Interacting with models¶

The GPy model class has a set of features which are designed to make it simple to explore the parameter space of the model. By default, the scipy optimisers are used to fit GPy models (via model.optimize()), for which we provide mechanisms for ‘free’ optimisation: GPy can ensure that naturally positive parameters (such as variances) remain positive. But these mechanisms are much more powerful than simple reparameterisation, as we shall see.

Along this tutorial we’ll use a sparse GP regression model as example. This example can be in GPy.examples.regression. All of the examples included in GPy return an instance of a model class, and therefore they can be called in the following way:

import numpy as np
import pylab as pb
pb.ion()
import GPy
m = GPy.examples.regression.sparse_GP_regression_1D()

Examining the model using print¶

To see the current state of the model parameters, and the model’s (marginal) likelihood just print the model

print m

The first thing displayed on the screen is the log-likelihood value of the model with its current parameters. Below the log-likelihood, a table with all the model’s parameters is shown. For each parameter, the table contains the name of the parameter, the current value, and in case there are defined: constraints, ties and prior distrbutions associated.

Name                 : sparse gp
Log-likelihood       : 588.947189413
Number of Parameters : 8
Parameters:
  sparse_gp.               |       Value        |  Constraint  |  Prior  |  Tied to
  inducing inputs          |            (5, 1)  |              |         |
  rbf.variance             |     1.91644016819  |     +ve      |         |
  rbf.lengthscale          |     2.62103621347  |     +ve      |         |
  Gaussian_noise.variance  |  0.00269870373421  |     +ve      |         |

In this case the kernel parameters (rbf.variance, rbf.lengthscale) as well as the likelihood noise parameter (Gaussian_noise.variance), are constrained to be positive, while the inducing inputs have no constraints associated. Also there are no ties or prior defined.

You can also print all subparts of the model, by printing the subcomponents individually:

print m.rbf

This will print the details of this particular parameter handle:

rbf.         |      Value      |  Constraint  |  Prior  |  Tied to
variance     |  1.91644016819  |     +ve      |         |
lengthscale  |  2.62103621347  |     +ve      |         |

When you want to get a closer look into multivalue parameters, print them directly:

print m.inducing_inputs

Index  |  sparse_gp.inducing_inputs  |  Constraint  |   Prior   |  Tied to
[0 0]  |                  2.7189499  |              |           |    N/A
[1 0]  |                 0.02006533  |              |           |    N/A
[2 0]  |                 -1.5299386  |              |           |    N/A
[3 0]  |                 -2.7001675  |              |           |    N/A
[4 0]  |                  1.4654162  |              |           |    N/A

Interacting with Parameters:¶

The preferred way of interacting with parameters is to act on the parameter handle itself. Interacting with parameter handles is simple. The names, printed by print m are accessible interactively and programatically. For example try to set kernels (rbf) lengthscale to .2 and print the result:

m.rbf.lengthscale = .2
print m

You should see this:

Name                 : sparse gp
Log-likelihood       : 588.947189413
Number of Parameters : 8
Parameters:
  sparse_gp.               |       Value        |  Constraint  |  Prior  |  Tied to
  inducing inputs          |            (5, 1)  |              |         |
  rbf.variance             |     1.91644016819  |     +ve      |         |
  rbf.lengthscale          |               0.2  |     +ve      |         |
  Gaussian_noise.variance  |  0.00269870373421  |     +ve      |         |

This will already have updated the model’s inner state, so you can plot it or see the changes in the posterior m.posterior of the model.

