mirror of
https://github.com/SheffieldML/GPy.git
synced 2026-05-15 06:52:39 +02:00
Merge branch 'newGP'
This commit is contained in:
Ricardo Andrade
commit
75b95619f2
12 changed files with 966 additions and 253 deletions
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@ -381,6 +381,43 @@ class model(parameterised):
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print grad_string
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print ''
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return False
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return True
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def EM(self,epsilon=.1,**kwargs):
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"""
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Expectation maximization for Expectation Propagation.
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kwargs are passed to the optimize function. They can be:
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:epsilon: convergence criterion
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:max_f_eval: maximum number of function evaluations
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:messages: whether to display during optimisation
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:param optimzer: whice optimizer to use (defaults to self.preferred optimizer)
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:type optimzer: string TODO: valid strings?
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"""
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assert self.EP, "EM not available for gaussian likelihood"
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log_change = epsilon + 1.
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self.log_likelihood_record = []
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self.gp_params_record = []
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self.ep_params_record = []
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iteration = 0
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last_value = -np.exp(1000)
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while log_change > epsilon or not iteration:
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print 'EM iteration %s' %iteration
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self.approximate_likelihood()
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self.optimize(**kwargs)
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new_value = self.log_likelihood()
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log_change = new_value - last_value
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if log_change > epsilon:
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self.log_likelihood_record.append(new_value)
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self.gp_params_record.append(self._get_params())
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self.ep_params_record.append((self.beta,self.Y,self.Z_ep))
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last_value = new_value
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else:
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convergence = False
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self.beta, self.Y, self.Z_ep = self.ep_params_record[-1]
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self._set_params(self.gp_params_record[-1])
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print "Log-likelihood decrement: %s \nLast iteration discarded." %log_change
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iteration += 1
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@ -76,11 +76,10 @@ def toy_linear_1d_classification(model_type='Full', inducing=4, seed=default_see
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# create simple GP model
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if model_type=='Full':
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m = GPy.models.simple_GP_EP(data['X'],likelihood)
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m = GPy.models.GP_EP(data['X'],likelihood)
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else:
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# create sparse GP EP model
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m = GPy.models.sparse_GP_EP(data['X'],likelihood=likelihood,inducing=inducing,ep_proxy=model_type)
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m.constrain_positive('var')
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m.constrain_positive('len')
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39
GPy/examples/ep_fix.py
Normal file
39
GPy/examples/ep_fix.py
Normal file
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@ -0,0 +1,39 @@
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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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model_type='Full'
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inducing=4
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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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data = GPy.util.datasets.toy_linear_1d_classification(seed=0)
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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'],likelihood.Y)
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m.ensure_default_constraints()
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# Optimize and plot
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#if not isinstance(m.likelihood,GPy.inference.likelihoods.gaussian):
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# m.approximate_likelihood()
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#m.optimize()
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m.EM()
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print m.log_likelihood()
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m.plot(samples=3)
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print(m)
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50
GPy/examples/poisson.py
Normal file
50
GPy/examples/poisson.py
Normal file
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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=6.)
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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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48
GPy/examples/sparse_ep_fix.py
Normal file
48
GPy/examples/sparse_ep_fix.py
Normal file
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@ -0,0 +1,48 @@
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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(X, Y, kernel, M=M)
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m1 = GPy.models.sparse_GP(X,Y=None, kernel=kernel, M=M,likelihood= likelihood)
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print m1.checkgrad()
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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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# 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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@ -1,9 +1,6 @@
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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 random
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import pylab as pb #TODO erase me
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from scipy import stats, linalg
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from .likelihoods import likelihood
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from ..core import model
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@ -11,21 +8,45 @@ from ..util.linalg import pdinv,mdot,jitchol
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from ..util.plot import gpplot
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from .. import kern
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class EP_base:
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"""
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Expectation Propagation.
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class EP:
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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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This is just the base class for expectation propagation. We'll extend it for full and sparse EP.
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"""
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def __init__(self,likelihood,epsilon=1e-3,powerep=[1.,1.]):
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Arguments
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---------
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X : input observations
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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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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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assert covariance.shape[0] == covariance.shape[1]
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if Kmn is not None:
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self.Kmm = covariance
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self.Kmn = Kmn
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self.M = self.Kmn.shape[0]
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self.N = self.Kmn.shape[1]
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assert self.M < self.N, 'The number of inducing inputs must be smaller than the number of observations'
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else:
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self.K = covariance
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self.N = self.K.shape[0]
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if Knn_diag is not None:
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self.Knn_diag = Knn_diag
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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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#Initial values - Likelihood approximation parameters:
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#p(y|f) = t(f|tau_tilde,v_tilde)
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self.restart_EP()
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"""
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Initial values - Likelihood approximation parameters:
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p(y|f) = t(f|tau_tilde,v_tilde)
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"""
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self.tau_tilde = np.zeros(self.N)
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self.v_tilde = np.zeros(self.N)
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def restart_EP(self):
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"""
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@ -35,26 +56,11 @@ class EP_base:
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self.v_tilde = np.zeros(self.N)
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self.mu = np.zeros(self.N)
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class Full(EP_base):
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"""
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param K: prior covariance matrix
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:type K: np.ndarray (N x N)
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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:param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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"""
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def __init__(self,K,likelihood,*args,**kwargs):
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assert K.shape[0] == K.shape[1]
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self.K = K
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self.N = self.K.shape[0]
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EP_base.__init__(self,likelihood,*args,**kwargs)
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def fit_EP(self,messages=False):
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class Full(EP):
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def fit_EP(self):
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"""
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The expectation-propagation algorithm.
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For nomenclature see Rasmussen & Williams 2006 (pag. 52-60)
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For nomenclature see Rasmussen & Williams 2006.
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"""
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#Prior distribution parameters: p(f|X) = N(f|0,K)
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#self.K = self.kernel.K(self.X,self.X)
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@ -69,16 +75,15 @@ class Full(EP_base):
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=np.float64)
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self.v_ = np.empty(self.N,dtype=np.float64)
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self.tau_ = np.empty(self.N,dtype=float)
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self.v_ = np.empty(self.N,dtype=float)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=np.float64)
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self.Z_hat = np.empty(self.N,dtype=np.float64)
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phi = np.empty(self.N,dtype=np.float64)
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mu_hat = np.empty(self.N,dtype=np.float64)
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sigma2_hat = np.empty(self.N,dtype=np.float64)
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z = np.empty(self.N,dtype=float)
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self.Z_hat = np.empty(self.N,dtype=float)
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phi = np.empty(self.N,dtype=float)
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mu_hat = np.empty(self.N,dtype=float)
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sigma2_hat = np.empty(self.N,dtype=float)
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#Approximation
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epsilon_np1 = self.epsilon + 1.
