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)
                [[-3.0, 2.0]]

        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.
        Returns:
            None

        Examples:
            >>> import numpy as np
                from spotpython.build.kriging import Kriging
                import logging
                logging.basicConfig(level=logging.DEBUG)
                # Define the number of theta and p parameters
                num_theta = 2
                num_p = 3
                # Initialize the Kriging model
                kriging_model = Kriging(
                    name='kriging',
                    seed=124,
                    n_theta=num_theta,
                    n_p=num_p,
                    optim_p=True,
                    noise=True
                )
                # Create bounds array
                bounds_array = np.array([1, 2, 3, 4, 5, 6])
                # Extract parameters from given bounds
                kriging_model.extract_from_bounds(new_theta_p_Lambda=bounds_array)
                # Assertions to check if parameters are correctly extracted
                assert np.array_equal(kriging_model.theta,
                    [1, 2]), f"Expected theta to be [1, 2] but got {kriging_model.theta}"
                assert np.array_equal(kriging_model.p,
                    [3, 4, 5]), f"Expected p to be [3, 4, 5] but got {kriging_model.p}"
                assert kriging_model.Lambda == 6, f"Expected Lambda to be 6 but got {kriging_model.Lambda}"
                print("All assertions passed!")
        """
        logger.debug("Extracting parameters from: %s", new_theta_p_Lambda)

        # Validate array length
        expected_length = self.n_theta
        if self.optim_p:
            expected_length += self.n_p
        if self.noise:
            expected_length += 1

        if len(new_theta_p_Lambda) < expected_length:
            logger.error("Input array is too short. Expected at least %d elements, got %d.",
                         expected_length, len(new_theta_p_Lambda))
            raise ValueError(f"Input array must have at least {expected_length} elements.")

        # Extract theta
        self.theta = new_theta_p_Lambda[:self.n_theta]
        logger.debug("Extracted theta: %s", self.theta)

        if self.optim_p:
            # Extract p if optim_p is True
            self.p = new_theta_p_Lambda[self.n_theta:self.n_theta + self.n_p]
            logger.debug("Extracted p: %s", self.p)

        if self.noise:
            # Extract Lambda
            lambda_index = self.n_theta + (self.n_p if self.optim_p else 0)
            self.Lambda = new_theta_p_Lambda[lambda_index]
            logger.debug("Extracted 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)
                [0.12167915 1.49467909 1.82808259 1.69648798 0.79564346]

        Returns:
            result["x"] (Union[List[float], Tuple[float]]):
                A list or tuple of optimized parameter values.
        """
        logger.debug("Entering optimize_model.")
        if not callable(self.model_optimizer):
            logger.error("model_optimizer is not callable.")
            raise ValueError("model_optimizer must be a callable function or method.")

        optimizer_strategies: Dict[str, Dict] = {
            'dual_annealing': {'func': self.fun_likelihood, 'bounds': self.de_bounds},
            'differential_evolution': {
                'func': self.fun_likelihood,
                'bounds': self.de_bounds,
                'maxiter': self.model_fun_evals,
                'seed': self.seed
            },
            'direct': {
                'func': self.fun_likelihood,
                'bounds': self.de_bounds,
                'eps': 1e-2
            },
            'shgo': {'func': self.fun_likelihood, 'bounds': self.de_bounds},
            'basinhopping': {'func': self.fun_likelihood, 'x0': np.mean(self.de_bounds, axis=1)}
        }

        optimizer_name = self.model_optimizer.__name__
        logger.debug("Optimizer selected: %s", optimizer_name)

        if optimizer_name not in optimizer_strategies:
            logger.info("Using default options for optimizer: %s", optimizer_name)
            optimizer_args = {'func': self.fun_likelihood, 'bounds': self.de_bounds}
        else:
            optimizer_args = optimizer_strategies[optimizer_name]

        logger.debug("Parameters for optimization: %s", optimizer_args)

        try:
            result = self.model_optimizer(**optimizer_args)
        except Exception as e:
            logger.error("Optimization failed due to error: %s", str(e))
            raise

        if "x" not in result:
            logger.error("Optimization result does not contain 'x'. Result: %s", result)
            raise ValueError("The optimization result does not contain the expected 'x' key.")
        logger.debug("Optimization result: %s", result)
        optimized_parameters = list(result["x"])
        logger.debug("Extracted optimized parameters: %s", optimized_parameters)
        return optimized_parameters

    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:
            negLnLike = self.negLnLike.copy()
            self.spot_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()
            self.spot_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()
                self.spot_writer.add_scalar("spot_Lambda", Lambda, self.counter+self.log_length)
            if self.optim_p:
                p = self.p.copy()
                self.spot_writer.add_scalars("spot_p",
                                             {f"p_{i}": p[i] for i in range(self.n_p)}, self.counter+self.log_length)
            self.spot_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 `buildPsi()`.
        3. U matrix via `buildU()`.

        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]

        """
        # Validate input dimensions
        if nat_X.ndim != 2 or nat_y.ndim != 1:
            raise ValueError("nat_X must be a 2D array and nat_y must be a 1D array.")
        if nat_X.shape[0] != nat_y.shape[0]:
            raise ValueError("The number of samples in nat_X and nat_y must be equal.")

        # Initialize instance variables
        self.nat_X = copy.deepcopy(nat_X)
        self.nat_y = copy.deepcopy(nat_y)
        self.n, self.k = self.nat_X.shape

        # Calculate and store min and max of X
        self.min_X = np.min(self.nat_X, axis=0)
        self.max_X = np.max(self.nat_X, axis=0)

        # Calculate the aggregated mean of y
        _, aggregated_mean_y, _ = aggregate_mean_var(X=self.nat_X, y=self.nat_y)
        self.aggregated_mean_y = np.copy(aggregated_mean_y)

        # Logging the initialized variables
        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: %d", self.n)
        logger.debug("In initialize_variables(): self.k: %d", 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)

        # Ensure var_type has appropriate length by defaulting to 'num'
        if len(self.var_type) < self.k:
            self.var_type = ['num'] * self.k  # Corrected to fill with 'num' instead of duplicating
            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)
        # Create masks for each type using numpy vectorized operations
        var_type_array = np.array(self.var_type)
        self.num_mask = (var_type_array == "num")
        self.factor_mask = (var_type_array == "factor")
        self.int_mask = (var_type_array == "int")
        self.ordered_mask = np.isin(var_type_array, ["int", "num", "float"])
        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)