Regular expressions¶

The model’s parameters can also be accessed through regular expressions, by ‘indexing’ the model with a regular expression, matching the parameter name. Through indexing by regular expression, you can only retrieve leafs of the hierarchy, and you can retrieve the values matched by calling values() on the returned object:

>>> print m['.*var']
  Index  |       sparse_gp.rbf.variance        |  Constraint  |    Prior     |  Tied to
   [0]   |                          2.1500132  |              |              |    N/A
  -----  |  sparse_gp.Gaussian_noise.variance  |  ----------  |  ----------  |  -------
   [0]   |                       0.0024268215  |              |              |    N/A
>>> print m['.*var'].values()
[ 2.1500132   0.00242682]
>>> print m['rbf']
  Index  |   sparse_gp.rbf.variance    |  Constraint  |    Prior     |  Tied to
   [0]   |                  2.1500132  |              |              |    N/A
  -----  |  sparse_gp.rbf.lengthscale  |  ----------  |  ----------  |  -------
   [0]   |                  2.6782803  |              |              |    N/A

There is access to setting parameters by regular expression, as well. Here are a few examples of how to set parameters by regular expression:

>>> m['.*var'] = .1
>>> print m['.*var']
  Index  |       sparse_gp.rbf.variance        |  Constraint  |    Prior     |  Tied to
   [0]   |                                0.1  |              |              |    N/A
  -----  |  sparse_gp.Gaussian_noise.variance  |  ----------  |  ----------  |  -------
   [0]   |                                0.1  |              |              |    N/A
>>> m['.*var'] = [.1, .2]
>>> print m['.*var']
  Index  |       sparse_gp.rbf.variance        |  Constraint  |    Prior     |  Tied to
   [0]   |                                0.1  |              |              |    N/A
  -----  |  sparse_gp.Gaussian_noise.variance  |  ----------  |  ----------  |  -------
   [0]   |                                0.2  |              |              |    N/A

The fact that only leaf nodes can be accesses we can print all parameters in a flattened view, by printing the regular expression match of matching all objects:

>>> print m['']
  Index  |      sparse_gp.inducing_inputs      |  Constraint  |    Prior     |  Tied to
  [0 0]  |                         -2.6716041  |              |              |    N/A
  [1 0]  |                         -1.4665111  |              |              |    N/A
  [2 0]  |                       -0.031010293  |              |              |    N/A
  [3 0]  |                          1.4563711  |              |              |    N/A
  [4 0]  |                          2.6803046  |              |              |    N/A
  -----  |       sparse_gp.rbf.variance        |  ----------  |  ----------  |  -------
   [0]   |                                0.1  |              |              |    N/A
  -----  |      sparse_gp.rbf.lengthscale      |  ----------  |  ----------  |  -------
   [0]   |                          2.6782803  |              |              |    N/A
  -----  |  sparse_gp.Gaussian_noise.variance  |  ----------  |  ----------  |  -------
   [0]   |                                0.2  |              |              |    N/A

Setting and fetching parameters parameter_array¶

Another way to interact with the model’s parameters is through the parameter_array. The Parameter array holds all the parameters of the model in one place and is editable. It can be accessed through indexing the model for example you can set all the parameters through this mechanism:

>>> new_params = np.r_[[-4,-2,0,2,4], [.5,2], [.3]]
>>> print new_params
array([-4. , -2. ,  0. ,  2. ,  4. ,  0.5,  2. ,  0.3])
>>> m[:] = new_params
>>> print m
Name                 : sparse gp
Log-likelihood       : -147.561160209
Number of Parameters : 8
Parameters:
  sparse_gp.               |  Value   |  Constraint  |  Prior  |  Tied to
  inducing inputs          |  (5, 1)  |              |         |
  rbf.variance             |     0.5  |     +sq      |         |
  rbf.lengthscale          |     2.0  |     +ve      |         |
  Gaussian_noise.variance  |     0.3  |     +sq      |         |

Parameters themselves (leafs of the hierarchy) can be indexed and used the same way as numpy arrays. First let us set a slice of the inducing_inputs:

>>> m.inducing_inputs[2:, 0] = [1,3,5]
>>> print m.inducing_indputs
  Index  |  sparse_gp.inducing_inputs  |  Constraint  |   Prior   |  Tied to
  [0 0]  |                         -4  |              |           |    N/A
  [1 0]  |                         -2  |              |           |    N/A
  [2 0]  |                          1  |              |           |    N/A
  [3 0]  |                          3  |              |           |    N/A
  [4 0]  |                          5  |              |           |    N/A

Or you use the parameters as normal numpy arrays for calculations:

>>> precision = 1./m.Gaussian_noise.variance
array([ 3.33333333])

