├── emptyfile.py ├── README ├── GPPOD.py ├── checkgrad.py ├── GP_kernel_experiments.py ├── GPDM.py ├── MLP.py ├── GPLVM.py ├── GP.py ├── kernels.py ├── PCA_EM.py └── COPYING /emptyfile.py: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /README: -------------------------------------------------------------------------------- 1 | pythonGPLVM 2 | 3 | Gaussian Process Latent Variable Modelling in python. 4 | 5 | Copyright 2009 James Hensman 6 | -------------------------------------------------------------------------------- /GPPOD.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | # 5 | # Gaussian Process Proper Orthogonal Decomposition. 6 | 7 | -------------------------------------------------------------------------------- /checkgrad.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | import numpy as np 5 | def checkgrad(f,fprime,x,step=1e-6, tolerance = 1e-4, *args): 6 | """check the gradient function fprime by comparing it to a numerical estiamte from the function f""" 7 | 8 | #choose a random direction to step in: 9 | dx = step*np.sign(np.random.uniform(-1,1,x.shape)) 10 | 11 | #evaulate around the point x 12 | f1 = f(x+dx,*args) 13 | f2 = f(x-dx,*args) 14 | 15 | numerical_gradient = (f1-f2)/(2*dx) 16 | gradient = fprime(x,*args) 17 | ratio = (f1-f2)/(2*np.dot(dx,gradient)) 18 | print "gradient = ",gradient 19 | print "numerical gradient = ",numerical_gradient 20 | print "ratio = ", ratio, '\n' 21 | 22 | if np.abs(1-ratio)>tolerance: 23 | print "Ratio far from unity. Testing individual gradients" 24 | for i in range(len(x)): 25 | dx = np.zeros(x.shape) 26 | dx[i] = step*np.sign(np.random.uniform(-1,1,x[i].shape)) 27 | 28 | f1 = f(x+dx,*args) 29 | f2 = f(x-dx,*args) 30 | 31 | numerical_gradient = (f1-f2)/(2*dx) 32 | gradient = fprime(x,*args) 33 | print i,"th element" 34 | #print "gradient = ",gradient 35 | #print "numerical gradient = ",numerical_gradient 36 | ratio = (f1-f2)/(2*np.dot(dx,gradient)) 37 | print "ratio = ",ratio,'\n' 38 | 39 | -------------------------------------------------------------------------------- /GP_kernel_experiments.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | 5 | from kernels import * 6 | 7 | def sample_1D(Nsamples=10): 8 | myRBF = RBF(1,.1) 9 | myLIN = linear(1,1) 10 | xlin = np.linspace(-10,10,100).reshape(100,1) 11 | K1 = myRBF(xlin,xlin)+np.eye(100)*1e-9 12 | Kchol1 = np.linalg.cholesky(K1) 13 | K2 = myLIN(xlin,xlin)+np.eye(100)*1e-9 14 | Kchol2 = np.linalg.cholesky(K2) 15 | samples1 = np.dot(np.random.randn(Nsamples,100),Kchol1.T) 16 | samples2 = np.dot(np.random.randn(Nsamples,100),Kchol2.T) 17 | pylab.figure() 18 | pylab.plot(xlin,samples1.T ) 19 | pylab.figure() 20 | pylab.plot(xlin,samples2.T ) 21 | pylab.figure() 22 | pylab.imshow(K1) 23 | pylab.figure() 24 | pylab.imshow(K2) 25 | 26 | 27 | def sample_2D(Nsamples=5): 28 | myRBF = RBF(1,.1) 29 | xx,yy = np.mgrid[-10:10:50j,-10:10:50j] 30 | XX = np.vstack((xx.flatten(),yy.flatten())).T 31 | K = myRBF(XX,XX) 32 | Kchol = np.linalg.cholesky(K+np.eye(2500,2500)*1e-9) 33 | samples = np.dot(np.random.randn(Nsamples,2500),Kchol.T) 34 | for sample in samples: 35 | pylab.figure() 36 | sample = sample.reshape(50,50) 37 | pylab.contourf(xx,yy,sample) 38 | pylab.figure() 39 | pylab.imshow(K) 40 | 41 | 42 | 43 | def sample_combined(Nsamples): 44 | mykernel = combined(1000,.1,.1,0.0) 45 | xx,yy = np.mgrid[-10:10:50j,-10:10:50j] 46 | XX = np.vstack((xx.flatten(),yy.flatten())).T 47 | K = mykernel(XX,XX) 48 | Kchol = np.linalg.cholesky(K+np.eye(2500,2500)*1e-9) 49 | samples = np.dot(np.random.randn(Nsamples,2500),Kchol.T) 50 | samples = [e.reshape(50,50) for e in samples] 51 | for sample in samples: 52 | pylab.figure() 53 | pylab.contourf(xx,yy,sample,40) 54 | pylab.figure() 55 | pylab.imshow(K) 56 | 57 | def sample_1D_poly(Nsamples=10): 58 | myPoly = polynomial(1,5) 59 | xlin = np.linspace(-2,2,100).reshape(100,1) 60 | K = myPoly(xlin,xlin)+np.eye(100)*1e-7 61 | Kchol = np.linalg.cholesky(K) 62 | samples = np.dot(np.random.randn(Nsamples,100),Kchol.T) 63 | pylab.figure() 64 | pylab.plot(xlin,samples.T ) 65 | pylab.figure() 66 | pylab.imshow(K) 67 | 68 | 69 | if __name__=="__main__": 70 | import numpy as np 71 | import pylab 72 | #sample_1D() 73 | #sample_2D() 74 | sample_1D_poly() 75 | #sample_combined(10) 76 | 77 | 78 | pylab.show() 79 | 80 | -------------------------------------------------------------------------------- /GPDM.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | # 5 | # Gaussian Process Dynamic Model 6 | import numpy as np 7 | import pylab 8 | from PCA_EM import PCA_EM 9 | import kernels 10 | import GP 11 | from scipy import optimize 12 | 13 | 14 | class GPDM: 15 | """ A Gaussian Process Dynamic Model. Two GPs are used - one for the observation mapping and one for the dynamic mapping. 16 | A particle filter is used for inference of the latent variables. 