scikit-learn · Dec 13, 2010 · Nov 29, 2010 · Dec 9, 2010 · Dec 12, 2010 · Dec 12, 2010
diff --git a/examples/gaussian_process/plot_gp_diabetes_dataset.py b/examples/gaussian_process/plot_gp_diabetes_dataset.py
@@ -2,69 +2,52 @@
 # -*- coding: utf-8 -*-
 
 """
-=========================================================================
+========================================================================
 Gaussian Processes regression: goodness-of-fit on the 'diabetes' dataset
-=========================================================================
+========================================================================
 
 This example consists in fitting a Gaussian Process model onto the diabetes
 dataset.
-WARNING: This is quite time consuming (about 2 minutes overall runtime).
 
-The corelation parameters are given in order to maximize the generalization
-capacity of the model. We assumed an anisotropic squared exponential
-correlation model with a constant regression model. We also used a
-nugget = 1e-2 in order to account for the (strong) noise in the targets.
+The correlation parameters are determined by means of maximum likelihood
+estimation (MLE). An anisotropic squared exponential correlation model with a
+constant regression model are assumed. We also used a nugget = 1e-2 in order to
+account for the (strong) noise in the targets.
 
-The figure is a goodness-of-fit plot obtained using leave-one-out predictions
-of the Gaussian Process model. Based on these predictions, we compute an
-explained variance error (Q2).
+We compute then compute a cross-validation estimate of the coefficient of
+determination (R2) without reperforming MLE, using the set of correlation
+parameters found on the whole dataset.
 """
+print __doc__
 
 # Author: Vincent Dubourg <[email protected]>
 # License: BSD style
 
-from scikits.learn import datasets, cross_val, metrics
+import numpy as np
+from scikits.learn import datasets
 from scikits.learn.gaussian_process import GaussianProcess
-from matplotlib import pyplot as pl
-
-# Print the docstring
-print __doc__
+from scikits.learn.cross_val import cross_val_score, KFold
+from scikits.learn.metrics import r2_score
 
 # Load the dataset from scikits' data sets
 diabetes = datasets.load_diabetes()
-X, y = diabetes['data'], diabetes['target']
+X, y = diabetes.data, diabetes.target
 
 # Instanciate a GP model
 gp = GaussianProcess(regr='constant', corr='absolute_exponential',
                      theta0=[1e-4] * 10, thetaL=[1e-12] * 10,
-                     thetaU=[1e-2] * 10, nugget=1e-2, optimizer='Welch',
-                     verbose=False)
+                     thetaU=[1e-2] * 10, nugget=1e-2, optimizer='Welch')
 
-# Fit the GP model to the data
+# Fit the GP model to the data performing maximum likelihood estimation
 gp.fit(X, y)
-gp.theta0 = gp.theta
-gp.thetaL = None
-gp.thetaU = None
-gp.verbose = False
-
-# Estimate the leave-one-out predictions using the cross_val module
-n_jobs = 2 # the distributing capacity available on the machine
-y_pred = y + cross_val.cross_val_score(gp, X, y=y,
-                                   cv=cross_val.LeaveOneOut(y.size),
-                                   n_jobs=n_jobs,
-                                ).ravel()
 
-# Compute the empirical explained variance
-Q2 = metrics.explained_variance_score(y, y_pred)
+# Deactivate maximum likelihood estimation for the cross-validation loop
+gp.theta0 = gp.theta # Given correlation parameter = MLE
+gp.thetaL, gp.thetaU = None, None # None bounds deactivate MLE
 
