Merge branch 'master' of git.fmrib.ox.ac.uk:fsl/pytreat-practicals-2020 into sean-migp

talks/matlab_vs_python/fitting/fit_model.m

+%% Play with model fitting
+
+% experimental parameters
+TEs = [10 40 50 60 80];
+TRs = [.8 1 1.5 2];
+[TEs,TRs] = meshgrid(TEs,TRs);
+TEs = TEs(:)'; TRs = TRs(:)';
+
+% forward model
+forward = @(p)( p(1)*exp(-p(3)*TEs).*(1-exp(-p(2)*TRs)));
+
+% simulate data
+
+true_p    = [100,1/.8,1/50];
+data      = forward(true_p);
+snr       = 50;
+noise_std = 100/snr;
+noise     = randn(size(data))*noise_std;
+data      = data+noise;
+
+plot(data)
+
+%%
+% cost function is mean squared error (MSE)
+cf = @(x)( mean( (forward(x)-data).^2 ) );
+
+% initial guess
+p0 = [200,1/1,1/70];
+
+
+% using fminsearch (Nelder-Mead)
+p  = fminsearch(@(x) cf(x),p0);
+
+% plot result
+figure,hold on
+plot(data,'.')
+plot(forward(p))
+
+%% The below uses fminunc, which allows morre flexibility 
+% (like choosing the algorithm or providing gradients and Hessians)
+
+options  = optimoptions('fminunc','Display','off','Algorithm','quasi-newton'); 
+
+[x,fval] = fminunc(cf,p0,options);
+
+figure,hold on
+plot(data,'.')
+plot(forward(x))
+

talks/matlab_vs_python/rbf/play_rbf.m

+%% Fitting Radial Basis Functions to some data
+
+% Generate a noisy sine wave and plot it
+x = linspace(0,10,100);
+y = sin(3*x) + randn(size(x))*.5;
+    
+figure,
+plot(x,y)
+
+%% Fit RBF
+
+% This defines a RBF atom function
+sig = 2;
+rbf = @(x,c)(exp(-(x-c).^2/sig^2));
+
+% design matrix for fitting
+xi     = linspace(0,10,20);
+desmat = zeros(length(x),length(xi));
+for i=1:length(xi)
+    % each column is an RBF centered around xi
+    desmat(:,i) = rbf(x,xi(i));
+end
+% fit model
+beta = desmat\y';
+
+% plot fit
+figure(1),hold on
+plot(x,y,'.','markersize',10)
+h = plot(x,desmat,'k'); for i =1:20;h(i).Color=[0,0,0,0.2];end
+plot(x,desmat*beta,'-','linewidth',2)
+% make it a little prettier
+grid on
+xlabel('x')
+ylabel('y')
+set(gca,'fontsize',16)
+title('RBF fitting')
+
+%% save figure
+print -depsc ~/Desktop/RBF.eps 
+close(1)
+!open  ~/Desktop/RBF.eps
+