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Sparse EP
This commit is contained in:
Ricardo
parent
a8738984b3
commit
fad0e07624
7 changed files with 399 additions and 8 deletions
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@ -10,6 +10,7 @@ import numpy as np
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import GPy
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pb.ion()
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pb.close('all')
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default_seed=10000
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model_type='Full'
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@ -26,11 +27,13 @@ data = GPy.util.datasets.toy_linear_1d_classification(seed=seed)
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likelihood = GPy.inference.likelihoods.probit(data['Y'][:, 0:1])
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m = GPy.models.GP(data['X'],likelihood=likelihood)
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#m = GPy.models.GP(data['X'],Y=likelihood.Y)
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m.constrain_positive('var')
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m.constrain_positive('len')
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m.tie_param('lengthscale')
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m.approximate_likelihood()
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if not isinstance(m.likelihood,GPy.inference.likelihoods.gaussian):
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m.approximate_likelihood()
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print m.checkgrad()
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# Optimize and plot
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m.optimize()
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50
GPy/examples/poisson.py
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50
GPy/examples/poisson.py
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@ -0,0 +1,50 @@
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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"""
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Simple Gaussian Processes classification
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"""
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import pylab as pb
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import numpy as np
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import GPy
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pb.ion()
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pb.close('all')
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default_seed=10000
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model_type='Full'
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inducing=4
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seed=default_seed
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"""Simple 1D classification example.
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:param model_type: type of model to fit ['Full', 'FITC', 'DTC'].
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:param seed : seed value for data generation (default is 4).
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:type seed: int
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:param inducing : number of inducing variables (only used for 'FITC' or 'DTC').
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:type inducing: int
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"""
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X = np.arange(0,100,5)[:,None]
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F = np.round(np.sin(X/18.) + .1*X)
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E = np.random.randint(-3,3,20)[:,None]
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Y = F + E
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pb.plot(X,F,'k-')
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pb.plot(X,Y,'ro')
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pb.figure()
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likelihood = GPy.inference.likelihoods.poisson(Y,scale=4.)
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m = GPy.models.GP(X,likelihood=likelihood)
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#m = GPy.models.GP(data['X'],Y=likelihood.Y)
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m.constrain_positive('var')
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m.constrain_positive('len')
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m.tie_param('lengthscale')
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if not isinstance(m.likelihood,GPy.inference.likelihoods.gaussian):
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m.approximate_likelihood()
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print m.checkgrad()
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# Optimize and plot
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m.optimize()
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#m.em(plot_all=False) # EM algorithm
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m.plot()
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print(m)
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76
GPy/examples/sparse_ep_fix.py
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GPy/examples/sparse_ep_fix.py
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@ -0,0 +1,76 @@
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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"""
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Sparse Gaussian Processes regression with an RBF kernel
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"""
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import pylab as pb
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import numpy as np
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import GPy
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np.random.seed(2)
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pb.ion()
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N = 500
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M = 5
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######################################
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## 1 dimensional example
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# sample inputs and outputs
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X = np.random.uniform(-3.,3.,(N,1))
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#Y = np.sin(X)+np.random.randn(N,1)*0.05
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F = np.sin(X)+np.random.randn(N,1)*0.05
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Y = np.ones([F.shape[0],1])
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Y[F<0] = -1
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likelihood = GPy.inference.likelihoods.probit(Y)
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# construct kernel
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rbf = GPy.kern.rbf(1)
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noise = GPy.kern.white(1)
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kernel = rbf + noise
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# create simple GP model
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#m1 = GPy.models.sparse_GP_regression(X, Y, kernel, M=M)
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m1 = GPy.models.sparse_GP(X, kernel, M=M,likelihood= likelihood)
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# contrain all parameters to be positive
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m1.constrain_positive('(variance|lengthscale|precision)')
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#m1.constrain_positive('(variance|lengthscale)')
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#m1.constrain_fixed('prec',10.)
