├── 1D Simple Harmonic Oscillator ├── README.md ├── SHO_Problem.mlx ├── SMCsampler.m ├── TMCMCsampler.m ├── box_function.m ├── desktop.ini ├── log_likelihood.m ├── model.m └── proposal_rnd.m ├── 1D Static Spring-Mass System ├── Linear_Problem.mlx ├── README.md ├── SMCsampler.m ├── TMCMCsampler.m ├── desktop.ini ├── log_likelihood.m └── model.m ├── 2D Eigen-value Problem ├── Eigenvalue_Problem.mlx ├── README.md ├── SMCsampler.m ├── TMCMCsampler.m ├── desktop.ini ├── log_likelihood.m └── model.m ├── Alternative_TMCMC_Transition_Criteria ├── DenHartogHarmonic.m ├── TMCMCsampler.m ├── TMCMCsampler2.m ├── angleCalc.m ├── areaMe.m ├── blackbox_model.m ├── example_SDOF_System_Coulomb_Friction_numerical.m ├── loglikelihood.m └── test ├── LICENSE └── README.md /1D Simple Harmonic Oscillator/README.md: -------------------------------------------------------------------------------- 1 | ## Instructions: 2 | 3 | * Run the tutorial MATLAB LIVE script: "SHO_Problem.mlx" 4 | -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/SHO_Problem.mlx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/1D Simple Harmonic Oscillator/SHO_Problem.mlx -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/SMCsampler.m: -------------------------------------------------------------------------------- 1 | function [output] = SMCsampler(varargin) 2 | %% Sequential Monte Carlo Dynamical (SMC) sampler 3 | % 4 | % This program implements the original Sequential Monte Carlo (SMC) sampling 5 | % class (see paper by N. Chopin (2002): A sequential particle filter method 6 | % for static models - https://www.jstor.org/stable/3879283) and employs 7 | % the use of the Affine-invariant Ensemble Sampler (AIES) proposed by 8 | % Goodman and Weare (2010) to update the samples at each iteration. 9 | % 10 | % This sampler function can be employed in Sequential Bayesian Model 11 | % Updating problems involving: 12 | % - Estimating time-invariant parameter(s) via Online Bayesian Model Updating; 13 | % - Estimating time-varying parameter(s), following a recursive dynamic model; 14 | % - Predicting the time-varying parameter(s) for the next time-step given 15 | % data/observations up to the previous time-step. 16 | % 17 | %-------------------------------------------------------------------------- 18 | % Author: 19 | % Adolphus Lye - adolphus.lye@liverpool.ac.uk 20 | %-------------------------------------------------------------------------- 21 | 22 | % Parse the information in the name/value pairs: 23 | pnames = {'nsamples','loglikelihoods','dynamic_model',... 24 | 'priorpdf','priorrnd','burnin','lastburnin','thinchain'}; 25 | 26 | % Define default values: 27 | dflts = {[], [], @(x) x, [], [], 0, 0, 3}; 28 | 29 | [nsamples,loglikelihoods,dynamic_model,priorpdf,prior_rnd,... 30 | burnin,lastBurnin,thinchain] = internal.stats.parseArgs(pnames, dflts, varargin{:}); 31 | 32 | %-------------------------------------------------------------------------- 33 | % 34 | % Inputs: 35 | % nsamples: Scalar value of the number of samples to be generated from the Posterior; 36 | % loglikelihoods: A M x 1 cell vector of likelihood functions containing the measurements at M different time-steps; 37 | % dynamic_model: A function-handle that relates theta(t+1) and theta(t), where t is the time-step. Output is N x dim; 38 | % priorpdf: Function-handle of the Prior PDF; 39 | % prior_rnd: Function-handle of the Prior random number generator; 40 | % burnin: Number of burn-in for all iterations up to M-1; 41 | % lastBurnin: Number of burn-in for the last iteration; 42 | % stepsize: The stepsize for the Ensemble sampler in the updating step (this is the tuning parameter); 43 | % thinchain: Thin all the chains of the Ensemble sampler by only storing every k'th step (default=3); 44 | % 45 | % Outputs: 46 | % output.samples: A N x dim matrix of Posterior samples; 47 | % output.allsamples: A N x dim x (M+1) array of samples from all iterations; 48 | % output.acceptance: A M x 1 vector of acceptance rates for all iterations; 49 | % output.log_evidence: A (M+1) x 1 vector of the logarithmic of the evidence; 50 | % output.step: A M x 1 vector of step-size; 51 | % output.indicator: A M x 1 vector of indicators denoting if resampling 52 | % has occured for any iterations (1 = Yes, 0 = No); 53 | % 54 | %-------------------------------------------------------------------------- 55 | 56 | %% Number of cores 57 | if ~isempty(gcp('nocreate')) 58 | pool = gcp; 59 | Ncores = pool.NumWorkers; 60 | fprintf('SMC is running on %d cores.\n', Ncores); 61 | end 62 | 63 | %% Initialize: Obtain N samples from the Prior PDF 64 | 65 | fprintf('Start SMC procedure ... \n'); 66 | 67 | prior_initial = priorpdf; % Define initial Prior PDF 68 | thetaj = prior_rnd(nsamples); % theta0 = N x dim 69 | Dimensions = size(thetaj, 2); % Dimensionality of theta, dim 70 | 71 | % Initialization of matrices and vectors: 72 | thetaj1 = zeros(nsamples, Dimensions); 73 | log_evidence = zeros(size(loglikelihoods,1)+1,1); % Initiate empty vector for log evidence 74 | log_evidence(1) = 0; 75 | 76 | acceptance = zeros(size(loglikelihoods,1),1); 77 | 78 | % Samples from filter distribution, P(theta(t)|Data(1:t)): 79 | allsamples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)+1); 80 | allsamples(:,:,1) = thetaj; 81 | 82 | % Statistics from predictive distribution, P(theta(t+1)|Data(1:t)): 83 | predictive_samples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)); 84 | 85 | % Resampling indicator vector: 86 | indicator = zeros(length(loglikelihoods), 1); 87 | % Note: This indicator vector returns a 1 for the iteration(s) where 88 | % resampling is initiated and 0 otherwise. 89 | 90 | %% Main sampling loop 91 | for iter = 1:length(loglikelihoods) 92 | 93 | fprintf('SMC: Iteration j = %2d \n', iter); 94 | 95 | loglikelihood = loglikelihoods{iter}; 96 | 97 | % Compute loglikelihood values for each sample: 98 | logL = zeros(nsamples,1); 99 | for l = 1:nsamples 100 | logL(l) = loglikelihood(thetaj(l,:)); 101 | end 102 | 103 | % Error check: 104 | if any(isinf(logL)) 105 | error('The prior distribution is too far from the true region'); 106 | end 107 | 108 | %% Compute weights of the samples, wj: 109 | 110 | % To compute the nominal weights: 111 | fprintf('Computing the weights ...\n'); 112 | wj = exp(logL); 113 | 114 | % To compute the log evidence for the current iteration: 115 | log_evidence(iter+1) = log(mean(wj)) + log_evidence(iter); 116 | 117 | % Check step for wj: 118 | for i = 1:nsamples 119 | if wj(i) == 0 120 | wj(i) = 1e-100; 121 | end 122 | end 123 | 124 | wj_norm = wj./sum(wj); % To normalise the weights 125 | 126 | %% Check step - Compute the sum of wj_norm and see if it is < nsamples/2: 127 | 128 | fprintf('Computing effective sample size ... \n'); 129 | Neff = 1/(sum(wj_norm.^2)); 130 | threshold = nsamples/2; 131 | 132 | %% Resampling step (conditional if Neff < threshold): 133 | 134 | if Neff < threshold 135 | fprintf('Resampling step initiated ... \n'); 136 | 137 | dx = randsample(nsamples, nsamples, true, wj_norm); 138 | 139 | thetaj_resampled = zeros(nsamples, Dimensions); 140 | for d = 1:nsamples 141 | thetaj_resampled(d,:) = thetaj(dx(d),:); 142 | end 143 | 144 | thetaj = thetaj_resampled; 145 | wj_norm = (1/nsamples).*ones(nsamples,1); 146 | indicator(iter) = 1; 147 | 148 | end 149 | 150 | %% Update the samples according to the current Posterior using MH sampler: 151 | 152 | % Define the logposterior: 153 | log_posterior = @(x) log(priorpdf(x)) + loglikelihood(x); 154 | 155 | % Weighted mean for Proposal distribution 156 | mu = zeros(1, Dimensions); 157 | for l = 1:nsamples 158 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 159 | end 160 | 161 | % Covariance matrix for Proposal distribution: 162 | cov_gauss = zeros(Dimensions); 163 | for k = 1:nsamples 164 | tk_mu = thetaj(k,:) - mu; 165 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 166 | end 167 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 168 | 169 | % Define the Proposal distribution: 170 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, prior_initial); % q(x,y) = q(x|y). 171 | proprnd = @(x) prop_rnd(x, cov_gauss, prior_initial); 172 | 173 | if iter == length(loglikelihoods) 174 | burnin = lastBurnin; 175 | end 176 | 177 | %% Start N different Markov chains 178 | fprintf('Markov chains ...\n\n'); 179 | 180 | idx = randsample(nsamples, nsamples, true, wj_norm); 181 | for i = 1:nsamples % For parallel, type: parfor 182 | [thetaj1(i,:), acceptance_rate(i)] = mhsample(thetaj(idx(i),:), 1, ... 183 | 'logpdf', log_posterior, ... 184 | 'proppdf', proppdf, ... 185 | 'proprnd', proprnd, ... 186 | 'thin', thinchain, ... 187 | 'burnin', burnin); 188 | end 189 | fprintf('\n'); 190 | acceptance(iter) = mean(acceptance_rate); % To store the acceptance rate values 191 | 192 | %% Prediction step: 193 | 194 | % Define the Predictive distribution of the samples, P(theta(t+1)|Data(t)): 195 | predictive_samples(:,:,iter) = dynamic_model(thetaj1); 196 | 197 | % Compute the Bandwidth vector for the kernel density function: 198 | pred_samps = predictive_samples(:,:,iter); 199 | 200 | bw = zeros(Dimensions,1); 201 | for dim = 1:Dimensions 202 | bw(dim) = std(pred_samps(:,dim)) .* (4/((Dimensions + 2) .* nsamples)).^(1/(Dimensions + 4)); 203 | end 204 | 205 | % Define the Predictive PDF, P(theta(t+1)|Data(t)) using mvksdensity: 206 | pred_pdf = @(x) mvksdensity(pred_samps, x, 'Bandwidth', bw); 207 | 208 | %% Prepare for the next iteration: 209 | 210 | allsamples(:,:,iter+1) = thetaj1; 211 | thetaj = pred_samps; 212 | priorpdf = @(x) pred_pdf(x); 213 | 214 | end 215 | 216 | %% Description of outputs: 217 | 218 | output.samples = thetaj; % To only show samples from the final filter distribution 219 | output.allsamples = allsamples; % To only show all filter samples across all iterations 220 | output.prediction = predictive_samples; % To only show all prediction samples across all iterations 221 | output.acceptance = acceptance; % To show the mean acceptance rates for all iterations 222 | output.log_evidence = log_evidence; % To show the (M+1) x 1 vector of the logarithmic of the evidence; 223 | output.indicator = indicator; % To indicate the iterations whereby resampling took place (denoted by 1s). 224 | 225 | fprintf('End of SMC procedure. \n\n'); 226 | 227 | return; % End 228 | 229 | function proppdf = prop_pdf(x, mu, covmat, box) 230 | % This is the Proposal PDF for the Markov Chain. 231 | 232 | % Box function is the Prior PDF in the feasible region. 233 | % So if a point is out of bounds, this function will 234 | % return 0. 235 | 236 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 237 | 238 | return; 239 | 240 | 241 | function proprnd = prop_rnd(mu, covmat, box) 242 | % Sampling from the proposal PDF for the Markov Chain. 243 | 244 | while true 245 | proprnd = mvnrnd(mu, covmat, 1); 246 | if box(proprnd) 247 | % The box function is the Prior PDF in the feasible region. 248 | % If a point is out of bounds, this function will return 0 = false. 249 | break; 250 | end 251 | end 252 | 253 | return -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/TMCMCsampler.m: -------------------------------------------------------------------------------- 1 | function [samples_fT_D, log_fD] = TMCMCsampler(varargin) 2 | %% Transitional Markov Chain Monte Carlo sampler 3 | % 4 | % This program implements a method described in: 5 | % Ching, J. and Chen, Y. (2007). "Transitional Markov Chain Monte Carlo 6 | % Method for Bayesian Model Updating, Model Class Selection, and Model 7 | % Averaging." J. Eng. Mech., 133(7), 816-832. 8 | % 9 | % Usage: 10 | % [samples_fT_D, fD] = tmcmc_v1(fD_T, fT, sample_from_fT, N); 11 | % 12 | % where: 13 | % 14 | % inputs: 15 | % log_fD_T = function handle of log(fD_T(t)), Loglikelihood 16 | % 17 | % fT = function handle of fT(t), Prior PDF 18 | % 19 | % sample_from_fT = handle to a function that samples from of fT(t), 20 | % Sampling rule function from Prior PDF 21 | % 22 | % nsamples = number of samples of fT_D, Posterior, to generate 23 | % 24 | % outputs: 25 | % samples_fT_D = samples of fT_D (N x D) = samples from Posterior 26 | % distribution 27 | % 28 | % log_fD = log(evidence) = log(normalization constant) 29 | 30 | % ------------------------------------------------------------------------ 31 | % who when observations 32 | %-------------------------------------------------------------------------- 33 | % Diego Andres Alvarez Jul-24-2013 First algorithm 34 | %-------------------------------------------------------------------------- 35 | % Diego Andres Alvarez - daalvarez@unal.edu.co 36 | % Edoardo Patelli - edoardo.patelli@strath.ac.uk 37 | 38 | % parse the information in the name/value pairs: 39 | pnames = {'nsamples','loglikelihood','priorpdf','priorrnd','burnin','lastburnin','beta'}; 40 | 41 | dflts = {[],[],[],[],[],0,0.2}; % define default values 42 | 43 | [nsamples,loglikelihood,priorpdf,prior_rnd,burnin,lastBurnin,beta] = ... 44 | internal.stats.parseArgs(pnames, dflts, varargin{:}); 45 | 46 | %% Obtain N samples from the prior pdf f(T) 47 | j = 0; % Initialise loop for the transitional likelihood 48 | thetaj = prior_rnd(nsamples); % theta0 = N x D 49 | pj = 0; % p0 = 0 (initial tempering parameter) 50 | Dimensions = size(thetaj, 2); % size of the vector theta 51 | 52 | %% Initialization of matrices and vectors 53 | thetaj1 = zeros(nsamples, Dimensions); 54 | %log_fD_T_thetaj = zeros(nsamples,1); 55 | 56 | %% Main loop 57 | while pj < 1 58 | j = j+1; 59 | 60 | %% Calculate the tempering parameter p(j+1): 61 | for l = 1:nsamples 62 | log_fD_T_thetaj(l) = loglikelihood(thetaj(l,:)); 63 | end 64 | if any(isinf(log_fD_T_thetaj)) 65 | error('The prior distribution is too far from the true region'); 66 | end 67 | pj1 = calculate_pj1(log_fD_T_thetaj, pj); 68 | fprintf('TMCMC: Iteration j = %2d, pj1 = %f\n', j, pj1); 69 | 70 | %% Compute the plausibility weight for each sample wrt f_{j+1} 71 | fprintf('Computing the weights ...\n'); 72 | % wj = fD_T(thetaj).