multifi_cokriging.py

"""
Integrates the Multi-Fidelity Co-Kriging method described in [LeGratiet2013].

(Author: Remi Vauclin vauclin.remi@gmail.com)

This code was implemented using the package scikit-learn as basis.
(Author: Vincent Dubourg, vincent.dubourg@gmail.com)

OpenMDAO adaptation. Regression and correlation functions were directly copied
from scikit-learn package here to avoid scikit-learn dependency.
(Author: Remi Lafage, remi.lafage@onera.fr)

ISAE/DMSM - ONERA/DCPS
"""
import numpy as np
from numpy import atleast_2d as array2d

from scipy import linalg
from scipy.optimize import minimize
from scipy.spatial.distance import squareform

from openmdao.surrogate_models.surrogate_model import MultiFiSurrogateModel

import logging
_logger = logging.getLogger()

MACHINE_EPSILON = np.finfo(np.double).eps  # machine precision
NUGGET = 10. * MACHINE_EPSILON  # nugget for robustness

INITIAL_RANGE_DEFAULT = 0.3  # initial range for optimizer
TOLERANCE_DEFAULT = 1e-6    # stopping criterion for MLE optimization

THETA0_DEFAULT = 0.5
THETAL_DEFAULT = 1e-5
THETAU_DEFAULT = 50

if hasattr(linalg, 'solve_triangular'):
    # only in scipy since 0.9
    solve_triangular = linalg.solve_triangular
else:
    # slower, but works
    def solve_triangular(x, y, lower=True):
        """Solve triangular."""
        return linalg.solve(x, y)


def constant_regression(x):
    """
    Zero order polynomial (constant, p = 1) regression model.

    x --> f(x) = 1

    Parameters
    ----------
    x : array_like
        Input data.

    Returns
    -------
    array_like
        Constant regression output.
    """
    x = np.asarray(x, dtype=np.float)
    n_eval = x.shape[0]
    f = np.ones([n_eval, 1])
    return f


def linear_regression(x):
    """
    First order polynomial (linear, p = n+1) regression model.

    x --> f(x) = [ 1, x_1, ..., x_n ].T

    Parameters
    ----------
    x : array_like
        Input data.

    Returns
    -------
    array_like
        Linear regression output.
    """
    x = np.asarray(x, dtype=np.float)
    n_eval = x.shape[0]
    f = np.hstack([np.ones([n_eval, 1]), x])
    return f


def squared_exponential_correlation(theta, d):
    """
    Squared exponential correlation model (Radial Basis Function).

    (Infinitely differentiable stochastic process, very smooth)::

                                            n
        theta, dx --> r(theta, dx) = exp(  sum  - theta_i * (dx_i)^2 )
                                          i = 1

    Parameters
    ----------
    theta : array_like
        An array with shape 1 (isotropic) or n (anisotropic) giving the
        autocorrelation parameter(s).
    d : array_like
        An array with shape (n_eval, n_features) giving the componentwise
        distances between locations x and x' at which the correlation model
        should be evaluated.

    Returns
    -------
    array_like
        An array with shape (n_eval, ) containing the values of the
        autocorrelation model.
    """
    theta = np.asarray(theta, dtype=np.float)
    d = np.asarray(d, dtype=np.float)

    if d.ndim > 1:
        n_features = d.shape[1]
    else:
        n_features = 1

    if theta.size == 1:
        return np.exp(-theta[0] * np.sum(d ** 2, axis=1))
    elif theta.size != n_features:
        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 l1_cross_distances(X, Y=None):
    """
    Compute the nonzero componentwise L1 cross-distances between the vectors in X and Y.

    Parameters
    ----------
    X : array_like
        An array with shape (n_samples_X, n_features).
    Y : array_like
        An array with shape (n_samples_Y, n_features).

    Returns
    -------
    array with shape (n_samples * (n_samples - 1) / 2, n_features)
        The array of componentwise L1 cross-distances.
    """
    X = array2d(X)

    if Y is None:
        n_samples, n_features = X.shape
        n_nonzero_cross_dist = n_samples * (n_samples - 1) // 2
        D = np.zeros((n_nonzero_cross_dist, n_features))
        ll_1 = 0
        for k in range(n_samples - 1):
            ll_0 = ll_1
            ll_1 = ll_0 + n_samples - k - 1
            D[ll_0:ll_1] = np.abs(X[k] - X[(k + 1):])
    else:
        Y = array2d(Y)
        n_samples_X, n_features_X = X.shape
        n_samples_Y, n_features_Y = Y.shape
        if n_features_X != n_features_Y:
            raise ValueError("X and Y must have the same dimensions.")
        n_features = n_features_X

        n_nonzero_cross_dist = n_samples_X * n_samples_Y
        D = np.zeros((n_nonzero_cross_dist, n_features))
        ll_1 = 0
        for k in range(n_samples_X):
            ll_0 = ll_1
            ll_1 = ll_0 + n_samples_Y  # - k - 1
            D[ll_0:ll_1] = np.abs(X[k] - Y)

    return D


class MultiFiCoKriging(object):
    """
    Integrate the Multi-Fidelity Co-Kriging method described in [LeGratiet2013].