        # Adjust `n_theta` if it exceeds `k`
        if 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 reset to: %s", self.n_theta)

        # Initialize theta values
        if hasattr(self, "theta_init_zero") and self.theta_init_zero:
            self.theta = np.zeros(self.n_theta, dtype=float)
            logger.debug("Theta initialized to zeros: %s", self.theta)
        else:
            logger.debug("In set_theta_values(): self.n: %s", self.n)
            self.theta = np.ones(self.n_theta, dtype=float) * self.n / (100 * self.k)
            logger.debug("Theta initialized based on n and k: %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)

        # Initialize p
        self.p = np.ones(self.n_p) * self.p_val
        logger.debug("In initialize_matrices(): self.p: %s", self.p)

        # Calculate variance of nat_y
        y_variance = var(self.nat_y)
        logger.debug("In initialize_matrices(): var(self.nat_y): %s", y_variance)

        # Set penalty value based on variance
        if y_variance > 0:
            self.pen_val = self.n * log(y_variance) + 1e4
        else:
            self.pen_val = self.n * y_variance + 1e4
        logger.debug("In initialize_matrices(): self.pen_val: %s", self.pen_val)

        # Initialize other attributes
        self.negLnLike = None
        self.LnDetPsi = None
        self.mu = None
        self.U = None
        self.SigmaSqr = None
        self.Lambda = None

        # Initialize generator
        self.gen = SpaceFilling(k=self.k, seed=self.seed)
        logger.debug("In initialize_matrices(): self.gen: %s", self.gen)

        # Initialize matrix Psi and vector psi
        self.Psi = np.zeros((self.n, self.n), dtype=np.float64)
        logger.debug("In initialize_matrices(): self.Psi shape: %s", self.Psi.shape)

        self.psi = np.zeros((self.n, 1), dtype=np.float64)
        logger.debug("In initialize_matrices(): self.psi shape: %s", self.psi.shape)

        # Initialize one
        self.one = np.ones(self.n, dtype=np.float64)
        logger.debug("In initialize_matrices(): self.one: %s", self.one)

    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) using several internal methods for matrix construction
        and likelihood evaluation. It handles potential errors by returning a
        penalty value for non-computable states.

        Args:
            new_theta_p_Lambda (np.ndarray): An array containing `theta`, `p`, and `Lambda` values.

        Returns:
            float: The negative log likelihood or the penalty value if computation fails.

        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.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}")
                S.likelihood()
                S.negLnLike
                    [[0]
                    [1]]
                    [0 1]
                    S.theta: [0.]
                    S.theta: [1.60036366]
                    S.Psi: [[1.00000001e+00 4.96525625e-18]
                    [4.96525625e-18 1.00000001e+00]]
                    S.U:[[1.00000001e+00 4.96525622e-18]
                    [0.00000000e+00 1.00000001e+00]]
                    -1.3862943611198906
        """
        # Extract hyperparameters
        self.extract_from_bounds(new_theta_p_Lambda)
        # Check transformed theta values
        theta_scaled = np.power(10.0, self.theta)
        if self.__is_any__(theta_scaled, 0):
            logger.warning("Failure in fun_likelihood: 10^theta == 0. Setting negLnLike to %s", self.pen_val)
            return self.pen_val
        # Build Psi matrix and check its condition
        self.build_Psi()
        if getattr(self, 'inf_Psi', False) or getattr(self, 'cnd_Psi', float('inf')) > 1e9:
            logger.warning("Failure in fun_likelihood: Psi is ill-conditioned: %s", getattr(self, 'cnd_Psi', 'unknown'))
            logger.warning("Setting negLnLike to: %s", self.pen_val)
            return self.pen_val
        # Build U matrix and handle exceptions
        try:
            self.build_U()
        except Exception as error:
            logger.error("Error in fun_likelihood(). Call to build_U() failed: %s", error)
            logger.error("Setting negLnLike to %.2f.", self.pen_val)
            return self.pen_val

        # Calculate likelihood
        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-like `x`
        is equal to the given value `v`. Converts inputs to numpy arrays as necessary.

        Args:
            x (Union[np.ndarray, Any]): The input array-like object to check.
            v (Any): The value to check for in `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, np.ndarray):
            x = np.array([x])  # Wrap scalar x in an array
        return np.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].

        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.

        Raises:
            LinAlgError: If building Psi fails.

        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: [1.60036366]
                    S.Psi: [[1.00000001e+00 4.96525625e-18]
                    [4.96525625e-18 1.00000001e+00]]
        """
        try:
            n = self.n
            k = self.k
            theta = np.power(10.0, self.theta)

            # Ensure theta has the correct length
            if self.n_theta == 1:
                theta = theta * np.ones(k)

            # Initialize the Psi matrix
            self.Psi = np.zeros((n, n), dtype=np.float64)

            # Calculate the distance matrix using ordered variables
            if self.ordered_mask.any():
                X_ordered = self.nat_X[:, self.ordered_mask]
                D_ordered = squareform(
                    pdist(X_ordered, metric='sqeuclidean', w=theta[self.ordered_mask])
                )
                self.Psi += D_ordered

            # Add the contribution of factor variables to the distance matrix
            if self.factor_mask.any():
                X_factor = self.nat_X[:, self.factor_mask]
                D_factor = squareform(
                    pdist(X_factor, metric=self.metric_factorial, w=theta[self.factor_mask])
                )
                self.Psi += D_factor

            # Calculate correlation from distance
            self.Psi = np.exp(-self.Psi)

            # Adjust diagonal elements for noise or minimum epsilon
            diag_indices = np.diag_indices_from(self.Psi)
            if self.noise:
                self.Psi[diag_indices] += self.Lambda
                logger.debug("Noise level Lambda applied to diagonal: %s", self.Lambda)
            else:
                self.Psi[diag_indices] += self.eps

            # Check for infinite values
            self.inf_Psi = np.isinf(self.Psi).any()

            # Calculate condition number
            self.cnd_Psi = cond(self.Psi)
            logger.debug("Condition number of Psi: %f", self.cnd_Psi)

        except LinAlgError as err:
            logger.error("Building Psi failed. Error: %s, Type: %s", err, type(err))
            raise

    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: [1.60036366]
                    S.Psi: [[1.00000001e+00 4.96525625e-18]
                    [4.96525625e-18 1.00000001e+00]]
                    S.U:[[1.00000001e+00 4.96525622e-18]
                    [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:
        """
        Calculate the negative concentrated log-likelihood.
        Implements equation (2.32) from [Forr08a] to compute the negative of the
        concentrated log-likelihood. Updates `mu`, `SigmaSqr`, `LnDetPsi`, and `negLnLike`.