^(pj1-pj); % N x 1 (eq 12) 73 | a = (pj1-pj)*log_fD_T_thetaj; 74 | wj = exp(a); 75 | wj_norm = wj./sum(wj); % normalization of the weights 76 | 77 | %% Compute S(j) = E[w{j}] (eq 15) 78 | S(j) = mean(wj); 79 | 80 | %% Do the resampling step to obtain N samples from f_{j+1}(theta) and 81 | % then perform Metropolis-Hastings on each of these samples using as a 82 | % stationary PDF "fj1" 83 | % fj1 = @(t) fT(t).*log_fD_T(t).^pj1; % stationary PDF (eq 11) f_{j+1}(theta) 84 | log_posterior = @(t) log(priorpdf(t)) + pj1*loglikelihood(t); 85 | 86 | 87 | % and using as proposal PDF a Gaussian centered at thetaj(idx,:) and 88 | % with covariance matrix equal to an scaled version of the covariance 89 | % matrix of fj1: 90 | 91 | % weighted mean 92 | mu = zeros(1, Dimensions); 93 | for l = 1:nsamples 94 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 95 | end 96 | 97 | % scaled covariance matrix of fj1 (eq 17) 98 | cov_gauss = zeros(Dimensions); 99 | for k = 1:nsamples 100 | % this formula is slightly different to eq 17 (the transpose) 101 | % because of the size of the vectors)m and because Ching and Chen 102 | % forgot to normalize the weight wj: 103 | tk_mu = thetaj(k,:) - mu; 104 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 105 | end 106 | cov_gauss = beta^2 * cov_gauss; 107 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 108 | 109 | % Define the Proposal distribution: 110 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, priorpdf); %q(x,y) = q(x|y). 111 | proprnd = @(x) prop_rnd(x, cov_gauss, priorpdf); 112 | 113 | %% During the last iteration we require to do a better burnin in order 114 | % to guarantee the quality of the samples: 115 | if pj1 == 1 116 | burnin = lastBurnin; 117 | end; 118 | 119 | %% Start N different Markov chains 120 | fprintf('Markov chains ...\n\n'); 121 | idx = randsample(nsamples, nsamples, true, wj_norm); 122 | for i = 1:nsamples % For parallel, type: parfor 123 | %% Sample one point with probability wj_norm 124 | 125 | % smpl = mhsample(start, nsamples, 126 | % 'pdf', pdf, 'proppdf', proppdf, 'proprnd', proprnd); 127 | % start = row vector containing the start value of the Markov Chain, 128 | % nsamples = number of samples to be generated 129 | [thetaj1(i,:), acceptance_rate] = mhsample(thetaj(idx(i), :), 1, ... 130 | 'logpdf', log_posterior, ... 131 | 'proppdf', proppdf, ... 132 | 'proprnd', proprnd, ... 133 | 'thin', 3, ... 134 | 'burnin', burnin); 135 | % According to Cheung and Beck (2009) - Bayesian model updating ..., 136 | % the initial samples from reweighting and the resample of samples of 137 | % fj, in general, do not exactly follow fj1, so that the Markov 138 | % chains must "burn-in" before samples follow fj1, requiring a large 139 | % amount of samples to be generated for each level. 140 | 141 | %% Adjust the acceptance rate (optimal = 23%) 142 | % See: http://www.dms.umontreal.ca/~bedard/Beyond_234.pdf 143 | %{ 144 | if acceptance_rate < 0.3 145 | % Many rejections means an inefficient chain (wasted computation 146 | %time), decrease the variance 147 | beta = 0.99*beta; 148 | elseif acceptance_rate > 0.5 149 | % High acceptance rate: Proposed jumps are very close to current 150 | % location, increase the variance 151 | beta = 1.01*beta; 152 | end 153 | %} 154 | end 155 | fprintf('\n'); 156 | 157 | %% Prepare for the next iteration 158 | thetaj = thetaj1; 159 | pj = pj1; 160 | end 161 | 162 | % TMCMC provides N samples distributed according to the Posterior distribution, f(T|D) 163 | samples_fT_D = thetaj; 164 | 165 | % estimation of f(D) -- this is the normalization constant in Bayes 166 | log_fD = sum(log(S(1:j))); 167 | 168 | return; % End 169 | 170 | 171 | %% Calculate the tempering parameter p(j+1) 172 | function pj1 = calculate_pj1(log_fD_T_thetaj, pj) 173 | % find pj1 such that COV <= threshold, that is 174 | % 175 | % std(wj) 176 | % --------- <= threshold 177 | % mean(wj) 178 | % 179 | % here 180 | % size(thetaj) = N x D, 181 | % wj = fD_T(thetaj).^(pj1 - pj) 182 | % e = pj1 - pj 183 | 184 | threshold = 1; % 100% = threshold on the COV 185 | 186 | % wj = @(e) fD_T_thetaj^e; % N x 1 187 | % Note the following trick in order to calculate e: 188 | % Take into account that e>=0 189 | wj = @(e) exp(abs(e)*log_fD_T_thetaj); % N x 1 190 | %fmin = @(e) std(wj(e))/mean(wj(e)) - threshold; 191 | fmin = @(e) std(wj(e)) - threshold*mean(wj(e)) + realmin; 192 | e = abs(fzero(fmin, 0)); % e is >= 0, and fmin is an even function 193 | if isnan(e) 194 | error('There is an error finding e'); 195 | end 196 | 197 | pj1 = min(1, pj + e); 198 | 199 | return; % End 200 | 201 | function proppdf = prop_pdf(x, mu, covmat, box) 202 | % This is the Proposal PDF for the Markov Chain. 203 | 204 | % Box function is the Prior PDF in the feasible region. 205 | % So if a point is out of bounds, this function will 206 | % return 0. 207 | 208 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 209 | 210 | return; 211 | 212 | 213 | function proprnd = prop_rnd(mu, covmat, box) 214 | % Sampling from the proposal PDF for the Markov Chain. 215 | 216 | while true 217 | proprnd = mvnrnd(mu, covmat, 1); 218 | if box(proprnd) 219 | % The box function is the Prior PDF in the feasible region. 220 | % If a point is out of bounds, this function will return 0 = false. 221 | break; 222 | end 223 | end 224 | 225 | return 226 | -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/box_function.m: -------------------------------------------------------------------------------- 1 | function [BoxFunction] = box_function(theta, parameters) 2 | 3 | % The Box Function serves as an indicator function such that should the 4 | % Candidate samples fall outside the range of values of the Uniform Prior, 5 | % the function returns a 0 and returns a 1 if the values of the Candidate 6 | % samples fall within the range of values of the Uniform Prior. 7 | 8 | 9 | if theta < parameters(1) || theta > parameters(2) 10 | 11 | BoxFunction = 0; 12 | 13 | else 14 | 15 | BoxFunction = 1; 16 | 17 | end 18 | 19 | end -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/desktop.ini: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/1D Simple Harmonic Oscillator/desktop.ini -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/log_likelihood.m: -------------------------------------------------------------------------------- 1 | function logl = log_likelihood(stiffness, modelInput, measurements, standard_deviation, ModelHandle) 2 | % Calculation of the log_likelihood for 1D Simple Harmonic Oscillator: 3 | % 4 | % USAGE: 5 | % logl = log_likelihood(stiffness, mass, measurements, standard_deviation, ModelHandle) 6 | % 7 | % INPUTS: 8 | % stiffness = epistemic parameter k [Nsamples x 1] 9 | % mass = model input [Nobservations x 1] 10 | % measurements = experimental observations [Nobservations x 1] 11 | % standard_deviation = the standard deviation of the log likelihood function [scalar] 12 | % ModelHandle = the function handle of the model (see file "model.m") 13 | % 14 | % OUTPUTS: 15 | % logl = loglikelihood function for the set of estimated stiffness values k and 16 | % the measurements 17 | % . 18 | % log1 = 1 x 1 19 | % 20 | 21 | %% Evaluate the model: 22 | nchains=size(stiffness,1); 23 | %ndims=size(stiffness,2); 24 | logl=zeros(nchains,1); 25 | 26 | for n=1:nchains 27 | modelOutput = ModelHandle(stiffness(n,:),modelInput); 28 | 29 | % Note: Details to the model can be found in the file: "model.m" 30 | 31 | %% Compute the log-likelihood: 32 | 33 | logl(n) = -0.5 * (1/standard_deviation)^2 *(measurements - modelOutput)' * (measurements - modelOutput); 34 | 35 | end 36 | 37 | -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/model.m: -------------------------------------------------------------------------------- 1 | function [frequency] = model(stiffness,mass) 2 | 3 | %MODEL: 1-Degree of freedom simple harmonic oscillator system 4 | 5 | frequency = sqrt(stiffness./mass); 6 | 7 | end -------------------------------------------------------------------------------- /1D Simple Harmonic Oscillator/proposal_rnd.m: -------------------------------------------------------------------------------- 1 | function [proprnd] = proposal_rnd(CurrentSample, Tuning_mcmc, NumberOfChains, BoxFunction) 2 | 3 | % This is a modified Proposal random number generator from the Normal 4 | % Proposal distribution. The normrnd function is now multiplied by the Box 5 | % function such that if the generated proposal sample values fall outide 6 | % the range of values as stipulated by the Prior distribution, the function 7 | % returns a 0 immediately and the sample value is rejected. This serves to 8 | % prevent any values which fall outside the range of Prior values from 9 | % being accepted. 10 | 11 | proprnd_nominal = normrnd(CurrentSample,Tuning_mcmc,NumberOfChains,1); 12 | 13 | % Initiate the storing of the array of Proposal sample values with an empty array: 14 | proprnd = zeros(NumberOfChains,1); 15 | 16 | % To store each array value of proprnd_nominal: 17 | for i = 1:NumberOfChains 18 | proprnd(i) = proprnd_nominal(i) .* BoxFunction(proprnd_nominal(i)); 19 | end 20 | 21 | end -------------------------------------------------------------------------------- /1D Static Spring-Mass System/Linear_Problem.mlx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/1D Static Spring-Mass System/Linear_Problem.mlx -------------------------------------------------------------------------------- /1D Static Spring-Mass System/README.md: -------------------------------------------------------------------------------- 1 | ## Instructions: 2 | 3 | * Run the tutorial MATLAB LIVE script: "Linear_Problem.mlx" 4 | -------------------------------------------------------------------------------- /1D Static Spring-Mass System/SMCsampler.m: -------------------------------------------------------------------------------- 1 | function [output] = SMCsampler(varargin) 2 | %% Sequential Monte Carlo Dynamical (SMC) sampler 3 | % 4 | % This program implements the original Sequential Monte Carlo (SMC) sampling 5 | % class (see paper by N. Chopin (2002): A sequential particle filter method 6 | % for static models - https://www.jstor.org/stable/3879283) and employs 7 | % the use of the Affine-invariant Ensemble Sampler (AIES) proposed by 8 | % Goodman and Weare (2010) to update the samples at each iteration. 9 | % 10 | % This sampler function can be employed in Sequential Bayesian Model 11 | % Updating problems involving: 12 | % - Estimating time-invariant parameter(s) via Online Bayesian Model Updating; 13 | % - Estimating time-varying parameter(s), following a recursive dynamic model; 14 | % - Predicting the time-varying parameter(s) for the next time-step given 15 | % data/observations up to the previous time-step. 16 | % 17 | %-------------------------------------------------------------------------- 18 | % Author: 19 | % Adolphus Lye - adolphus.lye@liverpool.ac.uk 20 | %-------------------------------------------------------------------------- 21 | 22 | % Parse the information in the name/value pairs: 23 | pnames = {'nsamples','loglikelihoods','dynamic_model',... 24 | 'priorpdf','priorrnd','burnin','lastburnin','thinchain'}; 25 | 26 | % Define default values: 27 | dflts = {[], [], @(x) x, [], [], 0, 0, 3}; 28 | 29 | [nsamples,loglikelihoods,dynamic_model,priorpdf,prior_rnd,... 30 | burnin,lastBurnin,thinchain] = internal.stats.parseArgs(pnames, dflts, varargin{:}); 31 | 32 | %-------------------------------------------------------------------------- 33 | % 34 | % Inputs: 35 | % nsamples: Scalar value of the number of samples to be generated from the Posterior; 36 | % loglikelihoods: A M x 1 cell vector of likelihood functions containing the measurements at M different time-steps; 37 | % dynamic_model: A function-handle that relates theta(t+1) and theta(t), where t is the time-step. Output is N x dim; 38 | % priorpdf: Function-handle of the Prior PDF; 39 | % prior_rnd: Function-handle of the Prior random number generator; 40 | % burnin: Number of burn-in for all iterations up to M-1; 41 | % lastBurnin: Number of burn-in for the last iteration; 42 | % stepsize: The stepsize for the Ensemble sampler in the updating step (this is the tuning parameter); 43 | % thinchain: Thin all the chains of the Ensemble sampler by only storing every k'th step (default=3); 44 | % 45 | % Outputs: 46 | % output.samples: A N x dim matrix of Posterior samples; 47 | % output.allsamples: A N x dim x (M+1) array of samples from all iterations; 48 | % output.acceptance: A M x 1 vector of acceptance rates for all iterations; 49 | % output.log_evidence: A (M+1) x 1 vector of the logarithmic of the evidence; 50 | % output.step: A M x 1 vector of step-size; 51 | % output.indicator: A M x 1 vector of indicators denoting if resampling 52 | % has occured for any iterations (1 = Yes, 0 = No); 53 | % 54 | %-------------------------------------------------------------------------- 55 | 56 | %% Number of cores 57 | if ~isempty(gcp('nocreate')) 58 | pool = gcp; 59 | Ncores = pool.NumWorkers; 60 | fprintf('SMC is running on %d cores.\n', Ncores); 61 | end 62 | 63 | %% Initialize: Obtain N samples from the Prior PDF 64 | 65 | fprintf('Start SMC procedure ... \n'); 66 | 67 | prior_initial = priorpdf; % Define initial Prior PDF 68 | thetaj = prior_rnd(nsamples); % theta0 = N x dim 69 | Dimensions = size(thetaj, 2); % Dimensionality of theta, dim 70 | 71 | % Initialization of matrices and vectors: 72 | thetaj1 = zeros(nsamples, Dimensions); 73 | log_evidence = zeros(size(loglikelihoods,1)+1,1); % Initiate empty vector for log evidence 74 | log_evidence(1) = 0; 75 | 76 | acceptance = zeros(size(loglikelihoods,1),1); 77 | 78 | % Samples from filter distribution, P(theta(t)|Data(1:t)): 79 | allsamples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)+1); 80 | allsamples(:,:,1) = thetaj; 81 | 82 | % Statistics from predictive distribution, P(theta(t+1)|Data(1:t)): 83 | predictive_samples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)); 84 | 85 | % Resampling indicator vector: 86 | indicator = zeros(length(loglikelihoods), 1); 87 | % Note: This indicator vector returns a 1 for the iteration(s) where 88 | % resampling is initiated and 0 otherwise. 