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211 lines
8.1 KiB
Python
211 lines
8.1 KiB
Python
# 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 .. import kern
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from ..core import model
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from ..util.linalg import pdinv,mdot
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from ..util.plot import gpplot, Tango
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class GP_regression(model):
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"""
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Gaussian Process model for regression
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:param X: input observations
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:param Y: observed values
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:param kernel: a GPy kernel, defaults to rbf+white
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:param normalize_X: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_X: False|True
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:param normalize_Y: whether to normalize the input data before computing (predictions will be in original scales)
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:type normalize_Y: False|True
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:param Xslices: how the X,Y data co-vary in the kernel (i.e. which "outputs" they correspond to). See (link:slicing)
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:rtype: model object
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.. Note:: Multiple independent outputs are allowed using columns of Y
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"""
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def __init__(self,X,Y,kernel=None,normalize_X=False,normalize_Y=False, Xslices=None):
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if kernel is None:
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kernel = kern.rbf(X.shape[1]) + kern.white(X.shape[1])
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# parse arguments
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self.Xslices = Xslices
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assert isinstance(kernel, kern.kern)
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self.kern = kernel
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self.X = X
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self.Y = Y
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assert len(self.X.shape)==2
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assert len(self.Y.shape)==2
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assert self.X.shape[0] == self.Y.shape[0]
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self.N, self.D = self.Y.shape
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self.N, self.Q = self.X.shape
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#here's some simple normalisation
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if normalize_X:
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self._Xmean = X.mean(0)[None,:]
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self._Xstd = X.std(0)[None,:]
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self.X = (X.copy()- self._Xmean) / self._Xstd
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else:
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self._Xmean = np.zeros((1,self.X.shape[1]))
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self._Xstd = np.ones((1,self.X.shape[1]))
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if normalize_Y:
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self._Ymean = Y.mean(0)[None,:]
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self._Ystd = Y.std(0)[None,:]
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self.Y = (Y.copy()- self._Ymean) / self._Ystd
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else:
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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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if self.D > self.N:
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# then it's more efficient to store Youter
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self.Youter = np.dot(self.Y, self.Y.T)
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else:
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self.Youter = None
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model.__init__(self)
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def set_param(self,p):
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self.kern.expand_param(p)
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self.K = self.kern.K(self.X,slices1=self.Xslices)
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self.Ki,self.hld = pdinv(self.K)
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def get_param(self):
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return self.kern.extract_param()
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def get_param_names(self):
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return self.kern.extract_param_names()
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def _model_fit_term(self):
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"""
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Computes the model fit using Youter if it's available
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"""
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if self.Youter is None:
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return -0.5*np.trace(mdot(self.Y.T,self.Ki,self.Y))
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else:
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return -0.5*np.sum(np.multiply(self.Ki, self.Y))
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def log_likelihood(self):
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complexity_term = -0.5*self.N*self.D*np.log(2.*np.pi) - self.D*self.hld
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return complexity_term + self._model_fit_term()
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def dL_dK(self):
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if self.Youter is None:
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alpha = np.dot(self.Ki,self.Y)
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dL_dK = 0.5*(np.dot(alpha,alpha.T)-self.D*self.Ki)
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else:
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dL_dK = 0.5*(mdot(self.Ki, self.Youter, self.Ki) - self.D*self.Ki)
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return dL_dK
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def log_likelihood_gradients(self):
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return self.kern.dK_dtheta(partial=self.dL_dK(),X=self.X)
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def predict(self,Xnew, slices=None):
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"""
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Predict the function(s) at the new point(s) Xnew.
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Arguments
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---------
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:param Xnew: The points at which to make a prediction
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:type Xnew: np.ndarray, Nnew x self.Q
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:param slices: specifies which outputs kernel(s) the Xnew correspond to (see below)
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:type slices: (None, list of slice objects, list of ints)
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:rtype: posterior mean, a Numpy array, Nnew x self.D
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:rtype: posterior variance, a Numpy array, Nnew x Nnew x (self.D)
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.. Note:: "slices" specifies how the the points X_new co-vary wich the training points.