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@ -87,7 +92,8 @@ class Full(EP_base):
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.random.permutation(self.N)
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update_order = np.arange(self.N)
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random.shuffle(update_order)
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma[i,i] - self.eta*self.tau_tilde[i]
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@ -111,35 +117,104 @@ class Full(EP_base):
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V,info = linalg.flapack.dtrtrs(L,Sroot_tilde_K,lower=1)
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self.Sigma = self.K - np.dot(V.T,V)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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epsilon_np1 = np.mean(self.tau_tilde-self.np1[-1]**2)
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epsilon_np2 = np.mean(self.v_tilde-self.np2[-1]**2)
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epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
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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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if messages:
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print "EP iteration %i, epsiolon %d"%(self.iterations,epsilon_np1)
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class FITC(EP_base):
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"""
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param Knn_diag: The diagonal elements of Knn is a 1D vector
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:param Kmn: The 'cross' variance between inducing inputs and data
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:param Kmm: the covariance matrix of the inducing inputs
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:param likelihood: Output's likelihood (e.g. probit)
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:type likelihood: GPy.inference.likelihood instance
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:param epsilon: Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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:param powerep: Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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"""
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def __init__(self,likelihood,Knn_diag,Kmn,Kmm,*args,**kwargs):
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self.Knn_diag = Knn_diag
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self.Kmn = Kmn
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self.Kmm = Kmm
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self.M = self.Kmn.shape[0]
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self.N = self.Kmn.shape[1]
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assert self.M <= self.N, 'The number of inducing inputs must be smaller than the number of observations'
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assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
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EP_base.__init__(self,likelihood,*args,**kwargs)
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#Variables to be called from GP
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mu_tilde = self.v_tilde/self.tau_tilde #When calling EP, this variable is used instead of Y in the GP model
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sigma_sum = 1./self.tau_ + 1./self.tau_tilde
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mu_diff_2 = (self.v_/self.tau_ - mu_tilde)**2
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Z_ep = np.sum(np.log(self.Z_hat)) + 0.5*np.sum(np.log(sigma_sum)) + 0.5*np.sum(mu_diff_2/sigma_sum) #Normalization constant
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return self.tau_tilde[:,None], mu_tilde[:,None], Z_ep
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class DTC(EP):
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def fit_EP(self):
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"""
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The expectation-propagation algorithm with sparse pseudo-input.
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For nomenclature see ... 2013.
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"""
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"""
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Prior approximation parameters:
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q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
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Sigma0 = Qnn = Knm*Kmmi*Kmn
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"""
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self.Kmmi, self.Kmm_hld = pdinv(self.Kmm)
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self.KmnKnm = np.dot(self.Kmn, self.Kmn.T)
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self.KmmiKmn = np.dot(self.Kmmi,self.Kmn)
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self.Qnn_diag = np.sum(self.Kmn*self.KmmiKmn,-2)
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self.LLT0 = self.Kmm.copy()
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"""
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Posterior approximation: q(f|y) = N(f| mu, Sigma)
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Sigma = Diag + P*R.T*R*P.T + K
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mu = w + P*gamma
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"""
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self.mu = np.zeros(self.N)
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self.LLT = self.Kmm.copy()
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self.Sigma_diag = self.Qnn_diag.copy()
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"""
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Initial values - Cavity distribution parameters:
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q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=float)
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self.v_ = np.empty(self.N,dtype=float)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=float)
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self.Z_hat = np.empty(self.N,dtype=float)
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phi = np.empty(self.N,dtype=float)
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mu_hat = np.empty(self.N,dtype=float)
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sigma2_hat = np.empty(self.N,dtype=float)
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#Approximation
|
||||
epsilon_np1 = 1
|
||||
epsilon_np2 = 1
|
||||
self.iterations = 0
|
||||
self.np1 = [self.tau_tilde.copy()]
|
||||
self.np2 = [self.v_tilde.copy()]
|
||||
while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
|
||||
update_order = np.arange(self.N)
|
||||
random.shuffle(update_order)
|
||||
for i in update_order:
|
||||
#Cavity distribution parameters
|
||||
self.tau_[i] = 1./self.Sigma_diag[i] - self.eta*self.tau_tilde[i]
|
||||
self.v_[i] = self.mu[i]/self.Sigma_diag[i] - self.eta*self.v_tilde[i]
|
||||
#Marginal moments
|
||||
self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
|
||||
#Site parameters update
|
||||
Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma_diag[i])
|
||||
Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma_diag[i])
|
||||
self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
|
||||
self.v_tilde[i] = self.v_tilde[i] + Delta_v
|
||||
#Posterior distribution parameters update
|
||||
self.LLT = self.LLT + np.outer(self.Kmn[:,i],self.Kmn[:,i])*Delta_tau
|
||||
L = jitchol(self.LLT)
|
||||
V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
|
||||
self.Sigma_diag = np.sum(V*V,-2)
|
||||
si = np.sum(V.T*V[:,i],-1)
|
||||
self.mu = self.mu + (Delta_v-Delta_tau*self.mu[i])*si
|
||||
self.iterations += 1
|
||||
#Sigma recomputation with Cholesky decompositon
|
||||
self.LLT0 = self.LLT0 + np.dot(self.Kmn*self.tau_tilde[None,:],self.Kmn.T)
|
||||
self.L = jitchol(self.LLT)
|
||||
V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
|
||||
V2,info = linalg.flapack.dtrtrs(L.T,V,lower=0)
|
||||
self.Sigma_diag = np.sum(V*V,-2)
|
||||
Knmv_tilde = np.dot(self.Kmn,self.v_tilde)
|
||||
self.mu = np.dot(V2.T,Knmv_tilde)
|
||||
epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
|
||||
epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
|
||||
self.np1.append(self.tau_tilde.copy())
|
||||
self.np2.append(self.v_tilde.copy())
|
||||
return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
|
||||
|
||||
class FITC(EP):
|
||||
def fit_EP(self):
|
||||
"""
|
||||
The expectation-propagation algorithm with sparse pseudo-input.