    Parameters
    ----------
    regr : str or callable, optional
        A regression function returning an array of outputs of the linear
        regression functional basis for Universal Kriging purpose.
        regr is assumed to be the same for all levels of code.
        Default assumes a simple constant regression trend.
        Available built-in regression models are:
        'constant', 'linear'.
    rho_regr : str or callable, optional
        A regression function returning an array of outputs of the linear
        regression functional basis. Defines the regression function for the
        autoregressive parameter rho.
        rho_regr is assumed to be the same for all levels of code.
        Default assumes a simple constant regression trend.
        Available built-in regression models are:
        'constant', 'linear'.
    normalize : bool, optional
        When true, normalize X and Y so that the mean is at zero.
    theta : double, array_like or list, optional
        Value of correlation parameters if they are known; no optimization is run.
        Default is None, so that optimization is run.
        if double: value is replicated for all features and all levels.
        if array_like: an array with shape (n_features, ) for
        isotropic calculation. It is replicated for all levels.
        if list: a list of nlevel arrays specifying value for each level.
    theta0 : double, array_like or list, optional
        Starting point for the maximum likelihood estimation of the
        best set of parameters.
        Default is None and meaning use of the default 0.5*np.ones(n_features)
        if double: value is replicated for all features and all levels.
        if array_like: an array with shape (n_features, ) for
        isotropic calculation. It is replicated for all levels.
        if list: a list of nlevel arrays specifying value for each level.
    thetaL : double, array_like or list, optional
        Lower bound on the autocorrelation parameters for maximum
        likelihood estimation.
        Default is None meaning use of the default 1e-5*np.ones(n_features).
        if double: value is replicated for all features and all levels.
        if array_like: An array with shape matching theta0's. It is replicated
        for all levels of code.
        if list: a list of nlevel arrays specifying value for each level.
    thetaU : double, array_like or list, optional
        Upper bound on the autocorrelation parameters for maximum
        likelihood estimation.
        Default is None meaning use of default value 50*np.ones(n_features).
        if double: value is replicated for all features and all levels.
        if array_like: An array with shape matching theta0's. It is replicated
        for all levels of code.
        if list: a list of nlevel arrays specifying value for each level.

    Attributes
    ----------
    corr : object
        Correlation function to use, default is squared_exponential_correlation.
    n_features : ndarray
        Number of features for each fidelity level.
    n_samples : ndarray
        Number of samples for each fidelity level.
    nlevel : int
        Number of fidelity levels.
    normalize : bool, optional
        When true, normalize X and Y so that the mean is at zero.
    regr : str or callable
        A regression function returning an array of outputs of the linear
        regression functional basis for Universal Kriging purpose.
        regr is assumed to be the same for all levels of code.
        Default assumes a simple constant regression trend.
        Available built-in regression models are:
        'constant', 'linear'
    rho_regr : str or callable or None
        A regression function returning an array of outputs of the linear
        regression functional basis. Defines the regression function for the
        autoregressive parameter rho.
        rho_regr is assumed to be the same for all levels of code.
        Default assumes a simple constant regression trend.
        Available built-in regression models are:
        'constant', 'linear'
    theta : double, array_like or list or None
        Value of correlation parameters if they are known; no optimization is run.
        Default is None, so that optimization is run.
        if double: value is replicated for all features and all levels.
        if array_like: an array with shape (n_features, ) for
        isotropic calculation. It is replicated for all levels.
        if list: a list of nlevel arrays specifying value for each level
    theta0 : double, array_like or list or None
        Starting point for the maximum likelihood estimation of the
        best set of parameters.
        Default is None and meaning use of the default 0.5*np.ones(n_features)
        if double: value is replicated for all features and all levels.
        if array_like: an array with shape (n_features, ) for
        isotropic calculation. It is replicated for all levels.
        if list: a list of nlevel arrays specifying value for each level
    thetaL : double, array_like or list or None
        Lower bound on the autocorrelation parameters for maximum
        likelihood estimation.
        Default is None meaning use of the default 1e-5*np.ones(n_features).
        if double: value is replicated for all features and all levels.
        if array_like: An array with shape matching theta0's. It is replicated
        for all levels of code.
        if list: a list of nlevel arrays specifying value for each level
    thetaU : double, array_like or list or None
        Upper bound on the autocorrelation parameters for maximum
        likelihood estimation.
        Default is None meaning use of default value 50*np.ones(n_features).
        if double: value is replicated for all features and all levels.
        if array_like: An array with shape matching theta0's. It is replicated
        for all levels of code.
        if list: a list of nlevel arrays specifying value for each level
    X_mean : float
        Mean of the low fidelity training data for X.
    X_std : float
        Standard deviation of the low fidelity training data for X.
    y_mean : float
        Mean of the low fidelity training data for y.
    y_std : float
        Standard deviation of the low fidelity training data for y.
    _nfev : int
        Number of function evaluations.

