uk3d.py

# coding: utf-8
"""
PyKrige
=======

Code by Benjamin S. Murphy and the PyKrige Developers
bscott.murphy@gmail.com

Summary
-------
Contains class UniversalKriging3D.

References
----------
.. [1] P.K. Kitanidis, Introduction to Geostatistcs: Applications in
    Hydrogeology, (Cambridge University Press, 1997) 272 p.
.. [2] N. Cressie, Statistics for spatial data,
   (Wiley Series in Probability and Statistics, 1993) 137 p.

Copyright (c) 2015-2020, PyKrige Developers
"""
import numpy as np
import scipy.linalg
from scipy.spatial.distance import cdist
from . import variogram_models
from . import core
from .core import (
    _adjust_for_anisotropy,
    _initialize_variogram_model,
    _make_variogram_parameter_list,
    _find_statistics,
    P_INV,
)
import warnings


class UniversalKriging3D:
    """Three-dimensional universal kriging.

    Parameters
    ----------
    x : array_like
        X-coordinates of data points.
    y : array_like
        Y-coordinates of data points.
    z : array_like
        Z-coordinates of data points.
    val : array_like
        Values at data points.
    variogram_model : str or GSTools CovModel, optional
        Specified which variogram model to use; may be one of the following:
        linear, power, gaussian, spherical, exponential, hole-effect.
        Default is linear variogram model. To utilize a custom variogram model,
        specify 'custom'; you must also provide variogram_parameters and
        variogram_function. Note that the hole-effect model is only
        technically correct for one-dimensional problems.
        You can also use a
        `GSTools <https://github.com/GeoStat-Framework/GSTools>`_ CovModel.
    variogram_parameters : list or dict, optional
        Parameters that define the specified variogram model. If not provided,
        parameters will be automatically calculated using a "soft" L1 norm
        minimization scheme. For variogram model parameters provided in a dict,
        the required dict keys vary according to the specified variogram
        model: ::

           # linear
               {'slope': slope, 'nugget': nugget}
           # power
               {'scale': scale, 'exponent': exponent, 'nugget': nugget}
           # gaussian, spherical, exponential and hole-effect:
               {'sill': s, 'range': r, 'nugget': n}
               # OR
               {'psill': p, 'range': r, 'nugget': n}

        Note that either the full sill or the partial sill
        (psill = sill - nugget) can be specified in the dict.
        For variogram model parameters provided in a list, the entries
        must be as follows: ::

           # linear
               [slope, nugget]
           # power
               [scale, exponent, nugget]
           # gaussian, spherical, exponential and hole-effect:
               [sill, range, nugget]

        Note that the full sill (NOT the partial sill) must be specified
        in the list format.
        For a custom variogram model, the parameters are required, as custom
        variogram models will not automatically be fit to the data.
        Furthermore, the parameters must be specified in list format, in the
        order in which they are used in the callable function (see
        variogram_function for more information). The code does not check
        that the provided list contains the appropriate number of parameters
        for the custom variogram model, so an incorrect parameter list in
        such a case will probably trigger an esoteric exception someplace
        deep in the code.
        NOTE that, while the list format expects the full sill, the code
        itself works internally with the partial sill.
    variogram_function : callable, optional
        A callable function that must be provided if variogram_model is
        specified as 'custom'. The function must take only two arguments:
        first, a list of parameters for the variogram model;
        second, the distances at which to calculate the variogram model.
        The list provided in variogram_parameters will be passed to the
        function as the first argument.
    nlags : int, optional
        Number of averaging bins for the semivariogram. Default is 6.
    weight : bool, optional
        Flag that specifies if semivariance at smaller lags should be weighted
        more heavily when automatically calculating variogram model.
        The routine is currently hard-coded such  that the weights are
        calculated from a logistic function, so weights at small lags are ~1
        and weights at the longest lags are ~0; the center of the logistic
        weighting is hard-coded to be at 70% of the distance from the shortest
        lag to the largest lag. Setting this parameter to True indicates that
        weights will be applied. Default is False.
        (Kitanidis suggests that the values at smaller lags are more
        important in fitting a variogram model, so the option is provided
        to enable such weighting.)
    anisotropy_scaling_y : float, optional
        Scalar stretching value to take into account anisotropy in
        the y direction. Default is 1 (effectively no stretching).
        Scaling is applied in the y direction in the rotated data frame
        (i.e., after adjusting for the anisotropy_angle_x/y/z,
        if anisotropy_angle_x/y/z is/are not 0).
    anisotropy_scaling_z : float, optional
        Scalar stretching value to take into account anisotropy in
        the z direction. Default is 1 (effectively no stretching).
        Scaling is applied in the z direction in the rotated data frame
        (i.e., after adjusting for the anisotropy_angle_x/y/z,
        if anisotropy_angle_x/y/z is/are not 0).
    anisotropy_angle_x : float, optional
        CCW angle (in degrees) by which to rotate coordinate system about
        the x axis in order to take into account anisotropy.
        Default is 0 (no rotation). Note that the coordinate system is rotated.
        X rotation is applied first, then y rotation, then z rotation.
        Scaling is applied after rotation.
    anisotropy_angle_y : float, optional
        CCW angle (in degrees) by which to rotate coordinate system about
        the y axis in order to take into account anisotropy.
        Default is 0 (no rotation). Note that the coordinate system is rotated.
        X rotation is applied first, then y rotation, then z rotation.
        Scaling is applied after rotation.
    anisotropy_angle_z : float, optional
        CCW angle (in degrees) by which to rotate coordinate system about
        the z axis in order to take into account anisotropy.
        Default is 0 (no rotation). Note that the coordinate system is rotated.
        X rotation is applied first, then y rotation, then z rotation.
        Scaling is applied after rotation.
    drift_terms : list of strings, optional
        List of drift terms to include in three-dimensional universal kriging.
        Supported drift terms are currently 'regional_linear', 'specified',
        and 'functional'.
    specified_drift : list of array-like objects, optional
        List of arrays that contain the drift values at data points.
        The arrays must be shape (N,) or (N, 1), where N is the number of
        data points. Any number of specified-drift terms may be used.
    functional_drift : list of callable objects, optional
        List of callable functions that will be used to evaluate drift terms.
        The function must be a function of only the three spatial coordinates
        and must return a single value for each coordinate triplet.
        It must be set up to be called with only three arguments,
        first an array of x values, the second an array of y values,
        and the third an array of z values. If the problem involves anisotropy,
        the drift values are calculated in the adjusted data frame.
    verbose : boolean, optional
        Enables program text output to monitor kriging process.
        Default is False (off).
    enable_plotting : boolean, optional
        Enables plotting to display variogram. Default is False (off).
    exact_values : bool, optional
        If True, interpolation provides input values at input locations.
        If False, interpolation accounts for variance/nugget within input
        values at input locations and does not behave as an
        exact-interpolator [2]. Note that this only has an effect if
        there is variance/nugget present within the input data since it is
        interpreted as measurement error. If the nugget is zero, the kriged
        field will behave as an exact interpolator.
    pseudo_inv : :class:`bool`, optional
        Whether the kriging system is solved with the pseudo inverted
        kriging matrix. If `True`, this leads to more numerical stability
        and redundant points are averaged. But it can take more time.
        Default: False
    pseudo_inv_type : :class:`str`, optional
        Here you can select the algorithm to compute the pseudo-inverse matrix:

            * `"pinv"`: use `pinv` from `scipy` which uses `lstsq`
            * `"pinv2"`: use `pinv2` from `scipy` which uses `SVD`
            * `"pinvh"`: use `pinvh` from `scipy` which uses eigen-values

        Default: `"pinv"`

    References
    ----------
    .. [1] P.K. Kitanidis, Introduction to Geostatistcs: Applications in
       Hydrogeology, (Cambridge University Press, 1997) 272 p.
    .. [2] N. Cressie, Statistics for spatial data,
       (Wiley Series in Probability and Statistics, 1993) 137 p.
    """

    UNBIAS = True  # This can be changed to remove the unbiasedness condition
    # Really for testing purposes only...
    eps = 1.0e-10  # Cutoff for comparison to zero
    variogram_dict = {
        "linear": variogram_models.linear_variogram_model,
        "power": variogram_models.power_variogram_model,
        "gaussian": variogram_models.gaussian_variogram_model,
        "spherical": variogram_models.spherical_variogram_model,
        "exponential": variogram_models.exponential_variogram_model,
        "hole-effect": variogram_models.hole_effect_variogram_model,
    }

    def __init__(
        self,
        x,
        y,
        z,
        val,
        variogram_model="linear",
        variogram_parameters=None,
        variogram_function=None,
        nlags=6,
        weight=False,
        anisotropy_scaling_y=1.0,
        anisotropy_scaling_z=1.0,
        anisotropy_angle_x=0.0,
        anisotropy_angle_y=0.0,
        anisotropy_angle_z=0.0,
        drift_terms=None,
        specified_drift=None,
        functional_drift=None,
        verbose=False,
        enable_plotting=False,
        exact_values=True,
        pseudo_inv=False,
        pseudo_inv_type="pinv",
    ):
        # config the pseudo inverse
        self.pseudo_inv = bool(pseudo_inv)
        self.pseudo_inv_type = str(pseudo_inv_type)
        if self.pseudo_inv_type not in P_INV:
            raise ValueError("pseudo inv type not valid: " + str(pseudo_inv_type))

        # Deal with mutable default argument
        if drift_terms is None:
            drift_terms = []
        if specified_drift is None:
            specified_drift = []
        if functional_drift is None:
            functional_drift = []

        # set up variogram model and parameters...
        self.variogram_model = variogram_model
        self.model = None

        if not isinstance(exact_values, bool):
            raise ValueError("exact_values has to be boolean True or False")
        self.exact_values = exact_values

        # check if a GSTools covariance model is given
        if hasattr(self.variogram_model, "pykrige_kwargs"):
            # save the model in the class
            self.model = self.variogram_model
            if self.model.dim < 3:
                raise ValueError("GSTools: model dim is not 3")
            self.variogram_model = "custom"
            variogram_function = self.model.pykrige_vario
            variogram_parameters = []
            anisotropy_scaling_y = self.model.pykrige_anis_y
            anisotropy_scaling_z = self.model.pykrige_anis_z
            anisotropy_angle_x = self.model.pykrige_angle_x
            anisotropy_angle_y = self.model.pykrige_angle_y
            anisotropy_angle_z = self.model.pykrige_angle_z
        if (
            self.variogram_model not in self.variogram_dict.keys()
            and self.variogram_model != "custom"
        ):
            raise ValueError(
                "Specified variogram model '%s' is not supported." % variogram_model
            )
        elif self.variogram_model == "custom":
            if variogram_function is None or not callable(variogram_function):
                raise ValueError(
                    "Must specify callable function for custom variogram model."
                )
            else:
                self.variogram_function = variogram_function
        else:
            self.variogram_function = self.variogram_dict[self.variogram_model]

