kriging

Kriging

Bases: surrogates

Kriging surrogate.

Source code in spotPython/build/kriging.py

class Kriging(surrogates):
    """Kriging surrogate.
    """
    def __init__(
            self: object,
            noise: bool = False,
            var_type: List[str] = ["num"],
            name: str = "kriging",
            seed: int = 124,
            model_optimizer=None,
            model_fun_evals: Optional[int] = None,
            min_theta: float = -3.0,
            max_theta: float = 2.0,
            n_theta: int = 1,
            theta_init_zero: bool = True,
            p_val: float = 2.0,
            n_p: int = 1,
            optim_p: bool = False,
            min_Lambda: float = 1e-9,
            max_Lambda: float = 1.,
            log_level: int = 50,
            spot_writer=None,
            counter=None,
            metric_factorial="canberra",
            **kwargs
    ):
        """
        Initialize the Kriging surrogate.

        Args:
            noise (bool): Use regression instead of interpolation kriging. Defaults to False.
            var_type (List[str]):
                Variable type. Can be either "num" (numerical) or "factor" (factor).
                Defaults to ["num"].
            name (str):
                Surrogate name. Defaults to "kriging".
            seed (int):
                Random seed. Defaults to 124.
            model_optimizer (Optional[object]):
                Optimizer on the surrogate. If None, differential_evolution is selected.
            model_fun_evals (Optional[int]):
                Number of iterations used by the optimizer on the surrogate.
            min_theta (float):
                Min log10 theta value. Defaults to -3.
            max_theta (float):
                Max log10 theta value. Defaults to 2.
            n_theta (int):
                Number of theta values. Defaults to 1.
            theta_init_zero (bool):
                Initialize theta with zero. Defaults to True.
            p_val (float):
                p value. Used as an initial value if optim_p = True. Otherwise as a constant. Defaults to 2.
            n_p (int):
                Number of p values. Defaults to 1.
            optim_p (bool):
                Determines whether p should be optimized. Deafults to False.
            min_Lambda (float):
                Min Lambda value. Defaults to 1e-9.
            max_Lambda (float):
                Max Lambda value. Defaults to 1.
            log_level (int):
                Logging level, e.g., 20 is "INFO". Defaults to 50 ("CRITICAL").
            spot_writer (Optional[object]):
                Spot writer. Defaults to None.
            counter (Optional[int]):
                Counter. Defaults to None.
            metric_factorial (str):
                Metric for factorial. Defaults to "canberra". Can be "euclidean",
                "cityblock", seuclidean", "sqeuclidean", "cosine",
                "correlation", "hamming", "jaccard", "jensenshannon",
                "chebyshev", "canberra", "braycurtis", "mahalanobis", "matching".

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                import matplotlib.pyplot as plt
                from numpy import linspace, arange
                rng = np.random.RandomState(1)
                X = linspace(start=0, stop=10, num=1_000).reshape(-1, 1)
                y = np.squeeze(X * np.sin(X))
                training_indices = rng.choice(arange(y.size), size=6, replace=False)
                X_train, y_train = X[training_indices], y[training_indices]
                S = Kriging(name='kriging', seed=124)
                S.fit(X_train, y_train)
                mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
                plt.plot(X, y, label=r"$f(x)$", linestyle="dotted")
                plt.scatter(X_train, y_train, label="Observations")
                plt.plot(X, mean_prediction, label="Mean prediction")
                plt.fill_between(
                    X.ravel(),
                    mean_prediction - 1.96 * std_prediction,
                    mean_prediction + 1.96 * std_prediction,
                    alpha=0.5,
                    label=r"95% confidence interval",
                    )
                plt.legend()
                plt.xlabel("$x$")
                plt.ylabel("$f(x)$")
                _ = plt.title("Gaussian process regression on noise-free dataset")
                plt.show()

        References:
            https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.pdist.html
            [[1](https://scikit-learn.org/stable/auto_examples/gaussian_process/plot_gpr_noisy_targets.html)]
            scikit-learn: Gaussian Processes regression: basic introductory example

        """
        super().__init__(name, seed, log_level)

        self.noise = noise
        self.var_type = var_type
        self.name = name
        self.seed = seed
        self.log_level = log_level
        self.spot_writer = spot_writer
        self.counter = counter
        self.metric_factorial = metric_factorial

        self.sigma = 0
        self.eps = sqrt(spacing(1))
        self.min_theta = min_theta
        self.max_theta = max_theta
        self.min_p = 1
        self.max_p = 2
        self.min_Lambda = min_Lambda
        self.max_Lambda = max_Lambda
        self.n_theta = n_theta
        self.p_val = p_val
        self.n_p = n_p
        self.optim_p = optim_p
        self.theta_init_zero = theta_init_zero
        # Psi matrix condition:
        self.cnd_Psi = 0
        self.inf_Psi = False

        self.model_optimizer = model_optimizer
        if self.model_optimizer is None:
            self.model_optimizer = differential_evolution
        self.model_fun_evals = model_fun_evals
        # differential evolution uses maxiter = 1000
        # and sets the number of function evaluations to
        # (maxiter + 1) * popsize * N, which results in
        # 1000 * 15 * k, because the default popsize is 15 and
        # N is the number of parameters. This seems to be quite large:
        # for k=2 these are 30 000 iterations. Therefore we set this value to
        # 100
        if self.model_fun_evals is None:
            self.model_fun_evals = 100

        # Logging information
        self.log["negLnLike"] = []
        self.log["theta"] = []
        self.log["p"] = []
        self.log["Lambda"] = []
        # Logger
        logger.setLevel(self.log_level)
        logger.info(f"Starting the logger at level {self.log_level} for module {__name__}:")

    def exp_imp(self, y0: float, s0: float) -> float:
        """
        Calculates the expected improvement for a given function value and error in coded units.

        Args:
            self (object): The Kriging object.
            y0 (float): The function value in coded units.
            s0 (float): The error value.

        Returns:
            float: The expected improvement value.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                S = Kriging(name='kriging', seed=124)
                S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
                S.exp_imp(1.0, 0.0)
                0.0
            >>> from spotPython.build.kriging import Kriging
                S = Kriging(name='kriging', seed=124)
                S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
                # assert S.exp_imp(0.0, 1.0) == 1/np.sqrt(2*np.pi)
                # which is approx. 0.3989422804014327
                S.exp_imp(0.0, 1.0)
                0.3989422804014327
        """
        # We do not use the min y values, but the aggragated mean values
        # y_min = min(self.nat_y)
        y_min = min(self.aggregated_mean_y)
        if s0 <= 0.0:
            EI = 0.0
        elif s0 > 0.0:
            EI_one = (y_min - y0) * (
                    0.5 + 0.5 * erf((1.0 / sqrt(2.0)) * ((y_min - y0) / s0))
            )
            EI_two = (s0 * (1.0 / sqrt(2.0 * pi))) * (
                exp(-(1.0 / 2.0) * ((y_min - y0) ** 2.0 / s0 ** 2.0))
            )
            EI = EI_one + EI_two
        return EI

    def set_de_bounds(self) -> None:
        """
        Determine search bounds for model_optimizer, e.g., differential evolution.

        This method sets the attribute `de_bounds` of the object to a list of lists,
        where each inner list represents the lower and upper bounds for a parameter
        being optimized. The number of inner lists is determined by the number of
        parameters being optimized (`n_theta` and `n_p`), as well as whether noise is
        being considered (`noise`).

        Args:
            self (object): The Kriging object.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                S = Kriging(name='kriging', seed=124)
                S.set_de_bounds()
                print(S.de_bounds)

        Returns:
            None
        """
        logger.debug("In set_de_bounds(): self.min_theta: %s", self.min_theta)
        logger.debug("In set_de_bounds(): self.max_theta: %s", self.max_theta)
        logger.debug("In set_de_bounds(): self.n_theta: %s", self.n_theta)
        logger.debug("In set_de_bounds(): self.optim_p: %s", self.optim_p)
        logger.debug("In set_de_bounds(): self.min_p: %s", self.min_p)
        logger.debug("In set_de_bounds(): self.max_p: %s", self.max_p)
        logger.debug("In set_de_bounds(): self.n_p: %s", self.n_p)
        logger.debug("In set_de_bounds(): self.noise: %s", self.noise)
        logger.debug("In set_de_bounds(): self.min_Lambda: %s", self.min_Lambda)
        logger.debug("In set_de_bounds(): self.max_Lambda: %s", self.max_Lambda)

        de_bounds = [[self.min_theta, self.max_theta] for _ in range(self.n_theta)]
        if self.optim_p:
            de_bounds += [[self.min_p, self.max_p] for _ in range(self.n_p)]
            if self.noise:
                de_bounds.append([self.min_Lambda, self.max_Lambda])
        else:
            if self.noise:
                de_bounds.append([self.min_Lambda, self.max_Lambda])
        self.de_bounds = de_bounds
        logger.debug("In set_de_bounds(): self.de_bounds: %s", self.de_bounds)

    def extract_from_bounds(self, new_theta_p_Lambda: np.ndarray) -> None:
        """
        Extract `theta`, `p`, and `Lambda` from bounds. The kriging object stores
        `theta` as an array,  `p` as an array, and `Lambda` as a float.

        Args:
            self (object): The Kriging object.
            new_theta_p_Lambda (np.ndarray):
                1d-array with theta, p, and Lambda values. Order is important.

        Examples:
            >>> import numpy as np
                from spotPython.build.kriging import Kriging
                n=2
                p=4
                S = Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
                S.extract_from_bounds(np.array([1, 2, 3]))
                print(S.theta)
                print(S.p)
                [1 2]
                [3]

        Returns:
            None
        """
        logger.debug("In extract_from_bounds(): new_theta_p_Lambda: %s", new_theta_p_Lambda)
        self.theta = new_theta_p_Lambda[:self.n_theta]
        logger.debug("In extract_from_bounds(): self.n_theta: %s", self.n_theta)
        if self.optim_p:
            self.p = new_theta_p_Lambda[self.n_theta:self.n_theta + self.n_p]
            logger.debug("In extract_from_bounds(): self.p: %s", self.p)
            if self.noise:
                self.Lambda = new_theta_p_Lambda[self.n_theta + self.n_p]
                logger.debug("In extract_from_bounds(): self.Lambda: %s", self.Lambda)
        else:
            if self.noise:
                self.Lambda = new_theta_p_Lambda[self.n_theta]
                logger.debug("In extract_from_bounds(): self.Lambda: %s", self.Lambda)

    def optimize_model(self) -> Union[List[float], Tuple[float]]:
        """
        Optimize the model using the specified model_optimizer.

        This method uses the specified model_optimizer to optimize the
        likelihood function (`fun_likelihood`) with respect to the model parameters.
        The optimization is performed within the bounds specified by the attribute
        `de_bounds`.
        The result of the optimization is returned as a list or tuple of optimized parameter values.

        Args:
            self (object): The Kriging object.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[1, 2], [3, 4]])
                nat_y = np.array([1, 2])
                n=2
                p=2
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                S.initialize_matrices()
                S.set_de_bounds()
                new_theta_p_Lambda = S.optimize_model()
                print(new_theta_p_Lambda)

        Returns:
            result["x"] (Union[List[float], Tuple[float]]):
                A list or tuple of optimized parameter values.
        """
        logger.debug("In optimize_model(): self.de_bounds passed to optimizer: %s", self.de_bounds)
        if self.model_optimizer.__name__ == 'dual_annealing':
            result = self.model_optimizer(func=self.fun_likelihood,
                                          bounds=self.de_bounds)
        elif self.model_optimizer.__name__ == 'differential_evolution':
            result = self.model_optimizer(func=self.fun_likelihood,
                                          bounds=self.de_bounds,
                                          maxiter=self.model_fun_evals,
                                          seed=self.seed)
        elif self.model_optimizer.__name__ == 'direct':
            result = self.model_optimizer(func=self.fun_likelihood,
                                          bounds=self.de_bounds,
                                          # maxfun=self.model_fun_evals,
                                          eps=1e-2)
        elif self.model_optimizer.__name__ == 'shgo':
            result = self.model_optimizer(func=self.fun_likelihood,
                                          bounds=self.de_bounds)
        elif self.model_optimizer.__name__ == 'basinhopping':
            result = self.model_optimizer(func=self.fun_likelihood,
                                          x0=mean(self.de_bounds, axis=1))
        else:
            result = self.model_optimizer(func=self.fun_likelihood, bounds=self.de_bounds)
        logger.debug("In optimize_model(): result: %s", result)
        logger.debug('In optimize_model(): returned result["x"]: %s', result["x"])
        return result["x"]

    def update_log(self) -> None:
        """
        Update the log with the current values of negLnLike, theta, p, and Lambda.

        This method appends the current values of negLnLike, theta, p (if optim_p is True),
        and Lambda (if noise is True)
        to their respective lists in the log dictionary.
        It also updates the log_length attribute with the current length
        of the negLnLike list in the log.

        If spot_writer is not None, this method also writes the current values of
        negLnLike, theta, p (if optim_p is True),
        and Lambda (if noise is True) to the spot_writer object.

        Args:
            self (object): The Kriging object.

        Returns:
            None

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[1, 2], [3, 4]])
                nat_y = np.array([1, 2])
                n=2
                p=2
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                S.initialize_matrices()
                S.set_de_bounds()
                new_theta_p_Lambda = S.optimize_model()
                S.update_log()
                print(S.log)
                {'negLnLike': array([-1.38629436]),
                 'theta': array([-1.14525993,  1.6123372 ]),
                  'p': array([1.84444406, 1.74590865]),
                  'Lambda': array([0.44268472])}

        """
        self.log["negLnLike"] = append(self.log["negLnLike"], self.negLnLike)
        self.log["theta"] = append(self.log["theta"], self.theta)
        if self.optim_p:
            self.log["p"] = append(self.log["p"], self.p)
        if self.noise:
            self.log["Lambda"] = append(self.log["Lambda"], self.Lambda)
        # get the length of the log
        self.log_length = len(self.log["negLnLike"])
        if self.spot_writer is not None:
            writer = self.spot_writer
            negLnLike = self.negLnLike.copy()
            writer.add_scalar("spot_negLnLike", negLnLike, self.counter+self.log_length)
            # add the self.n_theta theta values to the writer with one key "theta",
            # i.e, the same key for all theta values
            theta = self.theta.copy()
            writer.add_scalars("spot_theta", {f"theta_{i}": theta[i] for i in range(self.n_theta)},
                               self.counter+self.log_length)
            if self.noise:
                Lambda = self.Lambda.copy()
                writer.add_scalar("spot_Lambda", Lambda, self.counter+self.log_length)
            if self.optim_p:
                p = self.p.copy()
                writer.add_scalars("spot_p", {f"p_{i}": p[i] for i in range(self.n_p)}, self.counter+self.log_length)
            writer.flush()

    def fit(self, nat_X: np.ndarray, nat_y: np.ndarray) -> object:
        """
        Fits the hyperparameters (`theta`, `p`, `Lambda`) of the Kriging model.

        The function computes the following internal values:
        1. `theta`, `p`, and `Lambda` values via optimization of the function `fun_likelihood()`.
        2. Correlation matrix `Psi` via `rebuildPsi()`.