    Notes
    -----
    Implementation is based on the Package Scikit-Learn
    (Author: Vincent Dubourg, vincent.dubourg@gmail.com) which translates
    the DACE Matlab toolbox, see [NLNS2002]_.

    References
    ----------
    .. [NLNS2002] H. B. Nielsen, S. N. Lophaven, and J. Sondergaard.
       `DACE - A MATLAB Kriging Toolbox.` (2002)
       http://www2.imm.dtu.dk/~hbn/dace/dace.pdf

    .. [WBSWM1992] W. J. Welch, R. J. Buck, J. Sacks, H. P. Wynn, T. J. Mitchell,
       and M. D. Morris (1992). "Screening, predicting, and computer experiments."
       `Technometrics,` 34(1) 15--25.
       http://www.jstor.org/pss/1269548

    .. [LeGratiet2013] L. Le Gratiet (2013). "Multi-fidelity Gaussian process
       regression for computer experiments."
       PhD thesis, Universite Paris-Diderot-Paris VII.

    .. [TBKH2011] Toal, D. J., Bressloff, N. W., Keane, A. J., & Holden, C. M. E. (2011).
       "The development of a hybridized particle swarm for kriging hyperparameter
       tuning." `Engineering optimization`, 43(6), 675-699.

    Examples
    --------
    >>> from openmdao.surrogate_models.multifi_cokriging import MultiFiCoKriging
    >>> import numpy as np
    >>> # Xe: DOE for expensive code (nested in Xc)
    >>> # Xc: DOE for cheap code
    >>> # ye: expensive response
    >>> # yc: cheap response
    >>> Xe = np.array([[0],[0.4],[1]])
    >>> Xc = np.vstack((np.array([[0.1],[0.2],[0.3],[0.5],[0.6],[0.7],[0.8],[0.9]]),Xe))
    >>> ye = ((Xe*6-2)**2)*np.sin((Xe*6-2)*2)
    >>> yc = 0.5*((Xc*6-2)**2)*np.sin((Xc*6-2)*2)+(Xc-0.5)*10. - 5
    >>> model = MultiFiCoKriging(theta0=1, thetaL=1e-5, thetaU=50.)
    >>> model.fit([Xc, Xe], [yc, ye])
    >>> # Prediction on x=0.05
    >>> np.abs(float(model.predict([0.05])[0])- ((0.05*6-2)**2)*np.sin((0.05*6-2)*2)) < 0.05
    True
    """

    _regression_types = {
        'constant': constant_regression,
        'linear': linear_regression
    }

    def __init__(self, regr='constant', rho_regr='constant', normalize=True,
                 theta=None, theta0=None, thetaL=None, thetaU=None):
        """
        Initialize all attributes.
        """
        self.corr = squared_exponential_correlation
        self.regr = regr
        self.rho_regr = rho_regr
        self.theta = theta
        self.theta0 = theta0
        self.thetaL = thetaL
        self.thetaU = thetaU
        self.normalize = normalize
        self.X_mean = 0
        self.X_std = 1
        self.y_mean = 0
        self.y_std = 1

        self.n_features = None
        self.n_samples = None
        self.nlevel = None

        self._nfev = 0

    def _build_R(self, lvl, theta):
        """
        Build the correlation matrix with given theta for the specified level.

        Parameters
        ----------
        lvl : int
            Level of fidelity
        theta : array_like
            An array containing the autocorrelation parameters at which the
            Gaussian Process model parameters should be determined.
            Default uses the built-in autocorrelation parameters
            (ie ``theta = self.theta``).

        Returns
        -------
        ndarray
            Correlation matrix.
        """
        D = self.D[lvl]
        n_samples = self.n_samples[lvl]

        R = np.eye(n_samples) * (1. + NUGGET)

        corr = squareform(self.corr(theta, D))
        R = R + corr

        return R

    def fit(self, X, y, initial_range=INITIAL_RANGE_DEFAULT, tol=TOLERANCE_DEFAULT):
        """
        Implement the Multi-Fidelity co-kriging model fitting method.