        # Code assumes 1D input arrays. Ensures that any extraneous dimensions
        # don't get in the way. Copies are created to avoid any problems with
        # referencing the original passed arguments.
        self.X_ORIG = np.atleast_1d(
            np.squeeze(np.array(x, copy=True, dtype=np.float64))
        )
        self.Y_ORIG = np.atleast_1d(
            np.squeeze(np.array(y, copy=True, dtype=np.float64))
        )
        self.Z_ORIG = np.atleast_1d(
            np.squeeze(np.array(z, copy=True, dtype=np.float64))
        )
        self.VALUES = np.atleast_1d(
            np.squeeze(np.array(val, copy=True, dtype=np.float64))
        )

        self.verbose = verbose
        self.enable_plotting = enable_plotting
        if self.enable_plotting and self.verbose:
            print("Plotting Enabled\n")

        self.XCENTER = (np.amax(self.X_ORIG) + np.amin(self.X_ORIG)) / 2.0
        self.YCENTER = (np.amax(self.Y_ORIG) + np.amin(self.Y_ORIG)) / 2.0
        self.ZCENTER = (np.amax(self.Z_ORIG) + np.amin(self.Z_ORIG)) / 2.0
        self.anisotropy_scaling_y = anisotropy_scaling_y
        self.anisotropy_scaling_z = anisotropy_scaling_z
        self.anisotropy_angle_x = anisotropy_angle_x
        self.anisotropy_angle_y = anisotropy_angle_y
        self.anisotropy_angle_z = anisotropy_angle_z
        if self.verbose:
            print("Adjusting data for anisotropy...")
        self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED = _adjust_for_anisotropy(
            np.vstack((self.X_ORIG, self.Y_ORIG, self.Z_ORIG)).T,
            [self.XCENTER, self.YCENTER, self.ZCENTER],
            [self.anisotropy_scaling_y, self.anisotropy_scaling_z],
            [self.anisotropy_angle_x, self.anisotropy_angle_y, self.anisotropy_angle_z],
        ).T

        if self.verbose:
            print("Initializing variogram model...")

        vp_temp = _make_variogram_parameter_list(
            self.variogram_model, variogram_parameters
        )
        (
            self.lags,
            self.semivariance,
            self.variogram_model_parameters,
        ) = _initialize_variogram_model(
            np.vstack((self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)).T,
            self.VALUES,
            self.variogram_model,
            vp_temp,
            self.variogram_function,
            nlags,
            weight,
            "euclidean",
        )

        if self.verbose:
            if self.variogram_model == "linear":
                print("Using '%s' Variogram Model" % "linear")
                print("Slope:", self.variogram_model_parameters[0])
                print("Nugget:", self.variogram_model_parameters[1], "\n")
            elif self.variogram_model == "power":
                print("Using '%s' Variogram Model" % "power")
                print("Scale:", self.variogram_model_parameters[0])
                print("Exponent:", self.variogram_model_parameters[1])
                print("Nugget:", self.variogram_model_parameters[2], "\n")
            elif self.variogram_model == "custom":
                print("Using Custom Variogram Model")
            else:
                print("Using '%s' Variogram Model" % self.variogram_model)
                print("Partial Sill:", self.variogram_model_parameters[0])
                print(
                    "Full Sill:",
                    self.variogram_model_parameters[0]
                    + self.variogram_model_parameters[2],
                )
                print("Range:", self.variogram_model_parameters[1])
                print("Nugget:", self.variogram_model_parameters[2], "\n")
        if self.enable_plotting:
            self.display_variogram_model()

        if self.verbose:
            print("Calculating statistics on variogram model fit...")
        self.delta, self.sigma, self.epsilon = _find_statistics(
            np.vstack((self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)).T,
            self.VALUES,
            self.variogram_function,
            self.variogram_model_parameters,
            "euclidean",
            self.pseudo_inv,
        )
        self.Q1 = core.calcQ1(self.epsilon)
        self.Q2 = core.calcQ2(self.epsilon)
        self.cR = core.calc_cR(self.Q2, self.sigma)
        if self.verbose:
            print("Q1 =", self.Q1)
            print("Q2 =", self.Q2)
            print("cR =", self.cR, "\n")

        if self.verbose:
            print("Initializing drift terms...")