Getting the model’s log likelihood¶

Appart form the printing the model, the marginal log-likelihood can be obtained by using the function log_likelihood().:

>>> m.log_likelihood()
array([-152.83377316])

If you want to ensure the log likelihood as a float, call float() around it:

>>> float(m.log_likelihood())
-152.83377316356177

Getting the model parameter’s gradients¶

The gradients of a model can shed light on understanding the (possibly hard) optimization process. The gradients of each parameter handle can be accessed through their gradient field.:

>>> print m.gradient
[   5.51170031    9.71735112   -4.20282106   -3.45667035   -1.58828165
 -2.11549358   12.40292787 -627.75467803]
>>> print m.rbf.gradient
[ -2.11549358  12.40292787]
>>> m.optimize()
>>> print m.gradient
[ -5.98046560e-04  -3.64576085e-04   1.98005930e-04   3.43381219e-04
-6.85685104e-04  -1.28800748e-05   1.08552429e-03   2.74058081e-01]

Adjusting the model’s constraints¶

When we initially call the example, it was optimized and hence the log-likelihood gradients were close to zero. However, since we have been changing the parameters, the gradients are far from zero now. Next we are going to show how to optimize the model setting different restrictions on the parameters.

Once a constraint has been set on a parameter, it is possible to remove it with the command unconstrain(), which can be called on any parameter handle of the model. The methods constrain() and unconstrain() return the indices which were actually unconstrained, relative to the parameter handle the method was called on. This is particularly handy for reporting which parameters where reconstrained, when reconstraining a parameter, which was already constrained:

>>> m.rbf.variance.unconstrain()
array([0])
>>>m.unconstrain()
array([6, 7])

If you want to unconstrain only a specific constraint, you can pass it as an argument of unconstrain(Transformation) (Transformation), or call the respective method, such as unconstrain_fixed() (or unfix()) to only unfix fixed parameters.:

>>> m.inducing_input[0].fix()
>>> m.unfix()
>>> m.rbf.constrain_positive()
>>> print m
Name                 : sparse gp
Log-likelihood       : 620.741066698
Number of Parameters : 8
Parameters:
  sparse_gp.               |       Value        |  Constraint  |  Prior  |  Tied to
  inducing inputs          |            (5, 1)  |              |         |
  rbf.variance             |     1.48329711218  |     +ve      |         |
  rbf.lengthscale          |      2.5430947048  |     +ve      |         |
  Gaussian_noise.variance  |  0.00229714444128  |              |         |

As you can see, unfix() only unfixed the inducing_input, and did not change the positive constraint of the kernel.

The parameter handles come with default constraints, so you will rarely be needing to adjust the constraints of a model. In the rare cases of needing to adjust the constraints of a model, or in need of fixing some parameters, you can do so with the functions constrain_{positive|negative|bounded|fixed}().:

m['.*var'].constrain_positive()

Available Constraints¶

Logexp()
Exponent()
Square()
Logistic()
LogexpNeg()
NegativeExponent()
NegativeLogexp()

Tying Parameters¶

Not yet implemented for GPy version 0.6.0

Optimizing the model¶

Once we have finished defining the constraints, we can now optimize the model with the function optimize.:

m.Gaussian_noise.constrain_positive()
m.rbf.constrain_positive()
m.optimize()

By deafult, GPy uses the lbfgsb optimizer.

Some optional parameters may be discussed here.

optimizer: which optimizer to use, currently there are lbfgsb, fmin_tnc, scg, simplex or any unique identifier uniquely identifying an optimizer. Thus, you can say m.optimize('bfgs') for using the ``lbfgsb optimizer
messages: if the optimizer is verbose. Each optimizer has its own way of printing, so do not be confused by differing messages of different optimizers
max_iters: Maximum number of iterations to take. Some optimizers see iterations as function calls, others as iterations of the algorithm. Please be advised to look into scipy.optimize for more instructions, if the number of iterations matter, so you can give the right parameters to optimize()
gtol: only for some optimizers. Will determine the convergence criterion, as the tolerance of gradient to finish the optimization.