        Note:
            Requires prior calls to `build_Psi` and `build_U`.

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

        Raises:
            LinAlgError: If matrix operations fail.

        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 np.allclose(S.mu, 7.5, atol=1e-6)
                E = np.exp(1)
                sigma2 = E / (E**2 - 1) * (25/4 + 25/4*E)
                assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
                print(f"S.LnDetPsi:{S.LnDetPsi}")
                print(f"S.negLnLike:{S.negLnLike}")
                    S.LnDetPsi:-0.1454134234019476
                    S.negLnLike:2.2185498738611282
        """
        try:
            # Solving linear equations for needed components
            U_T_inv_one = solve(self.U.T, self.one)
            U_T_inv_nat_y = solve(self.U.T, self.nat_y)
            # Mean calculation: (2.20) in [Forr08a]
            self.mu = (self.one.T @ solve(self.U, U_T_inv_nat_y)) / (self.one.T @ solve(self.U, U_T_inv_one))
            # Residuals
            cod_y_minus_mu = self.nat_y - self.one * self.mu
            # Sigma squared calculation: (2.31) in [Forr08a]
            self.SigmaSqr = (cod_y_minus_mu.T @ solve(self.U, solve(self.U.T, cod_y_minus_mu))) / self.n
            # Log determinant of Psi: (2.32) in [Forr08a]
            self.LnDetPsi = 2.0 * np.sum(np.log(np.abs(np.diag(self.U))))
            # Negative log-likelihood calculation: simplified from (2.32)
            self.negLnLike = 0.5 * (self.n * np.log(self.SigmaSqr) + self.LnDetPsi)
            logger.debug("Likelihood calculated: mu=%s, SigmaSqr=%s, LnDetPsi=%s, negLnLike=%s",
                         self.mu, self.SigmaSqr, self.LnDetPsi, self.negLnLike)
        except LinAlgError as error:
            logger.error("LinAlgError in likelihood calculation: %s", error)
            raise

    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
                from spotpython.utils.init import fun_control_init, design_control_init
                # 1-dimensional example
                fun = analytical().fun_sphere
                fun_control=fun_control_init(lower = np.array([-1]),
                                            upper = np.array([1]),
                                            noise=False)
                design_control=design_control_init(init_size=10)
                S = spot.Spot(fun=fun,
                            fun_control=fun_control,
                            design_control=design_control)
                S.initialize_design()
                S.update_stats()
                S.fit_surrogate()
                S.surrogate.plot()
                # 2-dimensional example
                fun = analytical().fun_sphere
                fun_control=fun_control_init(lower = np.array([-1, -1]),
                                            upper = np.array([1, 1]),
                                            noise=False)
                design_control=design_control_init(init_size=10)
                S = spot.Spot(fun=fun,
                            fun_control=fun_control,
                            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): Specifies which prediction values to return. It can be "y", "s", "ei", or "all".

        Returns:
            Union[float, Tuple[float, float, float]]: Depending on `return_val`, returns the predicted value,
            predicted error, expected improvement, or all.

        Raises:
            TypeError: If `nat_X` is not an ndarray or doesn't match expected dimensions.

        Examples:
            >>> from spotpython.build.kriging import Kriging
                import numpy as np
                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}")
        """
        if not isinstance(nat_X, ndarray):
            raise TypeError(f"Expected an ndarray, got {type(nat_X)} instead.")

        try:
            X = nat_X.reshape(-1, self.nat_X.shape[1])
            X = repair_non_numeric(X, self.var_type)
        except Exception as e:
            raise TypeError("Input to predict was not convertible to the size of X") from e

        y, s, ei = self.predict_coded_batch(X)

        if return_val == "y":
            return y
        elif return_val == "s":
            return s
        elif return_val == "ei":
            return -ei
        elif return_val == "all":
            return y, s, -ei
        else:
            raise ValueError(f"Invalid return_val: {return_val}. Supported values are 'y', 's', 'ei', 'all'.")

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

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

        Returns:
            Tuple[float, float, float]: Predicted value, predicted error, and expected improvement.

        Note:
            Uses attributes such as `self.mu` and `self.SigmaSqr` that are expected
            to be calculated by `likelihood`.

        Examples:
            >>> from spotpython.build.kriging import Kriging
                import numpy as np
                from numpy import linspace, arange, empty
                rng = np.random.RandomState(1)
                X = linspace(start=0, stop=10, num=10).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)
                n = X.shape[0]
                y = empty(n, dtype=float)
                s = empty(n, dtype=float)
                ei = empty(n, dtype=float)
                for i in range(n):
                    y_coded, s_coded, ei_coded = S.predict_coded(X[i, :])
                    y[i] = y_coded if np.isscalar(y_coded) else y_coded.item()
                    s[i] = s_coded if np.isscalar(s_coded) else s_coded.item()
                    ei[i] = ei_coded if np.isscalar(ei_coded) else ei_coded.item()
                print(f"y: {y}")
                print(f"s: {s}")
                print(f"ei: {-1.0*ei}")
        """
        self.build_psi_vec(cod_x)
        mu_adj = self.mu
        psi = self.psi

        # Calculate the prediction
        U_T_inv = solve(self.U.T, self.nat_y - self.one.dot(mu_adj))
        f = mu_adj + psi.T.dot(solve(self.U, U_T_inv))[0]

        Lambda = self.Lambda if self.noise else 0.0

        # Calculate the estimated error
        SSqr = self.SigmaSqr * (1 + Lambda - psi.T.dot(solve(self.U, solve(self.U.T, psi))))
        SSqr = power(abs(SSqr), 0.5)[0]

        # Calculate expected improvement
        EI = self.exp_imp(y0=f, s0=SSqr)

        return f, SSqr, EI

    def predict_coded_batch(self, X: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        """
        Vectorized prediction for batch input using coded units.

