------------------------------------------------------------------
% Main loop
%------------------------------------------------------------------------------
for j = 1:(opt.nsamples+opt.burn_in)
if j > opt.burn_in
fprintf('Iter: %4d -------- Accept_ratio: %3d%% ---------- Minimum: %g ---------- \n',...
j, fix(accepted / j * 100), SS);
fprintf('%10.3g | ', OptAlgo.pTransfer('exp', oldpar)); fprintf('\n');
end
accept = false;
% Generate the new proposal point with R
newpar = oldpar + randn(1, opt.Nparams) * R;
% Check the boundary limiation
newpar( newpar < opt.bounds(1, :) ) = opt.bounds(1, newpar < opt.bounds(1, :));
newpar( newpar > opt.bounds(2, :) ) = opt.bounds(2, newpar > opt.bounds(2, :));
% Calculate the objective value of the new proposal
newSS = feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', newpar) );
% The Metropolis probability
if OptAlgo.logScale
rho12 = exp( -0.5 *((newSS - SS) / sigmaSqu) + sum(newpar) - sum(oldpar) ) * ...
OptAlgo.priorPDF(newpar) / OptAlgo.priorPDF(oldpar);
else
rho12 = exp( -0.5 * (newSS - SS) / sigmaSqu ) * ...
OptAlgo.priorPDF(newpar) / OptAlgo.priorPDF(oldpar);
end
% The new proposal is accepted with Metropolis probability
if rand <= min(1, rho12)
accept = true;
oldpar = newpar;
SS = newSS;
accepted = accepted + 1;
end
% If the poposal is denied, a Delayed Rejection procedure is adopted
% in order to increase the acceptance ratio
if ~accept && opt.delayReject
% Shrink the searching domain by the factor 1/10
newpar2 = oldpar + randn(1, opt.Nparams) * (R ./ 10);
% Check the boundary limitation of the new generated point
newpar2(newpar2 < opt.bounds(1, :)) = opt.bounds(1, newpar2 < opt.bounds(1, :));
newpar2(newpar2 > opt.bounds(2, :)) = opt.bounds(2, newpar2 > opt.bounds(2, :));
% Calculate the objective value of the new proposal
newSS2 = feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', newpar2) );
if OptAlgo.logScale
rho32 = exp( -0.5 *((newSS - newSS2) / sigmaSqu) + sum(newpar) - sum(newpar2) ) * ...
OptAlgo.priorPDF(newpar) / OptAlgo.priorPDF(newpar2);
% The conventional version of calculation
% q2 = exp( -0.5 *((newSS2 - SS) / sigmaSqu) + sum(newpar2) - sum(oldpar) ) * ...
% OptAlgo.priorPDF(newpar2) / OptAlgo.priorPDF(oldpar);
% q1 = exp( -0.5 * (norm((newpar2 - newpar) * inv(R))^2 - norm((oldpar - newpar) * inv(R))^2) );
% The speed-up version of above calculation
q1q2 = exp( -0.5 *( (newSS2 - SS) / sigmaSqu + ...
(newpar2 - newpar) * (R \ (R' \ (newpar2' - newpar'))) - ...
(oldpar - newpar) * (R \ (R' \ (oldpar' - newpar'))) ) + ...
sum(newpar2) - sum(oldpar) ) * ...
OptAlgo.priorPDF(newpar2) / OptAlgo.priorPDF(oldpar);
else
rho32 = exp( -0.5 * (newSS - newSS2) / sigmaSqu ) * ...
OptAlgo.priorPDF(newpar) / OptAlgo.priorPDF(newpar2);
% The conventional version of calculation
% q2 = exp( -0.5 * (newSS2 - SS) / sigmaSqu ) * ...
% OptAlgo.priorPDF(newpar2) / OptAlgo.priorPDF(oldpar);
% q1 = exp( -0.5 * (norm((newpar2 - newpar) * inv(R))^2 - norm((oldpar - newpar) * inv(R))^2) );
% The speed-up version of above calculation
q1q2 = exp( -0.5 *( (newSS2 - SS) / sigmaSqu + ...
(newpar2 - newpar) * (R \ (R' \ (newpar2' - newpar'))) - ...
(oldpar - newpar) * (R \ (R' \ (oldpar' - newpar'))) ) ) * ...
OptAlgo.priorPDF(newpar2) / OptAlgo.priorPDF(oldpar);
end
rho13 = q1q2 * (1 - rho32) / (1 - rho12);
if rand <= min(1, rho13)
oldpar = newpar2;
SS = newSS2;
accepted = accepted + 1;
end
end % if ~accept && delayReject
% During the burn-in period, if the acceptance rate is extremly high or low,
% the R matrix is manually adjusted
if j <= opt.burn_in
if mod(j, 50) == 0
if accepted/j < 0.05
fprintf('Acceptance ratio %3.2f smaller than 5 %%, scaled \n', accepted/j*100);
R = R ./ 5;
elseif accepted/j > 0.95
fprintf('Acceptance ratio %3.2f largeer than 95 %%, scaled \n', accepted/j*100);
R = R .* 5;
end
end
end
% After the burn-in period, the chain is stored
if j > opt.burn_in
chain(j-opt.burn_in, 1:opt.Nparams) = oldpar;
chain(j-opt.burn_in, opt.Nparams+1) = SS;
temp = chain(j-opt.burn_in, :);
save('chainData.dat', 'temp', '-ascii', '-append');
end
if mod(j-opt.burn_in, opt.convergInt) == 0 && (j-opt.burn_in) > 0
% Updata the R according to previous chain
[chaincov, chainmean, wsum] = OptAlgo.covUpdate( chain(lasti+1:j-opt.burn_in, 1:opt.Nparams), ...