        Parameters
        ----------
        X : list of double array_like elements
            A list of arrays with the input at which observations were made, from lowest
            fidelity to highest fidelity. Designs must be nested
            with X[i] = np.vstack([..., X[i+1]).
        y : list of double array_like elements
            A list of arrays with the observations of the scalar output to be predicted,
            from lowest fidelity to highest fidelity.
        initial_range : float
            Initial range for the optimizer.
        tol : float
            Optimizer terminates when the tolerance tol is reached.
        """
        # Run input checks
        # Transforms floats and arrays in lists to have a multifidelity
        # structure
        self._check_list_structure(X, y)
        # Checks if all parameters are structured as required
        self._check_params()

        X = self.X
        y = self.y
        nlevel = self.nlevel
        n_samples = self.n_samples

        # initialize lists
        self.beta = nlevel * [0]
        self.beta_rho = nlevel * [None]
        self.beta_regr = nlevel * [None]
        self.C = nlevel * [0]
        self.D = nlevel * [0]
        self.F = nlevel * [0]
        self.p = nlevel * [0]
        self.q = nlevel * [0]
        self.G = nlevel * [0]
        self.sigma2 = nlevel * [0]
        self._R_adj = nlevel * [None]

        # Training data will be normalized using statistical quantities from the low fidelity set.
        if self.normalize:
            self.X_mean = X_mean = np.mean(X[0], axis=0)
            self.X_std = X_std = np.std(X[0], axis=0)
            self.y_mean = y_mean = np.mean(y[0], axis=0)
            self.y_std = y_std = np.std(y[0], axis=0)

            X_std[X_std == 0.] = 1.
            y_std[y_std == 0.] = 1.

        for lvl in range(nlevel):

            if self.normalize:
                X[lvl] = (X[lvl] - X_mean) / X_std
                y[lvl] = (y[lvl] - y_mean) / y_std

            # Calculate matrix of distances D between samples
            self.D[lvl] = l1_cross_distances(X[lvl])
            if (np.min(np.sum(self.D[lvl], axis=1)) == 0.):
                raise ValueError("Multiple input features cannot have the same value.")

            # Regression matrix and parameters
            self.F[lvl] = self.regr(X[lvl])
            self.p[lvl] = self.F[lvl].shape[1]

            # Concatenate the autoregressive part for levels > 0
            if lvl > 0:
                F_rho = self.rho_regr(X[lvl])
                self.q[lvl] = F_rho.shape[1]
                self.F[lvl] = np.hstack((F_rho * np.dot((self.y[lvl - 1])[-n_samples[lvl]:],
                                                        np.ones((1, self.q[lvl]))), self.F[lvl]))
            else:
                self.q[lvl] = 0

            n_samples_F_i = self.F[lvl].shape[0]

            if n_samples_F_i != n_samples[lvl]:
                raise ValueError("Number of rows in F and X do not match. Most "
                                 "likely something is going wrong with the "
                                 "regression model.")

            if int(self.p[lvl] + self.q[lvl]) >= n_samples_F_i:
                raise ValueError(f"Ordinary least squares problem is undetermined "
                                 f"n_samples={n_samples[lvl]} must be greater than the regression"
                                 f" model size p+q={self.p[lvl] + self.q[lvl]}.")

        # Set attributes
        self.X = X
        self.y = y

        self.rlf_value = np.zeros(nlevel)

        for lvl in range(nlevel):
            # Determine Gaussian Process model parameters
            if self.theta[lvl] is None:
                # Maximum Likelihood Estimation of the parameters
                sol = self._max_rlf(lvl=lvl, initial_range=initial_range, tol=tol)
                self.theta[lvl] = sol['theta']
                self.rlf_value[lvl] = sol['rlf_value']

                if np.isinf(self.rlf_value[lvl]):
                    raise ValueError("Bad parameter region. Try increasing upper bound")
            else:
                self.rlf_value[lvl] = self.rlf(lvl=lvl)
                if np.isinf(self.rlf_value[lvl]):
                    raise ValueError("Bad point. Try increasing theta0.")

        return

    def rlf(self, lvl, theta=None):
        """
        Determine BLUP parameters and evaluate negative reduced likelihood function for theta.

        Maximizing this function wrt the autocorrelation parameters theta is
        equivalent to maximizing the likelihood of the assumed joint Gaussian
        distribution of the observations y evaluated onto the design of
        experiments X.

        Parameters
        ----------
        lvl : int
            Level of fidelity.
        theta : array_like, optional
            An array containing the autocorrelation parameters at which the
            Gaussian Process model parameters should be determined.
            Default uses the built-in autocorrelation parameters
            (ie ``theta = self.theta``).