        Args:
            X (np.ndarray): Input array of coded points.

        Returns:
            Tuple[np.ndarray, np.ndarray, np.ndarray]:
                Arrays of predicted values, predicted errors, and expected improvements.
        """
        n = X.shape[0]
        y = np.empty(n, dtype=float)
        s = np.empty(n, dtype=float)
        ei = np.empty(n, dtype=float)

        for i in range(n):
            y_coded, s_coded, ei_coded = self.predict_coded(X[i, :])
            y[i] = y_coded if np.isscalar(y_coded) else y_coded.item()
            s[i] = s_coded if np.isscalar(s_coded) else s_coded.item()
            ei[i] = ei_coded if np.isscalar(ei_coded) else ei_coded.item()

        return y, s, ei

    def build_psi_vec(self, cod_x: np.ndarray) -> None:
        """
        Build the psi vector required for predictive methods.

        Args:
            cod_x (ndarray): Point to calculate the psi vector for.

        Returns:
            None

        Modifies:
            self.psi (np.ndarray): Updates the 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}")
        """
        logger.debug("Building psi vector for point: %s", cod_x)
        try:
            self.psi = np.zeros((self.n, 1))
            theta_scaled = np.power(10.0, self.theta)
            if self.n_theta == 1:
                theta_scaled = theta_scaled * np.ones(self.k)

            D = np.zeros(self.n)

            # Compute ordered distance contributions
            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, -1),
                           X_ordered,
                           metric='sqeuclidean',
                           w=theta_scaled[self.ordered_mask]).ravel()
            logger.debug("Distance D after ordered mask: %s", D)
            # Compute factor distance contributions
            if self.factor_mask.any():
                X_factor = self.nat_X[:, self.factor_mask]
                x_factor = cod_x[self.factor_mask]
                D += cdist(x_factor.reshape(1, -1),
                           X_factor,
                           metric=self.metric_factorial,
                           w=theta_scaled[self.factor_mask]).ravel()
            logger.debug("Distance D after factor mask: %s", D)

            self.psi = np.exp(-D).reshape(-1, 1)

        except np.linalg.LinAlgError as err:
            logger.error("Building psi failed due to a linear algebra error: %s. Error type: %s", err, type(err))

    def weighted_exp_imp(self, cod_x: np.ndarray, w: float) -> float:
        """
        Weighted expected improvement. Currently not used in `spotpython`

        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-like x is equal to the given value v. Converts inputs to numpy arrays as necessary.

Parameters:

Name	Type	Description	Default
x	Union[ndarray, Any]	The input array-like object to check.	required
v	Any	The value to check for in 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-like `x`
    is equal to the given value `v`. Converts inputs to numpy arrays as necessary.

    Args:
        x (Union[np.ndarray, Any]): The input array-like object to check.
        v (Any): The value to check for in `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, np.ndarray):
        x = np.array([x])  # Wrap scalar x in an array
    return np.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].

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.

Raises:

Type	Description
LinAlgError	If building Psi fails.

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: [1.60036366]
        S.Psi: [[1.00000001e+00 4.96525625e-18]
        [4.96525625e-18 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].

    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.

    Raises:
        LinAlgError: If building Psi fails.

    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: [1.60036366]
                S.Psi: [[1.00000001e+00 4.96525625e-18]
                [4.96525625e-18 1.00000001e+00]]
    """
    try:
        n = self.n
        k = self.k
        theta = np.power(10.0, self.theta)

        # Ensure theta has the correct length
        if self.n_theta == 1:
            theta = theta * np.ones(k)

        # Initialize the Psi matrix
        self.Psi = np.zeros((n, n), dtype=np.float64)

        # Calculate the distance matrix using ordered variables
        if self.ordered_mask.any():
            X_ordered = self.nat_X[:, self.ordered_mask]
            D_ordered = squareform(
                pdist(X_ordered, metric='sqeuclidean', w=theta[self.ordered_mask])
            )
            self.Psi += D_ordered

        # Add the contribution of factor variables to the distance matrix
        if self.factor_mask.any():
            X_factor = self.nat_X[:, self.factor_mask]
            D_factor = squareform(
                pdist(X_factor, metric=self.metric_factorial, w=theta[self.factor_mask])
            )
            self.Psi += D_factor

        # Calculate correlation from distance
        self.Psi = np.exp(-self.Psi)

        # Adjust diagonal elements for noise or minimum epsilon
        diag_indices = np.diag_indices_from(self.Psi)
        if self.noise:
            self.Psi[diag_indices] += self.Lambda
            logger.debug("Noise level Lambda applied to diagonal: %s", self.Lambda)
        else:
            self.Psi[diag_indices] += self.eps

        # Check for infinite values
        self.inf_Psi = np.isinf(self.Psi).any()

        # Calculate condition number
        self.cnd_Psi = cond(self.Psi)
        logger.debug("Condition number of Psi: %f", self.cnd_Psi)

    except LinAlgError as err:
        logger.error("Building Psi failed. Error: %s, Type: %s", err, type(err))
        raise

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: [1.60036366]
        S.Psi: [[1.00000001e+00 4.96525625e-18]
        [4.96525625e-18 1.00000001e+00]]
        S.U:[[1.00000001e+00 4.96525622e-18]
        [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: [1.60036366]
                S.Psi: [[1.00000001e+00 4.96525625e-18]
                [4.96525625e-18 1.00000001e+00]]
                S.U:[[1.00000001e+00 4.96525622e-18]
                [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 required for predictive methods.

Parameters:

Name	Type	Description	Default
cod_x	ndarray	Point to calculate the psi vector for.	required

Returns:

Type	Description
None	None

Modifies

self.psi (np.ndarray): Updates the 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}")

Source code in spotpython/build/kriging.py

def build_psi_vec(self, cod_x: np.ndarray) -> None:
    """
    Build the psi vector required for predictive methods.