17 | 18 | In the EM-like procedure, the expected values of the latent variables are used for the GP data. This is obviously not strictly correct, but stochasic GP regression (i.e. with samples for the X data) is beyond me right now. """ 19 | def __init__(self,Y,dim,nparticles=100): 20 | self.Xdim = dim 21 | self.T,self.Ydim = Y.shape 22 | 23 | """Use PCA to initalise the problem. Uses EM version in this case...""" 24 | myPCA_EM = PCA_EM(Y,dim) 25 | myPCA_EM.learn(300) 26 | X = np.array(myPCA_EM.m_Z) 27 | 28 | self.observation_GP = GP.GP(X,Y) 29 | 30 | #create a linear kernel for the dynamics 31 | k = kernels.linear(-1,-1) 32 | self.dynamic_GP = GP.GP(X[:-1],X[1:],k) 33 | 34 | #initialise the samples from the state /latent variables 35 | self.particles = np.zeros((self.T,nparticles,self.Xdim)) 36 | 37 | #sample for x0 TODO: variable priors on X0 38 | self.particles[0,:,:] = np.random.randn(nparticles,self.Xdim) 39 | 40 | def filter(self,X,Y): 41 | """Inference of the state/latent variable using Sequential Monte Carlo. 42 | Currently uses a simple sample-importance-resample procedure. Could be update to include mcmc jumping or similar 43 | Takes arguments X and Y, so that is can be used for inference bith in learning (self.X,self.Y) and in use. 44 | 45 | X is a TxNxD array, T=time, N= number of particles, D= dimension of the state 46 | Y is a TxQ aray: T=time,Q is dimension of observed space""" 47 | 48 | weights = np.ones(X.shape[1]) 49 | for t in range(1,self.T): 50 | #sample 51 | mu,var = self.dynamic_GP.predict(X[t-1,:,:]) 52 | X[t,:,:] = np.random.randn(nparticles,self.Xdim)*np.sqrt(var) + mu # TODO check dimensions here? 53 | 54 | #importance 55 | ypred,predvar = self.observation_GP.predict(X[t,:,:]) 56 | weights = 0.5*np.sum(np.square(ypred-Y[t])/predvar) 57 | weights /= weights.sum() 58 | 59 | #resample 60 | # TODO: use some criteria as to whether resampling is necessary 61 | index = np.random.multinomial(X.shape[1],weights) 62 | X[t,:,:] = X[t,:,:].repeat(index,axis=0) 63 | return X 64 | 65 | def set_GP_params(self,params): 66 | """set the parameters of th two GPs to the passed values""" 67 | assert params.size == self.obsertation_GP.kernel.nparams_self.dynamic_GP.kernel.nparams +2, "Bad number fo parameters for setting" 68 | self.observation_GP.set_params(params[:self.observation_GP.kernel.nparams+1]) 69 | self.dynamic_GP.set_params(params[self.observation_GP.kernel.nparams+1:]) 70 | 71 | def get_GP__params(self): 72 | return np.hstack((self.observation_GP.get_params(),self.dynamic_GP.get_params())) 73 | 74 | def learn(self,iters): 75 | for i in range(iters): 76 | self.filter(self.particles,self.observation_GP.Y) 77 | # TODO reverse filter. 78 | EX = self.particles.mean(1) # the expected value of the latent variables. 79 | self.observation_GP.X = EX 80 | self.dynamic_GP.X,self.dynamic_GP.Y = EX[:-1],EX[1:] 81 | self.dynamic_GP.find_kernel_parameters() 82 | self.observation_GP.find_kernel_parameters() 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | -------------------------------------------------------------------------------- /MLP.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | import numpy as np 5 | from scipy import optimize 6 | 7 | class MLP: 8 | def __init__(self,structure,output='linear',alpha=1): 9 | assert len(structure) == 3 10 | self.structure = structure 11 | self.alpha = alpha # regulariser/prior on weights 12 | self.nweights = (structure[0]+1)*structure[1]+(structure[1]+1)*structure[2] 13 | 14 | self.initialise() 15 | 16 | if output=='linear': 17 | self.output_fn = lambda x:x 18 | self.error_fn = lambda y,t : 0.5*np.sum(np.square(y-t)) 19 | 20 | def initialise(self): 21 | """Initialise the weights of the network by sampling from the (Gaussian) prior""" 22 | nin,nhid,nout = self.structure 23 | s = 1./np.sqrt(self.alpha) 24 | self.w1 = np.random.randn(nin,nhid)*s 25 | self.b1 = np.random.randn(1,nhid)*s 26 | self.w2 = np.random.randn(nhid,nout)*s 27 | self.b2 = np.random.randn(1,nout)*s 28 | 29 | def forward(self,x): 30 | """calculate the outputs of the network given a set of inputs x""" 31 | n,d = x.shape 32 | assert d == self.structure[0], "Input dimension does not match this network" 33 | self.activations = np.tanh(np.dot(x,self.w1) + np.ones((n,1))*self.b1) 34 | self.out = np.dot(self.activations,self.w2) + np.ones((n,1))*self.b2 35 | return self.output_fn(self.out) 36 | 37 | def prior(self): 38 | """evaluate the current set of weights under the prior = return the log value""" 39 | return -0.5*self.alpha*np.sum(np.square(self.pack())) 40 | 41 | def prior_grad(self): 42 | """evaluate the gradient of the prior at the current set of weights""" 43 | return -self.alpha*self.pack() 44 | 45 | def gradient(self,weights,x,t): 46 | """used for training""" 47 | self.unpack(weights) 48 | y = self.forward(x) 49 | delta_out = y-t #gradient of the error wrt output, y 50 | return self.backpropagate(x,delta_out) - self.prior_grad() 51 | 52 | def backpropagate(self,x,delta_out): 53 | """'backpropagate' the gradeint of the error wrt to the output of the network to the gradient of the error wrt the weights 54 | Essentially doing de/dw = de/dy*dy/dw""" 55 | 56 | #Evaluate second-layer gradients. 