        Returns
        -------
        double
            The value of the negative concentrated reduced likelihood function
            associated to the given autocorrelation parameters theta.
        """
        if theta is None:
            # Use built-in autocorrelation parameters
            theta = self.theta[lvl]

        # Initialize output
        rlf_value = 1e20

        # Retrieve data
        n_samples = self.n_samples[lvl]
        y = self.y[lvl]
        F = self.F[lvl]
        p = self.p[lvl]
        q = self.q[lvl]

        R = self._build_R(lvl, theta)

        try:
            C = linalg.cholesky(R, lower=True)
        except linalg.LinAlgError:
            _logger.warning(('Cholesky decomposition of R at level %i failed' % lvl) +
                            ' with theta=' + str(theta))
            return rlf_value

        # Get generalized least squares solution
        Ft = solve_triangular(C, F, lower=True)
        Yt = solve_triangular(C, y, lower=True)
        try:
            Q, G = linalg.qr(Ft, econ=True)
        except TypeError:   # qr() got an unexpected keyword argument 'econ'
            # DeprecationWarning: qr econ argument will be removed after scipy
            # 0.7. The economy transform will then be available through the
            # mode='economic' argument.
            Q, G = linalg.qr(Ft, mode='economic')
            pass

        # Universal Kriging
        beta = solve_triangular(G, np.dot(Q.T, Yt))

        err = Yt - np.dot(Ft, beta)
        err2 = np.dot(err.T, err)[0, 0]
        self._err = err
        sigma2 = err2 / (n_samples - p - q)
        detR = ((np.diag(C))**(2. / n_samples)).prod()

        rlf_value = (n_samples - p - q) * np.log10(sigma2) \
            + n_samples * np.log10(detR)

        self.beta_rho[lvl] = beta[:q]
        self.beta_regr[lvl] = beta[q:]
        self.beta[lvl] = beta
        self.sigma2[lvl] = sigma2
        self.C[lvl] = C
        self.G[lvl] = G

        return rlf_value

    def _max_rlf(self, lvl, initial_range, tol):
        """
        Estimate autocorrelation parameter theta as maximizer of the reduced likelihood function.

        (Minimization of the negative reduced likelihood function is used for convenience.)

        Parameters
        ----------
        lvl : int
            Level of fidelity
        initial_range : float
            Initial range of the optimizer
        tol : float
            Optimizer terminates when the tolerance tol is reached.

        Returns
        -------
        array_like
            The optimal hyperparameters.
        double
            The optimal negative reduced likelihood function value.
        dict
            res['theta']: optimal theta
            res['rlf_value']: optimal value for likelihood
        """
        # Initialize input
        thetaL = self.thetaL[lvl]
        thetaU = self.thetaU[lvl]

        def rlf_transform(x):
            return self.rlf(theta=10.**x, lvl=lvl)

        # Use specified starting point as first guess
        theta0 = self.theta0[lvl]
        x0 = np.log10(theta0[0])

        constraints = []
        for i in range(theta0.size):
            constraints.append({'type': 'ineq', 'fun': lambda log10t, i=i:
                                log10t[i] - np.log10(thetaL[0][i])})
            constraints.append({'type': 'ineq', 'fun': lambda log10t, i=i:
                                np.log10(thetaU[0][i]) - log10t[i]})

        constraints = tuple(constraints)
        sol = minimize(rlf_transform, x0, method='COBYLA',
                       constraints=constraints,
                       options={'rhobeg': initial_range,
                                'tol': tol, 'disp': 0})

        log10_optimal_x = sol['x']
        optimal_rlf_value = sol['fun']
        self._nfev += sol['nfev']

        optimal_theta = 10. ** log10_optimal_x

        res = {}
        res['theta'] = optimal_theta
        res['rlf_value'] = optimal_rlf_value

        return res

    def predict(self, X, eval_MSE=True):
        """
        Perform the predictions of the kriging model on X.

        Parameters
        ----------
        X : array_like
            An array with shape (n_eval, n_features) giving the point(s) at
            which the prediction(s) should be made.
        eval_MSE : bool, optional
            A boolean specifying whether the Mean Squared Error should be
            evaluated or not. Default assumes evalMSE is True.