89 | 90 | %% Main sampling loop 91 | for iter = 1:length(loglikelihoods) 92 | 93 | fprintf('SMC: Iteration j = %2d \n', iter); 94 | 95 | loglikelihood = loglikelihoods{iter}; 96 | 97 | % Compute loglikelihood values for each sample: 98 | logL = zeros(nsamples,1); 99 | for l = 1:nsamples 100 | logL(l) = loglikelihood(thetaj(l,:)); 101 | end 102 | 103 | % Error check: 104 | if any(isinf(logL)) 105 | error('The prior distribution is too far from the true region'); 106 | end 107 | 108 | %% Compute weights of the samples, wj: 109 | 110 | % To compute the nominal weights: 111 | fprintf('Computing the weights ...\n'); 112 | wj = exp(logL); 113 | 114 | % To compute the log evidence for the current iteration: 115 | log_evidence(iter+1) = log(mean(wj)) + log_evidence(iter); 116 | 117 | % Check step for wj: 118 | for i = 1:nsamples 119 | if wj(i) == 0 120 | wj(i) = 1e-100; 121 | end 122 | end 123 | 124 | wj_norm = wj./sum(wj); % To normalise the weights 125 | 126 | %% Check step - Compute the sum of wj_norm and see if it is < nsamples/2: 127 | 128 | fprintf('Computing effective sample size ... \n'); 129 | Neff = 1/(sum(wj_norm.^2)); 130 | threshold = nsamples/2; 131 | 132 | %% Resampling step (conditional if Neff < threshold): 133 | 134 | if Neff < threshold 135 | fprintf('Resampling step initiated ... \n'); 136 | 137 | dx = randsample(nsamples, nsamples, true, wj_norm); 138 | 139 | thetaj_resampled = zeros(nsamples, Dimensions); 140 | for d = 1:nsamples 141 | thetaj_resampled(d,:) = thetaj(dx(d),:); 142 | end 143 | 144 | thetaj = thetaj_resampled; 145 | wj_norm = (1/nsamples).*ones(nsamples,1); 146 | indicator(iter) = 1; 147 | 148 | end 149 | 150 | %% Update the samples according to the current Posterior using MH sampler: 151 | 152 | % Define the logposterior: 153 | log_posterior = @(x) log(priorpdf(x)) + loglikelihood(x); 154 | 155 | % Weighted mean for Proposal distribution 156 | mu = zeros(1, Dimensions); 157 | for l = 1:nsamples 158 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 159 | end 160 | 161 | % Covariance matrix for Proposal distribution: 162 | cov_gauss = zeros(Dimensions); 163 | for k = 1:nsamples 164 | tk_mu = thetaj(k,:) - mu; 165 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 166 | end 167 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 168 | 169 | % Define the Proposal distribution: 170 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, prior_initial); % q(x,y) = q(x|y). 171 | proprnd = @(x) prop_rnd(x, cov_gauss, prior_initial); 172 | 173 | if iter == length(loglikelihoods) 174 | burnin = lastBurnin; 175 | end 176 | 177 | %% Start N different Markov chains 178 | fprintf('Markov chains ...\n\n'); 179 | 180 | idx = randsample(nsamples, nsamples, true, wj_norm); 181 | for i = 1:nsamples % For parallel, type: parfor 182 | [thetaj1(i,:), acceptance_rate(i)] = mhsample(thetaj(idx(i),:), 1, ... 183 | 'logpdf', log_posterior, ... 184 | 'proppdf', proppdf, ... 185 | 'proprnd', proprnd, ... 186 | 'thin', thinchain, ... 187 | 'burnin', burnin); 188 | end 189 | fprintf('\n'); 190 | acceptance(iter) = mean(acceptance_rate); % To store the acceptance rate values 191 | 192 | %% Prediction step: 193 | 194 | % Define the Predictive distribution of the samples, P(theta(t+1)|Data(t)): 195 | predictive_samples(:,:,iter) = dynamic_model(thetaj1); 196 | 197 | % Compute the Bandwidth vector for the kernel density function: 198 | pred_samps = predictive_samples(:,:,iter); 199 | 200 | bw = zeros(Dimensions,1); 201 | for dim = 1:Dimensions 202 | bw(dim) = std(pred_samps(:,dim)) .* (4/((Dimensions + 2) .* nsamples)).^(1/(Dimensions + 4)); 203 | end 204 | 205 | % Define the Predictive PDF, P(theta(t+1)|Data(t)) using mvksdensity: 206 | pred_pdf = @(x) mvksdensity(pred_samps, x, 'Bandwidth', bw); 207 | 208 | %% Prepare for the next iteration: 209 | 210 | allsamples(:,:,iter+1) = thetaj1; 211 | thetaj = pred_samps; 212 | priorpdf = @(x) pred_pdf(x); 213 | 214 | end 215 | 216 | %% Description of outputs: 217 | 218 | output.samples = thetaj; % To only show samples from the final filter distribution 219 | output.allsamples = allsamples; % To only show all filter samples across all iterations 220 | output.prediction = predictive_samples; % To only show all prediction samples across all iterations 221 | output.acceptance = acceptance; % To show the mean acceptance rates for all iterations 222 | output.log_evidence = log_evidence; % To show the (M+1) x 1 vector of the logarithmic of the evidence; 223 | output.indicator = indicator; % To indicate the iterations whereby resampling took place (denoted by 1s). 224 | 225 | fprintf('End of SMC procedure. \n\n'); 226 | 227 | return; % End 228 | 229 | function proppdf = prop_pdf(x, mu, covmat, box) 230 | % This is the Proposal PDF for the Markov Chain. 231 | 232 | % Box function is the Prior PDF in the feasible region. 233 | % So if a point is out of bounds, this function will 234 | % return 0. 235 | 236 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 237 | 238 | return; 239 | 240 | 241 | function proprnd = prop_rnd(mu, covmat, box) 242 | % Sampling from the proposal PDF for the Markov Chain. 243 | 244 | while true 245 | proprnd = mvnrnd(mu, covmat, 1); 246 | if box(proprnd) 247 | % The box function is the Prior PDF in the feasible region. 248 | % If a point is out of bounds, this function will return 0 = false. 249 | break; 250 | end 251 | end 252 | 253 | return -------------------------------------------------------------------------------- /1D Static Spring-Mass System/TMCMCsampler.m: -------------------------------------------------------------------------------- 1 | function [samples_fT_D, log_fD] = TMCMCsampler(varargin) 2 | %% Transitional Markov Chain Monte Carlo sampler 3 | % 4 | % This program implements a method described in: 5 | % Ching, J. and Chen, Y. (2007). "Transitional Markov Chain Monte Carlo 6 | % Method for Bayesian Model Updating, Model Class Selection, and Model 7 | % Averaging." J. Eng. Mech., 133(7), 816-832. 8 | % 9 | % Usage: 10 | % [samples_fT_D, fD] = tmcmc_v1(fD_T, fT, sample_from_fT, N); 11 | % 12 | % where: 13 | % 14 | % inputs: 15 | % log_fD_T = function handle of log(fD_T(t)), Loglikelihood 16 | % 17 | % fT = function handle of fT(t), Prior PDF 18 | % 19 | % sample_from_fT = handle to a function that samples from of fT(t), 20 | % Sampling rule function from Prior PDF 21 | % 22 | % nsamples = number of samples of fT_D, Posterior, to generate 23 | % 24 | % outputs: 25 | % samples_fT_D = samples of fT_D (N x D) = samples from Posterior 26 | % distribution 27 | % 28 | % log_fD = log(evidence) = log(normalization constant) 29 | 30 | % ------------------------------------------------------------------------ 31 | % who when observations 32 | %-------------------------------------------------------------------------- 33 | % Diego Andres Alvarez Jul-24-2013 First algorithm 34 | %-------------------------------------------------------------------------- 35 | % Diego Andres Alvarez - daalvarez@unal.edu.co 36 | % Edoardo Patelli - edoardo.patelli@strath.ac.uk 37 | 38 | % parse the information in the name/value pairs: 39 | pnames = {'nsamples','loglikelihood','priorpdf','priorrnd','burnin','lastburnin','beta'}; 40 | 41 | dflts = {[],[],[],[],[],0,0.2}; % define default values 42 | 43 | [nsamples,loglikelihood,priorpdf,prior_rnd,burnin,lastBurnin,beta] = ... 44 | internal.stats.parseArgs(pnames, dflts, varargin{:}); 45 | 46 | %% Obtain N samples from the prior pdf f(T) 47 | j = 0; % Initialise loop for the transitional likelihood 48 | thetaj = prior_rnd(nsamples); % theta0 = N x D 49 | pj = 0; % p0 = 0 (initial tempering parameter) 50 | Dimensions = size(thetaj, 2); % size of the vector theta 51 | 52 | %% Initialization of matrices and vectors 53 | thetaj1 = zeros(nsamples, Dimensions); 54 | %log_fD_T_thetaj = zeros(nsamples,1); 55 | 56 | %% Main loop 57 | while pj < 1 58 | j = j+1; 59 | 60 | %% Calculate the tempering parameter p(j+1): 61 | for l = 1:nsamples 62 | log_fD_T_thetaj(l) = loglikelihood(thetaj(l,:)); 63 | end 64 | if any(isinf(log_fD_T_thetaj)) 65 | error('The prior distribution is too far from the true region'); 66 | end 67 | pj1 = calculate_pj1(log_fD_T_thetaj, pj); 68 | fprintf('TMCMC: Iteration j = %2d, pj1 = %f\n', j, pj1); 69 | 70 | %% Compute the plausibility weight for each sample wrt f_{j+1} 71 | fprintf('Computing the weights ...\n'); 72 | % wj = fD_T(thetaj).^(pj1-pj); % N x 1 (eq 12) 73 | a = (pj1-pj)*log_fD_T_thetaj; 74 | wj = exp(a); 75 | wj_norm = wj./sum(wj); % normalization of the weights 76 | 77 | %% Compute S(j) = E[w{j}] (eq 15) 78 | S(j) = mean(wj); 79 | 80 | %% Do the resampling step to obtain N samples from f_{j+1}(theta) and 81 | % then perform Metropolis-Hastings on each of these samples using as a 82 | % stationary PDF "fj1" 83 | % fj1 = @(t) fT(t).*log_fD_T(t).^pj1; % stationary PDF (eq 11) f_{j+1}(theta) 84 | log_posterior = @(t) log(priorpdf(t)) + pj1*loglikelihood(t); 85 | 86 | 87 | % and using as proposal PDF a Gaussian centered at thetaj(idx,:) and 88 | % with covariance matrix equal to an scaled version of the covariance 89 | % matrix of fj1: 90 | 91 | % weighted mean 92 | mu = zeros(1, Dimensions); 93 | for l = 1:nsamples 94 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 95 | end 96 | 97 | % scaled covariance matrix of fj1 (eq 17) 98 | cov_gauss = zeros(Dimensions); 99 | for k = 1:nsamples 100 | % this formula is slightly different to eq 17 (the transpose) 101 | % because of the size of the vectors)m and because Ching and Chen 102 | % forgot to normalize the weight wj: 103 | tk_mu = thetaj(k,:) - mu; 104 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 105 | end 106 | cov_gauss = beta^2 * cov_gauss; 107 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 108 | 109 | % Define the Proposal distribution: 110 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, priorpdf); %q(x,y) = q(x|y). 111 | proprnd = @(x) prop_rnd(x, cov_gauss, priorpdf); 112 | 113 | %% During the last iteration we require to do a better burnin in order 114 | % to guarantee the quality of the samples: 115 | if pj1 == 1 116 | burnin = lastBurnin; 117 | end; 118 | 119 | %% Start N different Markov chains 120 | fprintf('Markov chains ...\n\n'); 121 | idx = randsample(nsamples, nsamples, true, wj_norm); 122 | for i = 1:nsamples % For parallel, type: parfor 123 | %% Sample one point with probability wj_norm 124 | 125 | % smpl = mhsample(start, nsamples, 126 | % 'pdf', pdf, 'proppdf', proppdf, 'proprnd', proprnd); 127 | % start = row vector containing the start value of the Markov Chain, 128 | % nsamples = number of samples to be generated 129 | [thetaj1(i,:), acceptance_rate] = mhsample(thetaj(idx(i), :), 1, ... 130 | 'logpdf', log_posterior, ... 131 | 'proppdf', proppdf, ... 132 | 'proprnd', proprnd, ... 133 | 'thin', 3, ... 134 | 'burnin', burnin); 135 | % According to Cheung and Beck (2009) - Bayesian model updating ..., 136 | % the initial samples from reweighting and the resample of samples of 137 | % fj, in general, do not exactly follow fj1, so that the Markov 138 | % chains must "burn-in" before samples follow fj1, requiring a large 139 | % amount of samples to be generated for each level. 140 | 141 | %% Adjust the acceptance rate (optimal = 23%) 142 | % See: http://www.dms.umontreal.ca/~bedard/Beyond_234.pdf 143 | %{ 144 | if acceptance_rate < 0.3 145 | % Many rejections means an inefficient chain (wasted computation 146 | %time), decrease the variance 147 | beta = 0.99*beta; 148 | elseif acceptance_rate > 0.5 149 | % High acceptance rate: Proposed jumps are very close to current 150 | % location, increase the variance 151 | beta = 1.01*beta; 152 | end 153 | %} 154 | end 155 | fprintf('\n'); 156 | 157 | %% Prepare for the next iteration 158 | thetaj = thetaj1; 159 | pj = pj1; 160 | end 161 | 162 | % TMCMC provides N samples distributed according to the Posterior distribution, f(T|D) 163 | samples_fT_D = thetaj; 164 | 165 | % estimation of f(D) -- this is the normalization constant in Bayes 166 | log_fD = sum(log(S(1:j))); 167 | 168 | return; % End 169 | 170 | 171 | %% Calculate the tempering parameter p(j+1) 172 | function pj1 = calculate_pj1(log_fD_T_thetaj, pj) 173 | % find pj1 such that COV <= threshold, that is 174 | % 175 | % std(wj) 176 | % --------- <= threshold 177 | % mean(wj) 178 | % 179 | % here 180 | % size(thetaj) = N x D, 181 | % wj = fD_T(thetaj).^(pj1 - pj) 182 | % e = pj1 - pj 183 | 184 | threshold = 1; % 100% = threshold on the COV 185 | 186 | % wj = @(e) fD_T_thetaj^e; % N x 1 187 | % Note the following trick in order to calculate e: 188 | % Take into account that e>=0 189 | wj = @(e) exp(abs(e)*log_fD_T_thetaj); % N x 1 190 | %fmin = @(e) std(wj(e))/mean(wj(e)) - threshold; 191 | fmin = @(e) std(wj(e)) - threshold*mean(wj(e)) + realmin; 192 | e = abs(fzero(fmin, 0)); % e is >= 0, and fmin is an even function 193 | if isnan(e) 194 | error('There is an error finding e'); 195 | end 196 | 197 | pj1 = min(1, pj + e); 198 | 199 | return; % End 200 | 201 | function proppdf = prop_pdf(x, mu, covmat, box) 202 | % This is the Proposal PDF for the Markov Chain. 203 | 204 | % Box function is the Prior PDF in the feasible region. 205 | % So if a point is out of bounds, this function will 206 | % return 0. 207 | 208 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 209 | 210 | return; 211 | 212 | 213 | function proprnd = prop_rnd(mu, covmat, box) 214 | % Sampling from the proposal PDF for the Markov Chain. 215 | 216 | while true 217 | proprnd = mvnrnd(mu, covmat, 1); 218 | if box(proprnd) 219 | % The box function is the Prior PDF in the feasible region. 220 | % If a point is out of bounds, this function will return 0 = false. 