GPy/GPy/kern/src/kern.py

# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import sys
import numpy as np
from ...core.parameterization.parameterized import Parameterized
from paramz.core.observable_array import ObsAr
from paramz.caching import Cache_this
from .kernel_slice_operations import KernCallsViaSlicerMeta
from functools import reduce
import six

@six.add_metaclass(KernCallsViaSlicerMeta)
class Kern(Parameterized):
    #===========================================================================
    # This adds input slice support. The rather ugly code for slicing can be
    # found in kernel_slice_operations
    # __meataclass__ is ignored in Python 3 - needs to be put in the function definiton
    #__metaclass__ = KernCallsViaSlicerMeta
    #Here, we use the Python module six to support Py3 and Py2 simultaneously
    #===========================================================================
    _support_GPU=False
    def __init__(self, input_dim, active_dims, name, useGPU=False, *a, **kw):
        """
        The base class for a kernel: a positive definite function
        which forms of a covariance function (kernel).

        input_dim:

            is the number of dimensions to work on. Make sure to give the
            tight dimensionality of inputs.
            You most likely want this to be the integer telling the number of
            input dimensions of the kernel.

        active_dims:

            is the active_dimensions of inputs X we will work on.
            All kernels will get sliced Xes as inputs, if _all_dims_active is not None
            Only positive integers are allowed in active_dims!
            if active_dims is None, slicing is switched off and all X will be passed through as given.

        :param int input_dim: the number of input dimensions to the function
        :param array-like|None active_dims: list of indices on which dimensions this kernel works on, or none if no slicing

        Do not instantiate.
        """
        super(Kern, self).__init__(name=name, *a, **kw)
        self.input_dim = int(input_dim)

        if active_dims is None:
            active_dims = np.arange(input_dim)

        self.active_dims = active_dims
        self._all_dims_active = np.atleast_1d(active_dims).astype(int)

        assert self._all_dims_active.size == self.input_dim, "input_dim={} does not match len(active_dim)={}, _all_dims_active={}".format(self.input_dim, self._all_dims_active.size, self._all_dims_active)

        self._sliced_X = 0
        self.useGPU = self._support_GPU and useGPU

        from .psi_comp import PSICOMP_GH
        self.psicomp = PSICOMP_GH()

    def __setstate__(self, state):
        self._all_dims_active = np.arange(0, max(state['active_dims'])+1)
        super(Kern, self).__setstate__(state)

    @property
    def _effective_input_dim(self):
        return np.size(self._all_dims_active)

    @Cache_this(limit=20)
    def _slice_X(self, X):
        return X[:, self._all_dims_active]

    def K(self, X, X2):
        """
        Compute the kernel function.

        .. math::
            K_{ij} = k(X_i, X_j)

        :param X: the first set of inputs to the kernel
        :param X2: (optional) the second set of arguments to the kernel. If X2
                   is None, this is passed throgh to the 'part' object, which
                   handLes this as X2 == X.
        """
        raise NotImplementedError
    def Kdiag(self, X):
        """
        The diagonal of the kernel matrix K

        .. math::
            Kdiag_{i} = k(X_i, X_i)
        """
        raise NotImplementedError
    def psi0(self, Z, variational_posterior):
        """
        .. math::
            \psi_0 = \sum_{i=0}^{n}E_{q(X)}[k(X_i, X_i)]
        """
        return self.psicomp.psicomputations(self, Z, variational_posterior)[0]
    def psi1(self, Z, variational_posterior):
        """
        .. math::
            \psi_1^{n,m} = E_{q(X)}[k(X_n, Z_m)]
        """
        return self.psicomp.psicomputations(self, Z, variational_posterior)[1]
    def psi2(self, Z, variational_posterior):
        """
        .. math::
            \psi_2^{m,m'} = \sum_{i=0}^{n}E_{q(X)}[ k(Z_m, X_i) k(X_i, Z_{m'})]
        """
        return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=False)[2]
    def psi2n(self, Z, variational_posterior):
        """
        .. math::
            \psi_2^{n,m,m'} = E_{q(X)}[ k(Z_m, X_n) k(X_n, Z_{m'})]

        Thus, we do not sum out n, compared to psi2
        """
        return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=True)[2]
    def gradients_X(self, dL_dK, X, X2):
        """
        .. math::

            \\frac{\partial L}{\partial X} = \\frac{\partial L}{\partial K}\\frac{\partial K}{\partial X}
        """
        raise NotImplementedError
    def gradients_X_X2(self, dL_dK, X, X2):
        return self.gradients_X(dL_dK, X, X2), self.gradients_X(dL_dK.T, X2, X)
    def gradients_XX(self, dL_dK, X, X2):
        """
        .. math::