|
||||
|
|
@ -178,15 +253,15 @@ class FITC(EP_base):
|
|||
sigma_ = 1./tau_
|
||||
mu_ = v_/tau_
|
||||
"""
|
||||
self.tau_ = np.empty(self.N,dtype=np.float64)
|
||||
self.v_ = np.empty(self.N,dtype=np.float64)
|
||||
self.tau_ = np.empty(self.N,dtype=float)
|
||||
self.v_ = np.empty(self.N,dtype=float)
|
||||
|
||||
#Initial values - Marginal moments
|
||||
z = np.empty(self.N,dtype=np.float64)
|
||||
self.Z_hat = np.empty(self.N,dtype=np.float64)
|
||||
phi = np.empty(self.N,dtype=np.float64)
|
||||
mu_hat = np.empty(self.N,dtype=np.float64)
|
||||
sigma2_hat = np.empty(self.N,dtype=np.float64)
|
||||
z = np.empty(self.N,dtype=float)
|
||||
self.Z_hat = np.empty(self.N,dtype=float)
|
||||
phi = np.empty(self.N,dtype=float)
|
||||
mu_hat = np.empty(self.N,dtype=float)
|
||||
sigma2_hat = np.empty(self.N,dtype=float)
|
||||
|
||||
#Approximation
|
||||
epsilon_np1 = 1
|
||||
|
|
@ -238,3 +313,4 @@ class FITC(EP_base):
|
|||
epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
|
||||
self.np1.append(self.tau_tilde.copy())
|
||||
self.np2.append(self.v_tilde.copy())
|
||||
return self.tau_tilde[:,None], self.v_tilde[:,None], self.Z_hat[:,None], self.tau_[:,None], self.v_[:,None]
|
||||
|
|
@ -21,7 +21,28 @@ class likelihood:
|
|||
self.location = location
|
||||
self.scale = scale
|
||||
|
||||
def plot1Da(self,X_new,Mean_new,Var_new,X_u,Mean_u,Var_u):
|
||||
def plot1D(self,X,mean,var,Z=None,mean_Z=None,var_Z=None,samples=0):
|
||||
"""
|
||||
Plot the predictive distribution of the GP model for 1-dimensional inputs
|
||||
|
||||
:param X: The points at which to make a prediction
|
||||
:param Mean: mean values at X
|
||||
:param Var: variance values at X
|
||||
:param Z: Set of points to be highlighted in the plot, i.e. inducing points
|
||||
:param mean_Z: mean values at Z
|
||||
:param var_Z: variance values at Z
|
||||
:samples: Number of samples to plot
|
||||
"""
|
||||
assert X.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X,mean,var.flatten())
|
||||
if samples: #NOTE why don't we put samples as a parameter of gpplot
|
||||
s = np.random.multivariate_normal(mean.flatten(),np.diag(var),samples)
|
||||
pb.plot(X.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
|
||||
#pb.subplot(211)
|
||||
#self.plot1Da(X,mean,var,Z,mean_Z,var_Z)
|
||||
|
||||
|
||||
def plot1Da(self,X,mean,var,Z=None,mean_Z=None,var_Z=None):
|
||||
"""
|
||||
Plot the predictive distribution of the GP model for 1-dimensional inputs
|
||||
|
||||
|
|
@ -32,11 +53,21 @@ class likelihood:
|
|||
:param Mean_u: mean values at X_u
|
||||
:param Var_new: variance values at X_u
|
||||
"""
|
||||
assert X_new.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X_new,Mean_new,Var_new)
|
||||
pb.errorbar(X_u,Mean_u,2*np.sqrt(Var_u),fmt='r+')
|
||||
pb.plot(X_u,Mean_u,'ro')
|
||||
assert X.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X,mean,var.flatten())
|
||||
pb.errorbar(Z.flatten(),mean_Z.flatten(),2*np.sqrt(var_Z.flatten()),fmt='r+')
|
||||
pb.plot(Z,mean_Z,'ro')
|
||||
|
||||
"""
|
||||
def plot1Db(self,X_obs,X,phi,Z=None):
|
||||
assert X_obs.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X,phi,np.zeros(X.shape[0]))
|
||||
pb.plot(X_obs,(self.Y+1)/2,'kx',mew=1.5)
|
||||
pb.ylim(-0.2,1.2)
|
||||
if Z is not None:
|
||||
pb.plot(Z,Z*0+.5,'r|',mew=1.5,markersize=12)
|
||||
|
||||
"""
|
||||
def plot2D(self,X,X_new,F_new,U=None):
|
||||
"""
|
||||
Predictive distribution of the fitted GP model for 2-dimensional inputs
|
||||
|
|
@ -90,20 +121,22 @@ class probit(likelihood):
|
|||
sigma2_hat = 1./tau_i - (phi/((tau_i**2+tau_i)*Z_hat))*(z+phi/Z_hat)
|
||||
return Z_hat, mu_hat, sigma2_hat
|
||||
|
||||
def plot1Db(self,X,X_new,F_new,U=None):
|
||||
assert X.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X_new,F_new,np.zeros(X_new.shape[0]))
|
||||
pb.plot(X,(self.Y+1)/2,'kx',mew=1.5)
|
||||
pb.ylim(-0.2,1.2)
|
||||
if U is not None:
|
||||
pb.plot(U,U*0+.5,'r|',mew=1.5,markersize=12)
|
||||
|
||||
def predictive_mean(self,mu,variance):
|
||||
return stats.norm.cdf(mu/np.sqrt(1+variance))
|
||||
def predictive_mean(self,mu,var):
|
||||
mu = mu.flatten()
|
||||
var = var.flatten()
|
||||
return stats.norm.cdf(mu/np.sqrt(1+var))
|
||||
|
||||
def _log_likelihood_gradients():
|
||||
raise NotImplementedError
|
||||
|
||||
def plot(self,X,phi,X_obs,Z=None):
|
||||
assert X_obs.shape[1] == 1, 'Number of dimensions must be 1'
|
||||
gpplot(X,phi,np.zeros(X.shape[0]))
|
||||
pb.plot(X_obs,(self.Y+1)/2,'kx',mew=1.5)
|
||||
if Z is not None:
|
||||
pb.plot(Z,Z*0+.5,'r|',mew=1.5,markersize=12)
|
||||
pb.ylim(-0.2,1.2)
|
||||
|
||||
class poisson(likelihood):
|
||||
"""
|
||||
Poisson likelihood
|
||||
|
|
|
|||
299
GPy/models/GP.py
Normal file
299
GPy/models/GP.py
Normal file
|
|
@ -0,0 +1,299 @@
|
|||
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
|
||||
# Licensed under the BSD 3-clause license (see LICENSE.txt)
|
||||
|
||||
|
||||
import numpy as np
|
||||
import pylab as pb
|
||||
from .. import kern
|
||||
from ..core import model
|
||||
from ..util.linalg import pdinv,mdot
|
||||
from ..util.plot import gpplot, Tango
|
||||
from ..inference.EP import Full
|
||||
from ..inference.likelihoods import likelihood,probit,poisson,gaussian
|
||||
|
||||
class GP(model):
|
||||
"""
|
||||
Gaussian Process model for regression and EP
|
||||
|
||||
:param X: input observations
|
||||
:param Y: observed values
|
||||
:param kernel: a GPy kernel, defaults to rbf+white
|
||||
:param normalize_X: whether to normalize the input data before computing (predictions will be in original scales)
|
||||
:type normalize_X: False|True
|
||||
:param normalize_Y: whether to normalize the input data before computing (predictions will be in original scales)
|
||||
:type normalize_Y: False|True
|
||||
:param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
|
||||
:rtype: model object
|
||||
:parm likelihood: a GPy likelihood, defaults to gaussian
|
||||
:param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 0.1
|
||||
:param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.]
|
||||
:type powerep: list
|
||||
|
||||
.. Note:: Multiple independent outputs are allowed using columns of Y
|
||||
|
||||
"""
|
||||
#TODO: make beta parameter explicit
|
||||
#TODO: when using EP, predict needs to return 3 values otherwise it just needs 2. At the moment predict returns 3 values in any case.