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#check gradient FIXME unit test please
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m1.checkgrad()
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# optimize and plot
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m1.optimize('tnc', messages = 1)
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m1.plot()
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# print(m1)
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######################################
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## 2 dimensional example
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# # sample inputs and outputs
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# X = np.random.uniform(-3.,3.,(N,2))
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# Y = np.sin(X[:,0:1]) * np.sin(X[:,1:2])+np.random.randn(N,1)*0.05
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# # construct kernel
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# rbf = GPy.kern.rbf(2)
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# noise = GPy.kern.white(2)
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# kernel = rbf + noise
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# # create simple GP model
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# m2 = GPy.models.sparse_GP_regression(X,Y,kernel, M = 50)
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# create simple GP model
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# # contrain all parameters to be positive (but not inducing inputs)
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# m2.constrain_positive('(variance|lengthscale|precision)')
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# #check gradient FIXME unit test please
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# m2.checkgrad()
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# # optimize and plot
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# pb.figure()
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# m2.optimize('tnc', messages = 1)
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# m2.plot()
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# print(m2)
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@ -9,7 +9,7 @@ from ..util.plot import gpplot
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from .. import kern
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class EP:
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def __init__(self,covariance,likelihood,Kmn=None,Knn_diag=None,epsilon=1e-3,powerep=[1.,1.]):
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def __init__(self,covariance,likelihood,Kmn=None,Knn_diag=None,epsilon=1e-3,power_ep=[1.,1.]):
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"""
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Expectation Propagation
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@ -19,7 +19,7 @@ class EP:
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likelihood : Output's likelihood (likelihood class)
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kernel : a GPy kernel (kern class)
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inducing : Either an array specifying the inducing points location or a sacalar defining their number. None value for using a non-sparse model is used.
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powerep : Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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power_ep : Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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epsilon : Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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"""
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self.likelihood = likelihood
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@ -38,7 +38,7 @@ class EP:
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assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
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self.epsilon = epsilon
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self.eta, self.delta = powerep
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self.eta, self.delta = power_ep
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self.jitter = 1e-12
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"""
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@ -110,6 +110,7 @@ class Full(EP):
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self.Sigma = self.Sigma - Delta_tau/(1.+ Delta_tau*self.Sigma[i,i])*np.dot(si,si.T)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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self.iterations += 1
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print self.tau_tilde[i] #TODO erase me
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#Sigma recomptutation with Cholesky decompositon
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Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*(self.K)
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B = np.eye(self.N) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K
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@ -206,6 +207,7 @@ class DTC(EP):
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
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class FITC(EP):
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def fit_EP(self):
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@ -306,3 +308,4 @@ class FITC(EP):
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
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@ -29,8 +29,8 @@ class GP(model):
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"""
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def __init__(self,X,Y=None,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,likelihood=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
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#TODO: specify beta parameter explicitely
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def __init__(self,X,Y=None,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,likelihood=None,epsilon_ep=1e-3,epsion_em=.1,power_ep=[1.,1.]):
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#TODO: make beta parameter explicit
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# parse arguments
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self.Xslices = Xslices
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@ -87,7 +87,7 @@ class GP(model):
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else:
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# Y is defined after approximating the likelihood
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self.EP = True
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self.eta,self.delta = powerep
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self.eta,self.delta = power_ep
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self.epsilon_ep = epsilon_ep
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self.tau_tilde = np.ones([self.N,self.D])
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self.v_tilde = np.zeros([self.N,self.D])
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@ -116,7 +116,7 @@ class GP(model):
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def approximate_likelihood(self):
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assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
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self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
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self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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self.tau_tilde, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
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# Y: EP likelihood is defined as a regression model for mu_tilde