57 | gw2 = np.dot(self.activations.T,delta_out) 58 | gb2 = np.sum(delta_out, 0) 59 | 60 | # Backpropagation to hidden layer. 61 | delta_hid = np.dot(delta_out,self.w2.T) 62 | delta_hid *= (1.0 - self.activations**2) 63 | 64 | # Finally, evaluate the first-layer gradients. 65 | gw1 = np.dot(x.T,delta_hid) 66 | gb1 = np.sum(delta_hid, 0) 67 | 68 | return np.hstack([e.flatten() for e in [gw1,gb1,gw2,gb2]]) 69 | 70 | def error(self,weights,x,t): 71 | "The thing to be optimised in training""" 72 | self.unpack(weights) 73 | y = self.forward(x) 74 | return self.error_fn(y,t) - self.prior() 75 | 76 | def train(self,x,t): 77 | w = optimize.fmin_cg(self.error,self.pack(),fprime=self.gradient,args=(x,t)) 78 | self.unpack(w) 79 | 80 | def unpack(self,weights): 81 | """take a np array and assign it to self.w1 etc.""" 82 | weights = weights.flatten() 83 | self.w1 = weights[:self.w1.size].reshape(self.w1.shape) 84 | self.b1 = weights[self.w1.size:self.w1.size+self.b1.size].reshape(self.b1.shape) 85 | self.w2 = weights[self.w1.size+self.b1.size:self.w1.size+self.b1.size+self.w2.size].reshape(self.w2.shape) 86 | self.b2 = weights[-self.b2.size:].reshape(self.b2.shape) 87 | 88 | def pack(self): 89 | """ 'Pack up' the weights and biases into a vector""" 90 | return np.hstack([e.flatten() for e in [self.w1,self.b1,self.w2,self.b2]]) 91 | 92 | 93 | 94 | 95 | if __name__=='__main__': 96 | import pylab 97 | 98 | N = 50 99 | x = np.random.randn(N,1) 100 | y = np.sin(2*x) + np.random.randn(N,1)*0.1 101 | 102 | xx = np.linspace(-2,2,100).reshape(100,1) 103 | myMLP = MLP((1,5,1),alpha=0.1) 104 | 105 | pylab.plot(x,y,'ro') 106 | pylab.plot(xx,myMLP.forward(xx)) 107 | 108 | w = optimize.fmin(myMLP.error,myMLP.pack(),args=(x,y)) 109 | myMLP.unpack(w) 110 | pylab.plot(xx,myMLP.forward(xx)) 111 | 112 | myMLP.initialise() 113 | w = optimize.fmin_cg(myMLP.error,myMLP.pack(),fprime=myMLP.gradient,args=(x,y)) 114 | myMLP.unpack(w) 115 | pylab.plot(xx,myMLP.forward(xx)) 116 | 117 | 118 | pylab.show() -------------------------------------------------------------------------------- /GPLVM.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | 5 | import numpy as np 6 | import pylab 7 | from PCA_EM import PCA_EM 8 | import kernels 9 | import GP 10 | from scipy import optimize 11 | import MLP 12 | 13 | 14 | class GPLVM: 15 | """ TODO: this should inherrit a GP, not contain an instance of it...""" 16 | def __init__(self,Y,dim): 17 | self.Xdim = dim 18 | self.N,self.Ydim = Y.shape 19 | 20 | """Use PCA to initalise the problem. Uses EM version in this case...""" 21 | myPCA_EM = PCA_EM(Y,dim) 22 | myPCA_EM.learn(100) 23 | X = np.array(myPCA_EM.m_Z) 24 | 25 | self.GP = GP.GP(X,Y)#choose particular kernel here if so desired. 26 | 27 | def learn(self,niters): 28 | for i in range(niters): 29 | self.optimise_latents() 30 | self.optimise_GP_kernel() 31 | 32 | def optimise_GP_kernel(self): 33 | """optimisation of the GP's kernel parameters""" 34 | self.GP.find_kernel_params() 35 | print self.GP.marginal(), 0.5*np.sum(np.square(self.GP.X)) 36 | 37 | def ll(self,xx,i): 38 | """The log likelihood function - used when changing the ith latent variable to xx""" 39 | self.GP.X[i] = xx 40 | self.GP.update() 41 | return -self.GP.marginal()+ 0.5*np.sum(np.square(xx)) 42 | 43 | def ll_grad(self,xx,i): 44 | """the gradient of the likelihood function for us in optimisation""" 45 | self.GP.X[i] = xx 46 | self.GP.update() 47 | self.GP.update_grad() 48 | matrix_grads = [self.GP.kernel.gradients_wrt_data(self.GP.X,i,jj) for jj in range(self.GP.Xdim)] 49 | grads = [-0.5*np.trace(np.dot(self.GP.alphalphK,e)) for e in matrix_grads] 50 | return np.array(grads) + xx 51 | 52 | def optimise_latents(self): 53 | """Direct optimisation of the latents variables.""" 54 | xtemp = np.zeros(self.GP.X.shape) 55 | for i,yy in enumerate(self.GP.Y): 56 | original_x = self.GP.X[i].copy() 57 | #xopt = optimize.fmin(self.ll,self.GP.X[i],disp=True,args=(i,)) 58 | xopt = optimize.fmin_cg(self.ll,self.GP.X[i],fprime=self.ll_grad,disp=True,args=(i,)) 59 | self.GP.X[i] = original_x 60 | xtemp[i] = xopt 61 | self.GP.X = xtemp.copy() 62 | 63 | 64 | 65 | 66 | class GPLVMC(GPLVM): 67 | """A(back) constrained version of the GPLVM""" 68 | def __init__(self,data,xdim,nhidden=5,mlp_alpha=2): 69 | GPLVM.__init__(self,data,xdim) 70 | 71 | self.MLP = MLP.MLP((self.Ydim,nhidden,self.Xdim),alpha=mlp_alpha) 72 | self.MLP.train(self.GP.Y,self.GP.X)#create an MLP initialised to the PCA solution... 