        Args:
            self (object): The Kriging object.
            nat_X (np.ndarray): Sample points.
            nat_y (np.ndarray): Function values.

        Returns:
            object: Fitted estimator.

        Attributes:
            theta (np.ndarray): Kriging theta values. Shape (k,).
            p (np.ndarray): Kriging p values. Shape (k,).
            LnDetPsi (np.float64): Determinant Psi matrix.
            Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
            psi (np.ndarray): psi vector. Shape (n,).
            one (np.ndarray): vector of ones. Shape (n,).
            mu (np.float64): Kriging expected mean value mu.
            U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).
            SigmaSqr (np.float64): Sigma squared value.
            Lambda (float): lambda noise value.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[1, 0], [1, 0]])
                nat_y = np.array([1, 2])
                S = Kriging()
                S.fit(nat_X, nat_y)
                print(S.Psi)
                [[1.00000001 1.        ]
                [1.         1.00000001]]

        """
        logger.debug("In fit(): nat_X: %s", nat_X)
        logger.debug("In fit(): nat_y: %s", nat_y)
        self.initialize_variables(nat_X, nat_y)
        self.set_variable_types()
        self.set_theta_values()
        self.initialize_matrices()
        # build_Psi() and build_U() are called in fun_likelihood
        self.set_de_bounds()
        # Finally, set new theta and p values and update the surrogate again
        # for new_theta_p_Lambda in de_results["x"]:
        new_theta_p_Lambda = self.optimize_model()
        self.extract_from_bounds(new_theta_p_Lambda)
        self.build_Psi()
        self.build_U()
        # TODO: check if the following line is necessary!
        self.likelihood()
        self.update_log()

    def initialize_variables(self, nat_X: np.ndarray, nat_y: np.ndarray) -> None:
        """
        Initialize variables for the class instance.

        This method takes in the independent and dependent variable data as input
        and initializes the class instance variables.
        It creates deep copies of the input data and stores them in the
        instance variables `nat_X` and `nat_y`.
        It also calculates the number of observations `n` and
        the number of independent variables `k` from the shape of `nat_X`.
        Finally, it creates empty arrays with the same shape as `nat_X`
        and `nat_y` and stores them in the instance variables `cod_X` and `cod_y`.

        Args:
            self (object): The Kriging object.
            nat_X (np.ndarray): The independent variable data.
            nat_y (np.ndarray): The dependent variable data.

        Returns:
            None

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[1, 2], [3, 4]])
                nat_y = np.array([1, 2])
                S = Kriging()
                S.initialize_variables(nat_X, nat_y)
                print(f"S.nat_X: {S.nat_X}")
                print(f"S.nat_y: {S.nat_y}")
                S.nat_X: [[1 2]
                          [3 4]]
                S.nat_y: [1 2]

        """
        self.nat_X = copy.deepcopy(nat_X)
        self.nat_y = copy.deepcopy(nat_y)
        self.n = self.nat_X.shape[0]
        self.k = self.nat_X.shape[1]

        self.min_X = min(self.nat_X, axis=0)
        self.max_X = max(self.nat_X, axis=0)

        Z = aggregate_mean_var(X=self.nat_X, y=self.nat_y)
        # aggregated y values:
        mu = Z[1]
        self.aggregated_mean_y = np.copy(mu)
        logger.debug("In initialize_variables(): self.nat_X: %s", self.nat_X)
        logger.debug("In initialize_variables(): self.nat_y: %s", self.nat_y)
        logger.debug("In initialize_variables(): self.aggregated_mean_y: %s", self.aggregated_mean_y)
        logger.debug("In initialize_variables(): self.min_X: %s", self.min_X)
        logger.debug("In initialize_variables(): self.max_X: %s", self.max_X)
        logger.debug("In initialize_variables(): self.n: %s", self.n)
        logger.debug("In initialize_variables(): self.k: %s", self.k)

    def set_variable_types(self) -> None:
        """
        Set the variable types for the class instance.

        This method sets the variable types for the class instance based
        on the `var_type` attribute. If the length of `var_type` is less
        than `k`, all variable types are forced to 'num' and a warning is logged.
        The method then creates Boolean masks for each variable
        type ('num', 'factor', 'int', 'ordered') using numpy arrays, e.g.,
        `num_mask = array([ True,  True])` if two numerical variables are present.

        Args:
            self (object): The Kriging object.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                nat_X = np.array([[1, 2], [3, 4]])
                nat_y = np.array([1, 2])
                n=2
                p=2
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                assert S.var_type == ['num', 'num']
                assert S.var_type == ['num', 'num']
                assert S.num_mask.all() == True
                assert S.factor_mask.all() == False
                assert S.int_mask.all() == False
                assert S.ordered_mask.all() == True

        Returns:
            None
        """
        logger.debug("In set_variable_types(): self.k: %s", self.k)
        logger.debug("In set_variable_types(): self.var_type: %s", self.var_type)
        # assume all variable types are "num" if "num" is
        # specified once:
        if len(self.var_type) < self.k:
            self.var_type = self.var_type * self.k
            logger.warning("In set_variable_types(): All variable types forced to 'num'.")
            logger.debug("In set_variable_types(): self.var_type: %s", self.var_type)
        self.num_mask = np.array(list(map(lambda x: x == "num", self.var_type)))
        self.factor_mask = np.array(list(map(lambda x: x == "factor", self.var_type)))
        self.int_mask = np.array(list(map(lambda x: x == "int", self.var_type)))
        self.ordered_mask = np.array(list(map(lambda x: x == "int" or x == "num" or x == "float", self.var_type)))
        logger.debug("In set_variable_types(): self.num_mask: %s", self.num_mask)
        logger.debug("In set_variable_types(): self.factor_mask: %s", self.factor_mask)
        logger.debug("In set_variable_types(): self.int_mask: %s", self.int_mask)
        logger.debug("In set_variable_types(): self.ordered_mask: %s", self.ordered_mask)

    def set_theta_values(self) -> None:
        """
        Set the theta values for the class instance.

        This method sets the theta values for the class instance based
        on the `n_theta` and `k` attributes. If `n_theta` is greater than
        `k`, `n_theta` is set to `k` and a warning is logged.
        The method then initializes the `theta` attribute as a list
        of zeros with length `n_theta`.
        The `x0_theta` attribute is also initialized as a list of ones
        with length `n_theta`, multiplied by `n / (100 * k)`.

        Args:
            self (object): The Kriging object.
        Returns:
            None

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                from numpy import array
                nat_X = np.array([[1, 2], [3, 4]])
                nat_y = np.array([1, 2])
                n=2
                p=2
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                assert S.theta.all() == array([0., 0.]).all()
        """
        logger.debug("In set_theta_values(): self.k: %s", self.k)
        logger.debug("In set_theta_values(): self.n_theta: %s", self.n_theta)
        if ((self.n_theta > 1) or (self.n_theta > self.k)) and (self.n_theta != self.k):
            self.n_theta = self.k
            logger.warning("Too few theta values or more theta values than dimensions. `n_theta` set to `k`.")
            logger.debug("In set_theta_values(): self.n_theta: %s", self.n_theta)
        if self.theta_init_zero:
            self.theta: List[float] = zeros(self.n_theta)
            logger.debug("In set_theta_values(): self.theta: %s", self.theta)
        else:
            logger.debug("In set_theta_values(): self.n: %s", self.n)
            self.theta: List[float] = ones((self.n_theta,)) * self.n / (100 * self.k)
            logger.debug("In set_theta_values(): self.theta: %s", self.theta)

    def initialize_matrices(self) -> None:
        """
        Initialize the matrices for the class instance.

        This method initializes several matrices and attributes for the class instance.
        The `p` attribute is initialized as a list of ones with length `n_p`, multiplied by 2.0.
        The `pen_val` attribute is initialized as the natural logarithm of the
        variance of `nat_y`, multiplied by `n`, plus 1e4.
        The `negLnLike`, `LnDetPsi`, `mu`, `U`, `SigmaSqr`, and `Lambda` attributes are all set to None.
        The `gen` attribute is initialized using the `spacefilling` function with arguments `k` and `seed`.
        The `Psi` attribute is initialized as a zero matrix with shape `(n, n)` and dtype `float64`.
        The `psi` attribute is initialized as a zero matrix with shape `(n, 1)`.
        The `one` attribute is initialized as a list of ones with length `n`.

        Args:
            self (object): The Kriging object.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                from numpy import log, var
                nat_X = np.array([[1, 2], [3, 4], [5, 6]])
                nat_y = np.array([1, 2, 3])
                n=3
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                S.initialize_matrices()
                # if var(self.nat_y) is > 0, then self.pen_val = self.n * log(var(self.nat_y)) + 1e4
                # else self.pen_val = self.n * var(self.nat_y) + 1e4
                assert S.pen_val == nat_X.shape[0] * log(var(S.nat_y)) + 1e4
                assert S.Psi.shape == (n, n)

        Returns:
            None
        """
        logger.debug("In initialize_matrices(): self.n_p: %s", self.n_p)
        self.p = ones(self.n_p) * self.p_val
        logger.debug("In initialize_matrices(): self.p: %s", self.p)
        # if var(self.nat_y) is > 0, then self.pen_val = self.n * log(var(self.nat_y)) + 1e4
        # else self.pen_val = self.n * var(self.nat_y) + 1e4
        logger.debug("In initialize_matrices(): var(self.nat_y): %s", var(self.nat_y))
        logger.debug("In initialize_matrices(): self.n: %s", self.n)
        if var(self.nat_y) > 0:
            self.pen_val = self.n * log(var(self.nat_y)) + 1e4
        else:
            self.pen_val = self.n * var(self.nat_y) + 1e4
        logger.debug("In initialize_matrices(): self.pen_val: %s", self.pen_val)
        self.negLnLike = None
        logger.debug("In initialize_matrices(): self.k: %s", self.k)
        logger.debug("In initialize_matrices(): self.seed: %s", self.seed)
        self.gen = spacefilling(k=self.k, seed=self.seed)
        logger.debug("In initialize_matrices(): self.gen: %s", self.gen)
        self.LnDetPsi = None
        self.Psi = zeros((self.n, self.n), dtype=float64)
        logger.debug("In initialize_matrices(): self.Psi: %s", self.Psi)
        self.psi = zeros((self.n, 1))
        logger.debug("In initialize_matrices(): self.psi: %s", self.psi)
        self.one = ones(self.n)
        logger.debug("In initialize_matrices(): self.one: %s", self.one)
        self.mu = None
        self.U = None
        self.SigmaSqr = None
        self.Lambda = None

    def fun_likelihood(self, new_theta_p_Lambda: np.ndarray) -> float:
        """
        Compute log likelihood for a set of hyperparameters (theta, p, Lambda).

        This method computes the log likelihood for a set of hyperparameters
        (theta, p, Lambda) by performing the following steps:
        1. Extracts the hyperparameters from the input array using `extract_from_bounds()`.
        2. Checks if any element in `10^theta` is equal to 0. If so, logs a warning and
        returns the penalty value (`pen_val`).
        3. Builds the `Psi` matrix using `build_Psi()`.
        4. Checks if `Psi` is ill-conditioned or infinite. If so, logs a warning and returns
        the penalty value (`pen_val`).
        5. Builds the `U` matrix using `build_U()`. If an exception occurs, logs an error and
        returns the penalty value (`pen_val`).
        6. Computes the negative log likelihood using `likelihood()`.
        7. Returns the computed negative log likelihood (`negLnLike`).

        Args:
            self (object): The Kriging object.
            new_theta_p_Lambda (np.ndarray):
                An array containing the `theta`, `p`, and `Lambda` values.

        Returns:
            float:
                The negative log likelihood of the surface at the specified hyperparameters.

        Attributes:
            theta (np.ndarray): Kriging theta values. Shape (k,).
            p (np.ndarray): Kriging p values. Shape (k,).
            Lambda (float): lambda noise value.
            Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
            U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).
            negLnLike (float): Negative log likelihood of the surface at the specified hyperparameters.
            pen_val (float): Penalty value.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[0], [1]])
                nat_y = np.array([0, 1])
                n=1
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                print(S.cod_X)
                print(S.cod_y)
                S.set_theta_values()
                print(f"S.theta: {S.theta}")
                S.initialize_matrices()
                S.set_de_bounds()
                new_theta_p_Lambda = S.optimize_model()
                S.extract_from_bounds(new_theta_p_Lambda)
                print(f"S.theta: {S.theta}")
                S.build_Psi()
                print(f"S.Psi: {S.Psi}")
                S.build_U()
                print(f"S.U:{S.U}")
                S.likelihood()
                S.negLnLike

        """
        self.extract_from_bounds(new_theta_p_Lambda)
        if self.__is_any__(power(10.0, self.theta), 0):
            logger.warning("Failure in fun_likelihood: 10^theta == 0. Setting negLnLike to %s", self.pen_val)
            return self.pen_val
        self.build_Psi()
        if (self.inf_Psi or self.cnd_Psi > 1e9):
            logger.warning("Failure in fun_likelihood: Psi is ill conditioned: %s", self.cnd_Psi)
            logger.warning("Setting negLnLike to: %s", self.pen_val)
            return self.pen_val

        try:
            self.build_U()
        except Exception as error:
            penalty_value = self.pen_val
            print("Error in fun_likelihood(). Call to build_U() failed.")
            print("error=%s, type(error)=%s" % (error, type(error)))
            print("Setting negLnLike to %.2f." % self.pen_val)
            return penalty_value
        self.likelihood()
        return self.negLnLike

    def __is_any__(self, x: Union[np.ndarray, Any], v: Any) -> bool:
        """
        Check if any element in `x` is equal to `v`.
        This method checks if any element in the input array `x` is equal to the value `v`.
        If `x` is not an instance of `ndarray`, it is first converted to a numpy array using
        the `array()` function.

        Args:
            self (object): The Kriging object.
            x (np.ndarray or array-like):
                The input array to check for the presence of value `v`.
            v (scalar):
                The value to check for in the input array `x`.

        Returns:
            bool:
                True if any element in `x` is equal to `v`, False otherwise.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                from numpy import power
                import numpy as np
                nat_X = np.array([[0], [1]])
                nat_y = np.array([0, 1])
                n=1
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                print(f"S.theta: {S.theta}")
                print(S.__is_any__(power(10.0, S.theta), 0))
                print(S.__is_any__(S.theta, 0))
                S.theta: [0.]
                False
                True

        """
        if not isinstance(x, ndarray):
            x = array([x])
        return any(x == v)

    def build_Psi(self) -> None:
        """
        Constructs a new (n x n) correlation matrix Psi to reflect new data
        or a change in hyperparameters.

        This method uses `theta`, `p`, and coded `X` values to construct the
        correlation matrix as described in [Forr08a, p.57].

        Args:
            self (object): The Kriging object.

        Returns:
            None

        Raises:
            LinAlgError: If building Psi fails.