        Returns
        -------
        array_like
            An array with shape (n_eval, ) with the Best Linear Unbiased
            Prediction at X. If all_levels is set to True, an array
            with shape (n_eval, nlevel) giving the BLUP for all levels.
        array_like, optional (if eval_MSE is True)
            An array with shape (n_eval, ) with the Mean Squared Error at X.
            If all_levels is set to True, an array with shape (n_eval, nlevel)
            giving the MSE for all levels.
        """
        X = array2d(X)
        nlevel = self.nlevel
        n_eval, n_features_X = X.shape

        # Normalize
        if self.normalize:
            X = (X - self.X_mean) / self.X_std

        # Calculate kriging mean and variance at level 0
        mu = np.zeros((n_eval, nlevel))

        f = self.regr(X)
        f0 = self.regr(X)
        dx = l1_cross_distances(X, Y=self.X[0])

        # Get regression function and correlation
        F = self.F[0]
        C = self.C[0]

        beta = self.beta[0]
        Ft = solve_triangular(C, F, lower=True)
        yt = solve_triangular(C, self.y[0], lower=True)
        r_ = self.corr(self.theta[0], dx).reshape(n_eval, self.n_samples[0])
        gamma = solve_triangular(C.T, yt - np.dot(Ft, beta), lower=False)

        # Scaled predictor
        mu[:, 0] = (np.dot(f, beta) + np.dot(r_, gamma)).ravel()

        if eval_MSE:
            MSE = np.zeros((n_eval, nlevel))
            r_t = solve_triangular(C, r_.T, lower=True)
            G = self.G[0]

            u_ = solve_triangular(G.T, f.T - np.dot(Ft.T, r_t), lower=True)
            MSE[:, 0] = self.sigma2[0] * \
                (1 - (r_t**2).sum(axis=0) + (u_**2).sum(axis=0))

        # Calculate recursively kriging mean and variance at level i
        for i in range(1, nlevel):
            C = self.C[i]
            F = self.F[i]
            g = self.rho_regr(X)
            dx = l1_cross_distances(X, Y=self.X[i])
            r_ = self.corr(self.theta[i], dx).reshape(
                n_eval, self.n_samples[i])
            f = np.vstack((g.T * mu[:, i - 1], f0.T))

            Ft = solve_triangular(C, F, lower=True)
            yt = solve_triangular(C, self.y[i], lower=True)
            r_t = solve_triangular(C, r_.T, lower=True)
            G = self.G[i]
            beta = self.beta[i]

            # scaled predictor
            mu[:, i] = (np.dot(f.T, beta)
                        + np.dot(r_t.T, yt - np.dot(Ft, beta))).ravel()

            if eval_MSE:
                Q_ = (np.dot((yt - np.dot(Ft, beta)).T,
                             yt - np.dot(Ft, beta)))[0, 0]
                u_ = solve_triangular(G.T, f - np.dot(Ft.T, r_t), lower=True)
                sigma2_rho = np.dot(g,
                                    self.sigma2[
                                        i] * linalg.inv(np.dot(G.T, G))[:self.q[i], :self.q[i]]
                                    + np.dot(beta[:self.q[i]], beta[:self.q[i]].T))
                sigma2_rho = (sigma2_rho * g).sum(axis=1)

                MSE[:, i] = sigma2_rho * MSE[:, i - 1] \
                    + Q_ / (2 * (self.n_samples[i] - self.p[i] - self.q[i])) \
                    * (1 - (r_t**2).sum(axis=0)) \
                    + self.sigma2[i] * (u_**2).sum(axis=0)

        # scaled predictor
        for i in range(nlevel):  # Predictor
            mu[:, i] = self.y_mean + self.y_std * mu[:, i]
            if eval_MSE:
                MSE[:, i] = self.y_std**2 * MSE[:, i]

        if eval_MSE:
            return mu[:, -1].reshape((n_eval, 1)), MSE[:, -1].reshape((n_eval, 1))
        else:
            return mu[:, -1].reshape((n_eval, 1))

    def _check_list_structure(self, X, y):
        """
        Transform floats and arrays in the training data lists to have a multifidelity structure.

        Parameters
        ----------
        X : list of double array_like elements
            A list of arrays with the input at which observations were made, from lowest
            fidelity to highest fidelity. Designs must be nested
            with X[i] = np.vstack([..., X[i+1])
        y : list of double array_like elements
            A list of arrays with the observations of the scalar output to be predicted,
            from lowest fidelity to highest fidelity.
        """
        if type(X) is not list:
            nlevel = 1
            X = [X]
        else:
            nlevel = len(X)

        if type(y) is not list:
            y = [y]

        if len(X) != len(y):
            raise ValueError("X and y must have the same length.")

        n_samples = np.zeros(nlevel, dtype=int)
        n_features = np.zeros(nlevel, dtype=int)
        n_samples_y = np.zeros(nlevel, dtype=int)
        for i in range(nlevel):
            n_samples[i], n_features[i] = X[i].shape
            if i > 0 and n_features[i] != n_features[i - 1]:
                raise ValueError("All X must have the same number of columns.")
            y[i] = np.asarray(y[i]).ravel()[:, np.newaxis]
            n_samples_y[i] = y[i].shape[0]
            if n_samples[i] != n_samples_y[i]:
                raise ValueError("X and y must have the same number of rows.")