|
||||
|
||||
def __init__(self,X,Y=None,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None,likelihood=None,epsilon_ep=1e-3,power_ep=[1.,1.]):
|
||||
|
||||
# parse arguments
|
||||
self.Xslices = Xslices
|
||||
self.X = X
|
||||
self.N, self.Q = self.X.shape
|
||||
assert len(self.X.shape)==2
|
||||
if kernel is None:
|
||||
kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
|
||||
else:
|
||||
assert isinstance(kernel, kern.kern)
|
||||
self.kern = kernel
|
||||
|
||||
#here's some simple normalisation
|
||||
if normalize_X:
|
||||
self._Xmean = X.mean(0)[None,:]
|
||||
self._Xstd = X.std(0)[None,:]
|
||||
self.X = (X.copy() - self._Xmean) / self._Xstd
|
||||
if hasattr(self,'Z'):
|
||||
self.Z = (self.Z - self._Xmean) / self._Xstd
|
||||
else:
|
||||
self._Xmean = np.zeros((1,self.X.shape[1]))
|
||||
self._Xstd = np.ones((1,self.X.shape[1]))
|
||||
|
||||
# Y - likelihood related variables, these might change whether using EP or not
|
||||
if likelihood is None:
|
||||
assert Y is not None, "Either Y or likelihood must be defined"
|
||||
self.likelihood = gaussian(Y)
|
||||
else:
|
||||
self.likelihood = likelihood
|
||||
assert len(self.likelihood.Y.shape)==2
|
||||
assert self.X.shape[0] == self.likelihood.Y.shape[0]
|
||||
self.N, self.D = self.likelihood.Y.shape
|
||||
|
||||
if isinstance(self.likelihood,gaussian):
|
||||
self.EP = False
|
||||
self.Y = Y
|
||||
self.beta = 100.#FIXME beta should be an explicit parameter for this model
|
||||
|
||||
# Here's some simple normalisation
|
||||
if normalize_Y:
|
||||
self._Ymean = Y.mean(0)[None,:]
|
||||
self._Ystd = Y.std(0)[None,:]
|
||||
self.Y = (Y.copy()- self._Ymean) / self._Ystd
|
||||
else:
|
||||
self._Ymean = np.zeros((1,self.Y.shape[1]))
|
||||
self._Ystd = np.ones((1,self.Y.shape[1]))
|
||||
|
||||
if self.D > self.N:
|
||||
# then it's more efficient to store YYT
|
||||
self.YYT = np.dot(self.Y, self.Y.T)
|
||||
else:
|
||||
self.YYT = None
|
||||
|
||||
else:
|
||||
# Y is defined after approximating the likelihood
|
||||
self.EP = True
|
||||
self.eta,self.delta = power_ep
|
||||
self.epsilon_ep = epsilon_ep
|
||||
self.beta = np.ones([self.N,self.D])
|
||||
self.Z_ep = 0
|
||||
self.Y = None
|
||||
self._Ymean = np.zeros((1,self.D))
|
||||
self._Ystd = np.ones((1,self.D))
|
||||
|
||||
model.__init__(self)
|
||||
|
||||
def _set_params(self,p):
|
||||
# TODO: add beta when not using EP
|
||||
self.kern._set_params_transformed(p)
|
||||
self.K = self.kern.K(self.X,slices1=self.Xslices)
|
||||
if self.EP:
|
||||
self.K += np.diag(1./self.beta.flatten())
|
||||
#else:
|
||||
# self.beta = p[-1]
|
||||
self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
|
||||
|
||||
def _get_params(self):
|
||||
# TODO: add beta when not using EP
|
||||
return self.kern._get_params_transformed()
|
||||
|
||||
def _get_param_names(self):
|
||||
# TODO: add beta when not using EP
|
||||
return self.kern._get_param_names_transformed()
|
||||
|
||||
def approximate_likelihood(self):
|
||||
"""
|
||||
Approximates a non-gaussian likelihood using Expectation Propagation
|
||||
"""
|
||||
assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
|
||||
self.ep_approx = Full(self.K,self.likelihood,epsilon = self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
self.beta, self.Y, self.Z_ep = self.ep_approx.fit_EP()
|
||||
if self.D > self.N:
|
||||
# then it's more efficient to store YYT
|
||||
self.YYT = np.dot(self.Y, self.Y.T)
|
||||
else:
|
||||
self.YYT = None
|
||||
# Kernel plus noise variance term
|
||||
self.K = self.kern.K(self.X,slices1=self.Xslices) + np.diag(1./self.beta.flatten())
|
||||
self.Ki, self.L, self.Li, self.K_logdet = pdinv(self.K)
|
||||
|
||||
def _model_fit_term(self):
|
||||
"""
|
||||
Computes the model fit using YYT if it's available
|
||||
"""
|
||||
if self.YYT is None:
|
||||
return -0.5*np.sum(np.square(np.dot(self.Li,self.Y)))
|
||||
else:
|
||||
return -0.5*np.sum(np.multiply(self.Ki, self.YYT))
|
||||
|
||||
def log_likelihood(self):
|
||||
"""
|
||||
The log marginal likelihood for an EP model can be written as the log likelihood of
|
||||
a regression model for a new variable Y* = v_tilde/tau_tilde, with a covariance
|
||||
matrix K* = K + diag(1./tau_tilde) plus a normalization term.
|
||||
"""
|
||||
L = -0.5*selff.D*self.K_logdet + self.model_fit_term()
|
||||
if self.EP:
|
||||
L += self.normalisation_term()
|
||||
return L
|
||||
|
||||
def log_likelihood(self):
|
||||
complexity_term = -0.5*self.N*self.D*np.log(2.*np.pi) - 0.5*self.D*self.K_logdet
|
||||
return complexity_term + self._model_fit_term()
|
||||
|
||||
def dL_dK(self):
|
||||
if self.YYT is None:
|
||||
alpha = np.dot(self.Ki,self.Y)
|
||||
dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Ki)
|
||||
else:
|
||||
dL_dK = 0.5*(mdot(self.Ki, self.YYT, self.Ki) - self.D*self.Ki)
|
||||
return dL_dK
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
|
||||
|
||||
def predict(self,Xnew, slices=None, full_cov=False):
|
||||
"""
|
||||
Predict the function(s) at the new point(s) Xnew.
|
||||
|
||||
Arguments
|
||||
---------
|
||||
:param Xnew: The points at which to make a prediction
|
||||
:type Xnew: np.ndarray, Nnew x self.Q
|
||||
:param slices: specifies which outputs kernel(s) the Xnew correspond to (see below)
|
||||
:type slices: (None, list of slice objects, list of ints)
|
||||
:param full_cov: whether to return the folll covariance matrix, or just the diagonal
|
||||
:type full_cov: bool
|
||||
:rtype: posterior mean, a Numpy array, Nnew x self.D
|
||||
:rtype: posterior variance, a Numpy array, Nnew x Nnew x (self.D)
|
||||
|
||||
.. Note:: "slices" specifies how the the points X_new co-vary wich the training points.