        Attributes:
            Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
            cnd_Psi (float): Condition number of Psi.
            inf_Psi (bool): True if Psi is infinite, False otherwise.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[0], [1]])
                nat_y = np.array([0, 1])
                n=1
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                print(S.nat_X)
                print(S.nat_y)
                S.set_theta_values()
                print(f"S.theta: {S.theta}")
                S.initialize_matrices()
                S.set_de_bounds()
                new_theta_p_Lambda = S.optimize_model()
                S.extract_from_bounds(new_theta_p_Lambda)
                print(f"S.theta: {S.theta}")
                S.build_Psi()
                print(f"S.Psi: {S.Psi}")
                [[0]
                [1]]
                [0 1]
                S.theta: [0.]
                S.theta: [1.72284258]
                S.Psi: [[1.00000001e+00 1.14348852e-23]
                [1.14348852e-23 1.00000001e+00]]

        """
        self.Psi = zeros((self.n, self.n), dtype=float64)
        theta = power(10.0, self.theta)
        if self.n_theta == 1:
            theta = theta * ones(self.k)
        try:
            D = zeros((self.n, self.n))
            if self.ordered_mask.any():
                X_ordered = self.nat_X[:, self.ordered_mask]
                D = squareform(
                    pdist(
                        X_ordered, metric='sqeuclidean', out=None, w=theta[self.ordered_mask]))
            if self.factor_mask.any():
                X_factor = self.nat_X[:, self.factor_mask]
                D = (D + squareform(
                    pdist(X_factor,
                          metric=self.metric_factorial,
                          out=None,
                          w=theta[self.factor_mask])))
            self.Psi = exp(-D)
        except LinAlgError as err:
            print(f"Building Psi failed:\n {self.Psi}. {err=}, {type(err)=}")
        if self.noise:
            logger.debug("In build_Psi(): self.Lambda: %s", self.Lambda)
            self.Psi[diag_indices_from(self.Psi)] += self.Lambda
        else:
            self.Psi[diag_indices_from(self.Psi)] += self.eps
        if (isinf(self.Psi)).any():
            self.inf_Psi = True
        self.cnd_Psi = cond(self.Psi)

    def build_U(self, scipy: bool = True) -> None:
        """
        Performs Cholesky factorization of Psi as U as described in [Forr08a, p.57].
        This method uses either `scipy_cholesky` or numpy's `cholesky` to perform the Cholesky factorization of Psi.

        Args:
            self (object):
                The Kriging object.
            scipy (bool):
                If True, use `scipy_cholesky`.
                If False, use numpy's `cholesky`.
                Defaults to True.

        Returns:
            None

        Raises:
            LinAlgError:
                If Cholesky factorization fails for Psi.

        Attributes:
            U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[0], [1]])
                nat_y = np.array([0, 1])
                n=1
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                print(S.nat_X)
                print(S.nat_y)
                S.set_theta_values()
                print(f"S.theta: {S.theta}")
                S.initialize_matrices()
                S.set_de_bounds()
                new_theta_p_Lambda = S.optimize_model()
                S.extract_from_bounds(new_theta_p_Lambda)
                print(f"S.theta: {S.theta}")
                S.build_Psi()
                print(f"S.Psi: {S.Psi}")
                S.build_U()
                print(f"S.U:{S.U}")
                [[0]
                [1]]
                [0 1]
                S.theta: [0.]
                S.theta: [1.72284258]
                S.Psi: [[1.00000001e+00 1.14348852e-23]
                [1.14348852e-23 1.00000001e+00]]
                S.U:[[1.00000001e+00 1.14348851e-23]
                [0.00000000e+00 1.00000001e+00]]
        """
        try:
            self.U = scipy_cholesky(self.Psi, lower=True) if scipy else cholesky(self.Psi)
            self.U = self.U.T
        except LinAlgError as err:
            print(f"build_U() Cholesky failed for Psi:\n {self.Psi}. {err=}, {type(err)=}")

    def likelihood(self) -> None:
        """
        Calculates the negative of the concentrated log-likelihood.

        This method implements equation (2.32) in [Forr08a] to calculate
        the negative of the concentrated log-likelihood. It also modifies `mu`,
        `SigmaSqr`, `LnDetPsi`, and `negLnLike`.

        Note:
            `build_Psi` and `build_U` should be called first.

        Args:
            self (object):
                The Kriging object.

        Returns:
            None

        Attributes:
            mu (np.float64): Kriging expected mean value mu.
            SigmaSqr (np.float64): Sigma squared value.
            LnDetPsi (np.float64): Determinant Psi matrix.
            negLnLike (float): Negative log likelihood of the surface at the specified hyperparameters.

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                nat_X = np.array([[1], [2]])
                nat_y = np.array([5, 10])
                n=2
                p=1
                S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False, theta_init_zero=True)
                S.initialize_variables(nat_X, nat_y)
                S.set_variable_types()
                S.set_theta_values()
                S.initialize_matrices()
                S.build_Psi()
                S.build_U()
                S.likelihood()
                # assert S.mu is close to 7.5 with a tolerance of 1e-6
                assert np.allclose(S.mu, 7.5, atol=1e-6)
                E = np.exp(1)
                sigma2 = E/(E**2 -1) * (25/4 + 25/4*E)
                # asssert S.SigmaSqr is close to sigma2 with a tolerance of 1e-6
                assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
                print(f"S.LnDetPsi:{S.LnDetPsi}")
                print(f"S.self.negLnLike:{S.negLnLike}")
        """
        # (2.20) in [Forr08a]:
        U_T_inv_one = solve(self.U.T, self.one)
        U_T_inv_cod_y = solve(self.U.T, self.nat_y)
        mu = self.one.T.dot(solve(self.U, U_T_inv_cod_y)) / self.one.T.dot(solve(self.U, U_T_inv_one))
        self.mu = mu
        # (2.31) in [Forr08a]
        cod_y_minus_mu = self.nat_y - self.one.dot(self.mu)
        self.SigmaSqr = cod_y_minus_mu.T.dot(solve(self.U, solve(self.U.T, cod_y_minus_mu))) / self.n
        # (2.32) in [Forr08a]
        self.LnDetPsi = 2.0 * sum(log(abs(diag(self.U))))
        self.negLnLike = -1.0 * (-(self.n / 2.0) * log(self.SigmaSqr) - 0.5 * self.LnDetPsi)

    def plot(self, show: Optional[bool] = True) -> None:
        """
        This function plots 1D and 2D surrogates.

        Args:
            self (object):
                The Kriging object.
            show (bool):
                If `True`, the plots are displayed.
                If `False`, `plt.show()` should be called outside this function.

        Returns:
            None

        Note:
            * This method provides only a basic plot. For more advanced plots,
                use the `plot_contour()` method of the `Spot` class.

        Examples:
            >>> import numpy as np
                from spotPython.fun.objectivefunctions import analytical
                from spotPython.spot import spot
                # 1-dimensional example
                fun = analytical().fun_sphere
                lower = np.array([-1])
                upper = np.array([1])
                design_control={"init_size": 10}
                S = spot.Spot(fun=fun,
                            noise=False,
                            lower = lower,
                            upper= upper,
                            design_control=design_control,)
                S.initialize_design()
                S.update_stats()
                S.fit_surrogate()
                S.surrogate.plot()
                # 2-dimensional example
                fun = analytical().fun_sphere
                lower = np.array([-1, -1])
                upper = np.array([1, 1])
                design_control={"init_size": 10}
                S = spot.Spot(fun=fun,
                            noise=False,
                            lower = lower,
                            upper= upper,
                            design_control=design_control,)
                S.initialize_design()
                S.update_stats()
                S.fit_surrogate()
                S.surrogate.plot()
        """
        if self.k == 1:
            # TODO: Improve plot (add conf. interval etc.)
            fig = pylab.figure(figsize=(9, 6))
            n_grid = 100
            x = linspace(
                self.min_X[0], self.max_X[0], num=n_grid
            )
            y = self.predict(x)
            plt.figure()
            plt.plot(x, y, "k")
            if show:
                plt.show()

        if self.k == 2:
            fig = pylab.figure(figsize=(9, 6))
            n_grid = 100
            x = linspace(
                self.min_X[0], self.max_X[0], num=n_grid
            )
            y = linspace(
                self.min_X[1], self.max_X[1], num=n_grid
            )
            X, Y = meshgrid(x, y)
            # Predict based on the optimized results
            zz = array(
                [self.predict(array([x, y]), return_val="all") for x, y in zip(ravel(X), ravel(Y))]
            )
            zs = zz[:, 0, :]
            zse = zz[:, 1, :]
            Z = zs.reshape(X.shape)
            Ze = zse.reshape(X.shape)

            nat_point_X = self.nat_X[:, 0]
            nat_point_Y = self.nat_X[:, 1]
            contour_levels = 30
            ax = fig.add_subplot(224)
            # plot predicted values:
            pylab.contourf(X, Y, Ze, contour_levels, cmap="jet")
            pylab.title("Error")
            pylab.colorbar()
            # plot observed points:
            pylab.plot(nat_point_X, nat_point_Y, "ow")
            #
            ax = fig.add_subplot(223)
            # plot predicted values:
            plt.contourf(X, Y, Z, contour_levels, zorder=1, cmap="jet")
            plt.title("Surrogate")
            # plot observed points:
            pylab.plot(nat_point_X, nat_point_Y, "ow", zorder=3)
            pylab.colorbar()
            #
            ax = fig.add_subplot(221, projection="3d")
            ax.plot_surface(X, Y, Z, rstride=3, cstride=3, alpha=0.9, cmap="jet")
            #
            ax = fig.add_subplot(222, projection="3d")
            ax.plot_surface(X, Y, Ze, rstride=3, cstride=3, alpha=0.9, cmap="jet")
            #
            pylab.show()

    def predict(self, nat_X: ndarray, return_val: str = "y") -> Union[float,
                                                                      Tuple[float, float]]:
        """
        This function returns the prediction (in natural units) of the surrogate at the natural coordinates of X.

        Args:
            self (object):
                The Kriging object.
            nat_X (ndarray):
                Design variable to evaluate in natural units.
            return_val (str):
                whether `y`, `s`, neg. `ei` (negative expected improvement),
                or all three values are returned.
                Default is (for compatibility with sklearn) "y". To return `s`, select "s",
                to return neg. `ei`, select "ei".
                To return the tuple `(y, s, ei)`, select "all".

        Returns:
            float:
                The predicted value in natural units if return_val is "y".
            float:
                predicted error if return_val is "s".
            float:
                expected improvement if return_val is "ei".
            Tuple[float, float, float]:
                The predicted value in natural units, predicted error
                and expected improvement if return_val is "all".

        Examples:
            >>> from spotPython.build.kriging import Kriging
                import numpy as np
                import matplotlib.pyplot as plt
                from numpy import linspace, arange
                rng = np.random.RandomState(1)
                X = linspace(start=0, stop=10, num=1_0).reshape(-1, 1)
                y = np.squeeze(X * np.sin(X))
                training_indices = rng.choice(arange(y.size), size=6, replace=False)
                X_train, y_train = X[training_indices], y[training_indices]
                S = Kriging(name='kriging', seed=124)
                S.fit(X_train, y_train)
                mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
                print(f"mean_prediction: {mean_prediction}")
                print(f"std_prediction: {std_prediction}")
                print(f"s_ei: {s_ei}")
                mean_prediction: [-1.41991225e-08  6.48310037e-01  1.76715565e+00 -6.35226564e-01
                                  -4.28585379e+00 -1.22301198e+00  2.49434148e+00  5.61900501e-01
                                  -3.04558205e+00 -5.44021104e+00]
                std_prediction: [3.69706811e-04 2.07958787e+00 3.69706810e-04 3.69706807e-04
                                3.69706809e-04 2.07958584e+00 3.69706811e-04 2.60615408e+00
                                2.60837033e+00 3.69706811e-04]
                s_ei: [-0.00000000e+00 -1.02341235e-03 -0.00000000e+00 -0.00000000e+00
                       -0.00000000e+00 -1.63799181e-02 -0.00000000e+00 -9.45766290e-03
                       -2.53405666e-01 -1.47459347e-04]

        """
        # Check for the shape and the type of the Input
        if isinstance(nat_X, ndarray):
            try:
                X = nat_X.reshape(-1, self.nat_X.shape[1])
                X = repair_non_numeric(X, self.var_type)
            except Exception:
                raise TypeError("13.1: Input to predict was not convertible to the size of X")
        else:
            raise TypeError(f"type of the given input is an {type(nat_X)} instead of an ndarray")
        n = X.shape[0]
        y = empty(n, dtype=float)
        s = empty(n, dtype=float)
        ei = empty(n, dtype=float)
        for i in range(n):
            x = X[i, :]
            y[i], s[i], ei[i] = self.predict_coded(x)
        if return_val == "y":
            return y
        elif return_val == "s":
            return s
        elif return_val == "ei":
            return -1.0 * ei
        else:
            return y, s, -1.0 * ei

    def predict_coded(self, cod_x: np.ndarray) -> Tuple[float, float, float]:
        """
        Kriging prediction of one point in the coded units as described in (2.20) in [Forr08a].
        The error is returned as well.

        Args:
            self (object):
                The Kriging object.
            cod_x (np.ndarray):
                Point in coded units to make prediction at.

        Returns:
            f (float): Predicted value in coded units.
            SSqr (float): Predicted error.
            EI (float): Expected improvement.

        Note:
            `self.mu` and `self.SigmaSqr` are computed in `likelihood`, not here.
            See also [Forr08a, p.60].
        """
        self.build_psi_vec(cod_x)
        U_T_inv = solve(self.U.T, self.nat_y - self.one.dot(self.mu))
        f = self.mu + self.psi.T.dot(solve(self.U, U_T_inv))
        if self.noise:
            Lambda = self.Lambda
        else:
            Lambda = 0.0
        # Error in [Forr08a, p.87]:
        SSqr = self.SigmaSqr * (1 + Lambda - self.psi.T.dot(solve(self.U, solve(self.U.T, self.psi))))
        SSqr = power(abs(SSqr[0]), 0.5)[0]
        EI = self.exp_imp(y0=f[0], s0=SSqr)
        return f[0], SSqr, EI

    def build_psi_vec(self, cod_x: ndarray) -> None:
        """
        Build the psi vector. Needed by `predict_cod`, `predict_err_coded`,
        `regression_predict_coded`. Modifies `self.psi`.