    Args:
        cod_x (ndarray): Point to calculate the psi vector for.

    Returns:
        None

    Modifies:
        self.psi (np.ndarray): Updates the 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}")
    """
    logger.debug("Building psi vector for point: %s", cod_x)
    try:
        self.psi = np.zeros((self.n, 1))
        theta_scaled = np.power(10.0, self.theta)
        if self.n_theta == 1:
            theta_scaled = theta_scaled * np.ones(self.k)

        D = np.zeros(self.n)

        # Compute ordered distance contributions
        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, -1),
                       X_ordered,
                       metric='sqeuclidean',
                       w=theta_scaled[self.ordered_mask]).ravel()
        logger.debug("Distance D after ordered mask: %s", D)
        # Compute factor distance contributions
        if self.factor_mask.any():
            X_factor = self.nat_X[:, self.factor_mask]
            x_factor = cod_x[self.factor_mask]
            D += cdist(x_factor.reshape(1, -1),
                       X_factor,
                       metric=self.metric_factorial,
                       w=theta_scaled[self.factor_mask]).ravel()
        logger.debug("Distance D after factor mask: %s", D)

        self.psi = np.exp(-D).reshape(-1, 1)

    except np.linalg.LinAlgError as err:
        logger.error("Building psi failed due to a linear algebra error: %s. Error type: %s", 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

Returns: None

Examples:

>>> import numpy as np
    from spotpython.build.kriging import Kriging
    import logging
    logging.basicConfig(level=logging.DEBUG)
    # Define the number of theta and p parameters
    num_theta = 2
    num_p = 3
    # Initialize the Kriging model
    kriging_model = Kriging(
        name='kriging',
        seed=124,
        n_theta=num_theta,
        n_p=num_p,
        optim_p=True,
        noise=True
    )
    # Create bounds array
    bounds_array = np.array([1, 2, 3, 4, 5, 6])
    # Extract parameters from given bounds
    kriging_model.extract_from_bounds(new_theta_p_Lambda=bounds_array)
    # Assertions to check if parameters are correctly extracted
    assert np.array_equal(kriging_model.theta,
        [1, 2]), f"Expected theta to be [1, 2] but got {kriging_model.theta}"
    assert np.array_equal(kriging_model.p,
        [3, 4, 5]), f"Expected p to be [3, 4, 5] but got {kriging_model.p}"
    assert kriging_model.Lambda == 6, f"Expected Lambda to be 6 but got {kriging_model.Lambda}"
    print("All assertions passed!")

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.
    Returns:
        None

    Examples:
        >>> import numpy as np
            from spotpython.build.kriging import Kriging
            import logging
            logging.basicConfig(level=logging.DEBUG)
            # Define the number of theta and p parameters
            num_theta = 2
            num_p = 3
            # Initialize the Kriging model
            kriging_model = Kriging(
                name='kriging',
                seed=124,
                n_theta=num_theta,
                n_p=num_p,
                optim_p=True,
                noise=True
            )
            # Create bounds array
            bounds_array = np.array([1, 2, 3, 4, 5, 6])
            # Extract parameters from given bounds
            kriging_model.extract_from_bounds(new_theta_p_Lambda=bounds_array)
            # Assertions to check if parameters are correctly extracted
            assert np.array_equal(kriging_model.theta,
                [1, 2]), f"Expected theta to be [1, 2] but got {kriging_model.theta}"
            assert np.array_equal(kriging_model.p,
                [3, 4, 5]), f"Expected p to be [3, 4, 5] but got {kriging_model.p}"
            assert kriging_model.Lambda == 6, f"Expected Lambda to be 6 but got {kriging_model.Lambda}"
            print("All assertions passed!")
    """
    logger.debug("Extracting parameters from: %s", new_theta_p_Lambda)

    # Validate array length
    expected_length = self.n_theta
    if self.optim_p:
        expected_length += self.n_p
    if self.noise:
        expected_length += 1

    if len(new_theta_p_Lambda) < expected_length:
        logger.error("Input array is too short. Expected at least %d elements, got %d.",
                     expected_length, len(new_theta_p_Lambda))
        raise ValueError(f"Input array must have at least {expected_length} elements.")

    # Extract theta
    self.theta = new_theta_p_Lambda[:self.n_theta]
    logger.debug("Extracted theta: %s", self.theta)

    if self.optim_p:
        # Extract p if optim_p is True
        self.p = new_theta_p_Lambda[self.n_theta:self.n_theta + self.n_p]
        logger.debug("Extracted p: %s", self.p)

    if self.noise:
        # Extract Lambda
        lambda_index = self.n_theta + (self.n_p if self.optim_p else 0)
        self.Lambda = new_theta_p_Lambda[lambda_index]
        logger.debug("Extracted 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 buildPsi(). 3. U matrix via buildU().

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 `buildPsi()`.
    3. U matrix via `buildU()`.

    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) using several internal methods for matrix construction and likelihood evaluation. It handles potential errors by returning a penalty value for non-computable states.

Parameters:

Name	Type	Description	Default
new_theta_p_Lambda	ndarray	An array containing theta, p, and Lambda values.	required

Returns:

Name	Type	Description
float	float	The negative log likelihood or the penalty value if computation fails.

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.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}")
    S.likelihood()
    S.negLnLike
        [[0]
        [1]]
        [0 1]
        S.theta: [0.]
        S.theta: [1.60036366]
        S.Psi: [[1.00000001e+00 4.96525625e-18]
        [4.96525625e-18 1.00000001e+00]]
        S.U:[[1.00000001e+00 4.96525622e-18]
        [0.00000000e+00 1.00000001e+00]]
        -1.3862943611198906

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) using several internal methods for matrix construction
    and likelihood evaluation. It handles potential errors by returning a
    penalty value for non-computable states.

    Args:
        new_theta_p_Lambda (np.ndarray): An array containing `theta`, `p`, and `Lambda` values.

    Returns:
        float: The negative log likelihood or the penalty value if computation fails.