221 | break; 222 | end 223 | end 224 | 225 | return 226 | -------------------------------------------------------------------------------- /1D Static Spring-Mass System/desktop.ini: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/1D Static Spring-Mass System/desktop.ini -------------------------------------------------------------------------------- /1D Static Spring-Mass System/log_likelihood.m: -------------------------------------------------------------------------------- 1 | function logl = log_likelihood(stiffness, modelInput, measurements, standard_deviation, ModelHandle) 2 | % Calculation of the log_likelihood for 1D Linear Static Problem: 3 | % 4 | % USAGE: 5 | % logl = log_likelihood(stiffness, displacement, measurements, standard_deviation, ModelHandle) 6 | % 7 | % INPUTS: 8 | % stiffness = epistemic parameter k [Nsamples x 1] 9 | % displacement = model input [Nobservations x 1] 10 | % measurements = experimental observations [Nobservations x 1] 11 | % standard_deviation = the standard deviation of the log likelihood function [scalar] 12 | % ModelHandle = the function handle of the model (see file "model.m") 13 | % 14 | % OUTPUTS: 15 | % logl = loglikelihood function for the set of estimated stiffness values k and 16 | % the measurements 17 | 18 | 19 | %% Evaluate the model: 20 | nchains=size(stiffness,1); 21 | %ndims=size(stiffness,2); 22 | logl=zeros(nchains,1); 23 | 24 | for n=1:nchains 25 | modelOutput = ModelHandle(stiffness(n,:),modelInput); 26 | 27 | % Note: Details to the model can be found in the file: "model.m" 28 | 29 | %% Compute the log-likelihood: 30 | 31 | logl(n) = -0.5 * (1/standard_deviation)^2 *(measurements - modelOutput)' * (measurements - modelOutput); 32 | 33 | end 34 | 35 | -------------------------------------------------------------------------------- /1D Static Spring-Mass System/model.m: -------------------------------------------------------------------------------- 1 | function [force] = model(stiffness,displacement) 2 | 3 | %MODEL: 1-Degree of freedom mass-spring system 4 | 5 | force = - stiffness.*displacement; 6 | 7 | end 8 | 9 | -------------------------------------------------------------------------------- /2D Eigen-value Problem/Eigenvalue_Problem.mlx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/2D Eigen-value Problem/Eigenvalue_Problem.mlx -------------------------------------------------------------------------------- /2D Eigen-value Problem/README.md: -------------------------------------------------------------------------------- 1 | ## Instructions: 2 | 3 | * Run the tutorial MATLAB LIVE script: "Eigenvalue_Problem.mlx" 4 | -------------------------------------------------------------------------------- /2D Eigen-value Problem/SMCsampler.m: -------------------------------------------------------------------------------- 1 | function [output] = SMCsampler(varargin) 2 | %% Sequential Monte Carlo Dynamical (SMC) sampler 3 | % 4 | % This program implements the original Sequential Monte Carlo (SMC) sampling 5 | % class (see paper by N. Chopin (2002): A sequential particle filter method 6 | % for static models - https://www.jstor.org/stable/3879283) and employs 7 | % the use of the Affine-invariant Ensemble Sampler (AIES) proposed by 8 | % Goodman and Weare (2010) to update the samples at each iteration. 9 | % 10 | % This sampler function can be employed in Sequential Bayesian Model 11 | % Updating problems involving: 12 | % - Estimating time-invariant parameter(s) via Online Bayesian Model Updating; 13 | % - Estimating time-varying parameter(s), following a recursive dynamic model; 14 | % - Predicting the time-varying parameter(s) for the next time-step given 15 | % data/observations up to the previous time-step. 16 | % 17 | %-------------------------------------------------------------------------- 18 | % Author: 19 | % Adolphus Lye - adolphus.lye@liverpool.ac.uk 20 | %-------------------------------------------------------------------------- 21 | 22 | % Parse the information in the name/value pairs: 23 | pnames = {'nsamples','loglikelihoods','dynamic_model',... 24 | 'priorpdf','priorrnd','burnin','lastburnin','thinchain'}; 25 | 26 | % Define default values: 27 | dflts = {[], [], @(x) x, [], [], 0, 0, 3}; 28 | 29 | [nsamples,loglikelihoods,dynamic_model,priorpdf,prior_rnd,... 30 | burnin,lastBurnin,thinchain] = internal.stats.parseArgs(pnames, dflts, varargin{:}); 31 | 32 | %-------------------------------------------------------------------------- 33 | % 34 | % Inputs: 35 | % nsamples: Scalar value of the number of samples to be generated from the Posterior; 36 | % loglikelihoods: A M x 1 cell vector of likelihood functions containing the measurements at M different time-steps; 37 | % dynamic_model: A function-handle that relates theta(t+1) and theta(t), where t is the time-step. Output is N x dim; 38 | % priorpdf: Function-handle of the Prior PDF; 39 | % prior_rnd: Function-handle of the Prior random number generator; 40 | % burnin: Number of burn-in for all iterations up to M-1; 41 | % lastBurnin: Number of burn-in for the last iteration; 42 | % stepsize: The stepsize for the Ensemble sampler in the updating step (this is the tuning parameter); 43 | % thinchain: Thin all the chains of the Ensemble sampler by only storing every k'th step (default=3); 44 | % 45 | % Outputs: 46 | % output.samples: A N x dim matrix of Posterior samples; 47 | % output.allsamples: A N x dim x (M+1) array of samples from all iterations; 48 | % output.acceptance: A M x 1 vector of acceptance rates for all iterations; 49 | % output.log_evidence: A (M+1) x 1 vector of the logarithmic of the evidence; 50 | % output.step: A M x 1 vector of step-size; 51 | % output.indicator: A M x 1 vector of indicators denoting if resampling 52 | % has occured for any iterations (1 = Yes, 0 = No); 53 | % 54 | %-------------------------------------------------------------------------- 55 | 56 | %% Number of cores 57 | if ~isempty(gcp('nocreate')) 58 | pool = gcp; 59 | Ncores = pool.NumWorkers; 60 | fprintf('SMC is running on %d cores.\n', Ncores); 61 | end 62 | 63 | %% Initialize: Obtain N samples from the Prior PDF 64 | 65 | fprintf('Start SMC procedure ... \n'); 66 | 67 | prior_initial = priorpdf; % Define initial Prior PDF 68 | thetaj = prior_rnd(nsamples); % theta0 = N x dim 69 | Dimensions = size(thetaj, 2); % Dimensionality of theta, dim 70 | 71 | % Initialization of matrices and vectors: 72 | thetaj1 = zeros(nsamples, Dimensions); 73 | log_evidence = zeros(size(loglikelihoods,1)+1,1); % Initiate empty vector for log evidence 74 | log_evidence(1) = 0; 75 | 76 | acceptance = zeros(size(loglikelihoods,1),1); 77 | 78 | % Samples from filter distribution, P(theta(t)|Data(1:t)): 79 | allsamples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)+1); 80 | allsamples(:,:,1) = thetaj; 81 | 82 | % Statistics from predictive distribution, P(theta(t+1)|Data(1:t)): 83 | predictive_samples = zeros(size(thetaj,1), size(thetaj,2), size(loglikelihoods,1)); 84 | 85 | % Resampling indicator vector: 86 | indicator = zeros(length(loglikelihoods), 1); 87 | % Note: This indicator vector returns a 1 for the iteration(s) where 88 | % resampling is initiated and 0 otherwise. 89 | 90 | %% Main sampling loop 91 | for iter = 1:length(loglikelihoods) 92 | 93 | fprintf('SMC: Iteration j = %2d \n', iter); 94 | 95 | loglikelihood = loglikelihoods{iter}; 96 | 97 | % Compute loglikelihood values for each sample: 98 | logL = zeros(nsamples,1); 99 | for l = 1:nsamples 100 | logL(l) = loglikelihood(thetaj(l,:)); 101 | end 102 | 103 | % Error check: 104 | if any(isinf(logL)) 105 | error('The prior distribution is too far from the true region'); 106 | end 107 | 108 | %% Compute weights of the samples, wj: 109 | 110 | % To compute the nominal weights: 111 | fprintf('Computing the weights ...\n'); 112 | wj = exp(logL); 113 | 114 | % To compute the log evidence for the current iteration: 115 | log_evidence(iter+1) = log(mean(wj)) + log_evidence(iter); 116 | 117 | % Check step for wj: 118 | for i = 1:nsamples 119 | if wj(i) == 0 120 | wj(i) = 1e-100; 121 | end 122 | end 123 | 124 | wj_norm = wj./sum(wj); % To normalise the weights 125 | 126 | %% Check step - Compute the sum of wj_norm and see if it is < nsamples/2: 127 | 128 | fprintf('Computing effective sample size ... \n'); 129 | Neff = 1/(sum(wj_norm.^2)); 130 | threshold = nsamples/2; 131 | 132 | %% Resampling step (conditional if Neff < threshold): 133 | 134 | if Neff < threshold 135 | fprintf('Resampling step initiated ... \n'); 136 | 137 | dx = randsample(nsamples, nsamples, true, wj_norm); 138 | 139 | thetaj_resampled = zeros(nsamples, Dimensions); 140 | for d = 1:nsamples 141 | thetaj_resampled(d,:) = thetaj(dx(d),:); 142 | end 143 | 144 | thetaj = thetaj_resampled; 145 | wj_norm = (1/nsamples).*ones(nsamples,1); 146 | indicator(iter) = 1; 147 | 148 | end 149 | 150 | %% Update the samples according to the current Posterior using MH sampler: 151 | 152 | % Define the logposterior: 153 | log_posterior = @(x) log(priorpdf(x)) + loglikelihood(x); 154 | 155 | % Weighted mean for Proposal distribution 156 | mu = zeros(1, Dimensions); 157 | for l = 1:nsamples 158 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 159 | end 160 | 161 | % Covariance matrix for Proposal distribution: 162 | cov_gauss = zeros(Dimensions); 163 | for k = 1:nsamples 164 | tk_mu = thetaj(k,:) - mu; 165 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 166 | end 167 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 168 | 169 | % Define the Proposal distribution: 170 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, prior_initial); % q(x,y) = q(x|y). 171 | proprnd = @(x) prop_rnd(x, cov_gauss, prior_initial); 172 | 173 | if iter == length(loglikelihoods) 174 | burnin = lastBurnin; 175 | end 176 | 177 | %% Start N different Markov chains 178 | fprintf('Markov chains ...\n\n'); 179 | 180 | idx = randsample(nsamples, nsamples, true, wj_norm); 181 | for i = 1:nsamples % For parallel, type: parfor 182 | [thetaj1(i,:), acceptance_rate(i)] = mhsample(thetaj(idx(i),:), 1, ... 183 | 'logpdf', log_posterior, ... 184 | 'proppdf', proppdf, ... 185 | 'proprnd', proprnd, ... 186 | 'thin', thinchain, ... 187 | 'burnin', burnin); 188 | end 189 | fprintf('\n'); 190 | acceptance(iter) = mean(acceptance_rate); % To store the acceptance rate values 191 | 192 | %% Prediction step: 193 | 194 | % Define the Predictive distribution of the samples, P(theta(t+1)|Data(t)): 195 | predictive_samples(:,:,iter) = dynamic_model(thetaj1); 196 | 197 | % Compute the Bandwidth vector for the kernel density function: 198 | pred_samps = predictive_samples(:,:,iter); 199 | 200 | bw = zeros(Dimensions,1); 201 | for dim = 1:Dimensions 202 | bw(dim) = std(pred_samps(:,dim)) .* (4/((Dimensions + 2) .* nsamples)).^(1/(Dimensions + 4)); 203 | end 204 | 205 | % Define the Predictive PDF, P(theta(t+1)|Data(t)) using mvksdensity: 206 | pred_pdf = @(x) mvksdensity(pred_samps, x, 'Bandwidth', bw); 207 | 208 | %% Prepare for the next iteration: 209 | 210 | allsamples(:,:,iter+1) = thetaj1; 211 | thetaj = pred_samps; 212 | priorpdf = @(x) pred_pdf(x); 213 | 214 | end 215 | 216 | %% Description of outputs: 217 | 218 | output.samples = thetaj; % To only show samples from the final filter distribution 219 | output.allsamples = allsamples; % To only show all filter samples across all iterations 220 | output.prediction = predictive_samples; % To only show all prediction samples across all iterations 221 | output.acceptance = acceptance; % To show the mean acceptance rates for all iterations 222 | output.log_evidence = log_evidence; % To show the (M+1) x 1 vector of the logarithmic of the evidence; 223 | output.indicator = indicator; % To indicate the iterations whereby resampling took place (denoted by 1s). 224 | 225 | fprintf('End of SMC procedure. \n\n'); 226 | 227 | return; % End 228 | 229 | function proppdf = prop_pdf(x, mu, covmat, box) 230 | % This is the Proposal PDF for the Markov Chain. 231 | 232 | % Box function is the Prior PDF in the feasible region. 233 | % So if a point is out of bounds, this function will 234 | % return 0. 235 | 236 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 237 | 238 | return; 239 | 240 | 241 | function proprnd = prop_rnd(mu, covmat, box) 242 | % Sampling from the proposal PDF for the Markov Chain. 243 | 244 | while true 245 | proprnd = mvnrnd(mu, covmat, 1); 246 | if box(proprnd) 247 | % The box function is the Prior PDF in the feasible region. 248 | % If a point is out of bounds, this function will return 0 = false. 249 | break; 250 | end 251 | end 252 | 253 | return -------------------------------------------------------------------------------- /2D Eigen-value Problem/TMCMCsampler.m: -------------------------------------------------------------------------------- 1 | function [samples_fT_D, log_fD] = TMCMCsampler(varargin) 2 | %% Transitional Markov Chain Monte Carlo sampler 3 | % 4 | % This program implements a method described in: 5 | % Ching, J. and Chen, Y. (2007). "Transitional Markov Chain Monte Carlo 6 | % Method for Bayesian Model Updating, Model Class Selection, and Model 7 | % Averaging." J. Eng. Mech., 133(7), 816-832. 8 | % 9 | % Usage: 10 | % [samples_fT_D, fD] = tmcmc_v1(fD_T, fT, sample_from_fT, N); 11 | % 12 | % where: 13 | % 14 | % inputs: 15 | % log_fD_T = function handle of log(fD_T(t)), Loglikelihood 16 | % 17 | % fT = function handle of fT(t), Prior PDF 18 | % 19 | % sample_from_fT = handle to a function that samples from of fT(t), 20 | % Sampling rule function from Prior PDF 21 | % 22 | % nsamples = number of samples of fT_D, Posterior, to generate 23 | % 24 | % outputs: 25 | % samples_fT_D = samples of fT_D (N x D) = samples from Posterior 26 | % distribution 27 | % 28 | % log_fD = log(evidence) = log(normalization constant) 29 | 30 | % ------------------------------------------------------------------------ 31 | % who when observations 32 | %-------------------------------------------------------------------------- 33 | % Diego Andres Alvarez Jul-24-2013 First algorithm 34 | %-------------------------------------------------------------------------- 35 | % Diego Andres Alvarez - daalvarez@unal.edu.co 36 | % Edoardo Patelli - edoardo.patelli@strath.ac.uk 37 | 38 | % parse the information in the name/value pairs: 39 | pnames = {'nsamples','loglikelihood','priorpdf','priorrnd','burnin','lastburnin','beta'}; 40 | 41 | dflts = {[],[],[],[],[],0,0.2}; % define default values 42 | 43 | [nsamples,loglikelihood,priorpdf,prior_rnd,burnin,lastBurnin,beta] = ... 