        Args:
            self (object):
                The Kriging object.
            cod_x (ndarray):
                point to calculate psi

        Returns:
            None

        Attributes:
            self.psi (ndarray):
                psi vector

        Examples:
            >>> import numpy as np
                from spotPython.build.kriging import Kriging
                X_train = np.array([[1., 2.],
                                    [2., 4.],
                                    [3., 6.]])
                y_train = np.array([1., 2., 3.])
                S = Kriging(name='kriging',
                            seed=123,
                            log_level=50,
                            n_theta=1,
                            noise=False,
                            cod_type="norm")
                S.fit(X_train, y_train)
                # force theta to simple values:
                S.theta = np.array([0.0])
                nat_X = np.array([1., 0.])
                S.psi = np.zeros((S.n, 1))
                S.build_psi_vec(nat_X)
                res = np.array([[np.exp(-4)],
                    [np.exp(-17)],
                    [np.exp(-40)]])
                assert np.array_equal(S.psi, res)
                print(f"S.psi: {S.psi}")
                print(f"Control value res: {res}")
                S.psi:
                [[1.83156389e-02]
                [4.13993772e-08]
                [4.24835426e-18]]
                Control value res:
                [[1.83156389e-02]
                [4.13993772e-08]
                [4.24835426e-18]]
        """
        self.psi = zeros((self.n))
        theta = power(10.0, self.theta)
        if self.n_theta == 1:
            theta = theta * ones(self.k)
        try:
            D = zeros((self.n))
            if self.ordered_mask.any():
                X_ordered = self.nat_X[:, self.ordered_mask]
                x_ordered = cod_x[self.ordered_mask]
                D = cdist(x_ordered.reshape(-1, sum(self.ordered_mask)),
                          X_ordered.reshape(-1, sum(self.ordered_mask)),
                          metric='sqeuclidean',
                          out=None,
                          w=theta[self.ordered_mask])
            if self.factor_mask.any():
                X_factor = self.nat_X[:, self.factor_mask]
                x_factor = cod_x[self.factor_mask]
                D = (D + cdist(x_factor.reshape(-1, sum(self.factor_mask)),
                               X_factor.reshape(-1, sum(self.factor_mask)),
                               metric=self.metric_factorial,
                               out=None,
                               w=theta[self.factor_mask]))
            self.psi = exp(-D).T
        except LinAlgError as err:
            print(f"Building psi failed:\n {self.psi}. {err=}, {type(err)=}")

    def weighted_exp_imp(self, cod_x: np.ndarray, w: float) -> float:
        """
        Weighted expected improvement.

        Args:
            self (object): The Kriging object.
            cod_x (np.ndarray): A coded design vector.
            w (float): Weight.

        Returns:
            EI (float): Weighted expected improvement.

        References:
            [Sobester et al. 2005].
        """
        y0, s0 = self.predict_coded(cod_x)
        y_min = min(self.nat_y)
        if s0 <= 0.0:
            EI = 0.0
        else:
            y_min_y0 = y_min - y0
            EI_one = w * (
                    y_min_y0
                    * (0.5 + 0.5 * erf((1.0 / sqrt(2.0)) * (y_min_y0 / s0)))
            )
            EI_two = (
                    (1.0 - w)
                    * (s0 * (1.0 / sqrt(2.0 * pi)))
                    * (exp(-(1.0 / 2.0) * ((y_min_y0) ** 2.0 / s0 ** 2.0)))
            )
            EI = EI_one + EI_two
        return EI

__init__(noise=False, var_type=['num'], name='kriging', seed=124, model_optimizer=None, model_fun_evals=None, min_theta=-3.0, max_theta=2.0, n_theta=1, theta_init_zero=True, p_val=2.0, n_p=1, optim_p=False, min_Lambda=1e-09, max_Lambda=1.0, log_level=50, spot_writer=None, counter=None, metric_factorial='canberra', **kwargs)

Initialize the Kriging surrogate.

Parameters:

Name	Type	Description	Default
noise	bool	Use regression instead of interpolation kriging. Defaults to False.	False
var_type	List[str]	Variable type. Can be either “num” (numerical) or “factor” (factor). Defaults to [“num”].	['num']
name	str	Surrogate name. Defaults to “kriging”.	'kriging'
seed	int	Random seed. Defaults to 124.	124
model_optimizer	Optional[object]	Optimizer on the surrogate. If None, differential_evolution is selected.	None
model_fun_evals	Optional[int]	Number of iterations used by the optimizer on the surrogate.	None
min_theta	float	Min log10 theta value. Defaults to -3.	-3.0
max_theta	float	Max log10 theta value. Defaults to 2.	2.0
n_theta	int	Number of theta values. Defaults to 1.	1
theta_init_zero	bool	Initialize theta with zero. Defaults to True.	True
p_val	float	p value. Used as an initial value if optim_p = True. Otherwise as a constant. Defaults to 2.	2.0
n_p	int	Number of p values. Defaults to 1.	1
optim_p	bool	Determines whether p should be optimized. Deafults to False.	False
min_Lambda	float	Min Lambda value. Defaults to 1e-9.	1e-09
max_Lambda	float	Max Lambda value. Defaults to 1.	1.0
log_level	int	Logging level, e.g., 20 is “INFO”. Defaults to 50 (“CRITICAL”).	50
spot_writer	Optional[object]	Spot writer. Defaults to None.	None
counter	Optional[int]	Counter. Defaults to None.	None
metric_factorial	str	Metric for factorial. Defaults to “canberra”. Can be “euclidean”, “cityblock”, seuclidean”, “sqeuclidean”, “cosine”, “correlation”, “hamming”, “jaccard”, “jensenshannon”, “chebyshev”, “canberra”, “braycurtis”, “mahalanobis”, “matching”.	'canberra'

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    import matplotlib.pyplot as plt
    from numpy import linspace, arange
    rng = np.random.RandomState(1)
    X = linspace(start=0, stop=10, num=1_000).reshape(-1, 1)
    y = np.squeeze(X * np.sin(X))
    training_indices = rng.choice(arange(y.size), size=6, replace=False)
    X_train, y_train = X[training_indices], y[training_indices]
    S = Kriging(name='kriging', seed=124)
    S.fit(X_train, y_train)
    mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
    plt.plot(X, y, label=r"$f(x)$", linestyle="dotted")
    plt.scatter(X_train, y_train, label="Observations")
    plt.plot(X, mean_prediction, label="Mean prediction")
    plt.fill_between(
        X.ravel(),
        mean_prediction - 1.96 * std_prediction,
        mean_prediction + 1.96 * std_prediction,
        alpha=0.5,
        label=r"95% confidence interval",
        )
    plt.legend()
    plt.xlabel("$x$")
    plt.ylabel("$f(x)$")
    _ = plt.title("Gaussian process regression on noise-free dataset")
    plt.show()

References

https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.pdist.html [1] scikit-learn: Gaussian Processes regression: basic introductory example

Source code in spotPython/build/kriging.py

def __init__(
        self: object,
        noise: bool = False,
        var_type: List[str] = ["num"],
        name: str = "kriging",
        seed: int = 124,
        model_optimizer=None,
        model_fun_evals: Optional[int] = None,
        min_theta: float = -3.0,
        max_theta: float = 2.0,
        n_theta: int = 1,
        theta_init_zero: bool = True,
        p_val: float = 2.0,
        n_p: int = 1,
        optim_p: bool = False,
        min_Lambda: float = 1e-9,
        max_Lambda: float = 1.,
        log_level: int = 50,
        spot_writer=None,
        counter=None,
        metric_factorial="canberra",
        **kwargs
):
    """
    Initialize the Kriging surrogate.

    Args:
        noise (bool): Use regression instead of interpolation kriging. Defaults to False.
        var_type (List[str]):
            Variable type. Can be either "num" (numerical) or "factor" (factor).
            Defaults to ["num"].
        name (str):
            Surrogate name. Defaults to "kriging".
        seed (int):
            Random seed. Defaults to 124.
        model_optimizer (Optional[object]):
            Optimizer on the surrogate. If None, differential_evolution is selected.
        model_fun_evals (Optional[int]):
            Number of iterations used by the optimizer on the surrogate.
        min_theta (float):
            Min log10 theta value. Defaults to -3.
        max_theta (float):
            Max log10 theta value. Defaults to 2.
        n_theta (int):
            Number of theta values. Defaults to 1.
        theta_init_zero (bool):
            Initialize theta with zero. Defaults to True.
        p_val (float):
            p value. Used as an initial value if optim_p = True. Otherwise as a constant. Defaults to 2.
        n_p (int):
            Number of p values. Defaults to 1.
        optim_p (bool):
            Determines whether p should be optimized. Deafults to False.
        min_Lambda (float):
            Min Lambda value. Defaults to 1e-9.
        max_Lambda (float):
            Max Lambda value. Defaults to 1.
        log_level (int):
            Logging level, e.g., 20 is "INFO". Defaults to 50 ("CRITICAL").
        spot_writer (Optional[object]):
            Spot writer. Defaults to None.
        counter (Optional[int]):
            Counter. Defaults to None.
        metric_factorial (str):
            Metric for factorial. Defaults to "canberra". Can be "euclidean",
            "cityblock", seuclidean", "sqeuclidean", "cosine",
            "correlation", "hamming", "jaccard", "jensenshannon",
            "chebyshev", "canberra", "braycurtis", "mahalanobis", "matching".

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            import matplotlib.pyplot as plt
            from numpy import linspace, arange
            rng = np.random.RandomState(1)
            X = linspace(start=0, stop=10, num=1_000).reshape(-1, 1)
            y = np.squeeze(X * np.sin(X))
            training_indices = rng.choice(arange(y.size), size=6, replace=False)
            X_train, y_train = X[training_indices], y[training_indices]
            S = Kriging(name='kriging', seed=124)
            S.fit(X_train, y_train)
            mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
            plt.plot(X, y, label=r"$f(x)$", linestyle="dotted")
            plt.scatter(X_train, y_train, label="Observations")
            plt.plot(X, mean_prediction, label="Mean prediction")
            plt.fill_between(
                X.ravel(),
                mean_prediction - 1.96 * std_prediction,
                mean_prediction + 1.96 * std_prediction,
                alpha=0.5,
                label=r"95% confidence interval",
                )
            plt.legend()
            plt.xlabel("$x$")
            plt.ylabel("$f(x)$")
            _ = plt.title("Gaussian process regression on noise-free dataset")
            plt.show()

    References:
        https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.distance.pdist.html
        [[1](https://scikit-learn.org/stable/auto_examples/gaussian_process/plot_gpr_noisy_targets.html)]
        scikit-learn: Gaussian Processes regression: basic introductory example

    """
    super().__init__(name, seed, log_level)

    self.noise = noise
    self.var_type = var_type
    self.name = name
    self.seed = seed
    self.log_level = log_level
    self.spot_writer = spot_writer
    self.counter = counter
    self.metric_factorial = metric_factorial

    self.sigma = 0
    self.eps = sqrt(spacing(1))
    self.min_theta = min_theta
    self.max_theta = max_theta
    self.min_p = 1
    self.max_p = 2
    self.min_Lambda = min_Lambda
    self.max_Lambda = max_Lambda
    self.n_theta = n_theta
    self.p_val = p_val
    self.n_p = n_p
    self.optim_p = optim_p
    self.theta_init_zero = theta_init_zero
    # Psi matrix condition:
    self.cnd_Psi = 0
    self.inf_Psi = False

    self.model_optimizer = model_optimizer
    if self.model_optimizer is None:
        self.model_optimizer = differential_evolution
    self.model_fun_evals = model_fun_evals
    # differential evolution uses maxiter = 1000
    # and sets the number of function evaluations to
    # (maxiter + 1) * popsize * N, which results in
    # 1000 * 15 * k, because the default popsize is 15 and
    # N is the number of parameters. This seems to be quite large:
    # for k=2 these are 30 000 iterations. Therefore we set this value to
    # 100
    if self.model_fun_evals is None:
        self.model_fun_evals = 100

    # Logging information
    self.log["negLnLike"] = []
    self.log["theta"] = []
    self.log["p"] = []
    self.log["Lambda"] = []
    # Logger
    logger.setLevel(self.log_level)
    logger.info(f"Starting the logger at level {self.log_level} for module {__name__}:")

__is_any__(x, v)

Check if any element in x is equal to v. This method checks if any element in the input array x is equal to the value v. If x is not an instance of ndarray, it is first converted to a numpy array using the array() function.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
x	ndarray or array - like	The input array to check for the presence of value v.	required
v	scalar	The value to check for in the input array x.	required

Returns:

Name	Type	Description
bool	bool	True if any element in x is equal to v, False otherwise.

Examples:

>>> from spotPython.build.kriging import Kriging
    from numpy import power
    import numpy as np
    nat_X = np.array([[0], [1]])
    nat_y = np.array([0, 1])
    n=1
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    print(f"S.theta: {S.theta}")
    print(S.__is_any__(power(10.0, S.theta), 0))
    print(S.__is_any__(S.theta, 0))
    S.theta: [0.]
    False
    True

Source code in spotPython/build/kriging.py

def __is_any__(self, x: Union[np.ndarray, Any], v: Any) -> bool:
    """
    Check if any element in `x` is equal to `v`.
    This method checks if any element in the input array `x` is equal to the value `v`.
    If `x` is not an instance of `ndarray`, it is first converted to a numpy array using
    the `array()` function.

    Args:
        self (object): The Kriging object.
        x (np.ndarray or array-like):
            The input array to check for the presence of value `v`.
        v (scalar):
            The value to check for in the input array `x`.

    Returns:
        bool:
            True if any element in `x` is equal to `v`, False otherwise.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            from numpy import power
            import numpy as np
            nat_X = np.array([[0], [1]])
            nat_y = np.array([0, 1])
            n=1
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            print(f"S.theta: {S.theta}")
            print(S.__is_any__(power(10.0, S.theta), 0))
            print(S.__is_any__(S.theta, 0))
            S.theta: [0.]
            False
            True

    """
    if not isinstance(x, ndarray):
        x = array([x])
    return any(x == v)

build_Psi()

Constructs a new (n x n) correlation matrix Psi to reflect new data or a change in hyperparameters.

This method uses theta, p, and coded X values to construct the correlation matrix as described in [Forr08a, p.57].

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Returns:

Type	Description
None	None

Raises:

Type	Description
LinAlgError	If building Psi fails.

Attributes:

Name	Type	Description
Psi	matrix	Correlation matrix Psi. Shape (n,n).
cnd_Psi	float	Condition number of Psi.
inf_Psi	bool	True if Psi is infinite, False otherwise.

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[0], [1]])
    nat_y = np.array([0, 1])
    n=1
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    print(S.nat_X)
    print(S.nat_y)
    S.set_theta_values()
    print(f"S.theta: {S.theta}")
    S.initialize_matrices()
    S.set_de_bounds()
    new_theta_p_Lambda = S.optimize_model()
    S.extract_from_bounds(new_theta_p_Lambda)
    print(f"S.theta: {S.theta}")
    S.build_Psi()
    print(f"S.Psi: {S.Psi}")
    [[0]
    [1]]
    [0 1]
    S.theta: [0.]
    S.theta: [1.72284258]
    S.Psi: [[1.00000001e+00 1.14348852e-23]
    [1.14348852e-23 1.00000001e+00]]

Source code in spotPython/build/kriging.py

def build_Psi(self) -> None:
    """
    Constructs a new (n x n) correlation matrix Psi to reflect new data
    or a change in hyperparameters.

    This method uses `theta`, `p`, and coded `X` values to construct the
    correlation matrix as described in [Forr08a, p.57].

    Args:
        self (object): The Kriging object.

    Returns:
        None

    Raises:
        LinAlgError: If building Psi fails.