-# Goodness-of-fit plot
-pl.figure()
-pl.title('Goodness-of-fit plot (Q2 = %1.2e)' % Q2)
-pl.plot(y, y_pred, 'r.', label='Leave-one-out')
-pl.plot(y, gp.predict(X), 'k.', label='Whole dataset (nugget=1e-2)')
-pl.plot([y.min(), y.max()], [y.min(), y.max()], 'k--')
-pl.xlabel('Observations')
-pl.ylabel('Predictions')
-pl.legend(loc='upper left')
-pl.axis('tight')
-pl.show()
+# Perform a cross-validation estimate of the coefficient of determination using
+# the cross_val module using all CPUs available on the machine
+K = 20 # folds
+R2 = cross_val_score(gp, X, y=y, cv=KFold(y.size, K), n_jobs=-1).mean()
+print("The %d-Folds estimate of the coefficient of determination is R2 = %s"
+    % (K, R2))
diff --git a/examples/gaussian_process/plot_gp_probabilistic_classification_after_regression.py b/examples/gaussian_process/plot_gp_probabilistic_classification_after_regression.py
@@ -14,6 +14,7 @@
 corresponds to the 95% confidence interval on the prediction of the zero level
 set.
 """
+print __doc__
 
 # Author: Vincent Dubourg <[email protected]>
 # License: BSD style
@@ -24,22 +25,19 @@
 from matplotlib import pyplot as pl
 from matplotlib import cm
 
-# Print the docstring
-print __doc__
-
 # Standard normal distribution functions
-Grv = stats.distributions.norm()
-phi = lambda x: Grv.pdf(x)
-PHI = lambda x: Grv.cdf(x)
-PHIinv = lambda x: Grv.ppf(x)
+phi = stats.distributions.norm().pdf
+PHI = stats.distributions.norm().cdf
+PHIinv = stats.distributions.norm().ppf
 
 # A few constants
 lim = 8
-b, kappa, e = 5, .5, .1
 
-# The function to predict (classification will then consist in predicting
-# wheter g(x) <= 0 or not)
-g = lambda x: b - x[:, 1] - kappa * (x[:, 0] - e) ** 2.
+
+def g(x):
+    """The function to predict (classification will then consist in predicting
+    whether g(x) <= 0 or not)"""
+    return 5. - x[:, 1] - .5 * x[:, 0] ** 2.
 
 # Design of experiments
 X = np.array([[-4.61611719, -6.00099547],

diff --git a/examples/gaussian_process/plot_gp_regression.py b/examples/gaussian_process/plot_gp_regression.py
@@ -2,9 +2,9 @@
 # -*- coding: utf-8 -*-
 
 """
-=================================================================
+=========================================================
 Gaussian Processes regression: basic introductory example
-=================================================================
+=========================================================
 
 A simple one-dimensional regression exercise with a cubic correlation
 model whose parameters are estimated using the maximum likelihood principle.
@@ -13,6 +13,7 @@
 model as well as its probabilistic nature in the form of a pointwise 95%
 confidence interval.
 """
+print __doc__
 
 # Author: Vincent Dubourg <[email protected]>
 # License: BSD style
@@ -21,11 +22,10 @@
 from scikits.learn.gaussian_process import GaussianProcess
 from matplotlib import pyplot as pl
 
-# Print the docstring
-print __doc__
 
-# The function to predict
-f = lambda x: x * np.sin(x)
+def f(x):
+    """The function to predict."""
+    return x * np.sin(x)
 
 # The design of experiments
 X = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T

diff --git a/scikits/learn/gaussian_process/correlation_models.py b/scikits/learn/gaussian_process/correlation_models.py
@@ -39,23 +39,20 @@ def absolute_exponential(theta, d):
         An array with shape (n_eval, ) containing the values of the
         autocorrelation model.
     """
-
     theta = np.asanyarray(theta, dtype=np.float)
-    d = np.asanyarray(d, dtype=np.float)
+    d = np.abs(np.asanyarray(d, dtype=np.float))
 
