Method predict

numpy_ml/nonparametric/gp.py:64–154

Return the MAP estimate for :math:`y^*`, corresponding the mean/mode of the posterior predictive distribution, :math:`p(y^* \mid x^*, X, y)`. Notes ----- Under the GP regression model, the posterior predictive distribution is .. math::

(self, X, conf_interval=0.95, return_cov=False)

        self.parameters["GP_mean"] = mu

    def predict(self, X, conf_interval=0.95, return_cov=False):
        """
        Return the MAP estimate for :math:`y^*`, corresponding the mean/mode of
        the posterior predictive distribution, :math:`p(y^* \mid x^*, X, y)`.

        Notes
        -----
        Under the GP regression model, the posterior predictive distribution is

        .. math::

            y^* \mid x^*, X, y \sim \mathcal{N}(\mu^*, \\text{cov}^*)

        where

        .. math::

            \mu^*  &=  K^* (K + \\alpha I)^{-1} y \\\\
            \\text{cov}^*  &=  K^{**} - K^{*'} (K + \\alpha I)^{-1} K^*

        and

        .. math::

            K  &=  \\text{kernel}(X, X) \\\\
            K^*  &=  \\text{kernel}(X, X^*) \\\\
            K^{**}  &=  \\text{kernel}(X^*, X^*)

        NB. This implementation uses the inefficient but general purpose
        `np.linalg.inv` routine to invert :math:`(K + \\alpha I)`. A more
        efficient way is to rely on the fact that `K` (and hence also :math:`K
        + \\alpha I`) is symmetric positive (semi-)definite and take the inner
        product of the inverse of its (lower) Cholesky decompositions:

        .. math::

            Q^{-1} = \\text{cholesky}(Q)^{-1 \\top} \\text{cholesky}(Q)^{-1}

        For more details on a production-grade implementation, see Algorithm
        2.1 in Rasmussen & Williams (2006).

        Parameters
        ----------
        X : :py:class:`ndarray <numpy.ndarray>` of shape (N, M)
            The collection of datapoints to generate predictions on
        conf_interval : float in (0, 1)
            The percentage confidence bound to return for each prediction. If
            the scipy package is not available, this value is always set to
            0.95. Default is 0.95.
        return_cov : bool
            If True, also return the covariance (`cov*`) of the posterior
            predictive distribution for the points in `X`. Default is False.

        Returns
        -------
        y_pred : :py:class:`ndarray <numpy.ndarray>` of shape `(N, O)`
            The predicted values for each point in `X`, each with
            dimensionality `O`.