73 | self.GP.X = self.MLP.forward(self.GP.Y) 74 | 75 | def unpack(self,w): 76 | """ Unpack the np array into the free variables of the current instance""" 77 | assert w.size == self.MLP.nweights + self.GP.kernel.nparams + 1,"bad number of parameters for unpacking" 78 | self.MLP.unpack(w[:self.MLP.nweights]) 79 | self.GP.X = self.MLP.forward(self.GP.Y) 80 | self.GP.set_params(w[self.MLP.nweights:]) 81 | 82 | def pack(self): 83 | """ 'Pack up' all of the free variables in the model into a np array""" 84 | return np.hstack((self.MLP.pack(),self.GP.get_params())) 85 | 86 | def ll(self,w): 87 | """Calculate and return the -ve log likelihood of the model (actually, the log probabiulity of the model). To be used in optimisation routine""" 88 | self.unpack(w) 89 | self.GP.update() 90 | return self.GP.ll() + 0.5*np.sum(np.square(self.GP.X)) -self.MLP.prior() 91 | 92 | 93 | def ll_grad(self,w): 94 | """The gradient of the ll function - used for quicker optimisation via fmin_cg""" 95 | self.unpack(w) 96 | 97 | #gradients wrt the GP parameters can be done inside the GP class. This also updates the GP, computes alphalphK. 98 | GP_grads = self.GP.ll_grad(w[self.MLP.nweights:]) 99 | 100 | #gradient matrices (gradients of the kernel matrix wrt data) 101 | gradient_matrices = self.GP.kernel.gradients_wrt_data(self.GP.X) 102 | 103 | #gradients of the error function wrt 'network outputs', i.e. latent variables 104 | x_gradients = np.array([-0.5*np.trace(np.dot(self.GP.alphalphK,e)) for e in gradient_matrices]).reshape(self.GP.X.shape) + self.GP.X 105 | 106 | #backpropagate... 107 | weight_gradients = self.MLP.backpropagate(self.GP.Y,x_gradients) - self.MLP.prior_grad() 108 | return np.hstack((weight_gradients,GP_grads)) 109 | 110 | def learn(self,callback=None,gtol=1e-4): 111 | """'Learn' by optimising the weights of the MLP and the GP hyper parameters together. """ 112 | w_opt = optimize.fmin_cg(self.ll,np.hstack((self.MLP.pack(),self.GP.kernel.get_params(),np.log(self.GP.beta))),self.ll_grad,args=(),callback=callback,gtol=gtol) 113 | final_cost = self.ll(w_opt)#sets all the parameters... 114 | 115 | 116 | 117 | if __name__=="__main__": 118 | N = 20 119 | colours = np.arange(N)#something to colour the dots with... 120 | theta = np.linspace(2,6,N) 121 | Y = np.vstack((np.sin(theta)*(1+theta),np.cos(theta)*theta)).T 122 | Y += 0.1*np.random.randn(N,2) 123 | 124 | thetanorm = (theta-theta.mean())/theta.std() 125 | 126 | xlin = np.linspace(-1,1,1000).reshape(1000,1) 127 | 128 | myGPLVM = GPLVMC(Y,1,nhidden=3) 129 | 130 | def plot_current(): 131 | pylab.figure() 132 | ax = pylab.axes([0.05,0.8,0.9,0.15]) 133 | pylab.scatter(myGPLVM.GP.X[:,0]/myGPLVM.GP.X.std(),np.zeros(N),40,colours) 134 | pylab.scatter(thetanorm,np.ones(N)/2,40,colours) 135 | pylab.yticks([]);pylab.ylim(-0.5,1) 136 | ax = pylab.axes([0.05,0.05,0.9,0.7]) 137 | pylab.scatter(Y[:,0],Y[:,1],40,colours) 138 | Y_pred = myGPLVM.GP.predict(xlin)[0] 139 | pylab.plot(Y_pred[:,0],Y_pred[:,1],'b') 140 | 141 | class callback: 142 | def __init__(self,print_interval): 143 | self.counter = 0 144 | self.print_interval = print_interval 145 | def __call__(self,w): 146 | self.counter +=1 147 | if not self.counter%self.print_interval: 148 | print self.counter, 'iterations, cost: ',myGPLVM.GP.get_params() 149 | plot_current() 150 | 151 | cb = callback(100) 152 | 153 | myGPLVM.learn(callback=cb) 154 | plot_current() 155 | 156 | pylab.show() 157 | 158 | 159 | 160 | 161 | -------------------------------------------------------------------------------- /GP.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | import numpy as np 5 | import pylab 6 | from scipy.optimize import fmin, fmin_ncg, fmin_cg 7 | from scipy import linalg 8 | from sys import stdout 9 | import kernels 10 | 11 | class GP: 12 | def __init__(self,X,Y,kernel=None,parameter_priors=None): 13 | """ a simple GP with optimisation of the Hyper parameters via the marginal likelihood approach. 14 | There is a Univariate Gaussian Prior on the Hyper parameters (the kernel parameters and the noise parameter). 15 | SCG is used to optimise the parameters (MAP estimate)""" 16 | self.N = Y.shape[0] 17 | self.setX(X) 18 | self.setY(Y) 19 | 20 | if kernel==None: 21 | self.kernel = kernels.RBF_full(-1,-np.ones(self.Xdim)) 22 | else: 23 | self.kernel = kernel 24 | if parameter_priors==None: 25 | self.parameter_prior_widths = np.ones(self.kernel.nparams+1) 26 | else: 27 | assert parameter_priors.size==(self.kernel.nparams+1) 28 | self.parameter_prior_widths = np.array(parameter_priors).flatten() 29 | self.beta=0.1 30 | self.update() 31 | self.n2ln2pi = 0.5*self.Ydim*self.N*np.log(2*np.pi) # constant in the marginal. precompute for convenience. 32 | 33 | def setX(self,newX): 34 | self.X = newX.copy() 35 | N,self.Xdim = newX.shape 36 | assert N == self.N, "bad shape" 37 | #normalise... 38 | self.xmean = self.X.mean(0) 39 | self.xstd = self.X.std(0) 40 | self.X -= self.xmean 41 | self.X /= self.xstd 42 | 43 | def setY(self,newY): 44 | self.Y = newY.copy() 45 | N,self.Ydim = newY.shape 46 | assert N == self.N, "bad shape" 47 | #normalise... 