        # Note that the regional linear drift values will be based on the
        # adjusted coordinate system. Really, it doesn't actually matter
        # which coordinate system is used here.
        if "regional_linear" in drift_terms:
            self.regional_linear_drift = True
            if self.verbose:
                print("Implementing regional linear drift.")
        else:
            self.regional_linear_drift = False

        if "specified" in drift_terms:
            if type(specified_drift) is not list:
                raise TypeError(
                    "Arrays for specified drift terms must be "
                    "encapsulated in a list."
                )
            if len(specified_drift) == 0:
                raise ValueError(
                    "Must provide at least one drift-value array "
                    "when using the 'specified' drift capability."
                )
            self.specified_drift = True
            self.specified_drift_data_arrays = []
            for term in specified_drift:
                specified = np.squeeze(np.array(term, copy=True))
                if specified.size != self.X_ORIG.size:
                    raise ValueError(
                        "Must specify the drift values for each "
                        "data point when using the "
                        "'specified' drift capability."
                    )
                self.specified_drift_data_arrays.append(specified)
        else:
            self.specified_drift = False

        # The provided callable functions will be evaluated using
        # the adjusted coordinates.
        if "functional" in drift_terms:
            if type(functional_drift) is not list:
                raise TypeError(
                    "Callables for functional drift terms must "
                    "be encapsulated in a list."
                )
            if len(functional_drift) == 0:
                raise ValueError(
                    "Must provide at least one callable object "
                    "when using the 'functional' drift capability."
                )
            self.functional_drift = True
            self.functional_drift_terms = functional_drift
        else:
            self.functional_drift = False

    def update_variogram_model(
        self,
        variogram_model,
        variogram_parameters=None,
        variogram_function=None,
        nlags=6,
        weight=False,
        anisotropy_scaling_y=1.0,
        anisotropy_scaling_z=1.0,
        anisotropy_angle_x=0.0,
        anisotropy_angle_y=0.0,
        anisotropy_angle_z=0.0,
    ):
        """Changes the variogram model and variogram parameters
        for the kriging system.

        Parameters
        ----------
        variogram_model : str or GSTools CovModel
            May be any of the variogram models listed above.
            May also be 'custom', in which case variogram_parameters and
            variogram_function must be specified.
            You can also use a
            `GSTools <https://github.com/GeoStat-Framework/GSTools>`_ CovModel.
        variogram_parameters : list or dict, optional
            List or dict of variogram model parameters, as explained above.
            If not provided, a best fit model will be calculated as
            described above.
        variogram_function : callable, optional
            A callable function that must be provided if variogram_model is
            specified as 'custom'. See above for more information.
        nlags : int, optional)
            Number of averaging bins for the semivariogram. Default is 6.
        weight : boolean, optional
            Flag that specifies if semivariance at smaller lags should be
            weighted more heavily when automatically calculating variogram
            model. See above for more information. True indicates that
            weights will be applied. Default is False.
        anisotropy_scaling_y : float, optional
            Scalar stretching value to take into account anisotropy
            in y-direction. Default is 1 (effectively no stretching).
            See above for more information.
        anisotropy_scaling_z : float, optional
            Scalar stretching value to take into account anisotropy
            in z-direction. Default is 1 (effectively no stretching).
            See above for more information.
        anisotropy_angle_x : float, optional
            Angle (in degrees) by which to rotate coordinate system about
            the x axis in order to take into account anisotropy.
            Default is 0 (no rotation). See above for more information.
        anisotropy_angle_y : float, optional
            Angle (in degrees) by which to rotate coordinate system about
            the y axis in order to take into account anisotropy.
            Default is 0 (no rotation). See above for more information.
        anisotropy_angle_z : float, optional
            Angle (in degrees) by which to rotate coordinate system about
            the z axis in order to take into account anisotropy.
            Default is 0 (no rotation).
            See above for more information.
        """

        # set up variogram model and parameters...
        self.variogram_model = variogram_model
        self.model = None
        # check if a GSTools covariance model is given
        if hasattr(self.variogram_model, "pykrige_kwargs"):
            # save the model in the class
            self.model = self.variogram_model
            if self.model.dim < 3:
                raise ValueError("GSTools: model dim is not 3")
            self.variogram_model = "custom"
            variogram_function = self.model.pykrige_vario
            variogram_parameters = []
            anisotropy_scaling_y = self.model.pykrige_anis_y
            anisotropy_scaling_z = self.model.pykrige_anis_z
            anisotropy_angle_x = self.model.pykrige_angle_x
            anisotropy_angle_y = self.model.pykrige_angle_y
            anisotropy_angle_z = self.model.pykrige_angle_z
        if (
            self.variogram_model not in self.variogram_dict.keys()
            and self.variogram_model != "custom"
        ):
            raise ValueError(
                "Specified variogram model '%s' is not supported." % variogram_model
            )
        elif self.variogram_model == "custom":
            if variogram_function is None or not callable(variogram_function):
                raise ValueError(
                    "Must specify callable function for custom variogram model."
                )
            else:
                self.variogram_function = variogram_function
        else:
            self.variogram_function = self.variogram_dict[self.variogram_model]

        if (
            anisotropy_scaling_y != self.anisotropy_scaling_y
            or anisotropy_scaling_z != self.anisotropy_scaling_z
            or anisotropy_angle_x != self.anisotropy_angle_x
            or anisotropy_angle_y != self.anisotropy_angle_y
            or anisotropy_angle_z != self.anisotropy_angle_z
        ):
            if self.verbose:
                print("Adjusting data for anisotropy...")
            self.anisotropy_scaling_y = anisotropy_scaling_y
            self.anisotropy_scaling_z = anisotropy_scaling_z
            self.anisotropy_angle_x = anisotropy_angle_x
            self.anisotropy_angle_y = anisotropy_angle_y
            self.anisotropy_angle_z = anisotropy_angle_z
            self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED = _adjust_for_anisotropy(
                np.vstack((self.X_ORIG, self.Y_ORIG, self.Z_ORIG)).T,
                [self.XCENTER, self.YCENTER, self.ZCENTER],
                [self.anisotropy_scaling_y, self.anisotropy_scaling_z],
                [
                    self.anisotropy_angle_x,
                    self.anisotropy_angle_y,
                    self.anisotropy_angle_z,
                ],
            ).T

        if self.verbose:
            print("Updating variogram mode...")