    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.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}")
            S.likelihood()
            S.negLnLike
                [[0]
                [1]]
                [0 1]
                S.theta: [0.]
                S.theta: [1.60036366]
                S.Psi: [[1.00000001e+00 4.96525625e-18]
                [4.96525625e-18 1.00000001e+00]]
                S.U:[[1.00000001e+00 4.96525622e-18]
                [0.00000000e+00 1.00000001e+00]]
                -1.3862943611198906
    """
    # Extract hyperparameters
    self.extract_from_bounds(new_theta_p_Lambda)
    # Check transformed theta values
    theta_scaled = np.power(10.0, self.theta)
    if self.__is_any__(theta_scaled, 0):
        logger.warning("Failure in fun_likelihood: 10^theta == 0. Setting negLnLike to %s", self.pen_val)
        return self.pen_val
    # Build Psi matrix and check its condition
    self.build_Psi()
    if getattr(self, 'inf_Psi', False) or getattr(self, 'cnd_Psi', float('inf')) > 1e9:
        logger.warning("Failure in fun_likelihood: Psi is ill-conditioned: %s", getattr(self, 'cnd_Psi', 'unknown'))
        logger.warning("Setting negLnLike to: %s", self.pen_val)
        return self.pen_val
    # Build U matrix and handle exceptions
    try:
        self.build_U()
    except Exception as error:
        logger.error("Error in fun_likelihood(). Call to build_U() failed: %s", error)
        logger.error("Setting negLnLike to %.2f.", self.pen_val)
        return self.pen_val

    # Calculate likelihood
    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)

    # Initialize p
    self.p = np.ones(self.n_p) * self.p_val
    logger.debug("In initialize_matrices(): self.p: %s", self.p)

    # Calculate variance of nat_y
    y_variance = var(self.nat_y)
    logger.debug("In initialize_matrices(): var(self.nat_y): %s", y_variance)

    # Set penalty value based on variance
    if y_variance > 0:
        self.pen_val = self.n * log(y_variance) + 1e4
    else:
        self.pen_val = self.n * y_variance + 1e4
    logger.debug("In initialize_matrices(): self.pen_val: %s", self.pen_val)

    # Initialize other attributes
    self.negLnLike = None
    self.LnDetPsi = None
    self.mu = None
    self.U = None
    self.SigmaSqr = None
    self.Lambda = None

    # Initialize generator
    self.gen = SpaceFilling(k=self.k, seed=self.seed)
    logger.debug("In initialize_matrices(): self.gen: %s", self.gen)

    # Initialize matrix Psi and vector psi
    self.Psi = np.zeros((self.n, self.n), dtype=np.float64)
    logger.debug("In initialize_matrices(): self.Psi shape: %s", self.Psi.shape)

    self.psi = np.zeros((self.n, 1), dtype=np.float64)
    logger.debug("In initialize_matrices(): self.psi shape: %s", self.psi.shape)

    # Initialize one
    self.one = np.ones(self.n, dtype=np.float64)
    logger.debug("In initialize_matrices(): self.one: %s", self.one)

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]

    """
    # Validate input dimensions
    if nat_X.ndim != 2 or nat_y.ndim != 1:
        raise ValueError("nat_X must be a 2D array and nat_y must be a 1D array.")
    if nat_X.shape[0] != nat_y.shape[0]:
        raise ValueError("The number of samples in nat_X and nat_y must be equal.")

    # Initialize instance variables
    self.nat_X = copy.deepcopy(nat_X)
    self.nat_y = copy.deepcopy(nat_y)
    self.n, self.k = self.nat_X.shape

    # Calculate and store min and max of X
    self.min_X = np.min(self.nat_X, axis=0)
    self.max_X = np.max(self.nat_X, axis=0)

    # Calculate the aggregated mean of y
    _, aggregated_mean_y, _ = aggregate_mean_var(X=self.nat_X, y=self.nat_y)
    self.aggregated_mean_y = np.copy(aggregated_mean_y)

    # Logging the initialized variables
    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: %d", self.n)
    logger.debug("In initialize_variables(): self.k: %d", self.k)

likelihood()

Calculate the negative concentrated log-likelihood. Implements equation (2.32) from [Forr08a] to compute the negative of the concentrated log-likelihood. Updates mu, SigmaSqr, LnDetPsi, and negLnLike.

Note

Requires prior calls to build_Psi and build_U.

Attributes:

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

Raises:

Type	Description
LinAlgError	If matrix operations fail.

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 np.allclose(S.mu, 7.5, atol=1e-6)
    E = np.exp(1)
    sigma2 = E / (E**2 - 1) * (25/4 + 25/4*E)
    assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
    print(f"S.LnDetPsi:{S.LnDetPsi}")
    print(f"S.negLnLike:{S.negLnLike}")
        S.LnDetPsi:-0.1454134234019476
        S.negLnLike:2.2185498738611282

Source code in spotpython/build/kriging.py

def likelihood(self) -> None:
    """
    Calculate the negative concentrated log-likelihood.
    Implements equation (2.32) from [Forr08a] to compute the negative of the
    concentrated log-likelihood. Updates `mu`, `SigmaSqr`, `LnDetPsi`, and `negLnLike`.

    Note:
        Requires prior calls to `build_Psi` and `build_U`.

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

    Raises:
        LinAlgError: If matrix operations fail.

    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 np.allclose(S.mu, 7.5, atol=1e-6)
            E = np.exp(1)
            sigma2 = E / (E**2 - 1) * (25/4 + 25/4*E)
            assert np.allclose(S.SigmaSqr, sigma2, atol=1e-6)
            print(f"S.LnDetPsi:{S.LnDetPsi}")
            print(f"S.negLnLike:{S.negLnLike}")
                S.LnDetPsi:-0.1454134234019476
                S.negLnLike:2.2185498738611282
    """
    try:
        # Solving linear equations for needed components
        U_T_inv_one = solve(self.U.T, self.one)
        U_T_inv_nat_y = solve(self.U.T, self.nat_y)
        # Mean calculation: (2.20) in [Forr08a]
        self.mu = (self.one.T @ solve(self.U, U_T_inv_nat_y)) / (self.one.T @ solve(self.U, U_T_inv_one))
        # Residuals
        cod_y_minus_mu = self.nat_y - self.one * self.mu
        # Sigma squared calculation: (2.31) in [Forr08a]
        self.SigmaSqr = (cod_y_minus_mu.T @ solve(self.U, solve(self.U.T, cod_y_minus_mu))) / self.n
        # Log determinant of Psi: (2.32) in [Forr08a]
        self.LnDetPsi = 2.0 * np.sum(np.log(np.abs(np.diag(self.U))))
        # Negative log-likelihood calculation: simplified from (2.32)
        self.negLnLike = 0.5 * (self.n * np.log(self.SigmaSqr) + self.LnDetPsi)
        logger.debug("Likelihood calculated: mu=%s, SigmaSqr=%s, LnDetPsi=%s, negLnLike=%s",
                     self.mu, self.SigmaSqr, self.LnDetPsi, self.negLnLike)
    except LinAlgError as error:
        logger.error("LinAlgError in likelihood calculation: %s", error)
        raise