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self.Y = self.v_tilde/self.tau_tilde
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@ -10,3 +10,4 @@ from generalized_FITC import generalized_FITC
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from sparse_GPLVM import sparse_GPLVM
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from uncollapsed_sparse_GP import uncollapsed_sparse_GP
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from GP import GP
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from sparse_GP import sparse_GP
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258
GPy/models/sparse_GP.py
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258
GPy/models/sparse_GP.py
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@ -0,0 +1,258 @@
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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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import pylab as pb
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from ..util.linalg import mdot, jitchol, chol_inv, pdinv
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from ..util.plot import gpplot
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from .. import kern
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from GP import GP
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from ..inference.EP import Full
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from ..inference.likelihoods import likelihood,probit,poisson,gaussian
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#Still TODO:
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# make use of slices properly (kernel can now do this)
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# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
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class sparse_GP(GP):
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"""
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Variational sparse GP model (Regression)
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:param X: inputs
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:type X: np.ndarray (N x Q)
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:param Y: observed data
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:type Y: np.ndarray of observations (N x D)
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:param kernel : the kernel/covariance function. See link kernels
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:type kernel: a GPy kernel
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:param Z: inducing inputs (optional, see note)
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:type Z: np.ndarray (M x Q) | None
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:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
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:type X_uncertainty: np.ndarray (N x Q) | None
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:param Zslices: slices for the inducing inputs (see slicing TODO: link)
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:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
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:type M: int
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:param beta: noise precision. TODO> ignore beta if doing EP
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:type beta: float
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:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
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:type normalize_(X|Y): bool
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"""
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def __init__(self,X,Y,kernel=None, X_uncertainty=None, beta=100., Z=None,Zslices=None,M=10,normalize_X=False,normalize_Y=False,likelihood=None,method_ep='DTC',epsilon_ep=1e-3,epsilon_em=.1,power_ep=[1.,1.]):
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if Z is None:
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self.Z = np.random.permutation(X.copy())[:M]
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self.M = M
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else:
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assert Z.shape[1]==X.shape[1]
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self.Z = Z
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self.M = Z.shape[0]
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if X_uncertainty is None:
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self.has_uncertain_inputs=False
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else:
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assert X_uncertainty.shape==X.shape
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self.has_uncertain_inputs=True
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self.X_uncertainty = X_uncertainty
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self.beta = beta #FIXME
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GP.__init__(self, X, Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,epsion_em=epsilon_em,power_ep=power_ep)
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self.beta = beta if isinstance(likelihood,gaussian) else self.tau_tilde #TODO this should be defined in GP.__init__
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#normalise X uncertainty also
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if self.has_uncertain_inputs:
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self.X_uncertainty /= np.square(self._Xstd)
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def _set_params(self, p):
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if not self.EP:
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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self.beta = p[self.M*self.Q]
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self.kern._set_params(p[self.Z.size + 1:])
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self.beta2 = self.beta**2
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self._compute_kernel_matrices()
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self._computations()
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else:
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self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
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self.kern._set_params(p[self.Z.size:])
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#self._compute_kernel_matrices() this is replaced by _ep_covariance
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self._ep_covariance()
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self._ep_computations()
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def approximate_likelihood(self):
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assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
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if self.ep_proxy == 'DTC':
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self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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elif self.ep_proxy == 'FITC':
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self.Knn_diag = self.kern.psi0(self.Z,self.X, self.X_uncertainty) #TODO psi0 already calculates this
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self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.Knn_diag,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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else:
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self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
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self.beta, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
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# Y: EP likelihood is defined as a regression model for mu_tilde