        self.n_features = n_features[0]

        if type(self.theta) is not list:
            self.theta = nlevel * [self.theta]
        elif len(self.theta) != nlevel:
            raise ValueError(f"theta must be a list of {nlevel} element(s).")

        if type(self.theta0) is not list:
            self.theta0 = nlevel * [self.theta0]
        elif len(self.theta0) != nlevel:
            raise ValueError(f"theta0 must be a list of {nlevel} element(s).")

        if type(self.thetaL) is not list:
            self.thetaL = nlevel * [self.thetaL]
        elif len(self.thetaL) != nlevel:
            raise ValueError(f"thetaL must be a list of {nlevel} element(s).")

        if type(self.thetaU) is not list:
            self.thetaU = nlevel * [self.thetaU]
        elif len(self.thetaU) != nlevel:
            raise ValueError(f"thetaU must be a list of {nlevel} element(s).")

        self.nlevel = nlevel
        self.X = X[:]
        self.y = y[:]
        self.n_samples = n_samples

        return

    def _check_params(self):
        """
        Perform sanity checks on all parameters.
        """
        # Check regression model
        if not callable(self.regr):
            if self.regr in self._regression_types:
                self.regr = self._regression_types[self.regr]
            else:
                raise ValueError(f"regr should be one of {self._regression_types.keys()} or "
                                 f"callable, {self.regr} was given.")

        # Check rho regression model
        if not callable(self.rho_regr):
            if self.rho_regr in self._regression_types:
                self.rho_regr = self._regression_types[self.rho_regr]
            else:
                raise ValueError(f"regr should be one of {self._regression_types.keys()} or "
                                 f"callable, {self.rho_regr} was given.")

        for i in range(self.nlevel):
            # Check correlation parameters
            if self.theta[i] is not None:
                self.theta[i] = array2d(self.theta[i])
                if np.any(self.theta[i] <= 0):
                    raise ValueError("theta must be strictly positive.")

            if self.theta0[i] is not None:
                self.theta0[i] = array2d(self.theta0[i])
                if np.any(self.theta0[i] <= 0):
                    raise ValueError("theta0 must be strictly positive.")
            else:
                self.theta0[i] = array2d(self.n_features * [THETA0_DEFAULT])

            lth = self.theta0[i].size

            if self.thetaL[i] is not None:
                self.thetaL[i] = array2d(self.thetaL[i])
                if self.thetaL[i].size != lth:
                    raise ValueError("theta0 and thetaL must have the same length.")
            else:
                self.thetaL[i] = array2d(self.n_features * [THETAL_DEFAULT])

            if self.thetaU[i] is not None:
                self.thetaU[i] = array2d(self.thetaU[i])
                if self.thetaU[i].size != lth:
                    raise ValueError("theta0 and thetaU must have the same length.")
            else:
                self.thetaU[i] = array2d(self.n_features * [THETAU_DEFAULT])

            if np.any(self.thetaL[i] <= 0) or np.any(self.thetaU[i] < self.thetaL[i]):
                raise ValueError("The bounds must satisfy O < thetaL <= thetaU.")

        return


class MultiFiCoKrigingSurrogate(MultiFiSurrogateModel):
    """
    OpenMDAO adapter of multi-fidelity recursive cokriging method described in [LeGratiet2013].

    See MultiFiCoKriging class.

    Parameters
    ----------
    **kwargs : keyword args
        Some implementations of record_derivatives need additional args.

    Attributes
    ----------
    model : MultiFiCoKriging
        Contains MultiFiCoKriging surrogate.
    """

    def __init__(self, **kwargs):
        """
        Initialize all attributes.
        """
        super().__init__(**kwargs)
        self.model = None