            \\frac{\partial^2 L}{\partial X\partial X_2} = \\frac{\partial L}{\partial K}\\frac{\partial^2 K}{\partial X\partial X_2}
        """
        raise(NotImplementedError, "This is the second derivative of K wrt X and X2, and not implemented for this kernel")
    def gradients_XX_diag(self, dL_dKdiag, X):
        """
        The diagonal of the second derivative w.r.t. X and X2
        """
        raise(NotImplementedError, "This is the diagonal of the second derivative of K wrt X and X2, and not implemented for this kernel")
    def gradients_X_diag(self, dL_dKdiag, X):
        """
        The diagonal of the derivative w.r.t. X
        """
        raise NotImplementedError

    def update_gradients_diag(self, dL_dKdiag, X):
        """ update the gradients of all parameters when using only the diagonal elements of the covariance matrix"""
        raise NotImplementedError

    def update_gradients_full(self, dL_dK, X, X2):
        """Set the gradients of all parameters when doing full (N) inference."""
        raise NotImplementedError

    def update_gradients_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
        """
        Set the gradients of all parameters when doing inference with
        uncertain inputs, using expectations of the kernel.

        The essential maths is

        .. math::

            \\frac{\partial L}{\partial \\theta_i} & = \\frac{\partial L}{\partial \psi_0}\\frac{\partial \psi_0}{\partial \\theta_i}\\
                & \quad + \\frac{\partial L}{\partial \psi_1}\\frac{\partial \psi_1}{\partial \\theta_i}\\
                & \quad + \\frac{\partial L}{\partial \psi_2}\\frac{\partial \psi_2}{\partial \\theta_i}

        Thus, we push the different derivatives through the gradients of the psi
        statistics. Be sure to set the gradients for all kernel
        parameters here.
        """
        dtheta = self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[0]
        self.gradient[:] = dtheta

    def gradients_Z_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior,
                                psi0=None, psi1=None, psi2=None):
        """
        Returns the derivative of the objective wrt Z, using the chain rule
        through the expectation variables.
        """
        return self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[1]

    def gradients_qX_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
        """
        Compute the gradients wrt the parameters of the variational
        distruibution q(X), chain-ruling via the expectations of the kernel
        """
        return self.psicomp.psiDerivativecomputations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior)[2:]

    def plot(self, x=None, fignum=None, ax=None, title=None, plot_limits=None, resolution=None, **mpl_kwargs):
        """
        plot this kernel.
        :param x: the value to use for the other kernel argument (kernels are a function of two variables!)
        :param fignum: figure number of the plot
        :param ax: matplotlib axis to plot on
        :param title: the matplotlib title
        :param plot_limits: the range over which to plot the kernel
        :resolution: the resolution of the lines used in plotting
        :mpl_kwargs avalid keyword arguments to pass through to matplotlib (e.g. lw=7)
        """
        assert "matplotlib" in sys.modules, "matplotlib package has not been imported."
        from ...plotting.matplot_dep import kernel_plots
        kernel_plots.plot(self, x, fignum, ax, title, plot_limits, resolution, **mpl_kwargs)

    def input_sensitivity(self, summarize=True):
        """
        Returns the sensitivity for each dimension of this kernel.
        """
        return np.zeros(self.input_dim)

    def get_most_significant_input_dimensions(self, which_indices=None):
        """
        Determine which dimensions should be plotted

        Returns the top three most signification input dimensions

        if less then three dimensions, the non existing dimensions are
        labeled as None, so for a 1 dimensional input this returns
        (0, None, None).

        :param which_indices: force the indices to be the given indices.
        :type which_indices: int or tuple(int,int) or tuple(int,int,int)
        """
        if which_indices is None:
            which_indices = np.argsort(self.input_sensitivity())[::-1][:3]
        try:
            input_1, input_2, input_3 = which_indices
        except ValueError:
            # which indices is tuple or int
            try:
                input_3 = None
                input_1, input_2 = which_indices
            except TypeError:
                # which_indices is an int
                input_1, input_2 = which_indices, None
            except ValueError:
                # which_indices was a list or array like with only one int
                input_1, input_2 = which_indices[0], None
        return input_1, input_2, input_3


    def __add__(self, other):
        """ Overloading of the '+' operator. for more control, see self.add """
        return self.add(other)

    def __iadd__(self, other):
        return self.add(other)

    def add(self, other, name='sum'):
        """
        Add another kernel to this one.