    Attributes:
        Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
        cnd_Psi (float): Condition number of Psi.
        inf_Psi (bool): True if Psi is infinite, False otherwise.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[0], [1]])
            nat_y = np.array([0, 1])
            n=1
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            print(S.nat_X)
            print(S.nat_y)
            S.set_theta_values()
            print(f"S.theta: {S.theta}")
            S.initialize_matrices()
            S.set_de_bounds()
            new_theta_p_Lambda = S.optimize_model()
            S.extract_from_bounds(new_theta_p_Lambda)
            print(f"S.theta: {S.theta}")
            S.build_Psi()
            print(f"S.Psi: {S.Psi}")
            [[0]
            [1]]
            [0 1]
            S.theta: [0.]
            S.theta: [1.72284258]
            S.Psi: [[1.00000001e+00 1.14348852e-23]
            [1.14348852e-23 1.00000001e+00]]

    """
    self.Psi = zeros((self.n, self.n), dtype=float64)
    theta = power(10.0, self.theta)
    if self.n_theta == 1:
        theta = theta * ones(self.k)
    try:
        D = zeros((self.n, self.n))
        if self.ordered_mask.any():
            X_ordered = self.nat_X[:, self.ordered_mask]
            D = squareform(
                pdist(
                    X_ordered, metric='sqeuclidean', out=None, w=theta[self.ordered_mask]))
        if self.factor_mask.any():
            X_factor = self.nat_X[:, self.factor_mask]
            D = (D + squareform(
                pdist(X_factor,
                      metric=self.metric_factorial,
                      out=None,
                      w=theta[self.factor_mask])))
        self.Psi = exp(-D)
    except LinAlgError as err:
        print(f"Building Psi failed:\n {self.Psi}. {err=}, {type(err)=}")
    if self.noise:
        logger.debug("In build_Psi(): self.Lambda: %s", self.Lambda)
        self.Psi[diag_indices_from(self.Psi)] += self.Lambda
    else:
        self.Psi[diag_indices_from(self.Psi)] += self.eps
    if (isinf(self.Psi)).any():
        self.inf_Psi = True
    self.cnd_Psi = cond(self.Psi)

build_U(scipy=True)

Performs Cholesky factorization of Psi as U as described in [Forr08a, p.57]. This method uses either scipy_cholesky or numpy’s cholesky to perform the Cholesky factorization of Psi.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
scipy	bool	If True, use scipy_cholesky. If False, use numpy’s cholesky. Defaults to True.	True

Returns:

Type	Description
None	None

Raises:

Type	Description
LinAlgError	If Cholesky factorization fails for Psi.

Attributes:

Name	Type	Description
U	matrix	Kriging U matrix, Cholesky decomposition. Shape (n,n).

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[0], [1]])
    nat_y = np.array([0, 1])
    n=1
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    print(S.nat_X)
    print(S.nat_y)
    S.set_theta_values()
    print(f"S.theta: {S.theta}")
    S.initialize_matrices()
    S.set_de_bounds()
    new_theta_p_Lambda = S.optimize_model()
    S.extract_from_bounds(new_theta_p_Lambda)
    print(f"S.theta: {S.theta}")
    S.build_Psi()
    print(f"S.Psi: {S.Psi}")
    S.build_U()
    print(f"S.U:{S.U}")
    [[0]
    [1]]
    [0 1]
    S.theta: [0.]
    S.theta: [1.72284258]
    S.Psi: [[1.00000001e+00 1.14348852e-23]
    [1.14348852e-23 1.00000001e+00]]
    S.U:[[1.00000001e+00 1.14348851e-23]
    [0.00000000e+00 1.00000001e+00]]

Source code in spotPython/build/kriging.py

def build_U(self, scipy: bool = True) -> None:
    """
    Performs Cholesky factorization of Psi as U as described in [Forr08a, p.57].
    This method uses either `scipy_cholesky` or numpy's `cholesky` to perform the Cholesky factorization of Psi.

    Args:
        self (object):
            The Kriging object.
        scipy (bool):
            If True, use `scipy_cholesky`.
            If False, use numpy's `cholesky`.
            Defaults to True.

    Returns:
        None

    Raises:
        LinAlgError:
            If Cholesky factorization fails for Psi.

    Attributes:
        U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[0], [1]])
            nat_y = np.array([0, 1])
            n=1
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            print(S.nat_X)
            print(S.nat_y)
            S.set_theta_values()
            print(f"S.theta: {S.theta}")
            S.initialize_matrices()
            S.set_de_bounds()
            new_theta_p_Lambda = S.optimize_model()
            S.extract_from_bounds(new_theta_p_Lambda)
            print(f"S.theta: {S.theta}")
            S.build_Psi()
            print(f"S.Psi: {S.Psi}")
            S.build_U()
            print(f"S.U:{S.U}")
            [[0]
            [1]]
            [0 1]
            S.theta: [0.]
            S.theta: [1.72284258]
            S.Psi: [[1.00000001e+00 1.14348852e-23]
            [1.14348852e-23 1.00000001e+00]]
            S.U:[[1.00000001e+00 1.14348851e-23]
            [0.00000000e+00 1.00000001e+00]]
    """
    try:
        self.U = scipy_cholesky(self.Psi, lower=True) if scipy else cholesky(self.Psi)
        self.U = self.U.T
    except LinAlgError as err:
        print(f"build_U() Cholesky failed for Psi:\n {self.Psi}. {err=}, {type(err)=}")

build_psi_vec(cod_x)

Build the psi vector. Needed by predict_cod, predict_err_coded, regression_predict_coded. Modifies self.psi.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
cod_x	ndarray	point to calculate psi	required

Returns:

Type	Description
None	None

Attributes:

Name	Type	Description
self.psi	ndarray	psi vector

Examples:

>>> import numpy as np
    from spotPython.build.kriging import Kriging
    X_train = np.array([[1., 2.],
                        [2., 4.],
                        [3., 6.]])
    y_train = np.array([1., 2., 3.])
    S = Kriging(name='kriging',
                seed=123,
                log_level=50,
                n_theta=1,
                noise=False,
                cod_type="norm")
    S.fit(X_train, y_train)
    # force theta to simple values:
    S.theta = np.array([0.0])
    nat_X = np.array([1., 0.])
    S.psi = np.zeros((S.n, 1))
    S.build_psi_vec(nat_X)
    res = np.array([[np.exp(-4)],
        [np.exp(-17)],
        [np.exp(-40)]])
    assert np.array_equal(S.psi, res)
    print(f"S.psi: {S.psi}")
    print(f"Control value res: {res}")
    S.psi:
    [[1.83156389e-02]
    [4.13993772e-08]
    [4.24835426e-18]]
    Control value res:
    [[1.83156389e-02]
    [4.13993772e-08]
    [4.24835426e-18]]

Source code in spotPython/build/kriging.py

def build_psi_vec(self, cod_x: ndarray) -> None:
    """
    Build the psi vector. Needed by `predict_cod`, `predict_err_coded`,
    `regression_predict_coded`. Modifies `self.psi`.

    Args:
        self (object):
            The Kriging object.
        cod_x (ndarray):
            point to calculate psi

    Returns:
        None

    Attributes:
        self.psi (ndarray):
            psi vector

    Examples:
        >>> import numpy as np
            from spotPython.build.kriging import Kriging
            X_train = np.array([[1., 2.],
                                [2., 4.],
                                [3., 6.]])
            y_train = np.array([1., 2., 3.])
            S = Kriging(name='kriging',
                        seed=123,
                        log_level=50,
                        n_theta=1,
                        noise=False,
                        cod_type="norm")
            S.fit(X_train, y_train)
            # force theta to simple values:
            S.theta = np.array([0.0])
            nat_X = np.array([1., 0.])
            S.psi = np.zeros((S.n, 1))
            S.build_psi_vec(nat_X)
            res = np.array([[np.exp(-4)],
                [np.exp(-17)],
                [np.exp(-40)]])
            assert np.array_equal(S.psi, res)
            print(f"S.psi: {S.psi}")
            print(f"Control value res: {res}")
            S.psi:
            [[1.83156389e-02]
            [4.13993772e-08]
            [4.24835426e-18]]
            Control value res:
            [[1.83156389e-02]
            [4.13993772e-08]
            [4.24835426e-18]]
    """
    self.psi = zeros((self.n))
    theta = power(10.0, self.theta)
    if self.n_theta == 1:
        theta = theta * ones(self.k)
    try:
        D = zeros((self.n))
        if self.ordered_mask.any():
            X_ordered = self.nat_X[:, self.ordered_mask]
            x_ordered = cod_x[self.ordered_mask]
            D = cdist(x_ordered.reshape(-1, sum(self.ordered_mask)),
                      X_ordered.reshape(-1, sum(self.ordered_mask)),
                      metric='sqeuclidean',
                      out=None,
                      w=theta[self.ordered_mask])
        if self.factor_mask.any():
            X_factor = self.nat_X[:, self.factor_mask]
            x_factor = cod_x[self.factor_mask]
            D = (D + cdist(x_factor.reshape(-1, sum(self.factor_mask)),
                           X_factor.reshape(-1, sum(self.factor_mask)),
                           metric=self.metric_factorial,
                           out=None,
                           w=theta[self.factor_mask]))
        self.psi = exp(-D).T
    except LinAlgError as err:
        print(f"Building psi failed:\n {self.psi}. {err=}, {type(err)=}")

exp_imp(y0, s0)

Calculates the expected improvement for a given function value and error in coded units.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
y0	float	The function value in coded units.	required
s0	float	The error value.	required

Returns:

Name	Type	Description
float	float	The expected improvement value.

Examples:

>>> from spotPython.build.kriging import Kriging
    S = Kriging(name='kriging', seed=124)
    S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
    S.exp_imp(1.0, 0.0)
    0.0
>>> from spotPython.build.kriging import Kriging
    S = Kriging(name='kriging', seed=124)
    S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
    # assert S.exp_imp(0.0, 1.0) == 1/np.sqrt(2*np.pi)
    # which is approx. 0.3989422804014327
    S.exp_imp(0.0, 1.0)
    0.3989422804014327

Source code in spotPython/build/kriging.py

def exp_imp(self, y0: float, s0: float) -> float:
    """
    Calculates the expected improvement for a given function value and error in coded units.

    Args:
        self (object): The Kriging object.
        y0 (float): The function value in coded units.
        s0 (float): The error value.

    Returns:
        float: The expected improvement value.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            S = Kriging(name='kriging', seed=124)
            S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
            S.exp_imp(1.0, 0.0)
            0.0
        >>> from spotPython.build.kriging import Kriging
            S = Kriging(name='kriging', seed=124)
            S.aggregated_mean_y = [0.0, 0.0, 0.0, 0.0, 0.0]
            # assert S.exp_imp(0.0, 1.0) == 1/np.sqrt(2*np.pi)
            # which is approx. 0.3989422804014327
            S.exp_imp(0.0, 1.0)
            0.3989422804014327
    """
    # We do not use the min y values, but the aggragated mean values
    # y_min = min(self.nat_y)
    y_min = min(self.aggregated_mean_y)
    if s0 <= 0.0:
        EI = 0.0
    elif s0 > 0.0:
        EI_one = (y_min - y0) * (
                0.5 + 0.5 * erf((1.0 / sqrt(2.0)) * ((y_min - y0) / s0))
        )
        EI_two = (s0 * (1.0 / sqrt(2.0 * pi))) * (
            exp(-(1.0 / 2.0) * ((y_min - y0) ** 2.0 / s0 ** 2.0))
        )
        EI = EI_one + EI_two
    return EI

extract_from_bounds(new_theta_p_Lambda)

Extract theta, p, and Lambda from bounds. The kriging object stores theta as an array, p as an array, and Lambda as a float.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
new_theta_p_Lambda	ndarray	1d-array with theta, p, and Lambda values. Order is important.	required

Examples:

>>> import numpy as np
    from spotPython.build.kriging import Kriging
    n=2
    p=4
    S = Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
    S.extract_from_bounds(np.array([1, 2, 3]))
    print(S.theta)
    print(S.p)
    [1 2]
    [3]

Returns:

Type	Description
None	None

Source code in spotPython/build/kriging.py

def extract_from_bounds(self, new_theta_p_Lambda: np.ndarray) -> None:
    """
    Extract `theta`, `p`, and `Lambda` from bounds. The kriging object stores
    `theta` as an array,  `p` as an array, and `Lambda` as a float.

    Args:
        self (object): The Kriging object.
        new_theta_p_Lambda (np.ndarray):
            1d-array with theta, p, and Lambda values. Order is important.

    Examples:
        >>> import numpy as np
            from spotPython.build.kriging import Kriging
            n=2
            p=4
            S = Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
            S.extract_from_bounds(np.array([1, 2, 3]))
            print(S.theta)
            print(S.p)
            [1 2]
            [3]

    Returns:
        None
    """
    logger.debug("In extract_from_bounds(): new_theta_p_Lambda: %s", new_theta_p_Lambda)
    self.theta = new_theta_p_Lambda[:self.n_theta]
    logger.debug("In extract_from_bounds(): self.n_theta: %s", self.n_theta)
    if self.optim_p:
        self.p = new_theta_p_Lambda[self.n_theta:self.n_theta + self.n_p]
        logger.debug("In extract_from_bounds(): self.p: %s", self.p)
        if self.noise:
            self.Lambda = new_theta_p_Lambda[self.n_theta + self.n_p]
            logger.debug("In extract_from_bounds(): self.Lambda: %s", self.Lambda)
    else:
        if self.noise:
            self.Lambda = new_theta_p_Lambda[self.n_theta]
            logger.debug("In extract_from_bounds(): self.Lambda: %s", self.Lambda)

fit(nat_X, nat_y)

Fits the hyperparameters (theta, p, Lambda) of the Kriging model.

The function computes the following internal values: 1. theta, p, and Lambda values via optimization of the function fun_likelihood(). 2. Correlation matrix Psi via rebuildPsi().

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
nat_X	ndarray	Sample points.	required
nat_y	ndarray	Function values.	required

Returns:

Name	Type	Description
object	object	Fitted estimator.

Attributes:

Name	Type	Description
theta	ndarray	Kriging theta values. Shape (k,).
p	ndarray	Kriging p values. Shape (k,).
LnDetPsi	float64	Determinant Psi matrix.
Psi	matrix	Correlation matrix Psi. Shape (n,n).
psi	ndarray	psi vector. Shape (n,).
one	ndarray	vector of ones. Shape (n,).
mu	float64	Kriging expected mean value mu.
U	matrix	Kriging U matrix, Cholesky decomposition. Shape (n,n).
SigmaSqr	float64	Sigma squared value.
Lambda	float	lambda noise value.

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[1, 0], [1, 0]])
    nat_y = np.array([1, 2])
    S = Kriging()
    S.fit(nat_X, nat_y)
    print(S.Psi)
    [[1.00000001 1.        ]
    [1.         1.00000001]]

Source code in spotPython/build/kriging.py

def fit(self, nat_X: np.ndarray, nat_y: np.ndarray) -> object:
    """
    Fits the hyperparameters (`theta`, `p`, `Lambda`) of the Kriging model.

    The function computes the following internal values:
    1. `theta`, `p`, and `Lambda` values via optimization of the function `fun_likelihood()`.
    2. Correlation matrix `Psi` via `rebuildPsi()`.

    Args:
        self (object): The Kriging object.
        nat_X (np.ndarray): Sample points.
        nat_y (np.ndarray): Function values.