|
||||
|
||||
- If None, the new points covary throigh every kernel part (default)
|
||||
- If a list of slices, the i^th slice specifies which data are affected by the i^th kernel part
|
||||
- If a list of booleans, specifying which kernel parts are active
|
||||
|
||||
If full_cov and self.D > 1, the return shape of var is Nnew x Nnew x self.D. If self.D == 1, the return shape is Nnew x Nnew.
|
||||
This is to allow for different normalisations of the output dimensions.
|
||||
|
||||
"""
|
||||
|
||||
#normalise X values
|
||||
Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
|
||||
mu, var, phi = self._raw_predict(Xnew, slices, full_cov)
|
||||
|
||||
#un-normalise
|
||||
mu = mu*self._Ystd + self._Ymean
|
||||
if full_cov:
|
||||
if self.D==1:
|
||||
var *= np.square(self._Ystd)
|
||||
else:
|
||||
var = var[:,:,None] * np.square(self._Ystd)
|
||||
else:
|
||||
if self.D==1:
|
||||
var *= np.square(np.squeeze(self._Ystd))
|
||||
else:
|
||||
var = var[:,None] * np.square(self._Ystd)
|
||||
|
||||
return mu,var,phi
|
||||
|
||||
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.X,_Xnew, slices1=self.Xslices,slices2=slices)
|
||||
mu = np.dot(np.dot(Kx.T,self.Ki),self.Y)
|
||||
KiKx = np.dot(self.Ki,Kx)
|
||||
if full_cov:
|
||||
Kxx = self.kern.K(_Xnew, slices1=slices,slices2=slices)
|
||||
var = Kxx - np.dot(KiKx.T,Kx)
|
||||
else:
|
||||
Kxx = self.kern.Kdiag(_Xnew, slices=slices)
|
||||
var = Kxx - np.sum(np.multiply(KiKx,Kx),0)
|
||||
phi = None if not self.EP else self.likelihood.predictive_mean(mu,var)
|
||||
return mu, var, phi
|
||||
|
||||
def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
|
||||
"""
|
||||
:param samples: the number of a posteriori samples to plot
|
||||
:param which_data: which if the training data to plot (default all)
|
||||
:type which_data: 'all' or a slice object to slice self.X, self.Y
|
||||
:param plot_limits: The limits of the plot. If 1D [xmin,xmax], if 2D [[xmin,ymin],[xmax,ymax]]. Defaluts to data limits
|
||||
:param which_functions: which of the kernel functions to plot (additively)
|
||||
:type which_functions: list of bools
|
||||
:param resolution: the number of intervals to sample the GP on. Defaults to 200 in 1D and 50 (a 50x50 grid) in 2D
|
||||
|
||||
Plot the posterior of the GP.
|
||||
- In one dimension, the function is plotted with a shaded region identifying two standard deviations.
|
||||
- In two dimsensions, a contour-plot shows the mean predicted function
|
||||
- In higher dimensions, we've no implemented this yet !TODO!
|
||||
|
||||
Can plot only part of the data and part of the posterior functions using which_data and which_functions
|
||||
"""
|
||||
if which_functions=='all':
|
||||
which_functions = [True]*self.kern.Nparts
|
||||
if which_data=='all':
|
||||
which_data = slice(None)
|
||||
|
||||
X = self.X[which_data,:]
|
||||
Y = self.Y[which_data,:]
|
||||
|
||||
Xorig = X*self._Xstd + self._Xmean
|
||||
Yorig = Y*self._Ystd + self._Ymean #NOTE For EP this is v_tilde/beta
|
||||
|
||||
if plot_limits is None:
|
||||
xmin,xmax = Xorig.min(0),Xorig.max(0)
|
||||
xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
|
||||
elif len(plot_limits)==2:
|
||||
xmin, xmax = plot_limits
|
||||
else:
|
||||
raise ValueError, "Bad limits for plotting"
|
||||
|
||||
if self.X.shape[1]==1:
|
||||
Xnew = np.linspace(xmin,xmax,resolution or 200)[:,None]
|
||||
m,v,phi = self.predict(Xnew,slices=which_functions)
|
||||
if self.EP:
|
||||
pb.subplot(211)
|
||||
gpplot(Xnew,m,v)
|
||||
|
||||
if samples: #NOTE why don't we put samples as a parameter of gpplot
|
||||
s = np.random.multivariate_normal(m.flatten(),np.diag(v),samples)
|
||||
pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
|
||||
pb.plot(Xorig,Yorig,'kx',mew=1.5)
|
||||
pb.xlim(xmin,xmax)
|
||||
|
||||
if self.EP:
|
||||
pb.subplot(212)
|
||||
self.likelihood.plot(Xnew,phi,self.X)
|
||||
pb.xlim(xmin,xmax)
|
||||
|
||||
elif self.X.shape[1]==2:
|
||||
resolution = 50 or resolution
|
||||
xx,yy = np.mgrid[xmin[0]:xmax[0]:1j*resolution,xmin[1]:xmax[1]:1j*resolution]
|
||||
Xtest = np.vstack((xx.flatten(),yy.flatten())).T
|
||||
zz,vv,phi = self.predict(Xtest,slices=which_functions)
|
||||
zz = zz.reshape(resolution,resolution)
|
||||
pb.contour(xx,yy,zz,vmin=zz.min(),vmax=zz.max(),cmap=pb.cm.jet)
|
||||
pb.scatter(Xorig[:,0],Xorig[:,1],40,Yorig,linewidth=0,cmap=pb.cm.jet,vmin=zz.min(),vmax=zz.max())
|
||||
pb.xlim(xmin[0],xmax[0])
|
||||
pb.ylim(xmin[1],xmax[1])
|
||||
|
||||
else:
|
||||
raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
|
||||
|
|
@ -1,160 +0,0 @@
|
|||
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
|
||||
# Licensed under the BSD 3-clause license (see LICENSE.txt)
|
||||
|
||||
|
||||
import numpy as np
|
||||
import pylab as pb
|
||||
from scipy import stats, linalg
|
||||
from .. import kern
|
||||
from ..inference.Expectation_Propagation import Full
|
||||
from ..inference.likelihoods import likelihood,probit#,poisson,gaussian
|
||||
from ..core import model
|
||||
from ..util.linalg import pdinv,jitchol
|
||||
from ..util.plot import gpplot
|
||||
|
||||
class GP_EP(model):
|
||||
def __init__(self,X,likelihood,kernel=None,epsilon_ep=1e-3,epsion_em=.1,powerep=[1.,1.]):
|
||||
"""
|
||||
Simple Gaussian Process with Non-Gaussian likelihood
|
||||
|
||||
Arguments
|
||||
---------
|
||||
:param X: input observations (NxD numpy.darray)
|
||||
:param likelihood: a GPy likelihood (likelihood class)
|
||||
:param kernel: a GPy kernel (kern class)
|
||||
:param epsilon_ep: convergence criterion for the Expectation Propagation algorithm, defaults to 0.1 (float)
|
||||
:param powerep: power-EP parameters [$\eta$,$\delta$], defaults to [1.,1.] (list)
|
||||
:rtype: GPy model class.