44 | internal.stats.parseArgs(pnames, dflts, varargin{:}); 45 | 46 | %% Obtain N samples from the prior pdf f(T) 47 | j = 0; % Initialise loop for the transitional likelihood 48 | thetaj = prior_rnd(nsamples); % theta0 = N x D 49 | pj = 0; % p0 = 0 (initial tempering parameter) 50 | Dimensions = size(thetaj, 2); % size of the vector theta 51 | 52 | %% Initialization of matrices and vectors 53 | thetaj1 = zeros(nsamples, Dimensions); 54 | %log_fD_T_thetaj = zeros(nsamples,1); 55 | 56 | %% Main loop 57 | while pj < 1 58 | j = j+1; 59 | 60 | %% Calculate the tempering parameter p(j+1): 61 | for l = 1:nsamples 62 | log_fD_T_thetaj(l) = loglikelihood(thetaj(l,:)); 63 | end 64 | if any(isinf(log_fD_T_thetaj)) 65 | error('The prior distribution is too far from the true region'); 66 | end 67 | pj1 = calculate_pj1(log_fD_T_thetaj, pj); 68 | fprintf('TMCMC: Iteration j = %2d, pj1 = %f\n', j, pj1); 69 | 70 | %% Compute the plausibility weight for each sample wrt f_{j+1} 71 | fprintf('Computing the weights ...\n'); 72 | % wj = fD_T(thetaj).^(pj1-pj); % N x 1 (eq 12) 73 | a = (pj1-pj)*log_fD_T_thetaj; 74 | wj = exp(a); 75 | wj_norm = wj./sum(wj); % normalization of the weights 76 | 77 | %% Compute S(j) = E[w{j}] (eq 15) 78 | S(j) = mean(wj); 79 | 80 | %% Do the resampling step to obtain N samples from f_{j+1}(theta) and 81 | % then perform Metropolis-Hastings on each of these samples using as a 82 | % stationary PDF "fj1" 83 | % fj1 = @(t) fT(t).*log_fD_T(t).^pj1; % stationary PDF (eq 11) f_{j+1}(theta) 84 | log_posterior = @(t) log(priorpdf(t)) + pj1*loglikelihood(t); 85 | 86 | 87 | % and using as proposal PDF a Gaussian centered at thetaj(idx,:) and 88 | % with covariance matrix equal to an scaled version of the covariance 89 | % matrix of fj1: 90 | 91 | % weighted mean 92 | mu = zeros(1, Dimensions); 93 | for l = 1:nsamples 94 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 95 | end 96 | 97 | % scaled covariance matrix of fj1 (eq 17) 98 | cov_gauss = zeros(Dimensions); 99 | for k = 1:nsamples 100 | % this formula is slightly different to eq 17 (the transpose) 101 | % because of the size of the vectors)m and because Ching and Chen 102 | % forgot to normalize the weight wj: 103 | tk_mu = thetaj(k,:) - mu; 104 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 105 | end 106 | cov_gauss = beta^2 * cov_gauss; 107 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 108 | 109 | % Define the Proposal distribution: 110 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, priorpdf); %q(x,y) = q(x|y). 111 | proprnd = @(x) prop_rnd(x, cov_gauss, priorpdf); 112 | 113 | %% During the last iteration we require to do a better burnin in order 114 | % to guarantee the quality of the samples: 115 | if pj1 == 1 116 | burnin = lastBurnin; 117 | end; 118 | 119 | %% Start N different Markov chains 120 | fprintf('Markov chains ...\n\n'); 121 | idx = randsample(nsamples, nsamples, true, wj_norm); 122 | for i = 1:nsamples % For parallel, type: parfor 123 | %% Sample one point with probability wj_norm 124 | 125 | % smpl = mhsample(start, nsamples, 126 | % 'pdf', pdf, 'proppdf', proppdf, 'proprnd', proprnd); 127 | % start = row vector containing the start value of the Markov Chain, 128 | % nsamples = number of samples to be generated 129 | [thetaj1(i,:), acceptance_rate] = mhsample(thetaj(idx(i), :), 1, ... 130 | 'logpdf', log_posterior, ... 131 | 'proppdf', proppdf, ... 132 | 'proprnd', proprnd, ... 133 | 'thin', 3, ... 134 | 'burnin', burnin); 135 | % According to Cheung and Beck (2009) - Bayesian model updating ..., 136 | % the initial samples from reweighting and the resample of samples of 137 | % fj, in general, do not exactly follow fj1, so that the Markov 138 | % chains must "burn-in" before samples follow fj1, requiring a large 139 | % amount of samples to be generated for each level. 140 | 141 | %% Adjust the acceptance rate (optimal = 23%) 142 | % See: http://www.dms.umontreal.ca/~bedard/Beyond_234.pdf 143 | %{ 144 | if acceptance_rate < 0.3 145 | % Many rejections means an inefficient chain (wasted computation 146 | %time), decrease the variance 147 | beta = 0.99*beta; 148 | elseif acceptance_rate > 0.5 149 | % High acceptance rate: Proposed jumps are very close to current 150 | % location, increase the variance 151 | beta = 1.01*beta; 152 | end 153 | %} 154 | end 155 | fprintf('\n'); 156 | 157 | %% Prepare for the next iteration 158 | thetaj = thetaj1; 159 | pj = pj1; 160 | end 161 | 162 | % TMCMC provides N samples distributed according to the Posterior distribution, f(T|D) 163 | samples_fT_D = thetaj; 164 | 165 | % estimation of f(D) -- this is the normalization constant in Bayes 166 | log_fD = sum(log(S(1:j))); 167 | 168 | return; % End 169 | 170 | 171 | %% Calculate the tempering parameter p(j+1) 172 | function pj1 = calculate_pj1(log_fD_T_thetaj, pj) 173 | % find pj1 such that COV <= threshold, that is 174 | % 175 | % std(wj) 176 | % --------- <= threshold 177 | % mean(wj) 178 | % 179 | % here 180 | % size(thetaj) = N x D, 181 | % wj = fD_T(thetaj).^(pj1 - pj) 182 | % e = pj1 - pj 183 | 184 | threshold = 1; % 100% = threshold on the COV 185 | 186 | % wj = @(e) fD_T_thetaj^e; % N x 1 187 | % Note the following trick in order to calculate e: 188 | % Take into account that e>=0 189 | wj = @(e) exp(abs(e)*log_fD_T_thetaj); % N x 1 190 | %fmin = @(e) std(wj(e))/mean(wj(e)) - threshold; 191 | fmin = @(e) std(wj(e)) - threshold*mean(wj(e)) + realmin; 192 | e = abs(fzero(fmin, 0)); % e is >= 0, and fmin is an even function 193 | if isnan(e) 194 | error('There is an error finding e'); 195 | end 196 | 197 | pj1 = min(1, pj + e); 198 | 199 | return; % End 200 | 201 | function proppdf = prop_pdf(x, mu, covmat, box) 202 | % This is the Proposal PDF for the Markov Chain. 203 | 204 | % Box function is the Prior PDF in the feasible region. 205 | % So if a point is out of bounds, this function will 206 | % return 0. 207 | 208 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 209 | 210 | return; 211 | 212 | 213 | function proprnd = prop_rnd(mu, covmat, box) 214 | % Sampling from the proposal PDF for the Markov Chain. 215 | 216 | while true 217 | proprnd = mvnrnd(mu, covmat, 1); 218 | if box(proprnd) 219 | % The box function is the Prior PDF in the feasible region. 220 | % If a point is out of bounds, this function will return 0 = false. 221 | break; 222 | end 223 | end 224 | 225 | return 226 | -------------------------------------------------------------------------------- /2D Eigen-value Problem/desktop.ini: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/2D Eigen-value Problem/desktop.ini -------------------------------------------------------------------------------- /2D Eigen-value Problem/log_likelihood.m: -------------------------------------------------------------------------------- 1 | function logl = log_likelihood(Thetas, Eigenvalues, standard_deviations, ModelHandle) 2 | % Calculation of the log_likelihood for 2D Inverse Eigenvalue Problem: 3 | % 4 | % USAGE: 5 | % logl = log_likelihood(Thetas, Eigenvalues, standard_deviations, ModelHandle) 6 | % 7 | % INPUTS: 8 | % Thetas = Vector of epistemic parameters theta_1 and theta_2 [Nsamples x 2] 9 | % Eigenvalues = Observations of Eigenvalues from the 2 x 2 Square Matrix [Nobservations x 2] 10 | % standard_deviations = Vector of the standard deviations of the 2D log likelihood function [2 x 1] 11 | % ModelHandle = the function handle of the model (see file "model.m") 12 | % 13 | % OUTPUTS: 14 | % logl = loglikelihood function for the set of estimated values of theta_1 15 | % and theta_2 as well as the Eigenvalues 16 | 17 | 18 | %% Evaluate the model: 19 | nchains=size(Thetas,1); 20 | logl=zeros(nchains,1); 21 | 22 | for n=1:nchains 23 | modelOutput = ModelHandle(Thetas(n,:)); 24 | 25 | % Note: Details to the model can be found in the file: "model.m" 26 | 27 | %% Compute the overall 2D log-likelihood: 28 | 29 | logl_1 = - 0.5 * (1/standard_deviations(1))^2 *(Eigenvalues(:,1) - modelOutput(1))' * (Eigenvalues(:,1) - modelOutput(1)); 30 | 31 | % Note: logl_1 is the loglikelihood function in the theta_1 dimension. 32 | 33 | logl_2 = - 0.5 * (1/standard_deviations(2))^2 *(Eigenvalues(:,2) - modelOutput(2))' * (Eigenvalues(:,2) - modelOutput(2)); 34 | 35 | % Note: logl_2 is the loglikelihood function in the theta_2 dimension. 36 | 37 | logl(n) = logl_1 + logl_2; 38 | 39 | % logl is the overall 2D loglikelihood function. 40 | 41 | end -------------------------------------------------------------------------------- /2D Eigen-value Problem/model.m: -------------------------------------------------------------------------------- 1 | function [eigenvalues] = model(Thetas) 2 | 3 | %MODEL: Eigenvalues of 2 x 2 square matrix 4 | 5 | eigenvalues(:,1) = 0.5*((Thetas(:,1) + 2*Thetas(:,2)) + (Thetas(:,1).^2 + 4*(Thetas(:,2).^2)).^0.5); 6 | 7 | eigenvalues(:,2) = 0.5*((Thetas(:,1) + 2*Thetas(:,2)) - (Thetas(:,1).^2 + 4*(Thetas(:,2).^2)).^0.5); 8 | 9 | end -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/DenHartogHarmonic.m: -------------------------------------------------------------------------------- 1 | function [output] = DenHartogHarmonic(beta_v) 2 | %% Function-handle for Analytical plots of Tramissibility and Phase Angles: 3 | % 4 | % This set-up is based on the Single Degree-of-Freedom Dynamical System 5 | % subjected to Coulomb Friction Force that is presented in the literature: 6 | % 7 | % L. Marino and A. Cicirello (2020). Experimental investigation of a single- 8 | % degree-of-freedom system with Coulomb friction. Nonlinear Dynamics, 99(3), 9 | % 1781-1799. doi: 10.1007/s11071-019-05443-2 10 | % 11 | % This set-up is applicable only for the Base motion (with fixed wall) case. 12 | %-------------------------------------------------------------------------% 13 | %% Define key parameters: 14 | 15 | % Define the non-dimensional parameters: 16 | r_v = [0.01:0.01:0.9 0.901:0.001:0.999 1.001:0.001:1.099 1.1:0.025:2.6]; 17 | beta_v = [0 beta_v]; 18 | 19 | dt = pi/15; % Time-step 20 | t_half = 0:dt:pi; % Time half-period 21 | N_cyc = 30; % To be selected for FFT performance 22 | 23 | %% Obtain the boundary values of Displacement Transmissibility and Phase Angles: 24 | 25 | for ir = 1:length(r_v) % For each Frequency ratio value 26 | r = r_v(ir); 27 | 28 | % Response and damping functions: 29 | U = sin(pi/r)/(r*(1+cos(pi/r))); % See Eq. (7) 30 | V = 1/(1-r^2); % See Eq. (8) 31 | 32 | % Boundaries: 33 | sr = (r*sin(t_half/r)+U*r^2*(cos(t_half)-cos(t_half/r)))./sin(t_half); % See Eq. (9) 34 | S = max(sr(1:end-1)); 35 | 36 | beta_bound = sqrt(V^2/(U^2+(S/r^2)^2)); % See Eq. (6) 37 | X_bound = sqrt(V^2 - beta_bound^2*U^2); % Displacement Tranmissibility bound (see Eq. (11)) 38 | ph_bound = angleCalc(- beta_bound*U/V, X_bound/V, 'rad'); % Compute the bounds for Phase Angles 39 | 40 | x_bound_half = X_bound*cos(t_half) + beta_bound*U*sin(t_half) + ... 41 | beta_bound*(1 - cos(t_half/r) - U*r*sin(t_half/r)); % See Eq. (15) 42 | y_bound_half = cos(t_half + ph_bound); % See Eq. (16) 43 | 44 | t_period = [t_half t_half(2:end)+pi]; 45 | x_bound_period = [x_bound_half -x_bound_half(2:end)]; 46 | y_bound_period = [y_bound_half -y_bound_half(2:end)]; 47 | 48 | t = t_period; 49 | for in = 1:N_cyc - 1 50 | t = [t t_period(2:end)+2*in*pi]; 51 | end 52 | 53 | x_bound = [x_bound_period repmat(x_bound_period(2:end),1,N_cyc-1)]; 54 | y_bound = [y_bound_period repmat(y_bound_period(2:end),1,N_cyc-1)]; 55 | 56 | %% Fast-Fourier Transform Step: 57 | 58 | Nt = length(t); % No. of time-steps 59 | fs = 1/dt; % Frequency of data-collection 60 | df = 1/t(end); % Frequency steps 61 | omega = 2*pi*(0:df:fs); % Value of omegas = 2*pi*f 62 | 63 | % Compute the bounds in the time and frequency domains: 64 | x_bound_fft = fft(x_bound)/Nt; 65 | y_bound_fft = fft(y_bound)/Nt; 66 | 67 | i_peak = find(round(omega,3)>=1,1); 68 | % Note: round(omega,3) rounds the omega value to the nearest thousandth (10^-3). 69 | X_bound_peak = abs(x_bound_fft(i_peak)); 70 | Y_bound_peak = abs(y_bound_fft(i_peak)); 71 | 72 | % Identify the Den-Hartog's bound for Transmissibility (see Eq. (29)): 73 | Tr_bound(ir) = X_bound_peak/Y_bound_peak; 74 | 75 | % Identify the Den-Hartog's bound for Phase Angles (see Eq. (30)): 76 | phase_bound(ir) = rad2deg(angle(y_bound_fft(i_peak)/x_bound_fft(i_peak))); 77 | 78 | % Compute the Transmissibility and Phase Angle Response for different beta values: 79 | for ib = 1:length(beta_v) 80 | beta = beta_v(ib); 81 | 82 | if beta <= beta_bound || beta == 0 % In the continuous motion condition 83 | X = sqrt(V^2 - (beta^2 * U^2)); % See Eq. (11) 84 | ph = angleCalc(-beta*U/V,X/V,'rad'); % See Eq. (12) 85 | x_half = X*cos(t_half)+beta*U*sin(t_half)+beta*(1-cos(t_half/r)-U*r*sin(t_half/r)); % See Eq. (15) 86 | y_half = cos(t_half+ph); % See Eq. (16) 87 | 88 | % Post-processing step: 89 | x_period = [x_half -x_half(2:end)]; 90 | y_period = [y_half -y_half(2:end)]; 91 | 92 | x = [x_period repmat(x_period(2:end),1,N_cyc-1)]; 93 | y = [y_period repmat(y_period(2:end),1,N_cyc-1)]; 94 | 95 | y_fft = fft(y)/Nt; 96 | x_fft = fft(x)/Nt; 97 | 98 | Y_peak = abs(y_fft(i_peak)); 99 | X_peak = abs(x_fft(i_peak)); 100 | 101 | % Transimissibility response for r = r_v and beta = beta_v (see Eq. (29)): 102 | Tr(ib,ir) = X_peak/Y_peak; 103 | 104 | % Phase Angle response for r = r_v and beta = beta_v (see Eq. (30)): 105 | phase(ib,ir) = rad2deg(angle(y_fft(i_peak)/x_fft(i_peak))); 106 | 107 | else % In the stick-slip domain 108 | Tr(ib,ir) = NaN; 109 | phase(ib,ir) = NaN; 110 | end 111 | 112 | end 113 | end 114 | 115 | %% Consolidate the output: 116 | output.frequency_ratios = r_v; 117 | output.trans = Tr; 118 | output.phase_angles = phase; 119 | output.trans_bound = Tr_bound; 120 | output.phase_bound = phase_bound; 121 | end 122 | 123 | function theta = angleCalc(S,C,out_mode) 124 | %% This function computes the angle from sin and cos values (-180,180] degree. 125 | % 126 | % Usage: 127 | % theta = angleCalc(S,C,out_mode) 128 | % 129 | % Input: 130 | % S: Sine value of the angle 131 | % C: Cosine value of the angle 132 | % out_mode: 'deg' OR 'rad' 133 | % Note: default output mode is in degree 134 | % 135 | % Output: 136 | % theta: Angles in degrees or radians. 