        vp_temp = _make_variogram_parameter_list(
            self.variogram_model, variogram_parameters
        )
        (
            self.lags,
            self.semivariance,
            self.variogram_model_parameters,
        ) = _initialize_variogram_model(
            np.vstack((self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)).T,
            self.VALUES,
            self.variogram_model,
            vp_temp,
            self.variogram_function,
            nlags,
            weight,
            "euclidean",
        )

        if self.verbose:
            if self.variogram_model == "linear":
                print("Using '%s' Variogram Model" % "linear")
                print("Slope:", self.variogram_model_parameters[0])
                print("Nugget:", self.variogram_model_parameters[1], "\n")
            elif self.variogram_model == "power":
                print("Using '%s' Variogram Model" % "power")
                print("Scale:", self.variogram_model_parameters[0])
                print("Exponent:", self.variogram_model_parameters[1])
                print("Nugget:", self.variogram_model_parameters[2], "\n")
            elif self.variogram_model == "custom":
                print("Using Custom Variogram Model")
            else:
                print("Using '%s' Variogram Model" % self.variogram_model)
                print("Partial Sill:", self.variogram_model_parameters[0])
                print(
                    "Full Sill:",
                    self.variogram_model_parameters[0]
                    + self.variogram_model_parameters[2],
                )
                print("Range:", self.variogram_model_parameters[1])
                print("Nugget:", self.variogram_model_parameters[2], "\n")
        if self.enable_plotting:
            self.display_variogram_model()

        if self.verbose:
            print("Calculating statistics on variogram model fit...")
        self.delta, self.sigma, self.epsilon = _find_statistics(
            np.vstack((self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)).T,
            self.VALUES,
            self.variogram_function,
            self.variogram_model_parameters,
            "euclidean",
            self.pseudo_inv,
        )
        self.Q1 = core.calcQ1(self.epsilon)
        self.Q2 = core.calcQ2(self.epsilon)
        self.cR = core.calc_cR(self.Q2, self.sigma)
        if self.verbose:
            print("Q1 =", self.Q1)
            print("Q2 =", self.Q2)
            print("cR =", self.cR, "\n")

    def display_variogram_model(self):
        """Displays semivariogram and variogram model."""
        import matplotlib.pyplot as plt

        fig = plt.figure()
        ax = fig.add_subplot(111)
        ax.plot(self.lags, self.semivariance, "r*")
        ax.plot(
            self.lags,
            self.variogram_function(self.variogram_model_parameters, self.lags),
            "k-",
        )
        plt.show()

    def switch_verbose(self):
        """Enables/disables program text output. No arguments."""
        self.verbose = not self.verbose

    def switch_plotting(self):
        """Enables/disable variogram plot display. No arguments."""
        self.enable_plotting = not self.enable_plotting

    def get_epsilon_residuals(self):
        """Returns the epsilon residuals for the variogram fit. No arguments."""
        return self.epsilon

    def plot_epsilon_residuals(self):
        """Plots the epsilon residuals for the variogram fit. No arguments."""
        import matplotlib.pyplot as plt

        fig = plt.figure()
        ax = fig.add_subplot(111)
        ax.scatter(range(self.epsilon.size), self.epsilon, c="k", marker="*")
        ax.axhline(y=0.0)
        plt.show()

    def get_statistics(self):
        """Returns the Q1, Q2, and cR statistics for the
        variogram fit (in that order). No arguments.
        """
        return self.Q1, self.Q2, self.cR

    def print_statistics(self):
        """Prints out the Q1, Q2, and cR statistics for the variogram fit.
        NOTE that ideally Q1 is close to zero, Q2 is close to 1,
        and cR is as small as possible.
        """
        print("Q1 =", self.Q1)
        print("Q2 =", self.Q2)
        print("cR =", self.cR)

    def _get_kriging_matrix(self, n, n_withdrifts):
        """Assembles the kriging matrix."""

        xyz = np.concatenate(
            (
                self.X_ADJUSTED[:, np.newaxis],
                self.Y_ADJUSTED[:, np.newaxis],
                self.Z_ADJUSTED[:, np.newaxis],
            ),
            axis=1,
        )
        d = cdist(xyz, xyz, "euclidean")
        if self.UNBIAS:
            a = np.zeros((n_withdrifts + 1, n_withdrifts + 1))
        else:
            a = np.zeros((n_withdrifts, n_withdrifts))
        a[:n, :n] = -self.variogram_function(self.variogram_model_parameters, d)
        np.fill_diagonal(a, 0.0)

        i = n
        if self.regional_linear_drift:
            a[:n, i] = self.X_ADJUSTED
            a[i, :n] = self.X_ADJUSTED
            i += 1
            a[:n, i] = self.Y_ADJUSTED
            a[i, :n] = self.Y_ADJUSTED
            i += 1
            a[:n, i] = self.Z_ADJUSTED
            a[i, :n] = self.Z_ADJUSTED
            i += 1
        if self.specified_drift:
            for arr in self.specified_drift_data_arrays:
                a[:n, i] = arr
                a[i, :n] = arr
                i += 1
        if self.functional_drift:
            for func in self.functional_drift_terms:
                a[:n, i] = func(self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)
                a[i, :n] = func(self.X_ADJUSTED, self.Y_ADJUSTED, self.Z_ADJUSTED)
                i += 1
        if i != n_withdrifts:
            warnings.warn(
                "Error in creating kriging matrix. Kriging may fail.", RuntimeWarning
            )
        if self.UNBIAS:
            a[n_withdrifts, :n] = 1.0
            a[:n, n_withdrifts] = 1.0
            a[n : n_withdrifts + 1, n : n_withdrifts + 1] = 0.0

        return a

    def _exec_vector(self, a, bd, xyz, mask, n_withdrifts, spec_drift_grids):
        """Solves the kriging system as a vectorized operation. This method
        can take a lot of memory for large grids and/or large datasets."""