|
||||
"""
|
||||
if kernel is None:
|
||||
kernel = kern.rbf(X.shape[1]) + kern.bias(X.shape[1]) + kern.white(X.shape[1])
|
||||
|
||||
assert isinstance(kernel,kern.kern), 'kernel is not a kern instance'
|
||||
self.likelihood = likelihood
|
||||
self.Y = self.likelihood.Y
|
||||
self.kernel = kernel
|
||||
self.X = X
|
||||
self.N, self.D = self.X.shape
|
||||
self.eta,self.delta = powerep
|
||||
self.epsilon_ep = epsilon_ep
|
||||
self.jitter = 1e-12
|
||||
self.K = self.kernel.K(self.X)
|
||||
model.__init__(self)
|
||||
|
||||
def _set_params(self,p):
|
||||
self.kernel._set_params_transformed(p)
|
||||
|
||||
def _get_params(self):
|
||||
return self.kernel._get_params_transformed()
|
||||
|
||||
def _get_param_names(self):
|
||||
return self.kernel._get_param_names_transformed()
|
||||
|
||||
def approximate_likelihood(self):
|
||||
self.ep_approx = Full(self.K,self.likelihood,epsilon=self.epsilon_ep,powerep=[self.eta,self.delta])
|
||||
self.ep_approx.fit_EP()
|
||||
|
||||
def posterior_param(self):
|
||||
self.K = self.kernel.K(self.X)
|
||||
self.Sroot_tilde_K = np.sqrt(self.ep_approx.tau_tilde)[:,None]*self.K
|
||||
B = np.eye(self.N) + np.sqrt(self.ep_approx.tau_tilde)[None,:]*self.Sroot_tilde_K
|
||||
#self.L = np.linalg.cholesky(B)
|
||||
self.L = jitchol(B)
|
||||
V,info = linalg.flapack.dtrtrs(self.L,self.Sroot_tilde_K,lower=1)
|
||||
self.Sigma = self.K - np.dot(V.T,V)
|
||||
self.mu = np.dot(self.Sigma,self.ep_approx.v_tilde)
|
||||
|
||||
def log_likelihood(self):
|
||||
"""
|
||||
Returns
|
||||
-------
|
||||
The EP approximation to the log-marginal likelihood
|
||||
"""
|
||||
self.posterior_param()
|
||||
mu_ = self.ep_approx.v_/self.ep_approx.tau_
|
||||
L1 =.5*sum(np.log(1+self.ep_approx.tau_tilde*1./self.ep_approx.tau_))-sum(np.log(np.diag(self.L)))
|
||||
L2A =.5*np.sum((self.Sigma-np.diag(1./(self.ep_approx.tau_+self.ep_approx.tau_tilde))) * np.dot(self.ep_approx.v_tilde[:,None],self.ep_approx.v_tilde[None,:]))
|
||||
L2B = .5*np.dot(mu_*(self.ep_approx.tau_/(self.ep_approx.tau_tilde+self.ep_approx.tau_)),self.ep_approx.tau_tilde*mu_ - 2*self.ep_approx.v_tilde)
|
||||
L3 = sum(np.log(self.ep_approx.Z_hat))
|
||||
return L1 + L2A + L2B + L3
|
||||
|
||||
def _log_likelihood_gradients(self):
|
||||
dK_dp = self.kernel.dK_dtheta(self.X)
|
||||
self.dK_dp = dK_dp
|
||||
aux1,info_1 = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
|
||||
b = self.ep_approx.v_tilde - np.sqrt(self.ep_approx.tau_tilde)*linalg.flapack.dtrtrs(self.L.T,aux1)[0]
|
||||
U,info_u = linalg.flapack.dtrtrs(self.L,np.diag(np.sqrt(self.ep_approx.tau_tilde)),lower=1)
|
||||
dL_dK = 0.5*(np.outer(b,b)-np.dot(U.T,U))
|
||||
self.dL_dK = dL_dK
|
||||
return np.array([np.sum(dK_dpi*dL_dK) for dK_dpi in dK_dp.T])
|
||||
|
||||
def predict(self,X):
|
||||
#TODO: check output dimensions
|
||||
self.posterior_param()
|
||||
K_x = self.kernel.K(self.X,X)
|
||||
Kxx = self.kernel.K(X)
|
||||
aux1,info = linalg.flapack.dtrtrs(self.L,np.dot(self.Sroot_tilde_K,self.ep_approx.v_tilde),lower=1)
|
||||
aux2,info = linalg.flapack.dtrtrs(self.L.T, aux1,lower=0)
|
||||
zeta = np.sqrt(self.ep_approx.tau_tilde)*aux2
|
||||
f = np.dot(K_x.T,self.ep_approx.v_tilde-zeta)
|
||||
v,info = linalg.flapack.dtrtrs(self.L,np.sqrt(self.ep_approx.tau_tilde)[:,None]*K_x,lower=1)
|
||||
variance = Kxx - np.dot(v.T,v)
|
||||
vdiag = np.diag(variance)
|
||||
y=self.likelihood.predictive_mean(f,vdiag)
|
||||
return f,vdiag,y
|
||||
|
||||
def plot(self):
|
||||
"""
|
||||
Plot the fitted model: training function values, inducing points used, mean estimate and confidence intervals.
|
||||
"""
|
||||
if self.X.shape[1]==1:
|
||||
pb.figure()
|
||||
xmin,xmax = self.X.min(),self.X.max()
|
||||
xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
|
||||
Xnew = np.linspace(xmin,xmax,100)[:,None]
|
||||
mu_f, var_f, mu_phi = self.predict(Xnew)
|
||||
pb.subplot(211)
|
||||
self.likelihood.plot1Da(X_new=Xnew,Mean_new=mu_f,Var_new=var_f,X_u=self.X,Mean_u=self.mu,Var_u=np.diag(self.Sigma))
|
||||
pb.subplot(212)
|
||||
self.likelihood.plot1Db(self.X,Xnew,mu_phi)
|
||||
elif self.X.shape[1]==2:
|
||||
pb.figure()
|
||||
x1min,x1max = self.X[:,0].min(0),self.X[:,0].max(0)
|
||||
x2min,x2max = self.X[:,1].min(0),self.X[:,1].max(0)
|
||||
x1min, x1max = x1min-0.2*(x1max-x1min), x1max+0.2*(x1max-x1min)
|
||||
x2min, x2max = x2min-0.2*(x2max-x2min), x2max+0.2*(x1max-x1min)
|
||||
axis1 = np.linspace(x1min,x1max,50)
|
||||
axis2 = np.linspace(x2min,x2max,50)
|
||||
XX1, XX2 = [e.flatten() for e in np.meshgrid(axis1,axis2)]
|
||||
Xnew = np.c_[XX1.flatten(),XX2.flatten()]
|
||||
f,v,p = self.predict(Xnew)
|
||||
self.likelihood.plot2D(self.X,Xnew,p)
|
||||
else:
|
||||
raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
|
||||
|
||||
def em(self,max_f_eval=1e4,epsilon=.1,plot_all=False): #TODO check this makes sense
|
||||
"""
|
||||
Fits sparse_EP and optimizes the hyperparametes iteratively until convergence is achieved.