    def _declare_options(self):
        """
        Declare options before kwargs are processed in the init method.
        """
        opt = self.options
        opt.declare('normalize', default=True, types=bool,
                    desc="When true, normalize X and Y so that the mean is at zero.")
        opt.declare('regr', default='constant', types=(object, ),
                    desc="A regression function returning an array of outputs of the linear "
                    "regression functional basis for Universal Kriging purpose. regr is assumed "
                    "to be the same for all levels of code. Default assumes a simple constant "
                    "regression trend. Available built-in regression models can be accessed by "
                    "setting this option to the strings 'constant' or 'linear'")
        opt.declare('rho_regr', default='constant', types=(object, ),
                    desc="A regression function returning an array of outputs of the linear "
                    "regression functional basis. Defines the regression function for the "
                    "autoregressive parameter rho.. regr is assumed to be the same for all levels "
                    "of code. Default assumes a simple constant regression trend. Available "
                    "built-in regression models can be accessed by setting this option to the "
                    "strings 'constant' or 'linear'")
        opt.declare('theta', default=None, allow_none=True,
                    desc="Value of correlation parameters. If they are known, then no "
                    "optimization is run. Default is None, so that optimization is run. if double, "
                    "then value is replicated for all features and all levels. if array_like, "
                    "then an array with shape (n_features, ) for isotropic calculation. It is "
                    "replicated for all levels. if list, then a list of nlevel arrays specifying "
                    "value for each level")
        opt.declare('theta0', default=None, allow_none=True,
                    desc="Starting point for the maximum likelihood estimation of the best set "
                    "of parameters. "
                    "Default is None and meaning use of the default 0.5*np.ones(n_features) "
                    "if double: value is replicated for all features and all levels. "
                    "if array_like: an array with shape (n_features, ) for "
                    "isotropic calculation. It is replicated for all levels. "
                    "if list: a list of nlevel arrays specifying value for each level")
        opt.declare('thetaL', default=None, allow_none=True,
                    desc="Lower bound on the autocorrelation parameters for maximum "
                    "likelihood estimation."
                    "Default is None meaning use of the default 1e-5*np.ones(n_features). "
                    "if double: value is replicated for all features and all levels. "
                    "if array_like: An array with shape matching theta0s. It is replicate "
                    "for all levels of code. "
                    "if list: a list of nlevel arrays specifying value for each level")
        opt.declare('thetaU', default=None, allow_none=True,
                    desc="Upper bound on the autocorrelation parameters for maximum "
                    "likelihood estimation. "
                    "Default is None meaning use of default value 50*np.ones(n_features). "
                    "if double: value is replicated for all features and all levels. "
                    "if array_like: An array with shape matching theta0's. It is replicated "
                    "for all levels of code. "
                    "if list: a list of nlevel arrays specifying value for each level")
        opt.declare('tolerance', default=TOLERANCE_DEFAULT,
                    desc='Optimizer terminates when the tolerance tol is reached.')
        opt.declare('initial_range', default=INITIAL_RANGE_DEFAULT,
                    desc='Initial range for the optimizer.')

    def predict(self, new_x):
        """
        Calculate a predicted value of the response based on the current trained model.

        Parameters
        ----------
        new_x : array_like
            An array with shape (n_eval, n_features) giving the point(s) at
            which the prediction(s) should be made.

        Returns
        -------
        array_like
            An array with shape (n_eval, ) with the Best Linear Unbiased
            Prediction at X. If all_levels is set to True, an array
            with shape (n_eval, nlevel) giving the BLUP for all levels.

        array_like
            An array with shape (n_eval, ) with the square root of the Mean Squared Error at X.
        """
        Y_pred, MSE = self.model.predict([new_x])
        return Y_pred, np.sqrt(np.abs(MSE))

    def train_multifi(self, X, Y):
        """
        Train the surrogate model with the given set of inputs and outputs.

        Parameters
        ----------
        X : list of double array_like elements
            A list of arrays with the input at which observations were made, from highest
            fidelity to lowest fidelity. Designs must be nested
            with X[i] = np.vstack([..., X[i+1]).
        Y : list of double array_like elements
            A list of arrays with the observations of the scalar output to be predicted,
            from highest fidelity to lowest fidelity.
        """
        opt = self.options
        if not self.model:
            self.model = MultiFiCoKriging(regr=opt['regr'], rho_regr=opt['rho_regr'],
                                          theta=opt['theta'], theta0=opt['theta0'],
                                          thetaL=opt['thetaL'], thetaU=opt['thetaU'],
                                          normalize=opt['normalize'])

        X, Y = self._fit_adapter(X, Y)
        self.model.fit(X, Y, tol=opt['tolerance'],
                       initial_range=opt['initial_range'])

    def _fit_adapter(self, X, Y):
        """
        Manage special case with one fidelity.

        where can be called as [[xval1],[xval2]] instead of [[[xval1],[xval2]]]
        we detect if shape(X[0]) is like (m,) instead of (m, n)

        Parameters
        ----------
        X : array_like
            An array with shape (n_samples_X, n_features)
        Y : array_like
            An array with shape (n_samples_Y, n_features)

        Returns
        -------
        list of double array_like elements
            A list of arrays with the input at which observations were made, from lowest
            fidelity to highest fidelity. Designs must be nested
            with X[i] = np.vstack([..., X[i+1])
        list of double array_like elements
            A list of arrays with the observations of the scalar output to be predicted,
            from lowest fidelity to highest fidelity.
        """
        if len(np.shape(np.array(X[0]))) == 1:
            X = [X]
            Y = [Y]

        X = [np.array(x) for x in reversed(X)]
        Y = [np.array(y) for y in reversed(Y)]
        return (X, Y)