48 | self.ymean = self.Y.mean(0) 49 | self.ystd = self.Y.std(0) 50 | self.Y -= self.ymean 51 | self.Y /= self.ystd 52 | 53 | def hyper_prior(self): 54 | """return the log of the current hyper paramters under their prior""" 55 | return -0.5*np.dot(self.parameter_prior_widths,np.square(self.get_params())) 56 | 57 | def hyper_prior_grad(self): 58 | """return the gradient of the (log of the) hyper prior for the current parameters""" 59 | return -self.parameter_prior_widths*self.get_params() 60 | 61 | def get_params(self): 62 | """return the parameters of this GP: that is the kernel parameters and the beta value""" 63 | return np.hstack((self.kernel.get_params(),np.log(self.beta))) 64 | 65 | def set_params(self,params): 66 | """ set the kernel parameters and the noise parameter beta""" 67 | assert params.size==self.kernel.nparams+1 68 | self.beta = np.exp(params[-1]) 69 | self.kernel.set_params(params[:-1]) 70 | 71 | def ll(self,params=None): 72 | """ A cost function to optimise for setting the kernel parameters. Uses current parameter values if none are passed """ 73 | if not params == None: 74 | self.set_params(params) 75 | try: 76 | self.update() 77 | except: 78 | return np.inf 79 | return -self.marginal() - self.hyper_prior() 80 | 81 | def ll_grad(self,params=None): 82 | """ the gradient of the ll function, for use with conjugate gradient optimisation. uses current values of parameters if none are passed """ 83 | if not params == None: 84 | self.set_params(params) 85 | try: 86 | self.update() 87 | except: 88 | return np.ones(params.shape)*np.NaN 89 | self.update_grad() 90 | matrix_grads = [e for e in self.kernel.gradients(self.X)] 91 | matrix_grads.append(-np.eye(self.K.shape[0])/self.beta) #noise gradient matrix 92 | 93 | grads = [0.5*np.trace(np.dot(self.alphalphK,e)) for e in matrix_grads] 94 | 95 | return -np.array(grads) - self.hyper_prior_grad() 96 | 97 | def find_kernel_params(self,iters=1000): 98 | """Optimise the marginal likelihood. work with the log of beta - fmin works better that way. """ 99 | #new_params = fmin(self.ll,np.hstack((self.kernel.get_params(), np.log(self.beta))),maxiter=iters) 100 | new_params = fmin_cg(self.ll,np.hstack((self.kernel.get_params(), np.log(self.beta))),fprime=self.ll_grad,maxiter=iters) 101 | final_ll = self.ll(new_params) # sets variables - required! 102 | 103 | def update(self): 104 | """do the Cholesky decomposition as required to make predictions and calculate the marginal likelihood""" 105 | self.K = self.kernel(self.X,self.X) 106 | self.K += np.eye(self.K.shape[0])/self.beta 107 | self.L = np.linalg.cholesky(self.K) 108 | self.A = linalg.cho_solve((self.L,1),self.Y) 109 | 110 | def update_grad(self): 111 | """do the matrix manipulation required in order to calculate gradients""" 112 | self.Kinv = np.linalg.solve(self.L.T,np.linalg.solve(self.L,np.eye(self.L.shape[0]))) 113 | self.alphalphK = np.dot(self.A,self.A.T)-self.Ydim*self.Kinv 114 | 115 | def marginal(self): 116 | """The Marginal Likelihood. Useful for optimising Kernel parameters""" 117 | return -self.Ydim*np.sum(np.log(np.diag(self.L))) - 0.5*np.trace(np.dot(self.Y.T,self.A)) - self.n2ln2pi 118 | 119 | def predict(self,x_star): 120 | """Make a prediction upon new data points""" 121 | x_star = (np.asarray(x_star)-self.xmean)/self.xstd 122 | 123 | #Kernel matrix k(X_*,X) 124 | k_x_star_x = self.kernel(x_star,self.X) 125 | k_x_star_x_star = self.kernel(x_star,x_star) 126 | 127 | #find the means and covs of the projection... 128 | #means = np.dot(np.dot(k_x_star_x, self.K_inv), self.Y) 129 | means = np.dot(k_x_star_x, self.A) 130 | means *= self.ystd 131 | means += self.ymean 132 | 133 | v = np.linalg.solve(self.L,k_x_star_x.T) 134 | #covs = np.diag( k_x_star_x_star - np.dot(np.dot(k_x_star_x,self.K_inv),k_x_star_x.T)).reshape(x_star.shape[0],1) + self.beta 135 | variances = (np.diag( k_x_star_x_star - np.dot(v.T,v)).reshape(x_star.shape[0],1) + 1./self.beta) * self.ystd.reshape(1,self.Ydim) 136 | return means,variances 137 | 138 | if __name__=='__main__': 139 | #generate data: 140 | Ndata = 50 141 | X = np.linspace(-3,3,Ndata).reshape(Ndata,1) 142 | Y = np.sin(X) + np.random.standard_normal(X.shape)/20 143 | 144 | #create GP object 145 | myGP = GP(X,Y)#,kernels.linear(-1,-1)) 146 | 147 | #stuff for plotting 148 | xx = np.linspace(-4,4,200).reshape(200,1) 149 | def plot(): 150 | pylab.figure() 151 | pylab.plot(X,Y,'r.') 152 | yy,cc = myGP.predict(xx) 153 | pylab.plot(xx,yy,scaley=False) 154 | pylab.plot(xx,yy + 2*np.sqrt(cc),'k--',scaley=False) 155 | pylab.plot(xx,yy - 2*np.sqrt(cc),'k--',scaley=False) 156 | 157 | plot() 158 | myGP.find_kernel_params() 159 | plot() 160 | 161 | 162 | pylab.show() 163 | 164 | 165 | 166 | 167 | 168 | -------------------------------------------------------------------------------- /kernels.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # all kernels expect numpy arrays of data.i 3 | # Arrays must have two dimensions: the first for the Number of data, the second for the dimension of the data. 4 | 5 | import numpy as np 6 | 7 | class RBF: 8 | """Radial Basis Funcion (or 'Squared Exponential') kernel, with the same scale in all directions... 