1, chaincov, chainmean, wsum );
lasti = j;
R = chol(chaincov + eye(opt.Nparams)*1e-7);
% Check the convergence condition
criterion = OptAlgo.Geweke( chain(1:j-opt.burn_in, 1:opt.Nparams) );
if all( abs(criterion) < opt.criterion ) || j == opt.nsamples+opt.burn_in
maxIter = j;
break
end
end
% Updata the sigma^2 according to the current objective value
if SS < 0, sumSquare = exp(SS); else, sumSquare = SS; end
sigmaSqu = 1 / OptAlgo.GammarDistribution( 1, 1, (n0 + opt.nDataPoint)/2,...
2 / (n0 * sigmaSqu_0 + sumSquare) );
save('sigmaSqu.dat', 'sigmaSqu', '-ascii', '-append');
end % for j = 1:opt.nsamples
%------------------------------------------------------------------------------
% Post-process
%------------------------------------------------------------------------------
clear chain;
% Generate the population for figure plot
Population = OptAlgo.conversionDataMCMC(maxIter, opt);
OptAlgo.FigurePlot(Population, opt);
[yValue, row] = min(Population(:, opt.Nparams+1));
xValue = OptAlgo.pTransfer('exp', Population(row, 1:opt.Nparams));
% Gather some useful information and store them
result.optTime = etime(clock, startTime) / 3600;
result.covMatrix = R' * R;
result.Iteration = maxIter;
result.criterion = criterion;
result.accepted = fix(accepted/maxIter * 100);
result.xValue = xValue;
result.yValue = yValue;
result.sigma = sqrt(sigmaSqu);
fprintf('\n****************************************************** \n');
save(sprintf('result_%2d.mat', fix(rand*100)), 'result');
fprintf('Time %5.3g hours elapsed after %5d iterations \n', result.optTime, result.Iteration);
fprintf('The minimal objective value found during sampling is: %10.3g \n', yValue);
fprintf('The correspondingly optimal set is: ');
fprintf(' %g |', xValue);
fprintf('\nThe statistical information of MCMC is stored as result.mat \n');
fprintf('The historical chain of MCMC is stored as Population.dat \n');
fprintf('****************************************************** \n');
end % Markov_Chain_Monte_Carlo
function opt = getOptionsMCMC(obj)
%------------------------------------------------------------------------------
% The parameters for the Optimizer
%
% Return:
% - opt.
% + opt.Nparams. The number of optimized parameters
% + opt.bounds. 2*nCol matrix of the parameter limitation
% + opt.nsamples. The pre-defined maximal iteration
% + opt.criterion. The error tolerance to stop the algorithm
% + opt.burn_in. The burn-in period before adaptation begin
% + opt.convergInt. The integer for checking of the convergence
% + opt.rejectValue. This is specific used in the generation of R matrix
% when Jacobian information is not available. In the generated sample,
% the proposals whose objective value is larger than this will be rejected
% + opt.nDataPoint. The number of observations in the measured data
% + opt.deylayReject. By default DR= 1
% + opt.Jacobian. If Jacobian matrix is available, set it to true
%------------------------------------------------------------------------------
opt = [];
opt.Nparams = length(fieldnames(obj.params));
opt.bounds = OptAlgo.pTransfer('log', obj.paramBound)';
opt.nsamples = OptAlgo.sample;
opt.criterion = 0.0001;
opt.burn_in = 0;
opt.convergInt = 100;
opt.rejectValue = 1e5;
opt.nDataPoint = OptAlgo.dataPoint;
opt.Jacobian = false; % set it to true only when Jacobian matrix is available
opt.delayReject = true;
[row, col] = size(opt.Nparams);
if row > 1 || col > 1
error('OptAlgo.getOptionsMCMC: The initialized dimension of the parameter set might be wrong \n');
end
[row, col] = size(opt.bounds);
if col ~= opt.Nparams || row ~= 2
error('OptAlgo.getOptionsMCMC: Please check your setup of the range of parameters \n');
end
end % getOptionsMCMC
function [oldpar, SS, Jac]= initPointMCMC(opt)
%------------------------------------------------------------------------------
% Generate the initially guessing point for the MCMC algorithm
%------------------------------------------------------------------------------
if nargin < 1
error('OptAlgo.initPointMCMC: There are no enough input arguments \n');
end
oldpar = rand(1,opt.Nparams);
oldpar(:, 1:opt.Nparams) = opt.bounds(1,:) + ...
oldpar(:, 1:opt.Nparams) .* (opt.bounds(2,:) - opt.bounds(1,:));
% Check the boundary limiation
% oldpar(oldpar < opt.bounds(1, :)) = opt.bounds(1, oldpar < opt.bounds(1, :));
% oldpar(oldpar > opt.bounds(2, :)) = opt.bounds(2, oldpar > opt.bounds(2, :));
% Get the resudual value and Jacobian matrix of the guessing point
Jac = [];
if opt.Jacobian
[SS, ~, Jac] = feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', oldpar) );
else
SS = feval( OptAlgo.FUNC, OptAlgo.pTransfer('exp', oldpar) );
end
end % initPointMCMC
function [R, oldpar, SS] = burnInSamples(opt)
%------------------------------------------------------------------------------
% It is used for generating samples when Jocabian matrix is not available
%------------------------------------------------------------------------------