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)
    [0.12167915 1.49467909 1.82808259 1.69648798 0.79564346]

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)
            [0.12167915 1.49467909 1.82808259 1.69648798 0.79564346]

    Returns:
        result["x"] (Union[List[float], Tuple[float]]):
            A list or tuple of optimized parameter values.
    """
    logger.debug("Entering optimize_model.")
    if not callable(self.model_optimizer):
        logger.error("model_optimizer is not callable.")
        raise ValueError("model_optimizer must be a callable function or method.")

    optimizer_strategies: Dict[str, Dict] = {
        'dual_annealing': {'func': self.fun_likelihood, 'bounds': self.de_bounds},
        'differential_evolution': {
            'func': self.fun_likelihood,
            'bounds': self.de_bounds,
            'maxiter': self.model_fun_evals,
            'seed': self.seed
        },
        'direct': {
            'func': self.fun_likelihood,
            'bounds': self.de_bounds,
            'eps': 1e-2
        },
        'shgo': {'func': self.fun_likelihood, 'bounds': self.de_bounds},
        'basinhopping': {'func': self.fun_likelihood, 'x0': np.mean(self.de_bounds, axis=1)}
    }

    optimizer_name = self.model_optimizer.__name__
    logger.debug("Optimizer selected: %s", optimizer_name)

    if optimizer_name not in optimizer_strategies:
        logger.info("Using default options for optimizer: %s", optimizer_name)
        optimizer_args = {'func': self.fun_likelihood, 'bounds': self.de_bounds}
    else:
        optimizer_args = optimizer_strategies[optimizer_name]

    logger.debug("Parameters for optimization: %s", optimizer_args)

    try:
        result = self.model_optimizer(**optimizer_args)
    except Exception as e:
        logger.error("Optimization failed due to error: %s", str(e))
        raise

    if "x" not in result:
        logger.error("Optimization result does not contain 'x'. Result: %s", result)
        raise ValueError("The optimization result does not contain the expected 'x' key.")
    logger.debug("Optimization result: %s", result)
    optimized_parameters = list(result["x"])
    logger.debug("Extracted optimized parameters: %s", optimized_parameters)
    return optimized_parameters

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
    from spotpython.utils.init import fun_control_init, design_control_init
    # 1-dimensional example
    fun = analytical().fun_sphere
    fun_control=fun_control_init(lower = np.array([-1]),
                                upper = np.array([1]),
                                noise=False)
    design_control=design_control_init(init_size=10)
    S = spot.Spot(fun=fun,
                fun_control=fun_control,
                design_control=design_control)
    S.initialize_design()
    S.update_stats()
    S.fit_surrogate()
    S.surrogate.plot()
    # 2-dimensional example
    fun = analytical().fun_sphere
    fun_control=fun_control_init(lower = np.array([-1, -1]),
                                upper = np.array([1, 1]),
                                noise=False)
    design_control=design_control_init(init_size=10)
    S = spot.Spot(fun=fun,
                fun_control=fun_control,
                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
            from spotpython.utils.init import fun_control_init, design_control_init
            # 1-dimensional example
            fun = analytical().fun_sphere
            fun_control=fun_control_init(lower = np.array([-1]),
                                        upper = np.array([1]),
                                        noise=False)
            design_control=design_control_init(init_size=10)
            S = spot.Spot(fun=fun,
                        fun_control=fun_control,
                        design_control=design_control)
            S.initialize_design()
            S.update_stats()
            S.fit_surrogate()
            S.surrogate.plot()
            # 2-dimensional example
            fun = analytical().fun_sphere
            fun_control=fun_control_init(lower = np.array([-1, -1]),
                                        upper = np.array([1, 1]),
                                        noise=False)
            design_control=design_control_init(init_size=10)
            S = spot.Spot(fun=fun,
                        fun_control=fun_control,
                        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	Specifies which prediction values to return. It can be “y”, “s”, “ei”, or “all”.	'y'

Returns:

Type	Description
Union[float, Tuple[float, float]]	Union[float, Tuple[float, float, float]]: Depending on return_val, returns the predicted value,
Union[float, Tuple[float, float]]	predicted error, expected improvement, or all.

Raises:

Type	Description
TypeError	If nat_X is not an ndarray or doesn’t match expected dimensions.

Examples:

>>> from spotpython.build.kriging import Kriging
    import numpy as np
    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}")

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): Specifies which prediction values to return. It can be "y", "s", "ei", or "all".

    Returns:
        Union[float, Tuple[float, float, float]]: Depending on `return_val`, returns the predicted value,
        predicted error, expected improvement, or all.

    Raises:
        TypeError: If `nat_X` is not an ndarray or doesn't match expected dimensions.

    Examples:
        >>> from spotpython.build.kriging import Kriging
            import numpy as np
            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}")
    """
    if not isinstance(nat_X, ndarray):
        raise TypeError(f"Expected an ndarray, got {type(nat_X)} instead.")

    try:
        X = nat_X.reshape(-1, self.nat_X.shape[1])
        X = repair_non_numeric(X, self.var_type)
    except Exception as e:
        raise TypeError("Input to predict was not convertible to the size of X") from e

    y, s, ei = self.predict_coded_batch(X)

    if return_val == "y":
        return y
    elif return_val == "s":
        return s
    elif return_val == "ei":
        return -ei
    elif return_val == "all":
        return y, s, -ei
    else:
        raise ValueError(f"Invalid return_val: {return_val}. Supported values are 'y', 's', 'ei', 'all'.")

predict_coded(cod_x)

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

Parameters:

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

Returns:

Type	Description
Tuple[float, float, float]	Tuple[float, float, float]: Predicted value, predicted error, and expected improvement.