9 | k(x_i,x_j) = \alpha \exp \{ -\gamma ||x_1-x_2||^2 \} 10 | """ 11 | def __init__(self,alpha,gamma): 12 | self.alpha = np.exp(alpha) 13 | self.gamma = np.exp(gamma) 14 | self.nparams = 2 15 | 16 | def set_params(self,new_params): 17 | assert new_params.size == self.nparams 18 | self.alpha,self.gamma = np.exp(new_params).copy().flatten()#try to unpack np array safely. 19 | 20 | def get_params(self): 21 | #return np.array([self.alpha, self.gamma]) 22 | return np.log(np.array([self.alpha, self.gamma])) 23 | 24 | def __call__(self,x1,x2): 25 | N1,D1 = x1.shape 26 | N2,D2 = x2.shape 27 | assert D1==D2, "Vectors must be of matching dimension" 28 | #use broadcasting to avoid for loops. 29 | #should be uber fast 30 | diff = x1.reshape(N1,1,D1)-x2.reshape(1,N2,D2) 31 | diff = self.alpha*np.exp(-np.sum(np.square(diff),-1)*self.gamma) 32 | return diff 33 | 34 | def gradients(self,x1): 35 | """Calculate the gradient of the matrix K wrt the (log of the) free parameters""" 36 | N1,D1 = x1.shape 37 | diff = x1.reshape(N1,1,D1)-x1.reshape(1,N1,D1) 38 | diff = np.sum(np.square(diff),-1) 39 | #dalpha = np.exp(-diff*self.gamma) 40 | dalpha = self.alpha*np.exp(-diff*self.gamma) 41 | #dgamma = -self.alpha*diff*np.exp(-diff*self.gamma) 42 | dgamma = -self.alpha*self.gamma*diff*np.exp(-diff*self.gamma) 43 | return (dalpha, dgamma) 44 | 45 | def gradients_wrt_data(self,x1,indexn=None,indexd=None): 46 | """compute the derivative matrix of the kernel wrt the _data_. Crazy 47 | This returns a list of matices: each matrix is NxN, and there are N*D of them!""" 48 | N1,D1 = x1.shape 49 | diff = x1.reshape(N1,1,D1)-x1.reshape(1,N1,D1) 50 | diff = np.sum(np.square(diff),-1) 51 | expdiff = np.exp(-self.gamma*diff) 52 | 53 | if (indexn==None) and(indexd==None):#calculate all gradients 54 | rets = [] 55 | for n in range(N1): 56 | for d in range(D1): 57 | K = np.zeros((N1,N1)) 58 | K[n,:] = -2*self.alpha*self.gamma*(x1[n,d]-x1[:,d])*expdiff[n,:] 59 | K[:,n] = K[n,:] 60 | rets.append(K.copy()) 61 | return rets 62 | else: 63 | K = np.zeros((N1,N1)) 64 | K[indexn,:] = -2*self.alpha*self.gamma*(x1[indexn,indexd]-x1[:,indexd])*expdiff[indexn,:] 65 | K[:,indexn] = K[indexn,:] 66 | return K 67 | 68 | class RBF_full: 69 | def __init__(self,alpha,gammas): 70 | self.gammas = np.exp(gammas.flatten()) 71 | self.dim = gammas.size 72 | self.alpha = np.exp(alpha) 73 | self.nparams = self.dim+1 74 | 75 | def set_params(self,params): 76 | assert params.size==self.nparams 77 | self.alpha = np.exp(params.flatten()[0]) 78 | self.gammas = np.exp(params.flatten()[1:]) 79 | 80 | def get_params(self): 81 | return np.log(np.hstack((self.alpha,self.gammas))) 82 | 83 | def __call__(self,x1,x2): 84 | N1,D1 = x1.shape 85 | N2,D2 = x2.shape 86 | assert D1==D2, "Vectors must be of matching dimension" 87 | assert D1==self.dim, "That data does not match the dimensionality of this kernel" 88 | diff = x1.reshape(N1,1,D1)-x2.reshape(1,N2,D2) 89 | diff = self.alpha*np.exp(-np.sum(np.square(diff)*self.gammas,-1)) 90 | return diff 91 | 92 | def gradients(self,x1): 93 | """Calculate the gradient of the matrix K wrt the (log of the) free parameters""" 94 | N1,D1 = x1.shape 95 | diff = x1.reshape(N1,1,D1)-x1.reshape(1,N1,D1) 96 | sqdiff = np.sum(np.square(diff)*self.gammas,-1) 97 | expdiff = np.exp(-sqdiff) 98 | grads = [-g*np.square(diff[:,:,i])*self.alpha*expdiff for i,g in enumerate(self.gammas)] 99 | grads.insert(0, self.alpha*expdiff) 100 | return grads 101 | 102 | def gradients_wrt_data(self,x1,indexn=None,indexd=None): 103 | """compute the derivative matrix of the kernel wrt the _data_. Crazy 104 | This returns a list of matices: each matrix is NxN, and there are N*D of them!""" 105 | N1,D1 = x1.shape 106 | diff = x1.reshape(N1,1,D1)-x1.reshape(1,N1,D1) 107 | sqdiff = np.sum(np.square(diff)*self.gammas,-1) 108 | expdiff = np.exp(-sqdiff) 109 | 110 | if (indexn==None) and(indexd==None):#calculate all gradients 111 | rets = [] 112 | for n in range(N1): 113 | for d in range(D1): 114 | K = np.zeros((N1,N1)) 115 | K[n,:] = -2*self.alpha*self.gammas[d]*(x1[n,d]-x1[:,d])*expdiff[n,:] 116 | K[:,n] = K[n,:] 117 | rets.append(K.copy()) 118 | return rets 119 | else: 120 | K = np.zeros((N1,N1)) 121 | K[indexn,:] = -2*self.alpha*self.gammas[indexd]*(x1[indexn,indexd]-x1[:,indexd])*expdiff[indexn,:] 122 | K[:,indexn] = K[indexn,:] 123 | return K.copy() 124 | 125 | 126 | 127 | 128 | 129 | class linear: 130 | """effectively the inner product, I think""" 131 | def __init__(self,alpha,bias): 132 | self.alpha = np.exp(alpha) 133 | self.bias = np.exp(bias) 134 | self.nparams = 2 135 | def set_params(self,new_params): 136 | assert new_params.size == self.nparams 137 | self.alpha,self.bias = np.exp(new_params).flatten()#try to unpack np array safely. 