137 | % 138 | % Example: 139 | % theta = angleCalc(sin(-2*pi/3),cos(-2*pi/3)) 140 | % theta = -120; 141 | % theta= angleCalc(sin(2*pi/3),cos(2*pi/3),'rad') 142 | % theta= 2.0944 [rad] 143 | % --------------Disi A Jun 25, 2013 144 | %% Define the function: 145 | if nargin < 3 146 | out_mode='deg'; 147 | end 148 | 149 | if strcmp(out_mode,'deg') 150 | cons = 180/pi; 151 | else 152 | cons = 1; 153 | end 154 | 155 | for i = 1:length(S) 156 | theta(i) = asin(S(i)); 157 | if C(i) < 0 158 | if S(i) > 0 159 | theta(i) = pi - theta(i); 160 | elseif S(i) < 0 161 | theta(i) = - pi - theta(i); 162 | else % If S(i) = 0 163 | theta(i) = theta(i) + pi; 164 | end 165 | end 166 | 167 | theta(i) = theta(i) .* cons; 168 | end 169 | end 170 | 171 | 172 | -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/TMCMCsampler.m: -------------------------------------------------------------------------------- 1 | function [output] = TMCMCsampler(varargin) 2 | %% Transitional Markov Chain Monte Carlo sampler 3 | % 4 | % This program implements a method described in: 5 | % Ching, J. and Chen, Y. (2007). "Transitional Markov Chain Monte Carlo 6 | % Method for Bayesian Model Updating, Model Class Selection, and Model 7 | % Averaging." J. Eng. Mech., 133(7), 816-832. 8 | % 9 | % Usage: 10 | % [samples_fT_D, fD] = tmcmc_v1(fD_T, fT, sample_from_fT, N); 11 | % 12 | % where: 13 | % 14 | % inputs: 15 | % log_fD_T = function handle of log(fD_T(t)), Loglikelihood 16 | % 17 | % fT = function handle of fT(t), Prior PDF 18 | % 19 | % sample_from_fT = handle to a function that samples from of fT(t), 20 | % Sampling rule function from Prior PDF 21 | % 22 | % nsamples = number of samples of fT_D, Posterior, to generate 23 | % 24 | % outputs: 25 | % samples_fT_D = samples of fT_D (N x D) = samples from Posterior 26 | % distribution 27 | % 28 | % log_fD = log(evidence) = log(normalization constant) 29 | 30 | % ------------------------------------------------------------------------ 31 | % who when observations 32 | %-------------------------------------------------------------------------- 33 | % Diego Andres Alvarez Jul-24-2013 First algorithm 34 | %-------------------------------------------------------------------------- 35 | % Diego Andres Alvarez - daalvarez@unal.edu.co 36 | % Edoardo Patelli - edoardo.patelli@strath.ac.uk 37 | 38 | % parse the information in the name/value pairs: 39 | pnames = {'nsamples','loglikelihood','priorpdf','priorrnd','burnin','lastburnin','beta'}; 40 | 41 | dflts = {[],[],[],[],[],0,0.2}; % define default values 42 | 43 | [nsamples,loglikelihood,priorpdf,prior_rnd,burnin,lastBurnin,beta] = ... 44 | internal.stats.parseArgs(pnames, dflts, varargin{:}); 45 | 46 | %% Obtain N samples from the prior pdf f(T) 47 | j = 0; % Initialise loop for the transitional likelihood 48 | thetaj = prior_rnd(nsamples); % theta0 = N x D 49 | pj = 0; % p0 = 0 (initial tempering parameter) 50 | Dimensions = size(thetaj, 2); % size of the vector theta 51 | 52 | count = 1; % Counter 53 | samps(:,:,count) = thetaj; 54 | beta_j(count) = pj; 55 | 56 | %% Initialization of matrices and vectors 57 | thetaj1 = zeros(nsamples, Dimensions); 58 | %log_fD_T_thetaj = zeros(nsamples,1); 59 | 60 | %% Main loop 61 | while pj < 1 62 | j = j+1; 63 | 64 | %% Calculate the tempering parameter p(j+1): 65 | for l = 1:nsamples 66 | log_fD_T_thetaj(l) = loglikelihood(thetaj(l,:)); 67 | end 68 | if any(isinf(log_fD_T_thetaj)) 69 | error('The prior distribution is too far from the true region'); 70 | end 71 | pj1 = calculate_pj1(log_fD_T_thetaj, pj); 72 | fprintf('TMCMC: Iteration j = %2d, pj1 = %f\n', j, pj1); 73 | 74 | %% Compute the plausibility weight for each sample wrt f_{j+1} 75 | fprintf('Computing the weights ...\n'); 76 | % wj = fD_T(thetaj).^(pj1-pj); % N x 1 (eq 12) 77 | a = (pj1-pj)*log_fD_T_thetaj; 78 | wj = exp(a); 79 | wj_norm = wj./sum(wj); % normalization of the weights 80 | 81 | %% Compute S(j) = E[w{j}] (eq 15) 82 | S(j) = mean(wj); 83 | 84 | %% Do the resampling step to obtain N samples from f_{j+1}(theta) and 85 | % then perform Metropolis-Hastings on each of these samples using as a 86 | % stationary PDF "fj1" 87 | % fj1 = @(t) fT(t).*log_fD_T(t).^pj1; % stationary PDF (eq 11) f_{j+1}(theta) 88 | log_posterior = @(t) log(priorpdf(t)) + pj1*loglikelihood(t); 89 | 90 | 91 | % and using as proposal PDF a Gaussian centered at thetaj(idx,:) and 92 | % with covariance matrix equal to an scaled version of the covariance 93 | % matrix of fj1: 94 | 95 | % weighted mean 96 | mu = zeros(1, Dimensions); 97 | for l = 1:nsamples 98 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 99 | end 100 | 101 | % scaled covariance matrix of fj1 (eq 17) 102 | cov_gauss = zeros(Dimensions); 103 | for k = 1:nsamples 104 | % this formula is slightly different to eq 17 (the transpose) 105 | % because of the size of the vectors)m and because Ching and Chen 106 | % forgot to normalize the weight wj: 107 | tk_mu = thetaj(k,:) - mu; 108 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 109 | end 110 | cov_gauss = beta^2 * cov_gauss; 111 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 112 | 113 | % Define the Proposal distribution: 114 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, priorpdf); %q(x,y) = q(x|y). 115 | proprnd = @(x) prop_rnd(x, cov_gauss, priorpdf); 116 | 117 | %% During the last iteration we require to do a better burnin in order 118 | % to guarantee the quality of the samples: 119 | if pj1 == 1 120 | burnin = lastBurnin; 121 | end; 122 | 123 | %% Start N different Markov chains 124 | fprintf('Markov chains ...\n\n'); 125 | idx = randsample(nsamples, nsamples, true, wj_norm); 126 | 127 | for i = 1:nsamples % For parallel, type: parfor 128 | %% Sample one point with probability wj_norm 129 | 130 | % smpl = mhsample(start, nsamples, 131 | % 'pdf', pdf, 'proppdf', proppdf, 'proprnd', proprnd); 132 | % start = row vector containing the start value of the Markov Chain, 133 | % nsamples = number of samples to be generated 134 | [thetaj1(i,:), acceptance_rate(i)] = mhsample(thetaj(idx(i), :), 1, ... 135 | 'logpdf', log_posterior, ... 136 | 'proppdf', proppdf, ... 137 | 'proprnd', proprnd, ... 138 | 'thin', 3, ... 139 | 'burnin', burnin); 140 | % According to Cheung and Beck (2009) - Bayesian model updating ..., 141 | % the initial samples from reweighting and the resample of samples of 142 | % fj, in general, do not exactly follow fj1, so that the Markov 143 | % chains must "burn-in" before samples follow fj1, requiring a large 144 | % amount of samples to be generated for each level. 145 | 146 | %% Adjust the acceptance rate (optimal = 23%) 147 | % See: http://www.dms.umontreal.ca/~bedard/Beyond_234.pdf 148 | %{ 149 | if acceptance_rate < 0.3 150 | % Many rejections means an inefficient chain (wasted computation 151 | %time), decrease the variance 152 | beta = 0.99*beta; 153 | elseif acceptance_rate > 0.5 154 | % High acceptance rate: Proposed jumps are very close to current 155 | % location, increase the variance 156 | beta = 1.01*beta; 157 | end 158 | %} 159 | end 160 | fprintf('\n'); 161 | acceptance(count) = mean(acceptance_rate); 162 | %% Prepare for the next iteration 163 | count = count+1; 164 | samps(:,:,count) = thetaj1; 165 | thetaj = thetaj1; 166 | pj = pj1; 167 | beta_j(count) = pj; 168 | end 169 | 170 | % estimation of f(D) -- this is the normalization constant in Bayes 171 | log_fD = sum(log(S(1:j))); 172 | 173 | %% Description of outputs: 174 | 175 | output.allsamples = samps; % To show samples from all transitional distributions 176 | output.samples = samps(:,:,end); % To only show samples from the final posterior 177 | output.log_evidence = log_fD; % To generate the logarithmic of the evidence 178 | output.acceptance = acceptance; % To show the mean acceptance rates for all iterations 179 | output.beta = beta_j; % To show the values of temepring parameters, beta_j 180 | 181 | return; % End 182 | 183 | 184 | %% Calculate the tempering parameter p(j+1) 185 | function pj1 = calculate_pj1(log_fD_T_thetaj, pj) 186 | % find pj1 such that COV <= threshold, that is 187 | % 188 | % std(wj) 189 | % --------- <= threshold 190 | % mean(wj) 191 | % 192 | % here 193 | % size(thetaj) = N x D, 194 | % wj = fD_T(thetaj).^(pj1 - pj) 195 | % e = pj1 - pj 196 | 197 | threshold = 1; % 100% = threshold on the COV 198 | 199 | % wj = @(e) fD_T_thetaj^e; % N x 1 200 | % Note the following trick in order to calculate e: 201 | % Take into account that e>=0 202 | wj = @(e) exp(abs(e)*log_fD_T_thetaj); % N x 1 203 | %fmin = @(e) std(wj(e))/mean(wj(e)) - threshold; 204 | fmin = @(e) std(wj(e)) - threshold*mean(wj(e)) + realmin; 205 | e = abs(fzero(fmin, 0)); % e is >= 0, and fmin is an even function 206 | if isnan(e) 207 | error('There is an error finding e'); 208 | end 209 | 210 | pj1 = min(1, pj + e); 211 | 212 | return; % End 213 | 214 | function proppdf = prop_pdf(x, mu, covmat, box) 215 | % This is the Proposal PDF for the Markov Chain. 216 | 217 | % Box function is the Prior PDF in the feasible region. 218 | % So if a point is out of bounds, this function will 219 | % return 0. 220 | 221 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 222 | 223 | return; 224 | 225 | 226 | function proprnd = prop_rnd(mu, covmat, box) 227 | % Sampling from the proposal PDF for the Markov Chain. 228 | 229 | while true 230 | proprnd = mvnrnd(mu, covmat, 1); 231 | if box(proprnd) 232 | % The box function is the Prior PDF in the feasible region. 233 | % If a point is out of bounds, this function will return 0 = false. 234 | break; 235 | end 236 | end 237 | 238 | return 239 | -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/TMCMCsampler2.m: -------------------------------------------------------------------------------- 1 | function [output] = TMCMCsampler2(varargin) 2 | %% Transitional Markov Chain Monte Carlo sampler 3 | % 4 | % This program implements a method described in: 5 | % Ching, J. and Chen, Y. (2007). "Transitional Markov Chain Monte Carlo 6 | % Method for Bayesian Model Updating, Model Class Selection, and Model 7 | % Averaging." J. Eng. Mech., 133(7), 816-832. 8 | % 9 | % Usage: 10 | % [samples_fT_D, fD] = tmcmc_v1(fD_T, fT, sample_from_fT, N); 11 | % 12 | % where: 13 | % 14 | % inputs: 15 | % log_fD_T = function handle of log(fD_T(t)), Loglikelihood 16 | % 17 | % fT = function handle of fT(t), Prior PDF 18 | % 19 | % sample_from_fT = handle to a function that samples from of fT(t), 20 | % Sampling rule function from Prior PDF 21 | % 22 | % nsamples = number of samples of fT_D, Posterior, to generate 23 | % 24 | % outputs: 25 | % samples_fT_D = samples of fT_D (N x D) = samples from Posterior 26 | % distribution 27 | % 28 | % log_fD = log(evidence) = log(normalization constant) 29 | 30 | % ------------------------------------------------------------------------ 31 | % who when observations 32 | %-------------------------------------------------------------------------- 33 | % Diego Andres Alvarez Jul-24-2013 First algorithm 34 | %-------------------------------------------------------------------------- 35 | % Diego Andres Alvarez - daalvarez@unal.edu.co 36 | % Edoardo Patelli - edoardo.patelli@strath.ac.uk 37 | 38 | % parse the information in the name/value pairs: 39 | pnames = {'nsamples','loglikelihood','priorpdf','priorrnd','burnin','lastburnin','beta'}; 40 | 41 | dflts = {[],[],[],[],0,0,0.2}; % define default values 42 | 43 | [nsamples,loglikelihood,priorpdf,prior_rnd,burnin,lastBurnin,beta] = ... 44 | internal.stats.parseArgs(pnames, dflts, varargin{:}); 45 | 46 | %% Obtain N samples from the prior pdf f(T) 47 | j = 0; % Initialise loop for the transitional likelihood 48 | thetaj = prior_rnd(nsamples); % theta0 = N x D 49 | pj = 0; % p0 = 0 (initial tempering parameter) 50 | Dimensions = size(thetaj, 2); % size of the vector theta 51 | 52 | count = 1; % Counter 53 | samps(:,:,count) = thetaj; 54 | beta_j(count) = pj; 55 | 56 | %% Initialization of matrices and vectors 57 | thetaj1 = zeros(nsamples, Dimensions); 58 | 59 | %% Main loop 60 | while pj < 1 61 | j = j+1; 62 | 63 | %% Calculate the tempering parameter p(j+1): 64 | for l = 1:nsamples 65 | log_fD_T_thetaj(l) = loglikelihood(thetaj(l,:)); 66 | end 67 | if any(isinf(log_fD_T_thetaj)) 68 | error('The prior distribution is too far from the true region'); 69 | end 70 | pj1 = calculate_pj1(nsamples, log_fD_T_thetaj, pj); 71 | fprintf('TMCMC: Iteration j = %2d, pj1 = %f\n', j, pj1); 72 | 73 | %% Compute the plausibility weight for each sample wrt f_{j+1} 74 | fprintf('Computing the weights ...\n'); 75 | a = (pj1-pj)*log_fD_T_thetaj; 76 | wj = exp(a); 77 | wj_norm = wj./sum(wj); % normalization of the weights 78 | 79 | %% Compute S(j) = E[w{j}] (eq 15) 80 | S(j) = mean(wj); 81 | 82 | %% Do the resampling step to obtain N samples from f_{j+1}(theta) and 83 | % then perform Metropolis-Hastings on each of these samples using as a 84 | % stationary PDF "fj1" 85 | % fj1 = @(t) fT(t).*log_fD_T(t).^pj1; % stationary PDF (eq 11) f_{j+1}(theta) 86 | log_posterior = @(t) log(priorpdf(t)) + pj1*loglikelihood(t); 87 | 88 | 89 | % and using as proposal PDF a Gaussian centered at thetaj(idx,:) and 90 | % with covariance matrix equal to an scaled version of the covariance 91 | % matrix of fj1: 92 | 93 | % weighted mean 94 | mu = zeros(1, Dimensions); 95 | for l = 1:nsamples 96 | mu = mu + wj_norm(l)*thetaj(l,:); % 1 x N 97 | end 98 | 99 | % scaled covariance matrix of fj1 (eq 17) 100 | cov_gauss = zeros(Dimensions); 101 | for k = 1:nsamples 102 | % this formula is slightly different to eq 17 (the transpose) 103 | % because of the size of the vectors)m and because Ching and Chen 104 | % forgot to normalize the weight wj: 105 | tk_mu = thetaj(k,:) - mu; 106 | cov_gauss = cov_gauss + wj_norm(k)*(tk_mu'*tk_mu); 107 | end 108 | cov_gauss = beta^2 * cov_gauss; 109 | assert(~isinf(cond(cov_gauss)),'Something is wrong with the likelihood.') 110 | 111 | % Define the Proposal distribution: 112 | proppdf = @(x,y) prop_pdf(x, y, cov_gauss, priorpdf); %q(x,y) = q(x|y). 113 | proprnd = @(x) prop_rnd(x, cov_gauss, priorpdf); 114 | 115 | %% During the last iteration we require to do a better burnin in order 116 | % to guarantee the quality of the samples: 117 | if pj1 == 1 118 | burnin = lastBurnin; 119 | end 120 | 121 | %% Start N different Markov chains 122 | fprintf('Markov chains ...