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self.Y = self.v_tilde/self.beta
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self._Ymean = np.zeros((1,self.Y.shape[1]))
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self._Ystd = np.ones((1,self.Y.shape[1]))
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self.trbetaYYT = np.sum(self.beta*np.square(self.Y))
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if self.D > self.N:
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# then it's more efficient to store YYT
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self.YYT = np.dot(self.Y, self.Y.T)
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else:
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self.YYT = None
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self.mu_ = self.v_/self.tau_
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self._ep_covariance()
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self._computations()
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def _ep_covariance(self):
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self.Kmm = self.kern.K(self.Z)
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if self.has_uncertain_inputs:
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self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
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self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
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self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty) #FIXME include beta
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else:
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#self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
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self.Knn_diag = self.kern.Kdiag(self.X,slices=self.Xslices)
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self.psi0 = (self.beta*self.Knn_diag).sum() #TODO check dimensions
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self.psi1 = self.kern.K(self.Z,self.X)
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#self.psi2 = np.dot(self.psi1,self.psi1.T)
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self.psi2 = np.dot(self.psi1,self.beta*self.psi1.T)
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def _compute_kernel_matrices(self):
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# kernel computations, using BGPLVM notation
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#TODO: slices for psi statistics (easy enough)
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self.Kmm = self.kern.K(self.Z)
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if self.has_uncertain_inputs:
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self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
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self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
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self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
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else:
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self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
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self.psi1 = self.kern.K(self.Z,self.X)
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self.psi2 = np.dot(self.psi1,self.psi1.T)
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def _ep_computations(self):
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# TODO find routine to multiply triangular matrices
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self.V = self.beta*self.Y
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self.psi1V = np.dot(self.psi1, self.V)
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self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
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self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
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#self.A = mdot(self.Lmi, self.beta*self.psi2, self.Lmi.T)
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self.A = mdot(self.Lmi, self.psi2, self.Lmi.T)
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self.B = np.eye(self.M) + self.A
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self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
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self.LLambdai = np.dot(self.LBi, self.Lmi)
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#self.trace_K = self.psi0 - np.sum(np.dot(self.Lmi,self.psi1)**2,-1) #TODO check
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self.trace_K = self.psi0 - np.trace(self.A)
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self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
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self.C = mdot(self.LLambdai, self.psi1V)
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self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
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# Compute dL_dpsi
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#self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
|
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self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten() * np.ones(self.N) #TODO check
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self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
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#self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
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self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
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# Compute dL_dKmm
|
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self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
|
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self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*self.beta*mdot(self.LBL_inv, self.psi2, self.Kmmi) + self.Kmmi) # dC
|
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self.dL_dKmm += np.dot(np.dot(self.G,self.beta*self.psi2) - np.dot(self.LBL_inv, self.psi1VVpsi1), self.Kmmi) + 0.5*self.G # dE
|
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|
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def _get_params(self):
|
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if not self.EP:
|
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return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
|
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else:
|
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return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
|
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|
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def _get_param_names(self):
|
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if not self.EP:
|
||||
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + ['noise_precision']+self.kern._get_param_names_transformed()
|
||||
else:
|
||||
return sum([['iip_%i_%i'%(i,j) for i in range(self.Z.shape[0])] for j in range(self.Z.shape[1])],[]) + self.kern._get_param_names_transformed()
|
||||
|
||||
def log_likelihood(self):
|
||||
"""
|
||||
Compute the (lower bound on the) log marginal likelihood
|
||||
"""
|
||||
beta_logdet = self.N*self.D*np.log(self.beta) if not self.EP else self.D*np.sum(np.log(self.beta))
|
||||
A = -0.5*self.N*self.D*(np.log(2.*np.pi)) - 0.5*beta_logdet
|
||||
B = -0.5*self.beta*self.D*self.trace_K if not self.EP else -0.5*self.D*self.trace_K
|
||||
C = -0.5*self.D * self.B_logdet
|
||||
D = -0.5*self.beta*self.trYYT if not self.EP else -0.5*self.trbetaYYT
|
||||
E = +0.5*np.sum(self.psi1VVpsi1 * self.LBL_inv)
|
||||
return A+B+C+D+E
|
||||
|
||||
def dL_dbeta(self):
|
||||
"""
|
||||
Compute the gradient of the log likelihood wrt beta.