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- If None, the new points covary throigh every kernel part (default)
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- If a list of slices, the i^th slice specifies which data are affected by the i^th kernel part
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- If a list of booleans, specifying which kernel parts are active
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If 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.
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This is to allow for different normalisations of the output dimensions.
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"""
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#normalise X values
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Xnew = (Xnew.copy() - self._Xmean) / self._Xstd
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mu, var = self._raw_predict(Xnew,slices)
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#un-normalise
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mu = mu*self._Ystd + self._Ymean
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if self.D==1:
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var *= np.square(self._Ystd)
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else:
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var = var[:,:,None] * np.square(self._Ystd)
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return mu,var
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def _raw_predict(self,_Xnew,slices):
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"""Internal helper function for making predictions, does not account for normalisation"""
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Kx = self.kern.K(self.X,_Xnew, slices1=self.Xslices,slices2=slices)
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Kxx = self.kern.K(_Xnew, slices1=slices,slices2=slices)
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mu = np.dot(np.dot(Kx.T,self.Ki),self.Y)
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var = Kxx - np.dot(np.dot(Kx.T,self.Ki),Kx)
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return mu, var
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def plot(self,samples=0,plot_limits=None,which_data='all',which_functions='all',resolution=None):
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"""
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:param samples: the number of a posteriori samples to plot
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:param which_data: which if the training data to plot (default all)
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:type which_data: 'all' or a slice object to slice self.X, self.Y
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:param plot_limits: The limits of the plot. If 1D [xmin,xmax], if 2D [[xmin,ymin],[xmax,ymax]]. Defaluts to data limits
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:param which_functions: which of the kernel functions to plot (additively)
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:type which_functions: list of bools
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:param resolution: the number of intervals to sample the GP on. Defaults to 200 in 1D and 50 (a 50x50 grid) in 2D
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Plot the posterior of the GP.
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- In one dimension, the function is plotted with a shaded region identifying two standard deviations.
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- In two dimsensions, a contour-plot shows the mean predicted function
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- In higher dimensions, we've no implemented this yet !TODO!
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Can plot only part of the data and part of the posterior functions using which_data and which_functions
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"""
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if which_functions=='all':
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which_functions = [True]*self.kern.Nparts
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if which_data=='all':
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which_data = slice(None)
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X = self.X[which_data,:]
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Y = self.Y[which_data,:]
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Xorig = X*self._Xstd + self._Xmean
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Yorig = Y*self._Ystd + self._Ymean
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if plot_limits is None:
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xmin,xmax = Xorig.min(0),Xorig.max(0)
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xmin, xmax = xmin-0.2*(xmax-xmin), xmax+0.2*(xmax-xmin)
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elif len(plot_limits)==2:
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xmin, xmax = plot_limits
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else:
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raise ValueError, "Bad limits for plotting"
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if self.X.shape[1]==1:
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Xnew = np.linspace(xmin,xmax,resolution or 200)[:,None]
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m,v = self.predict(Xnew,slices=which_functions)
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gpplot(Xnew,m,v)
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if samples:
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s = np.random.multivariate_normal(m.flatten(),v,samples)
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pb.plot(Xnew.flatten(),s.T, alpha = 0.4, c='#3465a4', linewidth = 0.8)
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pb.plot(Xorig,Yorig,'kx',mew=1.5)
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pb.xlim(xmin,xmax)
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elif self.X.shape[1]==2:
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resolution = 50 or resolution
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xx,yy = np.mgrid[xmin[0]:xmax[0]:1j*resolution,xmin[1]:xmax[1]:1j*resolution]
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Xtest = np.vstack((xx.flatten(),yy.flatten())).T
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zz,vv = self.predict(Xtest,slices=which_functions)
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zz = zz.reshape(resolution,resolution)
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pb.contour(xx,yy,zz,vmin=zz.min(),vmax=zz.max(),cmap=pb.cm.jet)
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pb.scatter(Xorig[:,0],Xorig[:,1],40,Yorig,linewidth=0,cmap=pb.cm.jet,vmin=zz.min(),vmax=zz.max())
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pb.xlim(xmin[0],xmax[0])
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pb.ylim(xmin[1],xmax[1])
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else:
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raise NotImplementedError, "Cannot plot GPs with more than two input dimensions"
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