\n\n'); 123 | idx = randsample(nsamples, nsamples, true, wj_norm); 124 | 125 | for i = 1:nsamples % For parallel, type: parfor 126 | %% Sample one point with probability wj_norm 127 | 128 | % smpl = mhsample(start, nsamples, 129 | % 'pdf', pdf, 'proppdf', proppdf, 'proprnd', proprnd); 130 | % start = row vector containing the start value of the Markov Chain, 131 | % nsamples = number of samples to be generated 132 | [thetaj1(i,:), acceptance_rate(i)] = mhsample(thetaj(idx(i), :), 1, ... 133 | 'logpdf', log_posterior, ... 134 | 'proppdf', proppdf, ... 135 | 'proprnd', proprnd, ... 136 | 'thin', 3, ... 137 | 'burnin', burnin); 138 | % According to Cheung and Beck (2009) - Bayesian model updating ..., 139 | % the initial samples from reweighting and the resample of samples of 140 | % fj, in general, do not exactly follow fj1, so that the Markov 141 | % chains must "burn-in" before samples follow fj1, requiring a large 142 | % amount of samples to be generated for each level. 143 | 144 | %% Adjust the acceptance rate (optimal = 23%) 145 | % See: http://www.dms.umontreal.ca/~bedard/Beyond_234.pdf 146 | %{ 147 | if acceptance_rate < 0.3 148 | % Many rejections means an inefficient chain (wasted computation 149 | %time), decrease the variance 150 | beta = 0.99*beta; 151 | elseif acceptance_rate > 0.5 152 | % High acceptance rate: Proposed jumps are very close to current 153 | % location, increase the variance 154 | beta = 1.01*beta; 155 | end 156 | %} 157 | end 158 | fprintf('\n'); 159 | acceptance(count) = mean(acceptance_rate); 160 | %% Prepare for the next iteration 161 | count = count+1; 162 | samps(:,:,count) = thetaj1; 163 | thetaj = thetaj1; 164 | pj = pj1; 165 | beta_j(count) = pj; 166 | end 167 | 168 | % estimation of f(D) -- this is the normalization constant in Bayes 169 | log_fD = sum(log(S(1:j))); 170 | 171 | %% Description of outputs: 172 | 173 | output.allsamples = samps; % To show samples from all transitional distributions 174 | output.samples = samps(:,:,end); % To only show samples from the final posterior 175 | output.log_evidence = log_fD; % To generate the logarithmic of the evidence 176 | output.acceptance = acceptance; % To show the mean acceptance rates for all iterations 177 | output.beta = beta_j; % To show the values of temepring parameters, beta_j 178 | 179 | return; % End 180 | 181 | 182 | %% Calculate the tempering parameter p(j+1) 183 | function pj1 = calculate_pj1(nsamples, log_fD_T_thetaj, pj) 184 | % find pj1 such that the Effective Sample Size (ESS) equals N/2: 185 | % 186 | % 1 187 | % ------------------ >= threshold 188 | % sum[(hat_wj)^2] 189 | % 190 | % here: 191 | % hat_wj = wj/sum(wj) 192 | % size(thetaj) = N x D, 193 | % wj = fD_T(thetaj).^(pj1 - pj) 194 | % e = pj1 - pj 195 | 196 | threshold = nsamples; 197 | 198 | % Note the following trick in order to calculate e: 199 | % Take into account that e>=0 200 | wj = @(e) exp(abs(e)*log_fD_T_thetaj); % N x 1 201 | fmin = @(e) (sum(wj(e)))^2 - (threshold .* sum((wj(e)).^2)) + realmin; 202 | e = abs(fzero(fmin, 0)); % e is >= 0, and fmin is an even function 203 | if isnan(e) 204 | error('There is an error finding e'); 205 | end 206 | 207 | pj1 = min(1, pj + e); 208 | 209 | return; % End 210 | 211 | function proppdf = prop_pdf(x, mu, covmat, box) 212 | % This is the Proposal PDF for the Markov Chain. 213 | 214 | % Box function is the Prior PDF in the feasible region. 215 | % So if a point is out of bounds, this function will 216 | % return 0. 217 | 218 | proppdf = mvnpdf(x, mu, covmat).*box(x); %q(x,y) = q(x|y). 219 | 220 | return; 221 | 222 | 223 | function proprnd = prop_rnd(mu, covmat, box) 224 | % Sampling from the proposal PDF for the Markov Chain. 225 | 226 | while true 227 | proprnd = mvnrnd(mu, covmat, 1); 228 | if box(proprnd) 229 | % The box function is the Prior PDF in the feasible region. 230 | % If a point is out of bounds, this function will return 0 = false. 231 | break; 232 | end 233 | end 234 | 235 | return 236 | -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/angleCalc.m: -------------------------------------------------------------------------------- 1 | function theta = angleCalc(S,C,out_mode) 2 | %% This function computes the angle from sin and cos values (-180,180] degree. 3 | % 4 | % Usage: 5 | % theta = angleCalc(S,C,out_mode) 6 | % 7 | % Input: 8 | % S: Sine value of the angle 9 | % C: Cosine value of the angle 10 | % out_mode: 'deg' OR 'rad' 11 | % Note: default output mode is in degree 12 | % 13 | % Output: 14 | % theta: Angles in degrees or radians. 15 | % 16 | % Example: 17 | % theta = angleCalc(sin(-2*pi/3),cos(-2*pi/3)) 18 | % theta = -120; 19 | % theta= angleCalc(sin(2*pi/3),cos(2*pi/3),'rad') 20 | % theta= 2.0944 [rad] 21 | % --------------Disi A Jun 25, 2013 22 | %% Define the function: 23 | if nargin < 3 24 | out_mode='deg'; 25 | end 26 | 27 | if strcmp(out_mode,'deg') 28 | cons = 180/pi; 29 | else 30 | cons = 1; 31 | end 32 | 33 | for i = 1:length(S) 34 | theta(i) = asin(S(i)); 35 | if C(i) < 0 36 | if S(i) > 0 37 | theta(i) = pi - theta(i); 38 | elseif S(i) < 0 39 | theta(i) = - pi - theta(i); 40 | else % If S(i) = 0 41 | theta(i) = theta(i) + pi; 42 | end 43 | end 44 | 45 | theta(i) = theta(i) .* cons; 46 | end 47 | end -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/areaMe.m: -------------------------------------------------------------------------------- 1 | function [outputArg1] = areaMe(D1,D2) 2 | %AREAME Computes the area between two ECDFs 3 | % It does not work with a single datum. 4 | % 5 | % . 6 | % . by The Liverpool Git Pushers 7 | if length(D1)>length(D2) 8 | d1 = D2(:); 9 | d2 = D1(:); 10 | else 11 | d1 = D1(:); 12 | d2 = D2(:); 13 | end 14 | [Pxs,xs] = ecdf_Lpool(d1); % Compute the ecdf of the data sets 15 | [Pys,ys] = ecdf_Lpool(d2); 16 | Pys_eqx = Pxs; 17 | Pys_pure = Pys(2:end-1); % this does not work with a single datum 18 | Pall = sort([Pys_eqx;Pys_pure]); 19 | ys_eq_all = zeros(length(Pall),1); 20 | ys_eq_all(1)=ys(1); 21 | ys_eq_all(end)=ys(end); 22 | for k=2:length(Pall)-1 23 | ys_eq_all(k,1) = interpCDF_2(ys,Pys,Pall(k)); 24 | end 25 | xs_eq_all = zeros(length(Pall),1); 26 | xs_eq_all(1)=xs(1); 27 | xs_eq_all(end)=xs(end); 28 | for k=2:length(Pall)-1 29 | xs_eq_all(k,1) = interpCDF_2(xs,Pxs,Pall(k)); 30 | end 31 | diff_all_s = abs(ys_eq_all-xs_eq_all); 32 | diff_all_s = diff_all_s(2:end); 33 | diff_all_p = diff(Pall); 34 | area = diff_all_s' * diff_all_p; 35 | outputArg1 = area; 36 | end 37 | 38 | 39 | function [outputArg1] = interpCDF_2(xd,yd,pvalue) 40 | %INTERPCDF Summary of this function goes here 41 | % Detailed explanation goes here 42 | % 43 | % . 44 | % . by The Liverpool Git Pushers 45 | 46 | % [yd,xd]=ecdf_Lpool(data); 47 | beforr = diff(pvalue <= yd) == 1; % && diff(0.5>pv) == -1; 48 | beforrr = [0;beforr(:)]; 49 | if pvalue==0 50 | xvalue = xd(1); 51 | else 52 | xvalue = xd(beforrr==1); 53 | end 54 | outputArg1 = xvalue; 55 | end 56 | 57 | 58 | function [ps,xs] = ecdf_Lpool(x) 59 | 60 | xs = sort(x); 61 | xs = [xs(1);xs(:)]; 62 | ps = linspace(0,1,length(xs))'; 63 | 64 | end -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/blackbox_model.m: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Adolphus8/Bayesian-Model-Updating-Tutorials/f88cbbc1bac5e4caab6f5fd620d76a25c50defa5/Alternative_TMCMC_Transition_Criteria/blackbox_model.m -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/example_SDOF_System_Coulomb_Friction_numerical.m: -------------------------------------------------------------------------------- 1 | %% Numerical Example: SDOF System with Coulomb Friction 2 | % 3 | % This set-up is based on the Single Degree-of-Freedom Dynamical System 4 | % subjected to Coulomb Friction Force that is presented in the literature: 5 | % 6 | % L. Marino and A. Cicirello (2020). Experimental investigation of a single- 7 | % degree-of-freedom system with Coulomb friction. Nonlinear Dynamics, 99(3), 8 | % 1781-1799. doi: 10.1007/s11071-019-05443-2 9 | % 10 | % This set-up is applicable only for the Base motion (with fixed wall) case. 11 | %-------------------------------------------------------------------------% 12 | %% Load Numerical Data: 13 | clc; clear; 14 | load('noisy_data'); 15 | 16 | %% Define key parameters: 17 | omega_n = 3.086; % Natural frequency of the structure [Hz] 18 | driving_force = 2.5; % Driving force amplitude by the rotor [N] 19 | r_nom = data.frequency_ratio; % Nominal values of the dimensionless input frequency ratios 20 | coulomb_force = flip(driving_force .* data.force_ratio); % Nominal values of the Coulomb Frictions [N] 21 | 22 | %% Define the data-set: 23 | Ndata = 15; % Data size per time-step 24 | phase_angle_analytical = zeros(Ndata,1); 25 | phase_angle_analytical(:,1) = data.phase_angle.f010; 26 | phase_angle_analytical(:,2) = data.phase_angle.f025; 27 | phase_angle_analytical(:,3) = data.phase_angle.f040; 28 | phase_angle_analytical(:,4) = data.phase_angle.f055; 29 | 30 | % Consolidate the "noisy" Phase angle data: 31 | sigma_phi = 2; % True value of measurement noise for Phase angles 32 | % phase_angle_noisy = phase_angle_analytical + sigma_phi.*randn(Ndata, length(data.force_ratio)); 33 | phase_angle_noisy = data.noisy_phase_angles; 34 | data_phase_angle = phase_angle_noisy(2:11,:); % Take only 10 data for Bayesian model updating 35 | 36 | % Consolidate the "noisy" input Frequency data: 37 | sigma_r = 0.01; % True value of measurement noise for Frequency ratio 38 | r_v = data.noisy_frequency_ratios; 39 | frequency_data = r_v .* omega_n; 40 | data_frequency = frequency_data(2:11,:); % Take only 10 data for Bayesian model updating 41 | 42 | % Generate Analytical solution through Den-Hartog's solution: 43 | beta_nom = data.force_ratio; % Nominal force ratio 44 | output = DenHartogHarmonic(beta_nom'); 45 | r_an = output.frequency_ratios; % The output frequency ratios 46 | phase_an = output.phase_angles; % The output analytical phase angles 47 | phase_bound = output.phase_bound; % The phase angle bound defined by Den-Hartog's Boundary 48 | 49 | %% To plot the Phase angles vs Frequency ratio curves: 50 | colors = [0 0 1; 0 0.5 0; 1 0 0; 1 0 1]; 51 | 52 | figure; 53 | hold on; box on; grid on; 54 | for ib = 3 % To plot for different Friction Force ratio 55 | % Numerical scatterplot: 56 | plot([r_v(:,ib)], phase_angle_noisy(:,ib), 'o', 'color', colors(ib,:), 'linewidth', 2); 57 | % Analytical plot: 58 | plot([r_an], phase_an(ib+1,:), '--', 'color', colors(ib,:), 'linewidth', 1); 59 | end 60 | plot([r_an], phase_bound,'-- k'); 61 | legend(['Data for F_{\mu} = ',num2str(1.0864, '%.3f'), ' N'],... 62 | 'Analytical solution', 'Den-Hartog''s Boundary', 'linewidth', 2, 'location', 'southeast'); 63 | xlabel('$r$', 'Interpreter', 'latex'); ylabel('$\phi$ $[deg]$', 'Interpreter', 'latex'); 64 | xlim([0, 2]); ylim([0 180]); set(gca,'FontSize',20); 65 | 66 | %% Bayesian Model Updating Set-up: 67 | % The epistemic parameters to be inferred are the following: 68 | % {Coulomb Force, Natural Frequency, Frequency Ratio Noise, Phase Angle Noise} 69 | 70 | % Define the Prior distribution: 71 | lowerbound = [0.01, 0.001, 0.001]; upperbound = [10, 10, 1]; 72 | prior_coulomb = @(x) unifpdf(x, lowerbound(1), upperbound(1)); % Prior for Coulomb Friction 73 | prior_omega = @(x) unifpdf(x, lowerbound(2), upperbound(2)); % Prior for Natural Frequency 74 | prior_sigma_phi = @(x) unifpdf(x, lowerbound(2), upperbound(2)); % Prior for Phase Angle Noise 75 | prior_sigma_r = @(x) unifpdf(x, lowerbound(3), upperbound(3)); % Prior for Frequency Ratio Noise 76 | 77 | prior_pdf = @(x) prior_coulomb(x(:,1)) .* prior_omega(x(:,2)) .* ... 78 | prior_sigma_phi(x(:,3)) .* prior_sigma_r(x(:,4)); 79 | prior_rnd = @(N) [unifrnd(lowerbound(1), upperbound(1), N, 1), ... 80 | unifrnd(lowerbound(2), upperbound(2), N, 1), ... 81 | unifrnd(lowerbound(2), upperbound(2), N, 1), ... 82 | unifrnd(lowerbound(3), upperbound(3), N, 1)]; 83 | 84 | % Define the loglikelihood function: 85 | model = @(x,f) blackbox_model(x, f, driving_force); 86 | t = 3; 87 | logL = @(x) loglikelihood(x, model, data_phase_angle(:,t), data_frequency(:,t), r_v(2:11), r_nom(2:11)); 88 | 89 | %% Define Bayesian Model Updating Parameters: 90 | 91 | Nsamples = 1000; % No. of samples to generate from the Posterior 92 | Nbatch = 1; % No. of sample runs to perform 93 | Ncores = 12; 94 | TMCMC1 = cell(Nbatch,1); TMCMC2 = cell(Nbatch,1); 95 | timeTMCMC1 = zeros(Nbatch,1); timeTMCMC2 = zeros(Nbatch,1); 96 | 97 | %% Perform Bayesian Model Updating via TMCMC and TMCMC-II: 98 | 99 | % Initiate the samplers: 100 | 101 | parpool(Ncores) 102 | parfor r = 1:Nbatch 103 | 104 | fprintf('Batch no.: %d \n',r) 105 | 106 | tic; 107 | TMCMC1{r,1} = TMCMCsampler('nsamples',Nsamples,'loglikelihood',logL,'priorpdf',prior_pdf,... 108 | 'priorrnd',prior_rnd,'burnin',0,'lastburnin',0); 109 | timeTMCMC1(r,1) = toc; 110 | fprintf('Time elapsed is for the TMCMC sampler: %f \n',timeTMCMC1(r,1)) 111 | 112 | tic; 113 | TMCMC2{r,1} = TMCMCsampler2('nsamples',Nsamples,'loglikelihood',logL,'priorpdf',prior_pdf,... 