138 | def get_params(self): 139 | return np.log(np.array([self.alpha,self.bias])) 140 | def __call__(self,x1,x2): 141 | N1,D1 = x1.shape 142 | N2,D2 = x2.shape 143 | assert D1==D2, "Vectors must be of matching dimension" 144 | prod = x1.reshape(N1,1,D1)*x2.reshape(1,N2,D2) 145 | prod = self.alpha*np.sum(prod,-1) + self.bias 146 | #diff = self.alpha*np.sqrt(np.square(np.sum(diff,-1))) 147 | return prod 148 | def gradients(self,x1): 149 | """Calculate the gradient of the kernel matrix wrt the (log of the) parameters""" 150 | dalpha = self(x1,x1)-self.bias 151 | dbias = np.ones((x1.shape[0],x1.shape[0]))*self.bias 152 | return dalpha, dbias 153 | 154 | class combined: 155 | """ a combined kernel - linear in X and RBF in Y. 156 | treats first Dimensiona linearly, RBf on the remainder. 157 | TODO: specify which dimensions should be linear and which should be RBF""" 158 | def __init__(self,alpha_x,alpha_y,gamma,bias): 159 | self.linear_kernel = linear(alpha_x, bias) 160 | self.RBF_kernel = RBF(alpha_y, gamma) 161 | self.nparams = 4 162 | def set_params(self,new_params): 163 | assert new_params.size == self.nparams 164 | self.linear_kernel.set_params(new_params[:2]) 165 | self.RBF_kernel.set_params(new_params[2:]) 166 | 167 | def __call__(self,x1,x2): 168 | N1,D1 = x1.shape 169 | N2,D2 = x2.shape 170 | assert D1==D2, "Vectors must be of matching dimension" 171 | return self.linear_kernel(x1[:,0:1],x2[:,0:1])*self.RBF_kernel(x1[:,1:],x2[:,1:]) 172 | 173 | class polynomial: 174 | def __init__(self,alpha,order): 175 | """Order of the polynomila is considered fixed...TODO: make the order optimisable...""" 176 | self.alpha = alpha 177 | self.order = order 178 | self.nparams = 1 179 | def set_params(self,new_params): 180 | assert new_params.size == self.nparams 181 | self.alpha, = new_params.flatten() 182 | def __call__(self,x1,x2): 183 | N1,D1 = x1.shape 184 | N2,D2 = x2.shape 185 | assert D1==D2, "Vectors must be of matching dimension" 186 | prod = x1.reshape(N1,1,D1)*x2.reshape(1,N2,D2) 187 | prod = self.alpha*np.power(np.sum(prod,-1) + 1, self.order) 188 | return prod 189 | -------------------------------------------------------------------------------- /PCA_EM.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # Copyright 2009 James Hensman 3 | # Licensed under the Gnu General Public license, see COPYING 4 | #from numpy import matlib as ml 5 | import numpy as np 6 | from scipy import linalg 7 | 8 | class PCA_EM_matrix: 9 | def __init__(self,data,target_dim): 10 | """Maximum likelihood PCA by the EM algorithm""" 11 | self.X = ml.matrix(data) 12 | self.N,self.d = self.X.shape 13 | self.q = target_dim 14 | def learn(self,niters): 15 | self.mu = self.X.mean(0).reshape(self.d,1)#ML solution for mu 16 | self.X2 = self.X - self.mu.T 17 | self.xxTsum = ml.sum([x*x.T for x in self.X2])#precalculate for speed 18 | #initialise paramters: 19 | self.W = ml.randn(self.d,self.q) 20 | self.sigma2 = 1.2 21 | for i in range(niters): 22 | #print self.sigma2 23 | self.E_step() 24 | self.M_step() 25 | 26 | def E_step(self): 27 | M = self.W.T*self.W + ml.eye(self.q)*self.sigma2 28 | M_inv = ml.linalg.inv(M) 29 | self.m_Z = (M_inv*self.W.T*self.X2.T).T 30 | self.S_z = M_inv*self.sigma2 31 | def M_step(self): 32 | zzT = self.m_Z.T*self.m_Z + self.N*self.S_z 33 | self.W = self.X2.T*self.m_Z*ml.linalg.inv(zzT) 34 | WTW = self.W.T*self.W 35 | self.sigma2 = self.xxTsum - 2*ml.multiply(self.m_Z*self.W.T,self.X2).sum() + ml.trace(zzT*WTW) 36 | #self.sigma2 = self.xxTsum - 2*ml.trace(self.m_Z*self.W.T*self.X2.T) + ml.trace(zzT*WTW) 37 | #self.sigma2 = self.xxTsum + ml.sum([- 2*z*self.W.T*x.T + ml.trace((z.T*z + self.S_z)*WTW) for z,x in zip(self.m_Z, self.X2)]) 38 | self.sigma2 /= self.N*self.d 39 | 40 | class PCA_EM: 41 | def __init__(self,data,target_dim): 42 | """Maximum likelihood PCA by the EM algorithm""" 43 | self.X = np.array(data) 44 | self.N,self.d = self.X.shape 45 | self.q = target_dim 46 | def learn(self,niters): 47 | self.mu = self.X.mean(0).reshape(self.d,1)#ML solution for mu 48 | self.X2 = self.X - self.mu.T 49 | self.xxTsum = np.sum([np.dot(x,x.T) for x in self.X2])#precalculate for speed 50 | #initialise paramters: 51 | self.W = np.random.randn(self.d,self.q) 52 | self.sigma2 = 1.2 53 | for i in range(niters): 54 | #print self.sigma2 55 | self.E_step() 56 | self.M_step() 57 | 58 | def E_step(self): 59 | M = np.dot(self.W.T,self.W) + np.eye(self.q)*self.sigma2 60 | #M_inv = np.linalg.inv(M) 61 | #self.m_Z = np.dot(M_inv,np.dot(self.W.T,self.X2.T)).T 62 | #self.S_z = M_inv*self.sigma2 63 | M_chol = linalg.cholesky(M) 64 | M_inv = linalg.cho_solve((M_chol,1),np.eye(self.q)) 65 | self.m_Z = linalg.cho_solve((M_chol,1),np.dot(self.W.T,self.X2.T)).T 66 | self.S_z = M_inv*self.sigma2 67 | 68 | def M_step(self): 69 | zzT = np.dot(self.m_Z.T,self.m_Z) + self.N*self.S_z 70 | #self.W = np.dot(np.dot(self.X2.T,self.m_Z),np.linalg.inv(zzT)) 71 | zzT_chol = linalg.cholesky(zzT) 72 | self.W = linalg.cho_solve((zzT_chol,0),np.dot(self.m_Z.T,self.X2)).T 73 | WTW = np.dot(self.W.T,self.W) 74 | self.sigma2 = self.xxTsum - 2*np.sum(np.dot(self.m_Z,self.W.T)*self.X2) + np.trace(np.dot(zzT,WTW)) 75 | self.sigma2 /= self.N*self.d 76 | 77 | class PCA_EM_missing: 78 | def __init__(self,data,target_dim): 79 | """Maximum likelihood PCA by the EM algorithm, allows for missing data. uses a masked array to 'hide' the elements of X that are NaN""" 80 | self.X = np.array(data) 81 | self.imask,self.jmask = np.nonzero(np.isnan(self.X))#positions that are missing. 