        npt = bd.shape[0]
        n = self.X_ADJUSTED.shape[0]
        zero_index = None
        zero_value = False

        # use the desired method to invert the kriging matrix
        if self.pseudo_inv:
            a_inv = P_INV[self.pseudo_inv_type](a)
        else:
            a_inv = scipy.linalg.inv(a)

        if np.any(np.absolute(bd) <= self.eps):
            zero_value = True
            zero_index = np.where(np.absolute(bd) <= self.eps)

        if self.UNBIAS:
            b = np.zeros((npt, n_withdrifts + 1, 1))
        else:
            b = np.zeros((npt, n_withdrifts, 1))
        b[:, :n, 0] = -self.variogram_function(self.variogram_model_parameters, bd)
        if zero_value and self.exact_values:
            b[zero_index[0], zero_index[1], 0] = 0.0

        i = n
        if self.regional_linear_drift:
            b[:, i, 0] = xyz[:, 2]
            i += 1
            b[:, i, 0] = xyz[:, 1]
            i += 1
            b[:, i, 0] = xyz[:, 0]
            i += 1
        if self.specified_drift:
            for spec_vals in spec_drift_grids:
                b[:, i, 0] = spec_vals.flatten()
                i += 1
        if self.functional_drift:
            for func in self.functional_drift_terms:
                b[:, i, 0] = func(xyz[:, 2], xyz[:, 1], xyz[:, 0])
                i += 1
        if i != n_withdrifts:
            warnings.warn(
                "Error in setting up kriging system. Kriging may fail.",
                RuntimeWarning,
            )
        if self.UNBIAS:
            b[:, n_withdrifts, 0] = 1.0

        if (~mask).any():
            mask_b = np.repeat(
                mask[:, np.newaxis, np.newaxis], n_withdrifts + 1, axis=1
            )
            b = np.ma.array(b, mask=mask_b)

        if self.UNBIAS:
            x = (
                np.dot(a_inv, b.reshape((npt, n_withdrifts + 1)).T)
                .reshape((1, n_withdrifts + 1, npt))
                .T
            )
        else:
            x = (
                np.dot(a_inv, b.reshape((npt, n_withdrifts)).T)
                .reshape((1, n_withdrifts, npt))
                .T
            )
        kvalues = np.sum(x[:, :n, 0] * self.VALUES, axis=1)
        sigmasq = np.sum(x[:, :, 0] * -b[:, :, 0], axis=1)

        return kvalues, sigmasq

    def _exec_loop(self, a, bd_all, xyz, mask, n_withdrifts, spec_drift_grids):
        """Solves the kriging system by looping over all specified points.
        Less memory-intensive, but involves a Python-level loop."""

        npt = bd_all.shape[0]
        n = self.X_ADJUSTED.shape[0]
        kvalues = np.zeros(npt)
        sigmasq = np.zeros(npt)

        # use the desired method to invert the kriging matrix
        if self.pseudo_inv:
            a_inv = P_INV[self.pseudo_inv_type](a)
        else:
            a_inv = scipy.linalg.inv(a)

        for j in np.nonzero(~mask)[
            0
        ]:  # Note that this is the same thing as range(npt) if mask is not defined,
            bd = bd_all[j]  # otherwise it takes the non-masked elements.
            if np.any(np.absolute(bd) <= self.eps):
                zero_value = True
                zero_index = np.where(np.absolute(bd) <= self.eps)
            else:
                zero_value = False
                zero_index = None

            if self.UNBIAS:
                b = np.zeros((n_withdrifts + 1, 1))
            else:
                b = np.zeros((n_withdrifts, 1))
            b[:n, 0] = -self.variogram_function(self.variogram_model_parameters, bd)
            if zero_value and self.exact_values:
                b[zero_index[0], 0] = 0.0

            i = n
            if self.regional_linear_drift:
                b[i, 0] = xyz[j, 2]
                i += 1
                b[i, 0] = xyz[j, 1]
                i += 1
                b[i, 0] = xyz[j, 0]
                i += 1
            if self.specified_drift:
                for spec_vals in spec_drift_grids:
                    b[i, 0] = spec_vals.flatten()[i]
                    i += 1
            if self.functional_drift:
                for func in self.functional_drift_terms:
                    b[i, 0] = func(xyz[j, 2], xyz[j, 1], xyz[j, 0])
                    i += 1
            if i != n_withdrifts:
                warnings.warn(
                    "Error in setting up kriging system. Kriging may fail.",
                    RuntimeWarning,
                )
            if self.UNBIAS:
                b[n_withdrifts, 0] = 1.0

            x = np.dot(a_inv, b)
            kvalues[j] = np.sum(x[:n, 0] * self.VALUES)
            sigmasq[j] = np.sum(x[:, 0] * -b[:, 0])

        return kvalues, sigmasq

    def execute(
        self,
        style,
        xpoints,
        ypoints,
        zpoints,
        mask=None,
        backend="vectorized",
        specified_drift_arrays=None,
    ):
        """Calculates a kriged grid and the associated variance.