    Returns:
        object: Fitted estimator.

    Attributes:
        theta (np.ndarray): Kriging theta values. Shape (k,).
        p (np.ndarray): Kriging p values. Shape (k,).
        LnDetPsi (np.float64): Determinant Psi matrix.
        Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
        psi (np.ndarray): psi vector. Shape (n,).
        one (np.ndarray): vector of ones. Shape (n,).
        mu (np.float64): Kriging expected mean value mu.
        U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).
        SigmaSqr (np.float64): Sigma squared value.
        Lambda (float): lambda noise value.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[1, 0], [1, 0]])
            nat_y = np.array([1, 2])
            S = Kriging()
            S.fit(nat_X, nat_y)
            print(S.Psi)
            [[1.00000001 1.        ]
            [1.         1.00000001]]

    """
    logger.debug("In fit(): nat_X: %s", nat_X)
    logger.debug("In fit(): nat_y: %s", nat_y)
    self.initialize_variables(nat_X, nat_y)
    self.set_variable_types()
    self.set_theta_values()
    self.initialize_matrices()
    # build_Psi() and build_U() are called in fun_likelihood
    self.set_de_bounds()
    # Finally, set new theta and p values and update the surrogate again
    # for new_theta_p_Lambda in de_results["x"]:
    new_theta_p_Lambda = self.optimize_model()
    self.extract_from_bounds(new_theta_p_Lambda)
    self.build_Psi()
    self.build_U()
    # TODO: check if the following line is necessary!
    self.likelihood()
    self.update_log()

fun_likelihood(new_theta_p_Lambda)

Compute log likelihood for a set of hyperparameters (theta, p, Lambda).

This method computes the log likelihood for a set of hyperparameters (theta, p, Lambda) by performing the following steps: 1. Extracts the hyperparameters from the input array using extract_from_bounds(). 2. Checks if any element in 10^theta is equal to 0. If so, logs a warning and returns the penalty value (pen_val). 3. Builds the Psi matrix using build_Psi(). 4. Checks if Psi is ill-conditioned or infinite. If so, logs a warning and returns the penalty value (pen_val). 5. Builds the U matrix using build_U(). If an exception occurs, logs an error and returns the penalty value (pen_val). 6. Computes the negative log likelihood using likelihood(). 7. Returns the computed negative log likelihood (negLnLike).

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
new_theta_p_Lambda	ndarray	An array containing the theta, p, and Lambda values.	required

Returns:

Name	Type	Description
float	float	The negative log likelihood of the surface at the specified hyperparameters.

Attributes:

Name	Type	Description
theta	ndarray	Kriging theta values. Shape (k,).
p	ndarray	Kriging p values. Shape (k,).
Lambda	float	lambda noise value.
Psi	matrix	Correlation matrix Psi. Shape (n,n).
U	matrix	Kriging U matrix, Cholesky decomposition. Shape (n,n).
negLnLike	float	Negative log likelihood of the surface at the specified hyperparameters.
pen_val	float	Penalty value.

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[0], [1]])
    nat_y = np.array([0, 1])
    n=1
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    print(S.cod_X)
    print(S.cod_y)
    S.set_theta_values()
    print(f"S.theta: {S.theta}")
    S.initialize_matrices()
    S.set_de_bounds()
    new_theta_p_Lambda = S.optimize_model()
    S.extract_from_bounds(new_theta_p_Lambda)
    print(f"S.theta: {S.theta}")
    S.build_Psi()
    print(f"S.Psi: {S.Psi}")
    S.build_U()
    print(f"S.U:{S.U}")
    S.likelihood()
    S.negLnLike

Source code in spotPython/build/kriging.py

def fun_likelihood(self, new_theta_p_Lambda: np.ndarray) -> float:
    """
    Compute log likelihood for a set of hyperparameters (theta, p, Lambda).

    This method computes the log likelihood for a set of hyperparameters
    (theta, p, Lambda) by performing the following steps:
    1. Extracts the hyperparameters from the input array using `extract_from_bounds()`.
    2. Checks if any element in `10^theta` is equal to 0. If so, logs a warning and
    returns the penalty value (`pen_val`).
    3. Builds the `Psi` matrix using `build_Psi()`.
    4. Checks if `Psi` is ill-conditioned or infinite. If so, logs a warning and returns
    the penalty value (`pen_val`).
    5. Builds the `U` matrix using `build_U()`. If an exception occurs, logs an error and
    returns the penalty value (`pen_val`).
    6. Computes the negative log likelihood using `likelihood()`.
    7. Returns the computed negative log likelihood (`negLnLike`).

    Args:
        self (object): The Kriging object.
        new_theta_p_Lambda (np.ndarray):
            An array containing the `theta`, `p`, and `Lambda` values.

    Returns:
        float:
            The negative log likelihood of the surface at the specified hyperparameters.

    Attributes:
        theta (np.ndarray): Kriging theta values. Shape (k,).
        p (np.ndarray): Kriging p values. Shape (k,).
        Lambda (float): lambda noise value.
        Psi (np.matrix): Correlation matrix Psi. Shape (n,n).
        U (np.matrix): Kriging U matrix, Cholesky decomposition. Shape (n,n).
        negLnLike (float): Negative log likelihood of the surface at the specified hyperparameters.
        pen_val (float): Penalty value.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[0], [1]])
            nat_y = np.array([0, 1])
            n=1
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            print(S.cod_X)
            print(S.cod_y)
            S.set_theta_values()
            print(f"S.theta: {S.theta}")
            S.initialize_matrices()
            S.set_de_bounds()
            new_theta_p_Lambda = S.optimize_model()
            S.extract_from_bounds(new_theta_p_Lambda)
            print(f"S.theta: {S.theta}")
            S.build_Psi()
            print(f"S.Psi: {S.Psi}")
            S.build_U()
            print(f"S.U:{S.U}")
            S.likelihood()
            S.negLnLike

    """
    self.extract_from_bounds(new_theta_p_Lambda)
    if self.__is_any__(power(10.0, self.theta), 0):
        logger.warning("Failure in fun_likelihood: 10^theta == 0. Setting negLnLike to %s", self.pen_val)
        return self.pen_val
    self.build_Psi()
    if (self.inf_Psi or self.cnd_Psi > 1e9):
        logger.warning("Failure in fun_likelihood: Psi is ill conditioned: %s", self.cnd_Psi)
        logger.warning("Setting negLnLike to: %s", self.pen_val)
        return self.pen_val

    try:
        self.build_U()
    except Exception as error:
        penalty_value = self.pen_val
        print("Error in fun_likelihood(). Call to build_U() failed.")
        print("error=%s, type(error)=%s" % (error, type(error)))
        print("Setting negLnLike to %.2f." % self.pen_val)
        return penalty_value
    self.likelihood()
    return self.negLnLike

initialize_matrices()

Initialize the matrices for the class instance.

This method initializes several matrices and attributes for the class instance. The p attribute is initialized as a list of ones with length n_p, multiplied by 2.0. The pen_val attribute is initialized as the natural logarithm of the variance of nat_y, multiplied by n, plus 1e4. The negLnLike, LnDetPsi, mu, U, SigmaSqr, and Lambda attributes are all set to None. The gen attribute is initialized using the spacefilling function with arguments k and seed. The Psi attribute is initialized as a zero matrix with shape (n, n) and dtype float64. The psi attribute is initialized as a zero matrix with shape (n, 1). The one attribute is initialized as a list of ones with length n.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    from numpy import log, var
    nat_X = np.array([[1, 2], [3, 4], [5, 6]])
    nat_y = np.array([1, 2, 3])
    n=3
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    S.initialize_matrices()
    # if var(self.nat_y) is > 0, then self.pen_val = self.n * log(var(self.nat_y)) + 1e4
    # else self.pen_val = self.n * var(self.nat_y) + 1e4
    assert S.pen_val == nat_X.shape[0] * log(var(S.nat_y)) + 1e4
    assert S.Psi.shape == (n, n)

Returns:

Type	Description
None	None

Source code in spotPython/build/kriging.py

def initialize_matrices(self) -> None:
    """
    Initialize the matrices for the class instance.

    This method initializes several matrices and attributes for the class instance.
    The `p` attribute is initialized as a list of ones with length `n_p`, multiplied by 2.0.
    The `pen_val` attribute is initialized as the natural logarithm of the
    variance of `nat_y`, multiplied by `n`, plus 1e4.
    The `negLnLike`, `LnDetPsi`, `mu`, `U`, `SigmaSqr`, and `Lambda` attributes are all set to None.
    The `gen` attribute is initialized using the `spacefilling` function with arguments `k` and `seed`.
    The `Psi` attribute is initialized as a zero matrix with shape `(n, n)` and dtype `float64`.
    The `psi` attribute is initialized as a zero matrix with shape `(n, 1)`.
    The `one` attribute is initialized as a list of ones with length `n`.

    Args:
        self (object): The Kriging object.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            from numpy import log, var
            nat_X = np.array([[1, 2], [3, 4], [5, 6]])
            nat_y = np.array([1, 2, 3])
            n=3
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            S.initialize_matrices()
            # if var(self.nat_y) is > 0, then self.pen_val = self.n * log(var(self.nat_y)) + 1e4
            # else self.pen_val = self.n * var(self.nat_y) + 1e4
            assert S.pen_val == nat_X.shape[0] * log(var(S.nat_y)) + 1e4
            assert S.Psi.shape == (n, n)

    Returns:
        None
    """
    logger.debug("In initialize_matrices(): self.n_p: %s", self.n_p)
    self.p = ones(self.n_p) * self.p_val
    logger.debug("In initialize_matrices(): self.p: %s", self.p)
    # if var(self.nat_y) is > 0, then self.pen_val = self.n * log(var(self.nat_y)) + 1e4
    # else self.pen_val = self.n * var(self.nat_y) + 1e4
    logger.debug("In initialize_matrices(): var(self.nat_y): %s", var(self.nat_y))
    logger.debug("In initialize_matrices(): self.n: %s", self.n)
    if var(self.nat_y) > 0:
        self.pen_val = self.n * log(var(self.nat_y)) + 1e4
    else:
        self.pen_val = self.n * var(self.nat_y) + 1e4
    logger.debug("In initialize_matrices(): self.pen_val: %s", self.pen_val)
    self.negLnLike = None
    logger.debug("In initialize_matrices(): self.k: %s", self.k)
    logger.debug("In initialize_matrices(): self.seed: %s", self.seed)
    self.gen = spacefilling(k=self.k, seed=self.seed)
    logger.debug("In initialize_matrices(): self.gen: %s", self.gen)
    self.LnDetPsi = None
    self.Psi = zeros((self.n, self.n), dtype=float64)
    logger.debug("In initialize_matrices(): self.Psi: %s", self.Psi)
    self.psi = zeros((self.n, 1))
    logger.debug("In initialize_matrices(): self.psi: %s", self.psi)
    self.one = ones(self.n)
    logger.debug("In initialize_matrices(): self.one: %s", self.one)
    self.mu = None
    self.U = None
    self.SigmaSqr = None
    self.Lambda = None

initialize_variables(nat_X, nat_y)

Initialize variables for the class instance.

This method takes in the independent and dependent variable data as input and initializes the class instance variables. It creates deep copies of the input data and stores them in the instance variables nat_X and nat_y. It also calculates the number of observations n and the number of independent variables k from the shape of nat_X. Finally, it creates empty arrays with the same shape as nat_X and nat_y and stores them in the instance variables cod_X and cod_y.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
nat_X	ndarray	The independent variable data.	required
nat_y	ndarray	The dependent variable data.	required

Returns:

Type	Description
None	None

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[1, 2], [3, 4]])
    nat_y = np.array([1, 2])
    S = Kriging()
    S.initialize_variables(nat_X, nat_y)
    print(f"S.nat_X: {S.nat_X}")
    print(f"S.nat_y: {S.nat_y}")
    S.nat_X: [[1 2]
              [3 4]]
    S.nat_y: [1 2]

Source code in spotPython/build/kriging.py

def initialize_variables(self, nat_X: np.ndarray, nat_y: np.ndarray) -> None:
    """
    Initialize variables for the class instance.

    This method takes in the independent and dependent variable data as input
    and initializes the class instance variables.
    It creates deep copies of the input data and stores them in the
    instance variables `nat_X` and `nat_y`.
    It also calculates the number of observations `n` and
    the number of independent variables `k` from the shape of `nat_X`.
    Finally, it creates empty arrays with the same shape as `nat_X`
    and `nat_y` and stores them in the instance variables `cod_X` and `cod_y`.

    Args:
        self (object): The Kriging object.
        nat_X (np.ndarray): The independent variable data.
        nat_y (np.ndarray): The dependent variable data.

    Returns:
        None

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[1, 2], [3, 4]])
            nat_y = np.array([1, 2])
            S = Kriging()
            S.initialize_variables(nat_X, nat_y)
            print(f"S.nat_X: {S.nat_X}")
            print(f"S.nat_y: {S.nat_y}")
            S.nat_X: [[1 2]
                      [3 4]]
            S.nat_y: [1 2]

    """
    self.nat_X = copy.deepcopy(nat_X)
    self.nat_y = copy.deepcopy(nat_y)
    self.n = self.nat_X.shape[0]
    self.k = self.nat_X.shape[1]

    self.min_X = min(self.nat_X, axis=0)
    self.max_X = max(self.nat_X, axis=0)

    Z = aggregate_mean_var(X=self.nat_X, y=self.nat_y)
    # aggregated y values:
    mu = Z[1]
    self.aggregated_mean_y = np.copy(mu)
    logger.debug("In initialize_variables(): self.nat_X: %s", self.nat_X)
    logger.debug("In initialize_variables(): self.nat_y: %s", self.nat_y)
    logger.debug("In initialize_variables(): self.aggregated_mean_y: %s", self.aggregated_mean_y)
    logger.debug("In initialize_variables(): self.min_X: %s", self.min_X)
    logger.debug("In initialize_variables(): self.max_X: %s", self.max_X)
    logger.debug("In initialize_variables(): self.n: %s", self.n)
    logger.debug("In initialize_variables(): self.k: %s", self.k)

likelihood()

Calculates the negative of the concentrated log-likelihood.

This method implements equation (2.32) in [Forr08a] to calculate the negative of the concentrated log-likelihood. It also modifies mu, SigmaSqr, LnDetPsi, and negLnLike.

Note

build_Psi and build_U should be called first.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Returns:

Type	Description
None	None

Attributes:

Name	Type	Description
mu	float64	Kriging expected mean value mu.
SigmaSqr	float64	Sigma squared value.
LnDetPsi	float64	Determinant Psi matrix.
negLnLike	float	Negative log likelihood of the surface at the specified hyperparameters.

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[1], [2]])
    nat_y = np.array([5, 10])
    n=2
    p=1
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False, theta_init_zero=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    S.initialize_matrices()
    S.build_Psi()
    S.build_U()
    S.likelihood()
    # assert S.mu is close to 7.5 with a tolerance of 1e-6
    assert np.allclose(S.mu, 7.5, atol=1e-6)
    E = np.exp(1)
    sigma2 = E/(E**2 -1) * (25/4 + 25/4*E)
    # asssert S.SigmaSqr is close to sigma2 with a tolerance of 1e-6
    assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
    print(f"S.LnDetPsi:{S.LnDetPsi}")
    print(f"S.self.negLnLike:{S.negLnLike}")

Source code in spotPython/build/kriging.py

def likelihood(self) -> None:
    """
    Calculates the negative of the concentrated log-likelihood.