|
||||
"""
|
||||
#TODO: suport heteroscedatic noise
|
||||
dA_dbeta = 0.5 * self.N*self.D/self.beta
|
||||
dB_dbeta = - 0.5 * self.D * self.trace_K
|
||||
dC_dbeta = - 0.5 * self.D * np.sum(self.Bi*self.A)/self.beta
|
||||
dD_dbeta = - 0.5 * self.trYYT
|
||||
tmp = mdot(self.LBi.T, self.LLambdai, self.psi1V)
|
||||
dE_dbeta = (np.sum(np.square(self.C)) - 0.5 * np.sum(self.A * np.dot(tmp, tmp.T)))/self.beta
|
||||
|
||||
return np.squeeze(dA_dbeta + dB_dbeta + dC_dbeta + dD_dbeta + dE_dbeta)
|
||||
|
||||
def dL_dtheta(self):
|
||||
"""
|
||||
Compute and return the derivative of the log marginal likelihood wrt the parameters of the kernel
|
||||
"""
|
||||
dL_dtheta = self.kern.dK_dtheta(self.dL_dKmm,self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dtheta += self.kern.dpsi0_dtheta(self.dL_dpsi0, self.Z,self.X,self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi1_dtheta(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dtheta += self.kern.dpsi2_dtheta(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty) # for multiple_beta, dL_dpsi2 will be a different shape
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dtheta += self.kern.dK_dtheta(dL_dpsi1,self.Z,self.X)
|
||||
dL_dtheta += self.kern.dKdiag_dtheta(self.dL_dpsi0, self.X)
|
||||
|
||||
return dL_dtheta
|
||||
|
||||
def dL_dZ(self):
|
||||
"""
|
||||
The derivative of the bound wrt the inducing inputs Z
|
||||
"""
|
||||
dL_dZ = 2.*self.kern.dK_dX(self.dL_dKmm,self.Z,)#factor of two becase of vertical and horizontal 'stripes' in dKmm_dZ
|
||||
if self.has_uncertain_inputs:
|
||||
dL_dZ += self.kern.dpsi1_dZ(self.dL_dpsi1.T,self.Z,self.X, self.X_uncertainty)
|
||||
dL_dZ += self.kern.dpsi2_dZ(self.dL_dpsi2,self.Z,self.X, self.X_uncertainty)
|
||||
else:
|
||||
#re-cast computations in psi2 back to psi1:
|
||||
dL_dpsi1 = self.dL_dpsi1 + 2.*np.dot(self.dL_dpsi2,self.psi1)
|
||||
dL_dZ += self.kern.dK_dX(dL_dpsi1,self.Z,self.X)
|
||||
return dL_dZ
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
return np.hstack([self.dL_dZ().flatten(), self.dL_dbeta(), self.dL_dtheta()])
|
||||
|
||||
def _raw_predict(self, Xnew, slices, full_cov=False):
|
||||
"""Internal helper function for making predictions, does not account for normalisation"""
|
||||
Kx = self.kern.K(self.Z, Xnew)
|
||||
mu = mdot(Kx.T, self.LBL_inv, self.psi1V)
|
||||
if full_cov:
|
||||
noise_term = np.eye(Xnew.shape[0])/self.beta if not self.EP else 0
|
||||
Kxx = self.kern.K(Xnew)
|
||||
var = Kxx - mdot(Kx.T, (self.Kmmi - self.LBL_inv), Kx) + noise_term
|
||||
else:
|
||||
noise_term = 1./self.beta if not self.EP else 0
|
||||
Kxx = self.kern.Kdiag(Xnew)
|
||||
var = Kxx - np.sum(Kx*np.dot(self.Kmmi - self.LBL_inv, Kx),0) + noise_term
|
||||
return mu,var
|
||||
|
||||
def plot(self, *args, **kwargs):
|
||||
"""
|
||||
Plot the fitted model: just call the GP_regression plot function and then add inducing inputs
|
||||
"""
|
||||
GP_regression.plot(self,*args,**kwargs)
|
||||
if self.Q==1:
|
||||
pb.plot(self.Z,self.Z*0+pb.ylim()[0],'k|',mew=1.5,markersize=12)
|
||||
if self.has_uncertain_inputs:
|
||||
pb.errorbar(self.X[:,0], pb.ylim()[0]+np.zeros(self.N), xerr=2*np.sqrt(self.X_uncertainty.flatten()))
|
||||
if self.Q==2:
|
||||
pb.plot(self.Z[:,0],self.Z[:,1],'wo')
|
||||
Loading…
Add table
Add a link
Reference in a new issue