        This is now the method that performs the main kriging calculation.
        Note that currently measurements (i.e., z values) are
        considered 'exact'. This means that, when a specified coordinate for
        interpolation is exactly the same as one of the data points,
        the variogram evaluated at the point is forced to be zero. Also, the
        diagonal of the kriging matrix is also always forced to be zero.
        In forcing the variogram evaluated at data points to be zero, we are
        effectively saying that there is no variance at that point
        (no uncertainty, so the value is 'exact').

        In the future, the code may include an extra 'exact_values' boolean
        flag that can be adjusted to specify whether to treat the measurements
        as 'exact'. Setting the flag to false would indicate that the variogram
        should not be forced to be zero at zero distance (i.e., when evaluated
        at data points). Instead, the uncertainty in the point will be equal
        to the nugget. This would mean that the diagonal of the kriging matrix
        would be set to the nugget instead of to zero.

        Parameters
        ----------
        style : str
            Specifies how to treat input kriging points. Specifying 'grid'
            treats xpoints, ypoints, and zpoints as arrays of x, y, and z
            coordinates that define a rectangular grid. Specifying 'points'
            treats xpoints, ypoints, and zpoints as arrays that provide
            coordinates at which to solve the kriging system. Specifying
            'masked' treats xpoints, ypoints, and zpoints as arrays of x, y,
            and z coordinates that define a rectangular grid and uses mask
            to only evaluate specific points in the grid.
        xpoints : array_like, shape (N,) or (N, 1)
            If style is specific as 'grid' or 'masked', x-coordinates of
            LxMxN grid. If style is specified as 'points', x-coordinates of
            specific points at which to solve kriging system.
        ypoints : array_like, shape (M,) or (M, 1)
            If style is specified as 'grid' or 'masked', y-coordinates of
            LxMxN grid. If style is specified as 'points', y-coordinates of
            specific points at which to solve kriging system. Note that in this
            case, xpoints, ypoints, and zpoints must have the same dimensions
            (i.e., L = M = N).
        zpoints : array_like, shape (L,) or (L, 1)
            If style is specified as 'grid' or 'masked', z-coordinates of
            LxMxN grid. If style is specified as 'points', z-coordinates of
            specific points at which to solve kriging system. Note that in this
            case, xpoints, ypoints, and zpoints must have the same dimensions
            (i.e., L = M = N).
        mask : boolean array, shape (L, M, N), optional
            Specifies the points in the rectangular grid defined by xpoints,
            ypoints, zpoints that are to be excluded in the kriging
            calculations. Must be provided if style is specified as 'masked'.
            False indicates that the point should not be masked, so the kriging
            system will be solved at the point.
            True indicates that the point should be masked, so the kriging
            system will not be solved at the point.
        backend : string, optional
            Specifies which approach to use in kriging. Specifying 'vectorized'
            will solve the entire kriging problem at once in a vectorized
            operation. This approach is faster but also can consume a
            significant amount of memory for large grids and/or large datasets.
            Specifying 'loop' will loop through each point at which the kriging
            system is to be solved. This approach is slower but also less
            memory-intensive. Default is 'vectorized'.
        specified_drift_arrays : list of array-like objects, optional
            Specifies the drift values at the points at which the kriging
            system is to be evaluated. Required if 'specified' drift provided
            in the list of drift terms when instantiating the UniversalKriging3D
            class. Must be a list of arrays in the same order as the list
            provided when instantiating the kriging object. Array(s) must be
            the same dimension as the specified grid or have the same number
            of points as the specified points; i.e., the arrays either must be
            shape (L, M, N), where L is the number of z grid-points,
            M is the number of y grid-points, and N is the number of
            x grid-points, or shape (N,) or (N, 1), where N is the number of
            points at which to evaluate the kriging system.

        Returns
        -------
        kvalues : ndarray, shape (L, M, N) or (N,) or (N, 1)
            Interpolated values of specified grid or at the specified set
            of points. If style was specified as 'masked', kvalues will be a
            numpy masked array.
        sigmasq : ndarray, shape (L, M, N) or (N,) or (N, 1)
            Variance at specified grid points or at the specified set of points.
            If style was specified as 'masked', sigmasq will be a numpy
            masked array.
        """

        if self.verbose:
            print("Executing Ordinary Kriging...\n")

        if style != "grid" and style != "masked" and style != "points":
            raise ValueError("style argument must be 'grid', 'points', or 'masked'")

        xpts = np.atleast_1d(np.squeeze(np.array(xpoints, copy=True)))
        ypts = np.atleast_1d(np.squeeze(np.array(ypoints, copy=True)))
        zpts = np.atleast_1d(np.squeeze(np.array(zpoints, copy=True)))
        n = self.X_ADJUSTED.shape[0]
        n_withdrifts = n
        if self.regional_linear_drift:
            n_withdrifts += 3
        if self.specified_drift:
            n_withdrifts += len(self.specified_drift_data_arrays)
        if self.functional_drift:
            n_withdrifts += len(self.functional_drift_terms)
        nx = xpts.size
        ny = ypts.size
        nz = zpts.size
        a = self._get_kriging_matrix(n, n_withdrifts)