|
||||
"""
|
||||
self.epsilon_em = epsilon
|
||||
log_likelihood_change = self.epsilon_em + 1.
|
||||
self.parameters_path = [self.kernel._get_params()]
|
||||
self.approximate_likelihood()
|
||||
self.site_approximations_path = [[self.ep_approx.tau_tilde,self.ep_approx.v_tilde]]
|
||||
self.log_likelihood_path = [self.log_likelihood()]
|
||||
iteration = 0
|
||||
while log_likelihood_change > self.epsilon_em:
|
||||
print 'EM iteration', iteration
|
||||
self.optimize(max_f_eval = max_f_eval)
|
||||
log_likelihood_new = self.log_likelihood()
|
||||
log_likelihood_change = log_likelihood_new - self.log_likelihood_path[-1]
|
||||
if log_likelihood_change < 0:
|
||||
print 'log_likelihood decrement'
|
||||
self.kernel._set_params_transformed(self.parameters_path[-1])
|
||||
self.kernM._set_params_transformed(self.parameters_path[-1])
|
||||
else:
|
||||
self.approximate_likelihood()
|
||||
self.log_likelihood_path.append(self.log_likelihood())
|
||||
self.parameters_path.append(self.kernel._get_params())
|
||||
self.site_approximations_path.append([self.ep_approx.tau_tilde,self.ep_approx.v_tilde])
|
||||
iteration += 1
|
||||
|
|
@ -6,7 +6,8 @@ from GP_regression import GP_regression
|
|||
from sparse_GP_regression import sparse_GP_regression
|
||||
from GPLVM import GPLVM
|
||||
from warped_GP import warpedGP
|
||||
from GP_EP import GP_EP
|
||||
from generalized_FITC import generalized_FITC
|
||||
from sparse_GPLVM import sparse_GPLVM
|
||||
from uncollapsed_sparse_GP import uncollapsed_sparse_GP
|
||||
from GP import GP
|
||||
from sparse_GP import sparse_GP
|
||||
|
|
|
|||
|
|
@ -9,7 +9,8 @@ from .. import kern
|
|||
from ..core import model
|
||||
from ..util.linalg import pdinv,mdot
|
||||
from ..util.plot import gpplot
|
||||
from ..inference.Expectation_Propagation import FITC
|
||||
#from ..inference.Expectation_Propagation import FITC
|
||||
from ..inference.EP import FITC
|
||||
from ..inference.likelihoods import likelihood,probit
|
||||
|
||||
class generalized_FITC(model):
|
||||
|
|
|
|||
290
GPy/models/sparse_GP.py
Normal file
290
GPy/models/sparse_GP.py
Normal file
|
|
@ -0,0 +1,290 @@
|
|||
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
|
||||
# Licensed under the BSD 3-clause license (see LICENSE.txt)
|
||||
|
||||
import numpy as np
|
||||
import pylab as pb
|
||||
from ..util.linalg import mdot, jitchol, chol_inv, pdinv
|
||||
from ..util.plot import gpplot
|
||||
from .. import kern
|
||||
from GP import GP
|
||||
from ..inference.EP import Full
|
||||
from ..inference.likelihoods import likelihood,probit,poisson,gaussian
|
||||
|
||||
#Still TODO:
|
||||
# make use of slices properly (kernel can now do this)
|
||||
# enable heteroscedatic noise (kernel will need to compute psi2 as a (NxMxM) array)
|
||||
|
||||
class sparse_GP(GP):
|
||||
"""
|
||||
Variational sparse GP model (Regression)
|
||||
|
||||
:param X: inputs
|
||||
:type X: np.ndarray (N x Q)
|
||||
:param Y: observed data
|
||||
:type Y: np.ndarray of observations (N x D)
|
||||
:param kernel : the kernel/covariance function. See link kernels
|
||||
:type kernel: a GPy kernel
|
||||
:param Z: inducing inputs (optional, see note)
|
||||
:type Z: np.ndarray (M x Q) | None
|
||||
:param X_uncertainty: The uncertainty in the measurements of X (Gaussian variance)
|
||||
:type X_uncertainty: np.ndarray (N x Q) | None
|
||||
:param Zslices: slices for the inducing inputs (see slicing TODO: link)
|
||||
:param M : Number of inducing points (optional, default 10. Ignored if Z is not None)
|
||||
:type M: int
|
||||
:param beta: noise precision. TODO> ignore beta if doing EP
|
||||
:type beta: float
|
||||
:param normalize_(X|Y) : whether to normalize the data before computing (predictions will be in original scales)
|
||||
:type normalize_(X|Y): bool
|
||||
"""
|
||||
|
||||
def __init__(self,X,Y=None,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.]):
|
||||
|
||||
if Z is None:
|
||||
self.Z = np.random.permutation(X.copy())[:M]
|
||||
self.M = M
|
||||
else:
|
||||
assert Z.shape[1]==X.shape[1]
|
||||
self.Z = Z
|
||||
self.M = Z.shape[0]
|
||||
if X_uncertainty is None:
|
||||
self.has_uncertain_inputs=False
|
||||
else:
|
||||
assert X_uncertainty.shape==X.shape
|
||||
self.has_uncertain_inputs=True
|
||||
self.X_uncertainty = X_uncertainty
|
||||
|
||||
GP.__init__(self, X=X, Y=Y, kernel=kernel, normalize_X=normalize_X, normalize_Y=normalize_Y,likelihood=likelihood,epsilon_ep=epsilon_ep,epsilon_em=epsilon_em,power_ep=power_ep)
|
||||
self.trYYT = np.sum(np.square(self.Y)) if not self.EP else None
|
||||
|
||||
|
||||
#normalise X uncertainty also
|
||||
if self.has_uncertain_inputs:
|
||||
self.X_uncertainty /= np.square(self._Xstd)
|
||||
|
||||
def _set_params(self, p):
|
||||
if not self.EP:
|
||||
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
|
||||
self.beta = p[self.M*self.Q]
|
||||
self.kern._set_params(p[self.Z.size + 1:])
|
||||
self.beta2 = self.beta**2
|
||||
self._compute_kernel_matrices()
|
||||
self._computations()
|
||||
else:
|
||||
self.Z = p[:self.M*self.Q].reshape(self.M, self.Q)
|
||||
self.kern._set_params(p[self.Z.size:])
|
||||
#self._compute_kernel_matrices() this is replaced by _ep_kernel_matrices
|
||||
self._ep_kernel_matrices()
|
||||
self._ep_computations()
|
||||
|
||||
def _compute_kernel_matrices(self):
|
||||
# kernel computations, using BGPLVM notation
|
||||
#TODO: slices for psi statistics (easy enough)
|
||||
|
||||
self.Kmm = self.kern.K(self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
if self.hetero_noise:
|
||||
raise NotImplementedError, "uncertain ips and het noise not yet supported"
|
||||
else:
|
||||
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
|
||||
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
|
||||