        :param other: the other kernel to be added
        :type other: GPy.kern

        """
        assert isinstance(other, Kern), "only kernels can be added to kernels..."
        from .add import Add
        return Add([self, other], name=name)

    def __mul__(self, other):
        """ Here we overload the '*' operator. See self.prod for more information"""
        return self.prod(other)

    def __imul__(self, other):
        """ Here we overload the '*' operator. See self.prod for more information"""
        return self.prod(other)

    def __pow__(self, other):
        """
        Shortcut for tensor `prod`.
        """
        assert np.all(self._all_dims_active == range(self.input_dim)), "Can only use kernels, which have their input_dims defined from 0"
        assert np.all(other._all_dims_active == range(other.input_dim)), "Can only use kernels, which have their input_dims defined from 0"
        other._all_dims_active += self.input_dim
        return self.prod(other)

    def prod(self, other, name='mul'):
        """
        Multiply two kernels (either on the same space, or on the tensor
        product of the input space).

        :param other: the other kernel to be added
        :type other: GPy.kern

        """
        assert isinstance(other, Kern), "only kernels can be multiplied to kernels..."
        from .prod import Prod
        #kernels = []
        #if isinstance(self, Prod): kernels.extend(self.parameters)
        #else: kernels.append(self)
        #if isinstance(other, Prod): kernels.extend(other.parameters)
        #else: kernels.append(other)
        return Prod([self, other], name)

    def _check_input_dim(self, X):
        assert X.shape[1] == self.input_dim, "{} did not specify _all_dims_active and X has wrong shape: X_dim={}, whereas input_dim={}".format(self.name, X.shape[1], self.input_dim)

    def _check_active_dims(self, X):
        assert X.shape[1] >= len(self._all_dims_active), "At least {} dimensional X needed, X.shape={!s}".format(len(self._all_dims_active), X.shape)


class CombinationKernel(Kern):
    """
    Abstract super class for combination kernels.
    A combination kernel combines (a list of) kernels and works on those.
    Examples are the HierarchicalKernel or Add and Prod kernels.
    """
    def __init__(self, kernels, name, extra_dims=[]):
        """
        Abstract super class for combination kernels.
        A combination kernel combines (a list of) kernels and works on those.
        Examples are the HierarchicalKernel or Add and Prod kernels.

        :param list kernels: List of kernels to combine (can be only one element)
        :param str name: name of the combination kernel
        :param array-like extra_dims: if needed extra dimensions for the combination kernel to work on
        """
        assert all([isinstance(k, Kern) for k in kernels])
        extra_dims = np.array(extra_dims, dtype=int)
        input_dim, active_dims = self.get_input_dim_active_dims(kernels, extra_dims)
        # initialize the kernel with the full input_dim
        super(CombinationKernel, self).__init__(input_dim, active_dims, name)
        self.extra_dims = extra_dims
        self.link_parameters(*kernels)

    @property
    def parts(self):
        return self.parameters

    def get_input_dim_active_dims(self, kernels, extra_dims = None):
        self.active_dims = reduce(np.union1d, (np.r_[x.active_dims] for x in kernels), np.array([], dtype=int))
        #_all_dims_active = np.array(np.concatenate((_all_dims_active, extra_dims if extra_dims is not None else [])), dtype=int)
        input_dim = reduce(max, (k._all_dims_active.max() for k in kernels)) + 1

        if extra_dims is not None:
            input_dim += extra_dims.size

        _all_dims_active = np.arange(input_dim)
        return input_dim, _all_dims_active

    def input_sensitivity(self, summarize=True):
        """
        If summize is true, we want to get the summerized view of the sensitivities,
        otherwise put everything into an array with shape (#kernels, input_dim)
        in the order of appearance of the kernels in the parameterized object.
        """
        if not summarize:
            num_params = [0]
            parts = []
            def sum_params(x):
                if (not isinstance(x, CombinationKernel)) and isinstance(x, Kern):
                    num_params[0] += 1
                    parts.append(x)
            self.traverse(sum_params)
            i_s = np.zeros((num_params[0], self.input_dim))
            from operator import setitem
            [setitem(i_s, (i, k._all_dims_active), k.input_sensitivity(summarize)) for i, k in enumerate(parts)]
            return i_s
        else:
            raise NotImplementedError("Choose the kernel you want to get the sensitivity for. You need to override the default behaviour for getting the input sensitivity to be able to get the input sensitivity. For sum kernel it is the sum of all sensitivities, TODO: product kernel? Other kernels?, also TODO: shall we return all the sensitivities here in the combination kernel? So we can combine them however we want? This could lead to just plot all the sensitivities here...")

    def _check_active_dims(self, X):
        return

    def _check_input_dim(self, X):
        # As combination kernels cannot always know, what their inner kernels have as input dims, the check will be done inside them, respectively
        return