    This method implements equation (2.32) in [Forr08a] to calculate
    the negative of the concentrated log-likelihood. It also modifies `mu`,
    `SigmaSqr`, `LnDetPsi`, and `negLnLike`.

    Note:
        `build_Psi` and `build_U` should be called first.

    Args:
        self (object):
            The Kriging object.

    Returns:
        None

    Attributes:
        mu (np.float64): Kriging expected mean value mu.
        SigmaSqr (np.float64): Sigma squared value.
        LnDetPsi (np.float64): Determinant Psi matrix.
        negLnLike (float): Negative log likelihood of the surface at the specified hyperparameters.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[1], [2]])
            nat_y = np.array([5, 10])
            n=2
            p=1
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=False, theta_init_zero=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            S.initialize_matrices()
            S.build_Psi()
            S.build_U()
            S.likelihood()
            # assert S.mu is close to 7.5 with a tolerance of 1e-6
            assert np.allclose(S.mu, 7.5, atol=1e-6)
            E = np.exp(1)
            sigma2 = E/(E**2 -1) * (25/4 + 25/4*E)
            # asssert S.SigmaSqr is close to sigma2 with a tolerance of 1e-6
            assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
            print(f"S.LnDetPsi:{S.LnDetPsi}")
            print(f"S.self.negLnLike:{S.negLnLike}")
    """
    # (2.20) in [Forr08a]:
    U_T_inv_one = solve(self.U.T, self.one)
    U_T_inv_cod_y = solve(self.U.T, self.nat_y)
    mu = self.one.T.dot(solve(self.U, U_T_inv_cod_y)) / self.one.T.dot(solve(self.U, U_T_inv_one))
    self.mu = mu
    # (2.31) in [Forr08a]
    cod_y_minus_mu = self.nat_y - self.one.dot(self.mu)
    self.SigmaSqr = cod_y_minus_mu.T.dot(solve(self.U, solve(self.U.T, cod_y_minus_mu))) / self.n
    # (2.32) in [Forr08a]
    self.LnDetPsi = 2.0 * sum(log(abs(diag(self.U))))
    self.negLnLike = -1.0 * (-(self.n / 2.0) * log(self.SigmaSqr) - 0.5 * self.LnDetPsi)

optimize_model()

Optimize the model using the specified model_optimizer.

This method uses the specified model_optimizer to optimize the likelihood function (fun_likelihood) with respect to the model parameters. The optimization is performed within the bounds specified by the attribute de_bounds. The result of the optimization is returned as a list or tuple of optimized parameter values.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[1, 2], [3, 4]])
    nat_y = np.array([1, 2])
    n=2
    p=2
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    S.initialize_matrices()
    S.set_de_bounds()
    new_theta_p_Lambda = S.optimize_model()
    print(new_theta_p_Lambda)

Returns:

Type	Description
Union[List[float], Tuple[float]]	result[“x”] (Union[List[float], Tuple[float]]): A list or tuple of optimized parameter values.

Source code in spotPython/build/kriging.py

def optimize_model(self) -> Union[List[float], Tuple[float]]:
    """
    Optimize the model using the specified model_optimizer.

    This method uses the specified model_optimizer to optimize the
    likelihood function (`fun_likelihood`) with respect to the model parameters.
    The optimization is performed within the bounds specified by the attribute
    `de_bounds`.
    The result of the optimization is returned as a list or tuple of optimized parameter values.

    Args:
        self (object): The Kriging object.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[1, 2], [3, 4]])
            nat_y = np.array([1, 2])
            n=2
            p=2
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            S.initialize_matrices()
            S.set_de_bounds()
            new_theta_p_Lambda = S.optimize_model()
            print(new_theta_p_Lambda)

    Returns:
        result["x"] (Union[List[float], Tuple[float]]):
            A list or tuple of optimized parameter values.
    """
    logger.debug("In optimize_model(): self.de_bounds passed to optimizer: %s", self.de_bounds)
    if self.model_optimizer.__name__ == 'dual_annealing':
        result = self.model_optimizer(func=self.fun_likelihood,
                                      bounds=self.de_bounds)
    elif self.model_optimizer.__name__ == 'differential_evolution':
        result = self.model_optimizer(func=self.fun_likelihood,
                                      bounds=self.de_bounds,
                                      maxiter=self.model_fun_evals,
                                      seed=self.seed)
    elif self.model_optimizer.__name__ == 'direct':
        result = self.model_optimizer(func=self.fun_likelihood,
                                      bounds=self.de_bounds,
                                      # maxfun=self.model_fun_evals,
                                      eps=1e-2)
    elif self.model_optimizer.__name__ == 'shgo':
        result = self.model_optimizer(func=self.fun_likelihood,
                                      bounds=self.de_bounds)
    elif self.model_optimizer.__name__ == 'basinhopping':
        result = self.model_optimizer(func=self.fun_likelihood,
                                      x0=mean(self.de_bounds, axis=1))
    else:
        result = self.model_optimizer(func=self.fun_likelihood, bounds=self.de_bounds)
    logger.debug("In optimize_model(): result: %s", result)
    logger.debug('In optimize_model(): returned result["x"]: %s', result["x"])
    return result["x"]

plot(show=True)

This function plots 1D and 2D surrogates.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
show	bool	If True, the plots are displayed. If False, plt.show() should be called outside this function.	True

Returns:

Type	Description
None	None

Note

• This method provides only a basic plot. For more advanced plots, use the plot_contour() method of the Spot class.

Examples:

>>> import numpy as np
    from spotPython.fun.objectivefunctions import analytical
    from spotPython.spot import spot
    # 1-dimensional example
    fun = analytical().fun_sphere
    lower = np.array([-1])
    upper = np.array([1])
    design_control={"init_size": 10}
    S = spot.Spot(fun=fun,
                noise=False,
                lower = lower,
                upper= upper,
                design_control=design_control,)
    S.initialize_design()
    S.update_stats()
    S.fit_surrogate()
    S.surrogate.plot()
    # 2-dimensional example
    fun = analytical().fun_sphere
    lower = np.array([-1, -1])
    upper = np.array([1, 1])
    design_control={"init_size": 10}
    S = spot.Spot(fun=fun,
                noise=False,
                lower = lower,
                upper= upper,
                design_control=design_control,)
    S.initialize_design()
    S.update_stats()
    S.fit_surrogate()
    S.surrogate.plot()

Source code in spotPython/build/kriging.py

def plot(self, show: Optional[bool] = True) -> None:
    """
    This function plots 1D and 2D surrogates.

    Args:
        self (object):
            The Kriging object.
        show (bool):
            If `True`, the plots are displayed.
            If `False`, `plt.show()` should be called outside this function.

    Returns:
        None

    Note:
        * This method provides only a basic plot. For more advanced plots,
            use the `plot_contour()` method of the `Spot` class.

    Examples:
        >>> import numpy as np
            from spotPython.fun.objectivefunctions import analytical
            from spotPython.spot import spot
            # 1-dimensional example
            fun = analytical().fun_sphere
            lower = np.array([-1])
            upper = np.array([1])
            design_control={"init_size": 10}
            S = spot.Spot(fun=fun,
                        noise=False,
                        lower = lower,
                        upper= upper,
                        design_control=design_control,)
            S.initialize_design()
            S.update_stats()
            S.fit_surrogate()
            S.surrogate.plot()
            # 2-dimensional example
            fun = analytical().fun_sphere
            lower = np.array([-1, -1])
            upper = np.array([1, 1])
            design_control={"init_size": 10}
            S = spot.Spot(fun=fun,
                        noise=False,
                        lower = lower,
                        upper= upper,
                        design_control=design_control,)
            S.initialize_design()
            S.update_stats()
            S.fit_surrogate()
            S.surrogate.plot()
    """
    if self.k == 1:
        # TODO: Improve plot (add conf. interval etc.)
        fig = pylab.figure(figsize=(9, 6))
        n_grid = 100
        x = linspace(
            self.min_X[0], self.max_X[0], num=n_grid
        )
        y = self.predict(x)
        plt.figure()
        plt.plot(x, y, "k")
        if show:
            plt.show()

    if self.k == 2:
        fig = pylab.figure(figsize=(9, 6))
        n_grid = 100
        x = linspace(
            self.min_X[0], self.max_X[0], num=n_grid
        )
        y = linspace(
            self.min_X[1], self.max_X[1], num=n_grid
        )
        X, Y = meshgrid(x, y)
        # Predict based on the optimized results
        zz = array(
            [self.predict(array([x, y]), return_val="all") for x, y in zip(ravel(X), ravel(Y))]
        )
        zs = zz[:, 0, :]
        zse = zz[:, 1, :]
        Z = zs.reshape(X.shape)
        Ze = zse.reshape(X.shape)

        nat_point_X = self.nat_X[:, 0]
        nat_point_Y = self.nat_X[:, 1]
        contour_levels = 30
        ax = fig.add_subplot(224)
        # plot predicted values:
        pylab.contourf(X, Y, Ze, contour_levels, cmap="jet")
        pylab.title("Error")
        pylab.colorbar()
        # plot observed points:
        pylab.plot(nat_point_X, nat_point_Y, "ow")
        #
        ax = fig.add_subplot(223)
        # plot predicted values:
        plt.contourf(X, Y, Z, contour_levels, zorder=1, cmap="jet")
        plt.title("Surrogate")
        # plot observed points:
        pylab.plot(nat_point_X, nat_point_Y, "ow", zorder=3)
        pylab.colorbar()
        #
        ax = fig.add_subplot(221, projection="3d")
        ax.plot_surface(X, Y, Z, rstride=3, cstride=3, alpha=0.9, cmap="jet")
        #
        ax = fig.add_subplot(222, projection="3d")
        ax.plot_surface(X, Y, Ze, rstride=3, cstride=3, alpha=0.9, cmap="jet")
        #
        pylab.show()

predict(nat_X, return_val='y')

This function returns the prediction (in natural units) of the surrogate at the natural coordinates of X.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
nat_X	ndarray	Design variable to evaluate in natural units.	required
return_val	str	whether y, s, neg. ei (negative expected improvement), or all three values are returned. Default is (for compatibility with sklearn) “y”. To return s, select “s”, to return neg. ei, select “ei”. To return the tuple (y, s, ei), select “all”.	'y'

Returns:

Name	Type	Description
float	Union[float, Tuple[float, float]]	The predicted value in natural units if return_val is “y”.
float	Union[float, Tuple[float, float]]	predicted error if return_val is “s”.
float	Union[float, Tuple[float, float]]	expected improvement if return_val is “ei”.
	Union[float, Tuple[float, float]]	Tuple[float, float, float]: The predicted value in natural units, predicted error and expected improvement if return_val is “all”.

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    import matplotlib.pyplot as plt
    from numpy import linspace, arange
    rng = np.random.RandomState(1)
    X = linspace(start=0, stop=10, num=1_0).reshape(-1, 1)
    y = np.squeeze(X * np.sin(X))
    training_indices = rng.choice(arange(y.size), size=6, replace=False)
    X_train, y_train = X[training_indices], y[training_indices]
    S = Kriging(name='kriging', seed=124)
    S.fit(X_train, y_train)
    mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
    print(f"mean_prediction: {mean_prediction}")
    print(f"std_prediction: {std_prediction}")
    print(f"s_ei: {s_ei}")
    mean_prediction: [-1.41991225e-08  6.48310037e-01  1.76715565e+00 -6.35226564e-01
                      -4.28585379e+00 -1.22301198e+00  2.49434148e+00  5.61900501e-01
                      -3.04558205e+00 -5.44021104e+00]
    std_prediction: [3.69706811e-04 2.07958787e+00 3.69706810e-04 3.69706807e-04
                    3.69706809e-04 2.07958584e+00 3.69706811e-04 2.60615408e+00
                    2.60837033e+00 3.69706811e-04]
    s_ei: [-0.00000000e+00 -1.02341235e-03 -0.00000000e+00 -0.00000000e+00
           -0.00000000e+00 -1.63799181e-02 -0.00000000e+00 -9.45766290e-03
           -2.53405666e-01 -1.47459347e-04]

Source code in spotPython/build/kriging.py

def predict(self, nat_X: ndarray, return_val: str = "y") -> Union[float,
                                                                  Tuple[float, float]]:
    """
    This function returns the prediction (in natural units) of the surrogate at the natural coordinates of X.

    Args:
        self (object):
            The Kriging object.
        nat_X (ndarray):
            Design variable to evaluate in natural units.
        return_val (str):
            whether `y`, `s`, neg. `ei` (negative expected improvement),
            or all three values are returned.
            Default is (for compatibility with sklearn) "y". To return `s`, select "s",
            to return neg. `ei`, select "ei".
            To return the tuple `(y, s, ei)`, select "all".

    Returns:
        float:
            The predicted value in natural units if return_val is "y".
        float:
            predicted error if return_val is "s".
        float:
            expected improvement if return_val is "ei".
        Tuple[float, float, float]:
            The predicted value in natural units, predicted error
            and expected improvement if return_val is "all".

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            import matplotlib.pyplot as plt
            from numpy import linspace, arange
            rng = np.random.RandomState(1)
            X = linspace(start=0, stop=10, num=1_0).reshape(-1, 1)
            y = np.squeeze(X * np.sin(X))
            training_indices = rng.choice(arange(y.size), size=6, replace=False)
            X_train, y_train = X[training_indices], y[training_indices]
            S = Kriging(name='kriging', seed=124)
            S.fit(X_train, y_train)
            mean_prediction, std_prediction, s_ei = S.predict(X, return_val="all")
            print(f"mean_prediction: {mean_prediction}")
            print(f"std_prediction: {std_prediction}")
            print(f"s_ei: {s_ei}")
            mean_prediction: [-1.41991225e-08  6.48310037e-01  1.76715565e+00 -6.35226564e-01
                              -4.28585379e+00 -1.22301198e+00  2.49434148e+00  5.61900501e-01
                              -3.04558205e+00 -5.44021104e+00]
            std_prediction: [3.69706811e-04 2.07958787e+00 3.69706810e-04 3.69706807e-04
                            3.69706809e-04 2.07958584e+00 3.69706811e-04 2.60615408e+00
                            2.60837033e+00 3.69706811e-04]
            s_ei: [-0.00000000e+00 -1.02341235e-03 -0.00000000e+00 -0.00000000e+00
                   -0.00000000e+00 -1.63799181e-02 -0.00000000e+00 -9.45766290e-03
                   -2.53405666e-01 -1.47459347e-04]

    """
    # Check for the shape and the type of the Input
    if isinstance(nat_X, ndarray):
        try:
            X = nat_X.reshape(-1, self.nat_X.shape[1])
            X = repair_non_numeric(X, self.var_type)
        except Exception:
            raise TypeError("13.1: Input to predict was not convertible to the size of X")
    else:
        raise TypeError(f"type of the given input is an {type(nat_X)} instead of an ndarray")
    n = X.shape[0]
    y = empty(n, dtype=float)
    s = empty(n, dtype=float)
    ei = empty(n, dtype=float)
    for i in range(n):
        x = X[i, :]
        y[i], s[i], ei[i] = self.predict_coded(x)
    if return_val == "y":
        return y
    elif return_val == "s":
        return s
    elif return_val == "ei":
        return -1.0 * ei
    else:
        return y, s, -1.0 * ei

predict_coded(cod_x)

Kriging prediction of one point in the coded units as described in (2.20) in [Forr08a]. The error is returned as well.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
cod_x	ndarray	Point in coded units to make prediction at.	required

Returns:

Name	Type	Description
f	float	Predicted value in coded units.
SSqr	float	Predicted error.
EI	float	Expected improvement.