     if d.ndim > 1:
         n_features = d.shape[1]
     else:
         n_features = 1
+
     if theta.size == 1:
-        theta = np.repeat(theta, n_features)
+        return np.exp(- theta[0] * np.sum(d, axis=1))
     elif theta.size != n_features:
-        raise ValueError("Length of theta must be 1 or " + str(n_features))
-
-    td = - theta.reshape(1, n_features) * abs(d)
-    r = np.exp(np.sum(td, 1))
-
-    return r
+        raise ValueError("Length of theta must be 1 or %s" % n_features)
+    else:
+        return np.exp(- np.sum(theta.reshape(1, n_features) * d, axis=1))
 
 
 def squared_exponential(theta, d):
@@ -92,15 +89,13 @@ def squared_exponential(theta, d):
         n_features = d.shape[1]
     else:
         n_features = 1
+
     if theta.size == 1:
-        theta = np.repeat(theta, n_features)
+        return np.exp(- theta[0] * np.sum(d**2, axis=1))
     elif theta.size != n_features:
-        raise Exception("Length of theta must be 1 or " + str(n_features))
-
-    td = - theta.reshape(1, n_features) * d ** 2
-    r = np.exp(np.sum(td, 1))
-
-    return r
+        raise ValueError("Length of theta must be 1 or %s" % n_features)
+    else:
+        return np.exp(- np.sum(theta.reshape(1, n_features) * d**2, axis=1))
 
 
 def generalized_exponential(theta, d):
@@ -138,16 +133,17 @@ def generalized_exponential(theta, d):
         n_features = d.shape[1]
     else:
         n_features = 1
+
     lth = theta.size
     if n_features > 1 and lth == 2:
         theta = np.hstack([np.repeat(theta[0], n_features), theta[1]])
     elif lth != n_features + 1:
-        raise Exception("Length of theta must be 2 or " + str(n_features + 1))
+        raise Exception("Length of theta must be 2 or %s" % (n_features + 1))
     else:
         theta = theta.reshape(1, lth)
 
-    td = - theta[:, 0:-1].reshape(1, n_features) * abs(d) ** theta[:, -1]
-    r = np.exp(np.sum(td, 1))
+    td = theta[:, 0:-1].reshape(1, n_features) * np.abs(d) ** theta[:, -1]
+    r = np.exp(- np.sum(td, 1))
 
     return r
 
@@ -184,9 +180,7 @@ def pure_nugget(theta, d):
 
     n_eval = d.shape[0]
     r = np.zeros(n_eval)
-    # The ones on the diagonal of the correlation matrix are enforced within
-    # the KrigingModel instanciation to allow multiple design sites in this
-    # ordinary least squares context.
+    r[np.all(d == 0., axis=1)] = 1.
 
     return r
 
@@ -225,15 +219,15 @@ def cubic(theta, d):
         n_features = d.shape[1]
     else:
         n_features = 1
+
     lth = theta.size
     if  lth == 1:
-        theta = np.repeat(theta, n_features)[np.newaxis][:]
+        td = np.abs(d) * theta
     elif lth != n_features:
         raise Exception("Length of theta must be 1 or " + str(n_features))
     else:
-        theta = theta.reshape(1, n_features)
+        td = np.abs(d) * theta.reshape(1, n_features)
 
-    td = abs(d) * theta
     td[td > 1.] = 1.
     ss = 1. - td ** 2. * (3. - 2. * td)
     r = np.prod(ss, 1)
@@ -275,15 +269,15 @@ def linear(theta, d):
         n_features = d.shape[1]
     else:
         n_features = 1
+
     lth = theta.size
-    if  lth == 1:
-        theta = np.repeat(theta, n_features)[np.newaxis][:]
+    if lth == 1:
+        td = np.abs(d) * theta
     elif lth != n_features:
-        raise Exception("Length of theta must be 1 or " + str(n_features))
+        raise Exception("Length of theta must be 1 or %s" % n_features)
     else:
-        theta = theta.reshape(1, n_features)
+        td = np.abs(d) * theta.reshape(1, n_features)
 
-    td = abs(d) * theta
     td[td > 1.] = 1.
     ss = 1. - td
     r = np.prod(ss, 1)