82 | self.indices = [np.nonzero(np.isnan(x)-1)[0] for x in self.X] #positions that are not missing... 83 | self.N,self.d = self.X.shape 84 | self.q = target_dim 85 | 86 | def learn(self,niters): 87 | self.Xreconstruct = self.X.copy() 88 | self.Xreconstruct[self.imask,self.jmask] = 0 89 | self.mu = np.sum(self.Xreconstruct,0)/(self.X.shape[0]-np.sum(np.isnan(self.X),0)) 90 | 91 | self.X2 = self.X.copy()-self.mu 92 | self.X2reconstruct = self.X.copy() - self.mu 93 | #initialise paramters: 94 | self.W = np.random.randn(self.d,self.q) 95 | self.sigma2 = 1.2 96 | #pre-allocate self.m_Z and self.S_Z 97 | self.m_Z = np.zeros((self.X2.shape[0],self.q)) 98 | self.S_Z = np.zeros((self.X2.shape[0],self.q,self.q)) 99 | for i in range(niters): 100 | print i,self.sigma2 101 | self.E_step() 102 | self.M_step() 103 | self.Xreconstruct = self.X2reconstruct + self.mu 104 | 105 | def E_step(self): 106 | """ This should handle missing data, but needs testing (TODO)""" 107 | Ms = np.zeros((self.X.shape[0],self.q,self.q)) #M is going to be different for (potentially) every data point 108 | for m,x,i,mz,sz in zip(Ms,self.X2,self.indices,self.m_Z,self.S_Z): 109 | W = self.W.take(i,0)# get relevant bits of W 110 | x2 = np.array(x).take(i) # get relevant bits of x 111 | m[:,:] = np.dot(W.T,W) + np.eye(self.q)*self.sigma2 112 | mchol = linalg.cholesky(m) 113 | minv = linalg.cho_solve((mchol,1),np.eye(self.q)) 114 | mz[:] = linalg.cho_solve((mchol,1),np.dot(W.T,x2.reshape(i.size,1))).T 115 | sz[:,:] = minv*self.sigma2 116 | 117 | #calculate reconstructed X values 118 | self.X2reconstruct[self.imask,self.jmask] = np.dot(self.m_Z,self.W.T)[self.imask,self.jmask] 119 | self.xxTsum = np.sum(np.square(self.X2reconstruct))# can;t be pre-calculate in the missing data version :( 120 | 121 | def M_step(self): 122 | """ This should handle missing data - needs testing (TODO)""" 123 | zzT = np.dot(self.m_Z.T,self.m_Z) + np.sum(self.S_Z,0) 124 | #self.W = np.dot(np.dot(self.X2.T,self.m_Z),np.linalg.inv(zzT)) 125 | zzT_chol = linalg.cholesky(zzT) 126 | self.W = linalg.cho_solve((zzT_chol,0),np.dot(self.m_Z.T,self.X2reconstruct)).T 127 | WTW = np.dot(self.W.T,self.W) 128 | self.sigma2 = self.xxTsum - 2*np.sum(np.dot(self.m_Z,self.W.T)*self.X2reconstruct) + np.trace(np.dot(zzT,WTW)) 129 | self.sigma2 /= self.N*self.d 130 | 131 | if __name__=='__main__': 132 | q=5#latent dimensions 133 | d=15# observed dimensions 134 | N=500 135 | missing_pc = 100 # percentage of the data points to be 'missing' 136 | truesigma = .002 137 | niters = 300 138 | phases = np.random.rand(1,q)*2*np.pi 139 | frequencies = np.random.randn(1,q)*2 140 | latents = np.sin(np.linspace(0,12,N).reshape(N,1)*frequencies-phases) 141 | trueW = np.random.randn(d,q) 142 | observed = np.dot(latents,trueW.T) + np.random.randn(N,d)*truesigma 143 | 144 | #PCA without missing values 145 | a = PCA_EM(observed,q) 146 | a.learn(niters) 147 | 148 | #a missing data problem 149 | Nmissing = int(N*missing_pc/100) 150 | observed2 = observed.copy() 151 | missingi = np.argsort(np.random.rand(N))[:Nmissing] 152 | missingj = np.random.randint(0,d-q,Nmissing)#last q columns will be complete 153 | observed2[missingi,missingj] = np.NaN 154 | 155 | b = PCA_EM_missing(observed2,q) 156 | b.learn(niters) 157 | 158 | 159 | from hinton import hinton 160 | import pylab 161 | colours = np.arange(N)# to colour the dots with 162 | hinton(linalg.qr(trueW.T)[1].T) 163 | pylab.title('true transformation') 164 | pylab.figure() 165 | hinton(linalg.qr(a.W.T)[1].T) 166 | pylab.title('reconstructed transformation') 167 | pylab.figure() 168 | hinton(linalg.qr(b.W.T)[1].T) 169 | pylab.title('reconstructed transformation (missing data)') 170 | pylab.figure() 171 | pylab.subplot(3,1,1) 172 | pylab.plot(latents) 173 | pylab.title('true latents') 174 | pylab.subplot(3,1,2) 175 | pylab.plot(a.m_Z) 176 | pylab.title('reconstructed latents') 177 | pylab.subplot(3,1,3) 178 | pylab.plot(b.m_Z) 179 | pylab.title('reconstructed latents (missing data)') 180 | pylab.figure() 181 | pylab.subplot(2,1,1) 182 | pylab.plot(observed) 183 | pylab.title('Observed values') 184 | pylab.subplot(2,1,2) 185 | pylab.plot(observed2,linewidth=2,marker='.') 186 | pylab.plot(b.Xreconstruct) 187 | 188 | pylab.show() 189 | 190 | -------------------------------------------------------------------------------- /COPYING: -------------------------------------------------------------------------------- 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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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------