Note

Uses attributes such as self.mu and self.SigmaSqr that are expected to be calculated by likelihood.

Examples:

>>> from spotpython.build.kriging import Kriging
    import numpy as np
    from numpy import linspace, arange, empty
    rng = np.random.RandomState(1)
    X = linspace(start=0, stop=10, num=10).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)
    n = X.shape[0]
    y = empty(n, dtype=float)
    s = empty(n, dtype=float)
    ei = empty(n, dtype=float)
    for i in range(n):
        y_coded, s_coded, ei_coded = S.predict_coded(X[i, :])
        y[i] = y_coded if np.isscalar(y_coded) else y_coded.item()
        s[i] = s_coded if np.isscalar(s_coded) else s_coded.item()
        ei[i] = ei_coded if np.isscalar(ei_coded) else ei_coded.item()
    print(f"y: {y}")
    print(f"s: {s}")
    print(f"ei: {-1.0*ei}")

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 coded units as described in (2.20) in [Forr08a].
    The error is returned as well. The method is used in `predict`.

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

    Returns:
        Tuple[float, float, float]: Predicted value, predicted error, and expected improvement.

    Note:
        Uses attributes such as `self.mu` and `self.SigmaSqr` that are expected
        to be calculated by `likelihood`.

    Examples:
        >>> from spotpython.build.kriging import Kriging
            import numpy as np
            from numpy import linspace, arange, empty
            rng = np.random.RandomState(1)
            X = linspace(start=0, stop=10, num=10).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)
            n = X.shape[0]
            y = empty(n, dtype=float)
            s = empty(n, dtype=float)
            ei = empty(n, dtype=float)
            for i in range(n):
                y_coded, s_coded, ei_coded = S.predict_coded(X[i, :])
                y[i] = y_coded if np.isscalar(y_coded) else y_coded.item()
                s[i] = s_coded if np.isscalar(s_coded) else s_coded.item()
                ei[i] = ei_coded if np.isscalar(ei_coded) else ei_coded.item()
            print(f"y: {y}")
            print(f"s: {s}")
            print(f"ei: {-1.0*ei}")
    """
    self.build_psi_vec(cod_x)
    mu_adj = self.mu
    psi = self.psi

    # Calculate the prediction
    U_T_inv = solve(self.U.T, self.nat_y - self.one.dot(mu_adj))
    f = mu_adj + psi.T.dot(solve(self.U, U_T_inv))[0]

    Lambda = self.Lambda if self.noise else 0.0

    # Calculate the estimated error
    SSqr = self.SigmaSqr * (1 + Lambda - psi.T.dot(solve(self.U, solve(self.U.T, psi))))
    SSqr = power(abs(SSqr), 0.5)[0]

    # Calculate expected improvement
    EI = self.exp_imp(y0=f, s0=SSqr)

    return f, SSqr, EI

predict_coded_batch(X)

Vectorized prediction for batch input using coded units.

Parameters:

Name	Type	Description	Default
X	ndarray	Input array of coded points.	required

Returns:

Type	Description
Tuple[ndarray, ndarray, ndarray]	Tuple[np.ndarray, np.ndarray, np.ndarray]: Arrays of predicted values, predicted errors, and expected improvements.

Source code in spotpython/build/kriging.py

def predict_coded_batch(self, X: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Vectorized prediction for batch input using coded units.

    Args:
        X (np.ndarray): Input array of coded points.

    Returns:
        Tuple[np.ndarray, np.ndarray, np.ndarray]:
            Arrays of predicted values, predicted errors, and expected improvements.
    """
    n = X.shape[0]
    y = np.empty(n, dtype=float)
    s = np.empty(n, dtype=float)
    ei = np.empty(n, dtype=float)

    for i in range(n):
        y_coded, s_coded, ei_coded = self.predict_coded(X[i, :])
        y[i] = y_coded if np.isscalar(y_coded) else y_coded.item()
        s[i] = s_coded if np.isscalar(s_coded) else s_coded.item()
        ei[i] = ei_coded if np.isscalar(ei_coded) else ei_coded.item()

    return y, s, 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)
    [[-3.0, 2.0]]

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)
            [[-3.0, 2.0]]

    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)

    # Adjust `n_theta` if it exceeds `k`
    if 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 reset to: %s", self.n_theta)

    # Initialize theta values
    if hasattr(self, "theta_init_zero") and self.theta_init_zero:
        self.theta = np.zeros(self.n_theta, dtype=float)
        logger.debug("Theta initialized to zeros: %s", self.theta)
    else:
        logger.debug("In set_theta_values(): self.n: %s", self.n)
        self.theta = np.ones(self.n_theta, dtype=float) * self.n / (100 * self.k)
        logger.debug("Theta initialized based on n and k: %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)

    # Ensure var_type has appropriate length by defaulting to 'num'
    if len(self.var_type) < self.k:
        self.var_type = ['num'] * self.k  # Corrected to fill with 'num' instead of duplicating
        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)
    # Create masks for each type using numpy vectorized operations
    var_type_array = np.array(self.var_type)
    self.num_mask = (var_type_array == "num")
    self.factor_mask = (var_type_array == "factor")
    self.int_mask = (var_type_array == "int")
    self.ordered_mask = np.isin(var_type_array, ["int", "num", "float"])
    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:
        negLnLike = self.negLnLike.copy()
        self.spot_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()
        self.spot_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()
            self.spot_writer.add_scalar("spot_Lambda", Lambda, self.counter+self.log_length)
        if self.optim_p:
            p = self.p.copy()
            self.spot_writer.add_scalars("spot_p",
                                         {f"p_{i}": p[i] for i in range(self.n_p)}, self.counter+self.log_length)
        self.spot_writer.flush()

weighted_exp_imp(cod_x, w)

Weighted expected improvement. Currently not used in spotpython

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. Currently not used in `spotpython`

    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