        if style in ["grid", "masked"]:
            if style == "masked":
                if mask is None:
                    raise IOError(
                        "Must specify boolean masking array when style is 'masked'."
                    )
                if mask.ndim != 3:
                    raise ValueError("Mask is not three-dimensional.")
                if mask.shape[0] != nz or mask.shape[1] != ny or mask.shape[2] != nx:
                    if (
                        mask.shape[0] == nx
                        and mask.shape[2] == nz
                        and mask.shape[1] == ny
                    ):
                        mask = mask.swapaxes(0, 2)
                    else:
                        raise ValueError(
                            "Mask dimensions do not match specified grid dimensions."
                        )
                mask = mask.flatten()
            npt = nz * ny * nx
            grid_z, grid_y, grid_x = np.meshgrid(zpts, ypts, xpts, indexing="ij")
            xpts = grid_x.flatten()
            ypts = grid_y.flatten()
            zpts = grid_z.flatten()
        elif style == "points":
            if xpts.size != ypts.size and ypts.size != zpts.size:
                raise ValueError(
                    "xpoints and ypoints must have same "
                    "dimensions when treated as listing "
                    "discrete points."
                )
            npt = nx
        else:
            raise ValueError("style argument must be 'grid', 'points', or 'masked'")

        if specified_drift_arrays is None:
            specified_drift_arrays = []
        spec_drift_grids = []
        if self.specified_drift:
            if len(specified_drift_arrays) == 0:
                raise ValueError(
                    "Must provide drift values for kriging "
                    "points when using 'specified' drift "
                    "capability."
                )
            if type(specified_drift_arrays) is not list:
                raise TypeError(
                    "Arrays for specified drift terms must "
                    "be encapsulated in a list."
                )
            for spec in specified_drift_arrays:
                if style in ["grid", "masked"]:
                    if spec.ndim < 3:
                        raise ValueError(
                            "Dimensions of drift values array do "
                            "not match specified grid dimensions."
                        )
                    elif (
                        spec.shape[0] != nz
                        or spec.shape[1] != ny
                        or spec.shape[2] != nx
                    ):
                        if (
                            spec.shape[0] == nx
                            and spec.shape[2] == nz
                            and spec.shape[1] == ny
                        ):
                            spec_drift_grids.append(np.squeeze(spec.swapaxes(0, 2)))
                        else:
                            raise ValueError(
                                "Dimensions of drift values array "
                                "do not match specified grid "
                                "dimensions."
                            )
                    else:
                        spec_drift_grids.append(np.squeeze(spec))
                elif style == "points":
                    if spec.ndim != 1:
                        raise ValueError(
                            "Dimensions of drift values array do "
                            "not match specified grid dimensions."
                        )
                    elif spec.shape[0] != xpts.size:
                        raise ValueError(
                            "Number of supplied drift values in "
                            "array do not match specified number "
                            "of kriging points."
                        )
                    else:
                        spec_drift_grids.append(np.squeeze(spec))
            if len(spec_drift_grids) != len(self.specified_drift_data_arrays):
                raise ValueError(
                    "Inconsistent number of specified drift terms supplied."
                )
        else:
            if len(specified_drift_arrays) != 0:
                warnings.warn(
                    "Provided specified drift values, but "
                    "'specified' drift was not initialized during "
                    "instantiation of UniversalKriging3D class.",
                    RuntimeWarning,
                )

        xpts, ypts, zpts = _adjust_for_anisotropy(
            np.vstack((xpts, ypts, zpts)).T,
            [self.XCENTER, self.YCENTER, self.ZCENTER],
            [self.anisotropy_scaling_y, self.anisotropy_scaling_z],
            [self.anisotropy_angle_x, self.anisotropy_angle_y, self.anisotropy_angle_z],
        ).T

        if style != "masked":
            mask = np.zeros(npt, dtype="bool")

        xyz_points = np.concatenate(
            (zpts[:, np.newaxis], ypts[:, np.newaxis], xpts[:, np.newaxis]), axis=1
        )
        xyz_data = np.concatenate(
            (
                self.Z_ADJUSTED[:, np.newaxis],
                self.Y_ADJUSTED[:, np.newaxis],
                self.X_ADJUSTED[:, np.newaxis],
            ),
            axis=1,
        )
        bd = cdist(xyz_points, xyz_data, "euclidean")

        if backend == "vectorized":
            kvalues, sigmasq = self._exec_vector(
                a, bd, xyz_points, mask, n_withdrifts, spec_drift_grids
            )
        elif backend == "loop":
            kvalues, sigmasq = self._exec_loop(
                a, bd, xyz_points, mask, n_withdrifts, spec_drift_grids
            )
        else:
            raise ValueError(
                "Specified backend {} is not supported for "
                "3D ordinary kriging.".format(backend)
            )

        if style == "masked":
            kvalues = np.ma.array(kvalues, mask=mask)
            sigmasq = np.ma.array(sigmasq, mask=mask)

        if style in ["masked", "grid"]:
            kvalues = kvalues.reshape((nz, ny, nx))
            sigmasq = sigmasq.reshape((nz, ny, nx))

        return kvalues, sigmasq