114 | 'priorrnd',prior_rnd,'burnin',0,'lastburnin',0); 115 | timeTMCMC2(r,1) = toc; 116 | fprintf('Time elapsed is for the TMCMC-II sampler: %f \n',timeTMCMC2(r,1)) 117 | end 118 | %% Save the data: 119 | save('ESREL2023') 120 | 121 | %% 122 | %{ 123 | TMCMC1_cell = cell(50,1); TMCMC2_cell = cell(50,1); 124 | timeTMCMC1_vec = zeros(50,1); timeTMCMC2_vec = zeros(50,1); 125 | 126 | for i = 1:22 127 | load('ESREL2023_1.mat', 'TMCMC1', 'TMCMC2', 'timeTMCMC1', 'timeTMCMC2') 128 | TMCMC1_cell{i} = TMCMC1{i}; TMCMC2_cell{i} = TMCMC2{i}; 129 | timeTMCMC1_vec(i) = timeTMCMC1(i); timeTMCMC2_vec(i) = timeTMCMC2(i); 130 | end 131 | 132 | for i = 23:30 133 | load('ESREL2023_2.mat', 'TMCMC1', 'TMCMC2', 'timeTMCMC1', 'timeTMCMC2') 134 | TMCMC1_cell{i} = TMCMC1{i-22}; TMCMC2_cell{i} = TMCMC2{i-22}; 135 | timeTMCMC1_vec(i) = timeTMCMC1(i-22); timeTMCMC2_vec(i) = timeTMCMC2(i-22); 136 | end 137 | 138 | for i = 31:40 139 | load('ESREL2023_3.mat', 'TMCMC1', 'TMCMC2', 'timeTMCMC1', 'timeTMCMC2') 140 | TMCMC1_cell{i} = TMCMC1{i-30}; TMCMC2_cell{i} = TMCMC2{i-30}; 141 | timeTMCMC1_vec(i) = timeTMCMC1(i-30); timeTMCMC2_vec(i) = timeTMCMC2(i-30); 142 | end 143 | 144 | for i = 41:50 145 | load('ESREL2023_4.mat', 'TMCMC1', 'TMCMC2', 'timeTMCMC1', 'timeTMCMC2') 146 | TMCMC1_cell{i} = TMCMC1{i-40}; TMCMC2_cell{i} = TMCMC2{i-40}; 147 | timeTMCMC1_vec(i) = timeTMCMC1(i-40); timeTMCMC2_vec(i) = timeTMCMC2(i-40); 148 | end 149 | 150 | TMCMC1 = TMCMC1_cell; TMCMC2 = TMCMC2_cell; timeTMCMC1 = timeTMCMC1_vec; timeTMCMC2 = timeTMCMC2_vec; 151 | save('ESREL2023.mat', 'TMCMC1', 'TMCMC2', 'timeTMCMC1', 'timeTMCMC2') 152 | %} 153 | 154 | %% P-box Analysis: 155 | load('ESREL2023') 156 | 157 | sampsTMCMC1 = zeros(Nsamples, length(TMCMC1),2); sampsTMCMC2 = zeros(Nsamples, length(TMCMC2),2); 158 | pboxTMCMC1 = zeros(Nsamples,2,2); pboxTMCMC2 = zeros(Nsamples,2,2); 159 | for i = 1:length(TMCMC1) 160 | cell1 = TMCMC1{i}; cell2 = TMCMC2{i}; 161 | samps1 = cell1.samples; samps2 = cell2.samples; 162 | sampsTMCMC1(:,i,1) = sort(samps1(:,1), 'ascend'); sampsTMCMC1(:,i,2) = sort(samps1(:,2), 'ascend'); 163 | sampsTMCMC2(:,i,1) = sort(samps2(:,1), 'ascend'); sampsTMCMC2(:,i,2) = sort(samps2(:,2), 'ascend'); 164 | end 165 | 166 | for j = 1:Nsamples 167 | pboxTMCMC1(j,:,1) = [min(sampsTMCMC1(j,:,1)),max(sampsTMCMC1(j,:,1))]; pboxTMCMC1(j,:,2) = [min(sampsTMCMC1(j,:,2)),max(sampsTMCMC1(j,:,2))]; 168 | pboxTMCMC2(j,:,1) = [min(sampsTMCMC2(j,:,1)),max(sampsTMCMC2(j,:,1))]; pboxTMCMC2(j,:,2) = [min(sampsTMCMC2(j,:,2)),max(sampsTMCMC2(j,:,2))]; 169 | end 170 | 171 | % Pbox of estimates: 172 | figure; 173 | subplot(1,2,1) 174 | hold on; box on; grid on; 175 | [f1,x1] = ecdf(pboxTMCMC1(:,1,1)); [f2,x2] = ecdf(pboxTMCMC1(:,2,1)); 176 | stairs(x1,f1, 'b', 'linewidth', 2); stairs(x2,f2, 'b', 'linewidth', 2, 'handlevisibility', 'off'); 177 | plot([min(x1),min(x2)],[0,0], 'b', 'linewidth', 2, 'handlevisibility', 'off'); 178 | plot([max(x1),max(x2)],[1,1], 'b', 'linewidth', 2, 'handlevisibility', 'off'); 179 | 180 | [f1,x1] = ecdf(pboxTMCMC2(:,1,1)); [f2,x2] = ecdf(pboxTMCMC2(:,2,1)); 181 | stairs(x1,f1, 'r', 'linewidth', 2); stairs(x2,f2, 'r', 'linewidth', 2, 'handlevisibility', 'off'); 182 | plot([min(x1),min(x2)],[0,0], 'r', 'linewidth', 2, 'handlevisibility', 'off'); 183 | plot([max(x1),max(x2)],[1,1], 'r', 'linewidth', 2, 'handlevisibility', 'off'); 184 | xline(1.0864, 'k--', 'linewidth', 2); 185 | legend('P-box TMCMC', 'P-box TMCMC-II', 'True value F_{\mu} = 1.086 [N]', 'linewidth', 2) 186 | xlabel('$F_{\mu}$ $[N]$', 'Interpreter', 'latex'); ylabel('ECDF value'); set(gca, 'Fontsize', 18) 187 | xlim([0.7, 1.8]) 188 | 189 | subplot(1,2,2) 190 | hold on; box on; grid on; 191 | [f1,x1] = ecdf(pboxTMCMC1(:,1,2)); [f2,x2] = ecdf(pboxTMCMC1(:,2,2)); 192 | stairs(x1,f1, 'b', 'linewidth', 2); stairs(x2,f2, 'b', 'linewidth', 2, 'handlevisibility', 'off'); 193 | plot([min(x1),min(x2)],[0,0], 'b', 'linewidth', 2, 'handlevisibility', 'off'); 194 | plot([max(x1),max(x2)],[1,1], 'b', 'linewidth', 2, 'handlevisibility', 'off'); 195 | 196 | [f1,x1] = ecdf(pboxTMCMC2(:,1,2)); [f2,x2] = ecdf(pboxTMCMC2(:,2,2)); 197 | stairs(x1,f1, 'r', 'linewidth', 2); stairs(x2,f2, 'r', 'linewidth', 2, 'handlevisibility', 'off'); 198 | plot([min(x1),min(x2)],[0,0], 'r', 'linewidth', 2, 'handlevisibility', 'off'); 199 | plot([max(x1),max(x2)],[1,1], 'r', 'linewidth', 2, 'handlevisibility', 'off'); 200 | xline(3.086, 'k--', 'linewidth', 2); 201 | legend('P-box TMCMC', 'P-box TMCMC-II', 'True value \omega_n = 3.086 [Hz]', 'linewidth', 2) 202 | xlabel('$\omega_n$ $[Hz]$', 'Interpreter', 'latex'); ylabel('ECDF value'); set(gca, 'Fontsize', 18) 203 | 204 | area_mat = zeros(2,2); 205 | area_mat(1,1) = areaMe(pboxTMCMC1(:,1,1),pboxTMCMC1(:,2,1)); area_mat(1,2) = areaMe(pboxTMCMC1(:,1,2),pboxTMCMC1(:,2,2)); 206 | area_mat(2,1) = areaMe(pboxTMCMC2(:,1,1),pboxTMCMC2(:,2,1)); area_mat(2,2) = areaMe(pboxTMCMC2(:,1,2),pboxTMCMC2(:,2,2)); 207 | T = array2table(area_mat,'VariableNames', ... 208 | {'Coulomb_Friction_Pbox_area', 'Natural_Frequency_Pbox_area'},... 209 | 'RowNames', {'TMCMC', 'TMCMC-II'}); 210 | 211 | area_vec = [0.081508, 0.17135; 0.008878, 0.0089148]; 212 | x = categorical({'F_{\mu} [N]', '\omega_{n} [Hz]'}); x = reordercats(x,{'F_{\mu} [N]', '\omega_{n} [Hz]'}); 213 | 214 | figure; 215 | hold on; box on; grid on; 216 | b = bar(x, area_vec); 217 | xlabel('Parameter'); ylabel('Area of P-box'); 218 | set(gca, 'Fontsize', 18); legend('TMCMC', 'TMCMC-II', 'linewidth', 2) 219 | 220 | %% Mean Analysis: 221 | 222 | meanTMCMC1 = zeros(length(TMCMC1),2); meanTMCMC2 = zeros(length(TMCMC2),2); 223 | betaTMCMC1 = zeros(length(TMCMC1),1); betaTMCMC2 = zeros(length(TMCMC2),1); 224 | 225 | for i = 1:length(TMCMC1) 226 | cell1 = TMCMC1{i}; cell2 = TMCMC2{i}; 227 | samps1 = cell1.samples; samps2 = cell2.samples; 228 | meanTMCMC1(i,:) = mean(samps1(:,1:2)); meanTMCMC2(i,:) = mean(samps2(:,1:2)); 229 | betaTMCMC1(i) = length(cell1.beta)-1; betaTMCMC2(i) = length(cell2.beta)-1; 230 | end 231 | 232 | figure; 233 | subplot(2,2,1) 234 | hold on; box on; grid on; 235 | histogram(meanTMCMC1(:,1)) 236 | xlabel('$F_{\mu}$ $[N]$', 'Interpreter', 'latex'); ylabel('Count'); set(gca, 'Fontsize', 18) 237 | title('Mean values TMCMC') 238 | subplot(2,2,2) 239 | hold on; box on; grid on; 240 | histogram(meanTMCMC1(:,2)) 241 | xlabel('$\omega_n$ $[Hz]$', 'Interpreter', 'latex'); ylabel('Count'); set(gca, 'Fontsize', 18) 242 | title('Mean values TMCMC') 243 | 244 | subplot(2,2,3) 245 | hold on; box on; grid on; 246 | histogram(meanTMCMC2(:,1)) 247 | xlabel('$F_{\mu}$ $[N]$', 'Interpreter', 'latex'); ylabel('Count'); set(gca, 'Fontsize', 18) 248 | title('Mean values TMCMC2') 249 | subplot(2,2,4) 250 | hold on; box on; grid on; 251 | histogram(meanTMCMC2(:,2)) 252 | xlabel('$\omega_n$ $[Hz]$', 'Interpreter', 'latex'); ylabel('Count'); set(gca, 'Fontsize', 18) 253 | title('Mean values TMCMC2') 254 | 255 | figure; 256 | hold on; box on; grid on; 257 | [f1,x1] = ecdf(betaTMCMC1); [f2,x2] = ecdf(betaTMCMC2); 258 | stairs(x1,f1, 'b', 'linewidth', 2); stairs(x2,f2, 'r', 'linewidth', 2); 259 | plot([11,12], [1,1], 'b', 'linewidth',2, 'handlevisibility', 'off') 260 | xticks([10:12]); xlabel('No. of Transition steps'); ylabel('ECDF values'); set(gca, 'Fontsize', 18) 261 | 262 | beta_vec = [6,5 ; 44,36 ; 0,9]; 263 | iterations = [9, 10, 11]; 264 | 265 | figure; 266 | hold on; box on; grid on; 267 | b = bar(iterations, beta_vec); 268 | xlabel('No. of Iterations'); ylabel('Count'); xticks([9:11]) 269 | set(gca, 'Fontsize', 18); legend('TMCMC', 'TMCMC-II', 'linewidth', 2) 270 | 271 | -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/loglikelihood.m: -------------------------------------------------------------------------------- 1 | function logL = loglikelihood(theta, blackbox_model, phase_angle_data, frequency_data, frequency_ratio_exp, frequency_ratio_nom) 2 | %% Function-handle of the Loglikelihood function: 3 | % This function-handle computes the log-likelihood values. 4 | %-------------------------------------------------------------------------% 5 | % 6 | % Inputs: 7 | % theta: N x 4 input matrix of the epistemic parameters whereby: 8 | % - theta(:,1): First dimension is that of the time-varying Coulomb Friction [N]; 9 | % - theta(:,2): Second dimension is that of the static Natural Frequency [Hz]; 10 | % - theta(:,3): Third dimension is that of the static noise for Phase Angle measurements [deg]; 11 | % - theta(:,4): Fourth dimension is that of the static noise for Frequency ratio measurements; 12 | % Note: N is the sample size, the number of theta to generate from the posterior. 13 | % 14 | % blackbox_model: Blackbox function-handle (function of theta) used for model evaluation; 15 | % phase_angle_data: N_e x 1 input vector of the phase angles measured from the experiment [deg]; 16 | % frequency_data: N_e x 1 input vector of the driving frequencies used for the experiment [rad/s]; 17 | % frequency_ratio_nom: N_e x 1 input vector of nominal frequency ratio; 18 | % frequency_ratio_exp: N_e x 1 input vector of experimental frequency ratio; 19 | % Note: N_e is the number of experimental data taken. 20 | % 21 | % Output: 22 | % logL: N x 1 vector of loglikelihood output values; 23 | % 24 | %-------------------------------------------------------------------------% 25 | %% Define the Function-handle: 26 | 27 | % Initiate the empty vector of logL: 28 | logL = zeros(size(theta,1),1); 29 | 30 | for i = 1:size(theta,1) 31 | %% Generate the model output: 32 | model_output = blackbox_model(theta(i,1), frequency_ratio_exp); 33 | 34 | % Generate N_e x 1 model output of the Phase angles: 35 | phase_angle_model = model_output.phase_angles; 36 | 37 | %% Compute the loglikelihood for Phase Angles: 38 | 39 | logL_phi = - 0.5 .* (1./(theta(i,3)).^2) .* sum((phase_angle_data - phase_angle_model).^2) - ... 40 | size(phase_angle_data,1).*log(sqrt(2*pi).*theta(i,3)); 41 | 42 | %% Compute the loglikelihood for Frequency Ratios: 43 | 44 | % Generate N_e x 1 model output of the frequency ratios: 45 | freq_ratio_model = frequency_data./theta(i,2); 46 | 47 | logL_r = - 0.5 .* (1./(theta(i,4)).^2) .* sum((frequency_ratio_nom - freq_ratio_model).^2) - ... 48 | size(frequency_data,1).*log(sqrt(2*pi).*theta(i,4)); 49 | 50 | %% Compute the overall loglikelihood value: 51 | 52 | logL(i) = logL_phi + logL_r; 53 | 54 | % Set logL(i) = -1e10 if logL is NaN or Inf: 55 | if isnan(logL(i)) || isinf(logL(i)) 56 | logL(i) = -1e10; 57 | end 58 | 59 | end 60 | end 61 | 62 | -------------------------------------------------------------------------------- /Alternative_TMCMC_Transition_Criteria/test: -------------------------------------------------------------------------------- 1 | 2 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Bayesian Model Updating Tutorials: 2 | 3 | Bayesian Model Updating is a technique which casts the model updating problem in the form of a Bayesian Inference. There have been 3 popular advanced Monte Carlo sampling techniques which are adopted by researchers to address Bayesian Model Updating problems and make the necessary estimations of the epistemic parameter(s). These 3 techniques are: 4 | 5 | * Markov Chain Monte Carlo [(MCMC)](https://doi.org/10.1093/biomet/57.1.97) 6 | * Transitional Markov Chain Monte Carlo [(TMCMC)](https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(816)) 7 | * Sequential Monte Carlo [(SMC)](https://www.jstor.org/stable/3879283) 8 | 9 | In this repository, 3 tutorials are presented to enable users to understand how the advanced Monte Carlo techniques are implemented in addressing various Bayesian Model Updating problems. The following tutorials are (in order of increasing difficulty): 10 | 11 | * 1-Dimensional Linear Static System 12 | * 1-Dimensional Simple Harmonic Oscillator 13 | * 2-Dimensional Eigenvalue Problem 14 | 15 | ## Tutorials: 16 | 17 | ### 1) 1-Dimensional Linear Static System: 18 | 19 | This tutorial presents a simple static Spring-Mass system. In this set-up, the spring is assumed to obey [Hooke's Law](http://latex.codecogs.com/svg.latex?F%3D-k%5Ccdot%7Bd%7D) model whereby the restoring force of the spring, F, is linearly proportional to the length of its displacement from rest length, d. The elasticity constant of the spring is k. This study, seeks to realize two objectives: 20 | 21 | 1. To compare the estimation the epistemic parameter k; 22 | 23 | 2. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC. 24 | 25 | ### 2) 1-Dimensional Simple Harmonic Oscillator: 26 | 27 | This tutorial presents a simple harmonic oscillator system. In this set-up, the natural oscillating frequency of the ocillator, F, obeys the [Simple Harmonic Frequency](http://latex.codecogs.com/svg.latex?F%3D%5Csqrt%7B%5Cfrac%7Bk%7D%7Bm%7D%7D) model whereby F is defined as the square-root of the ratio between the elasticity constant of the spring, k, and the mass of the body attached to the oscillator, m. This study, seeks to realize two objectives: 28 | 29 | 1. To compare the estimation the epistemic parameter k; 30 | 31 | 2. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC. 32 | 33 | ### 3) 2-Dimensional Eigenvalue Problem: 34 | 35 | This tutorial presents a 2-by-2 square [matrix](http://latex.codecogs.com/svg.latex?%5Cbegin%7Bpmatrix%7D%0D%0A%7B%5Ctheta_1%7D%2B%7B%5Ctheta_2%7D%26-%7B%5Ctheta_2%7D%5C%5C-%7B%5Ctheta_2%7D%26%7B%5Ctheta_2%7D%5C%5C%0D%0A%5Cend%7Bpmatrix%7D) in which there exists two distinct real eignvalue solutions. The matrix elements here are defined by two epistemic parameters: Theta 1 and Theta 2. This tutorial seeks to achieve three objectives: 36 | 37 | 1. To observe the performance of each of the advanced Monte Carlo samplers in obtaining samples from a 2-dimensional, bi-modal posterior distribution; 38 | 39 | 2. To estimate the solutions to the epistemic parameters: Theta 1 and Theta 2; 40 | 41 | 3. To compare the model updating results obtained through the use of MCMC, TMCMC, and SMC. 42 | 43 | ### 4) Alternative TMCMC Transition criteria: 44 | 45 | The work explores a possible alternative transitional criteria for the TMCMC sampler involving the use of the Effective sample size metric. The alternative transitional criteria is such that in transiting from one transitional distribution to another, the Effective sample size has to be half the total sample size. The alternative TMCMC sampler is referred to as the TMCMC-II sampler and it will be implemented on a SDoF structure subjected to an unknown Coulomb friction. The code to be executed is named: "example_SDOF_System_Coulomb_Friction_numerical.m" 46 | 47 | The work was presented at the 33rd European Safety and Reliability Conference (ESREL 2023) held in Southampton, United Kingdom. 48 | 49 | ## Reference(s): 50 | * A. Lye, A. Cicirello, and E. Patelli (2021). Sampling methods for solving Bayesian model updating problems: A tutorial. *Mechanical Systems and Signal Processing, 159*, 107760. doi: [10.1016/j.ymssp.2021.107760](https://doi.org/10.1016/j.ymssp.2021.107760) 51 | * A. Lye, and L. Marino (2023). An investigation into an alternative transition criterion of the Transitional Markov Chain Monte Carlo method for Bayesian model updating. *In Proceedings of the 33rd European Safety and Reliability Conference, 1*. doi: [10.3850/978-981-18-8071-1_P331-cd](https://doi.org/10.3850/978-981-18-8071-1_P331-cd) 52 | 53 | ## Author: 54 | * Name: Adolphus Lye 55 | * Contact: adolphus.lye@liverpool.ac.uk 56 | * Affiliation: Insitute for Risk and Uncertainty, University of Liverpool 57 | --------------------------------------------------------------------------------