Note

self.mu and self.SigmaSqr are computed in likelihood, not here. See also [Forr08a, p.60].

Source code in spotPython/build/kriging.py

def predict_coded(self, cod_x: np.ndarray) -> Tuple[float, float, float]:
    """
    Kriging prediction of one point in the coded units as described in (2.20) in [Forr08a].
    The error is returned as well.

    Args:
        self (object):
            The Kriging object.
        cod_x (np.ndarray):
            Point in coded units to make prediction at.

    Returns:
        f (float): Predicted value in coded units.
        SSqr (float): Predicted error.
        EI (float): Expected improvement.

    Note:
        `self.mu` and `self.SigmaSqr` are computed in `likelihood`, not here.
        See also [Forr08a, p.60].
    """
    self.build_psi_vec(cod_x)
    U_T_inv = solve(self.U.T, self.nat_y - self.one.dot(self.mu))
    f = self.mu + self.psi.T.dot(solve(self.U, U_T_inv))
    if self.noise:
        Lambda = self.Lambda
    else:
        Lambda = 0.0
    # Error in [Forr08a, p.87]:
    SSqr = self.SigmaSqr * (1 + Lambda - self.psi.T.dot(solve(self.U, solve(self.U.T, self.psi))))
    SSqr = power(abs(SSqr[0]), 0.5)[0]
    EI = self.exp_imp(y0=f[0], s0=SSqr)
    return f[0], SSqr, EI

set_de_bounds()

Determine search bounds for model_optimizer, e.g., differential evolution.

This method sets the attribute de_bounds of the object to a list of lists, where each inner list represents the lower and upper bounds for a parameter being optimized. The number of inner lists is determined by the number of parameters being optimized (n_theta and n_p), as well as whether noise is being considered (noise).

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Examples:

>>> from spotPython.build.kriging import Kriging
    S = Kriging(name='kriging', seed=124)
    S.set_de_bounds()
    print(S.de_bounds)

Returns:

Type	Description
None	None

Source code in spotPython/build/kriging.py

def set_de_bounds(self) -> None:
    """
    Determine search bounds for model_optimizer, e.g., differential evolution.

    This method sets the attribute `de_bounds` of the object to a list of lists,
    where each inner list represents the lower and upper bounds for a parameter
    being optimized. The number of inner lists is determined by the number of
    parameters being optimized (`n_theta` and `n_p`), as well as whether noise is
    being considered (`noise`).

    Args:
        self (object): The Kriging object.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            S = Kriging(name='kriging', seed=124)
            S.set_de_bounds()
            print(S.de_bounds)

    Returns:
        None
    """
    logger.debug("In set_de_bounds(): self.min_theta: %s", self.min_theta)
    logger.debug("In set_de_bounds(): self.max_theta: %s", self.max_theta)
    logger.debug("In set_de_bounds(): self.n_theta: %s", self.n_theta)
    logger.debug("In set_de_bounds(): self.optim_p: %s", self.optim_p)
    logger.debug("In set_de_bounds(): self.min_p: %s", self.min_p)
    logger.debug("In set_de_bounds(): self.max_p: %s", self.max_p)
    logger.debug("In set_de_bounds(): self.n_p: %s", self.n_p)
    logger.debug("In set_de_bounds(): self.noise: %s", self.noise)
    logger.debug("In set_de_bounds(): self.min_Lambda: %s", self.min_Lambda)
    logger.debug("In set_de_bounds(): self.max_Lambda: %s", self.max_Lambda)

    de_bounds = [[self.min_theta, self.max_theta] for _ in range(self.n_theta)]
    if self.optim_p:
        de_bounds += [[self.min_p, self.max_p] for _ in range(self.n_p)]
        if self.noise:
            de_bounds.append([self.min_Lambda, self.max_Lambda])
    else:
        if self.noise:
            de_bounds.append([self.min_Lambda, self.max_Lambda])
    self.de_bounds = de_bounds
    logger.debug("In set_de_bounds(): self.de_bounds: %s", self.de_bounds)

set_theta_values()

Set the theta values for the class instance.

This method sets the theta values for the class instance based on the n_theta and k attributes. If n_theta is greater than k, n_theta is set to k and a warning is logged. The method then initializes the theta attribute as a list of zeros with length n_theta. The x0_theta attribute is also initialized as a list of ones with length n_theta, multiplied by n / (100 * k).

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Returns: None

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    from numpy import array
    nat_X = np.array([[1, 2], [3, 4]])
    nat_y = np.array([1, 2])
    n=2
    p=2
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    assert S.theta.all() == array([0., 0.]).all()

Source code in spotPython/build/kriging.py

def set_theta_values(self) -> None:
    """
    Set the theta values for the class instance.

    This method sets the theta values for the class instance based
    on the `n_theta` and `k` attributes. If `n_theta` is greater than
    `k`, `n_theta` is set to `k` and a warning is logged.
    The method then initializes the `theta` attribute as a list
    of zeros with length `n_theta`.
    The `x0_theta` attribute is also initialized as a list of ones
    with length `n_theta`, multiplied by `n / (100 * k)`.

    Args:
        self (object): The Kriging object.
    Returns:
        None

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            from numpy import array
            nat_X = np.array([[1, 2], [3, 4]])
            nat_y = np.array([1, 2])
            n=2
            p=2
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            assert S.theta.all() == array([0., 0.]).all()
    """
    logger.debug("In set_theta_values(): self.k: %s", self.k)
    logger.debug("In set_theta_values(): self.n_theta: %s", self.n_theta)
    if ((self.n_theta > 1) or (self.n_theta > self.k)) and (self.n_theta != self.k):
        self.n_theta = self.k
        logger.warning("Too few theta values or more theta values than dimensions. `n_theta` set to `k`.")
        logger.debug("In set_theta_values(): self.n_theta: %s", self.n_theta)
    if self.theta_init_zero:
        self.theta: List[float] = zeros(self.n_theta)
        logger.debug("In set_theta_values(): self.theta: %s", self.theta)
    else:
        logger.debug("In set_theta_values(): self.n: %s", self.n)
        self.theta: List[float] = ones((self.n_theta,)) * self.n / (100 * self.k)
        logger.debug("In set_theta_values(): self.theta: %s", self.theta)

set_variable_types()

Set the variable types for the class instance.

This method sets the variable types for the class instance based on the var_type attribute. If the length of var_type is less than k, all variable types are forced to ‘num’ and a warning is logged. The method then creates Boolean masks for each variable type (‘num’, ‘factor’, ‘int’, ‘ordered’) using numpy arrays, e.g., num_mask = array([ True, True]) if two numerical variables are present.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Examples:

>>> from spotPython.build.kriging import Kriging
    nat_X = np.array([[1, 2], [3, 4]])
    nat_y = np.array([1, 2])
    n=2
    p=2
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    assert S.var_type == ['num', 'num']
    assert S.var_type == ['num', 'num']
    assert S.num_mask.all() == True
    assert S.factor_mask.all() == False
    assert S.int_mask.all() == False
    assert S.ordered_mask.all() == True

Returns:

Type	Description
None	None

Source code in spotPython/build/kriging.py

def set_variable_types(self) -> None:
    """
    Set the variable types for the class instance.

    This method sets the variable types for the class instance based
    on the `var_type` attribute. If the length of `var_type` is less
    than `k`, all variable types are forced to 'num' and a warning is logged.
    The method then creates Boolean masks for each variable
    type ('num', 'factor', 'int', 'ordered') using numpy arrays, e.g.,
    `num_mask = array([ True,  True])` if two numerical variables are present.

    Args:
        self (object): The Kriging object.

    Examples:
        >>> from spotPython.build.kriging import Kriging
            nat_X = np.array([[1, 2], [3, 4]])
            nat_y = np.array([1, 2])
            n=2
            p=2
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            assert S.var_type == ['num', 'num']
            assert S.var_type == ['num', 'num']
            assert S.num_mask.all() == True
            assert S.factor_mask.all() == False
            assert S.int_mask.all() == False
            assert S.ordered_mask.all() == True

    Returns:
        None
    """
    logger.debug("In set_variable_types(): self.k: %s", self.k)
    logger.debug("In set_variable_types(): self.var_type: %s", self.var_type)
    # assume all variable types are "num" if "num" is
    # specified once:
    if len(self.var_type) < self.k:
        self.var_type = self.var_type * self.k
        logger.warning("In set_variable_types(): All variable types forced to 'num'.")
        logger.debug("In set_variable_types(): self.var_type: %s", self.var_type)
    self.num_mask = np.array(list(map(lambda x: x == "num", self.var_type)))
    self.factor_mask = np.array(list(map(lambda x: x == "factor", self.var_type)))
    self.int_mask = np.array(list(map(lambda x: x == "int", self.var_type)))
    self.ordered_mask = np.array(list(map(lambda x: x == "int" or x == "num" or x == "float", self.var_type)))
    logger.debug("In set_variable_types(): self.num_mask: %s", self.num_mask)
    logger.debug("In set_variable_types(): self.factor_mask: %s", self.factor_mask)
    logger.debug("In set_variable_types(): self.int_mask: %s", self.int_mask)
    logger.debug("In set_variable_types(): self.ordered_mask: %s", self.ordered_mask)

update_log()

Update the log with the current values of negLnLike, theta, p, and Lambda.

This method appends the current values of negLnLike, theta, p (if optim_p is True), and Lambda (if noise is True) to their respective lists in the log dictionary. It also updates the log_length attribute with the current length of the negLnLike list in the log.

If spot_writer is not None, this method also writes the current values of negLnLike, theta, p (if optim_p is True), and Lambda (if noise is True) to the spot_writer object.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required

Returns:

Type	Description
None	None

Examples:

>>> from spotPython.build.kriging import Kriging
    import numpy as np
    nat_X = np.array([[1, 2], [3, 4]])
    nat_y = np.array([1, 2])
    n=2
    p=2
    S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
    S.initialize_variables(nat_X, nat_y)
    S.set_variable_types()
    S.set_theta_values()
    S.initialize_matrices()
    S.set_de_bounds()
    new_theta_p_Lambda = S.optimize_model()
    S.update_log()
    print(S.log)
    {'negLnLike': array([-1.38629436]),
     'theta': array([-1.14525993,  1.6123372 ]),
      'p': array([1.84444406, 1.74590865]),
      'Lambda': array([0.44268472])}

Source code in spotPython/build/kriging.py

def update_log(self) -> None:
    """
    Update the log with the current values of negLnLike, theta, p, and Lambda.

    This method appends the current values of negLnLike, theta, p (if optim_p is True),
    and Lambda (if noise is True)
    to their respective lists in the log dictionary.
    It also updates the log_length attribute with the current length
    of the negLnLike list in the log.

    If spot_writer is not None, this method also writes the current values of
    negLnLike, theta, p (if optim_p is True),
    and Lambda (if noise is True) to the spot_writer object.

    Args:
        self (object): The Kriging object.

    Returns:
        None

    Examples:
        >>> from spotPython.build.kriging import Kriging
            import numpy as np
            nat_X = np.array([[1, 2], [3, 4]])
            nat_y = np.array([1, 2])
            n=2
            p=2
            S=Kriging(name='kriging', seed=124, n_theta=n, n_p=p, optim_p=True, noise=True)
            S.initialize_variables(nat_X, nat_y)
            S.set_variable_types()
            S.set_theta_values()
            S.initialize_matrices()
            S.set_de_bounds()
            new_theta_p_Lambda = S.optimize_model()
            S.update_log()
            print(S.log)
            {'negLnLike': array([-1.38629436]),
             'theta': array([-1.14525993,  1.6123372 ]),
              'p': array([1.84444406, 1.74590865]),
              'Lambda': array([0.44268472])}

    """
    self.log["negLnLike"] = append(self.log["negLnLike"], self.negLnLike)
    self.log["theta"] = append(self.log["theta"], self.theta)
    if self.optim_p:
        self.log["p"] = append(self.log["p"], self.p)
    if self.noise:
        self.log["Lambda"] = append(self.log["Lambda"], self.Lambda)
    # get the length of the log
    self.log_length = len(self.log["negLnLike"])
    if self.spot_writer is not None:
        writer = self.spot_writer
        negLnLike = self.negLnLike.copy()
        writer.add_scalar("spot_negLnLike", negLnLike, self.counter+self.log_length)
        # add the self.n_theta theta values to the writer with one key "theta",
        # i.e, the same key for all theta values
        theta = self.theta.copy()
        writer.add_scalars("spot_theta", {f"theta_{i}": theta[i] for i in range(self.n_theta)},
                           self.counter+self.log_length)
        if self.noise:
            Lambda = self.Lambda.copy()
            writer.add_scalar("spot_Lambda", Lambda, self.counter+self.log_length)
        if self.optim_p:
            p = self.p.copy()
            writer.add_scalars("spot_p", {f"p_{i}": p[i] for i in range(self.n_p)}, self.counter+self.log_length)
        writer.flush()

weighted_exp_imp(cod_x, w)

Weighted expected improvement.

Parameters:

Name	Type	Description	Default
self	object	The Kriging object.	required
cod_x	ndarray	A coded design vector.	required
w	float	Weight.	required

Returns:

Name	Type	Description
EI	float	Weighted expected improvement.

References

[Sobester et al. 2005].

Source code in spotPython/build/kriging.py

def weighted_exp_imp(self, cod_x: np.ndarray, w: float) -> float:
    """
    Weighted expected improvement.

    Args:
        self (object): The Kriging object.
        cod_x (np.ndarray): A coded design vector.
        w (float): Weight.

    Returns:
        EI (float): Weighted expected improvement.

    References:
        [Sobester et al. 2005].
    """
    y0, s0 = self.predict_coded(cod_x)
    y_min = min(self.nat_y)
    if s0 <= 0.0:
        EI = 0.0
    else:
        y_min_y0 = y_min - y0
        EI_one = w * (
                y_min_y0
                * (0.5 + 0.5 * erf((1.0 / sqrt(2.0)) * (y_min_y0 / s0)))
        )
        EI_two = (
                (1.0 - w)
                * (s0 * (1.0 / sqrt(2.0 * pi)))
                * (exp(-(1.0 / 2.0) * ((y_min_y0) ** 2.0 / s0 ** 2.0)))
        )
        EI = EI_one + EI_two
    return EI