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty)
|
||||
else:
|
||||
if self.hetero_noise:
|
||||
print "rick's stuff here"
|
||||
else:
|
||||
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices).sum()
|
||||
self.psi1 = self.kern.K(self.Z,self.X)
|
||||
self.psi2 = np.dot(self.psi1,self.psi1.T)
|
||||
|
||||
def _computations(self):
|
||||
# TODO find routine to multiply triangular matrices
|
||||
self.V = self.beta*self.Y
|
||||
self.psi1V = np.dot(self.psi1, self.V)
|
||||
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
|
||||
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
|
||||
self.A = mdot(self.Lmi, self.beta*self.psi2, self.Lmi.T)
|
||||
self.B = np.eye(self.M) + self.A
|
||||
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
|
||||
self.LLambdai = np.dot(self.LBi, self.Lmi)
|
||||
self.trace_K = self.psi0 - np.trace(self.A)/self.beta
|
||||
self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
|
||||
self.C = mdot(self.LLambdai, self.psi1V)
|
||||
self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
|
||||
|
||||
# Compute dL_dpsi
|
||||
self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
|
||||
self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
|
||||
self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
|
||||
# Compute dL_dKmm
|
||||
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
|
||||
self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*self.beta*mdot(self.LBL_inv, self.psi2, self.Kmmi) + self.Kmmi) # dC
|
||||
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
|
||||
|
||||
def approximate_likelihood(self):
|
||||
assert not isinstance(self.likelihood, gaussian), "EP is only available for non-gaussian likelihoods"
|
||||
if self.ep_proxy == 'DTC':
|
||||
self.ep_approx = DTC(self.Kmm,self.likelihood,self.psi1,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
elif self.ep_proxy == 'FITC':
|
||||
self.Knn_diag = self.kern.psi0(self.Z,self.X, self.X_uncertainty) #TODO psi0 already calculates this
|
||||
self.ep_approx = FITC(self.Kmm,self.likelihood,self.psi1,self.Knn_diag,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
else:
|
||||
self.ep_approx = Full(self.X,self.likelihood,self.kernel,inducing=None,epsilon=self.epsilon_ep,power_ep=[self.eta,self.delta])
|
||||
self.beta, self.v_tilde, self.Z_hat, self.tau_, self.v_=self.ep_approx.fit_EP()
|
||||
self._ep_kernel_matrices()
|
||||
self._computations()
|
||||
|
||||
def _ep_kernel_matrices(self):
|
||||
self.Kmm = self.kern.K(self.Z)
|
||||
if self.has_uncertain_inputs:
|
||||
self.psi0 = self.kern.psi0(self.Z,self.X, self.X_uncertainty).sum()
|
||||
self.psi1 = self.kern.psi1(self.Z,self.X, self.X_uncertainty).T
|
||||
self.psi2 = self.kern.psi2(self.Z,self.X, self.X_uncertainty) #FIXME include beta
|
||||
else:
|
||||
self.psi0 = self.kern.Kdiag(self.X,slices=self.Xslices)
|
||||
self.psi1 = self.kern.K(self.Z,self.X)
|
||||
self.psi2 = np.dot(self.psi1,self.psi1.T)
|
||||
self.psi2_beta_scaled = np.dot(self.psi1,self.beta*self.psi1.T)
|
||||
|
||||
def _ep_computations(self):
|
||||
# Y: EP likelihood is defined as a regression model for mu_tilde
|
||||
self.Y = self.v_tilde/self.beta
|
||||
self._Ymean = np.zeros((1,self.Y.shape[1]))
|
||||
self._Ystd = np.ones((1,self.Y.shape[1]))
|
||||
self.trbetaYYT = np.sum(self.beta*np.square(self.Y))
|
||||
if self.D > self.N:
|
||||
# then it's more efficient to store YYT
|
||||
self.YYT = np.dot(self.Y, self.Y.T)
|
||||
else:
|
||||
self.YYT = None
|
||||
self.mu_ = self.v_/self.tau_
|
||||
# TODO find routine to multiply triangular matrices
|
||||
self.V = self.beta*self.Y
|
||||
self.psi1V = np.dot(self.psi1, self.V)
|
||||
self.psi1VVpsi1 = np.dot(self.psi1V, self.psi1V.T)
|
||||
self.Kmmi, self.Lm, self.Lmi, self.Kmm_logdet = pdinv(self.Kmm)
|
||||
#self.A = mdot(self.Lmi, self.beta*self.psi2, self.Lmi.T)
|
||||
self.A = mdot(self.Lmi, self.psi2_beta_scaled, self.Lmi.T)
|
||||
self.B = np.eye(self.M) + self.A
|
||||
self.Bi, self.LB, self.LBi, self.B_logdet = pdinv(self.B)
|
||||
self.LLambdai = np.dot(self.LBi, self.Lmi)
|
||||
self.trace_K = self.psi0.sum() - np.trace(self.A)
|
||||
self.LBL_inv = mdot(self.Lmi.T, self.Bi, self.Lmi)
|
||||
self.C = mdot(self.LLambdai, self.psi1V)
|
||||
self.G = mdot(self.LBL_inv, self.psi1VVpsi1, self.LBL_inv.T)
|
||||
|
||||
# Compute dL_dpsi
|
||||
#self.dL_dpsi0 = - 0.5 * self.D * self.beta * np.ones(self.N)
|
||||
self.dL_dpsi0 = - 0.5 * self.D * self.beta.flatten() * np.ones(self.N) #TODO check
|
||||
self.dL_dpsi1 = mdot(self.LLambdai.T,self.C,self.V.T)
|
||||
#self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
self.dL_dpsi2 = - 0.5 * self.beta * (self.D*(self.LBL_inv - self.Kmmi) + self.G)
|
||||
|
||||
# Compute dL_dKmm
|
||||
self.dL_dKmm = -0.5 * self.D * mdot(self.Lmi.T, self.A, self.Lmi) # dB
|
||||
self.dL_dKmm += -0.5 * self.D * (- self.LBL_inv - 2.*self.beta*mdot(self.LBL_inv, self.psi2, self.Kmmi) + self.Kmmi) # dC
|
||||
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
|
||||
|
||||
def _get_params(self):
|
||||
if not self.EP:
|
||||
return np.hstack([self.Z.flatten(),self.beta,self.kern._get_params_transformed()])
|
||||
else:
|
||||
return np.hstack([self.Z.flatten(),self.kern._get_params_transformed()])
|
||||
|
||||
def _get_param_names(self):
|
||||
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))
|
||||
if self.hetero_noise:
|
||||
A = foo
|
||||
B = bar
|
||||
D = -0.5*self.trbetaYYT
|
||||
else:
|
||||
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
|
||||
D = -0.5*self.beta*self.trYYT
|
||||
C = -0.5*self.D * self.B_logdet
|
||||
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,None#TODO add phi for EP
|
||||
|
||||
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)
|
||||
GP.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