objectivefunctions

analytical

Class for analytical test functions.

Parameters:

Name	Type	Description	Default
offset	float	Offset value. Defaults to 0.0.	0.0
hz	float	Horizontal value. Defaults to 0.	0
seed	int	Seed value for random number generation. Defaults to 126.	126

Notes

See Numpy Random Sampling

Attributes:

Name	Type	Description
offset	float	Offset value.
hz	float	Horizontal value.
sigma	float	Noise level.
seed	int	Seed value for random number generation.
rng	Generator	Numpy random number generator object.
fun_control	dict	Dictionary containing control parameters for the function.

Source code in spotpython/fun/objectivefunctions.py

class analytical:
    """
    Class for analytical test functions.

    Args:
        offset (float):
            Offset value. Defaults to 0.0.
        hz (float):
            Horizontal value. Defaults to 0.
        seed (int):
            Seed value for random number generation. Defaults to 126.

    Notes:
        See [Numpy Random Sampling](https://numpy.org/doc/stable/reference/random/index.html#random-quick-start)

    Attributes:
        offset (float):
            Offset value.
        hz (float):
            Horizontal value.
        sigma (float):
            Noise level.
        seed (int):
            Seed value for random number generation.
        rng (Generator):
            Numpy random number generator object.
        fun_control (dict):
            Dictionary containing control parameters for the function.
    """

    def __init__(self, offset: float = 0.0, hz: float = 0, sigma=0.0, seed: int = 126) -> None:
        self.offset = offset
        self.hz = hz
        self.sigma = sigma
        self.seed = seed
        self.rng = default_rng(seed=self.seed)
        self.fun_control = {"sigma": sigma, "seed": None, "sel_var": None}

    def __repr__(self) -> str:
        return f"analytical(offset={self.offset}, hz={self.hz}, seed={self.seed})"

    def add_noise(self, y: List[float]) -> np.ndarray:
        """
        Adds noise to the input data.
        This method takes in a list of float values y as input and adds noise to
        the data using a random number generator. The method returns a numpy array
        containing the noisy data.

        Args:
            self (analytical): analytical class object.
            y (List[float]): Input data.

        Returns:
            np.ndarray: Noisy data.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
                import numpy as np
                y = np.array([1, 2, 3, 4, 5])
                fun = analytical(sigma=1.0, seed=123)
                fun.add_noise(y)
            array([0.01087865, 1.63221335, 4.28792526, 4.19397442, 5.9202309 ])

        """
        # Use own rng:
        if self.fun_control["seed"] is not None:
            rng = default_rng(seed=self.fun_control["seed"])
        # Use class rng:
        else:
            rng = self.rng
        noise_y = np.array([], dtype=float)
        for y_i in y:
            noise_y = np.append(
                noise_y,
                y_i + rng.normal(loc=0, scale=self.fun_control["sigma"], size=1),
            )
        return noise_y

    def fun_branin_factor(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """
        Calculates the Branin function of (x1, x2) with an additional factor based on the value of x3.
        If x3 = 1, the value of the Branin function is increased by 10.
        If x3 = 2, the value of the Branin function is decreased by 10.
        Otherwise, the value of the Branin function is not changed.

        Args:
            X (np.ndarray):
                A 2D numpy array with shape (n, 3) where n is the number of samples.
            fun_control (Optional[Dict]):
                A dictionary containing control parameters for the function.
                If None, self.fun_control is used. Defaults to None.

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
                import numpy as np
                X = np.array([[0, 0, 0], [0, 0, 1], [0, 0, 2]])
                fun = analytical()
                fun.fun_branin_factor(X)
                array([55.60211264, 65.60211264, 45.60211264])
        """
        if fun_control is None:
            fun_control = self.fun_control
        if len(X.shape) == 1:
            X = np.array([X])
        if X.shape[1] != 3:
            raise Exception("X must have shape (n, 3)")
        x1 = X[:, 0]
        x2 = X[:, 1]
        x3 = X[:, 2]
        a = 1
        b = 5.1 / (4 * np.pi**2)
        c = 5 / np.pi
        r = 6
        s = 10
        t = 1 / (8 * np.pi)
        y = a * (x2 - b * x1**2 + c * x1 - r) ** 2 + s * (1 - t) * np.cos(x1) + s
        for j in range(X.shape[0]):
            if x3[j] == 1:
                y[j] = y[j] + 10
            elif x3[j] == 2:
                y[j] = y[j] - 10
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_linear(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Linear function.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_linear(X)
            array([ 6., 15.])

        """
        if fun_control is not None:
            self.fun_control = fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        X = np.atleast_2d(X)
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, np.sum(X[i]))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_sphere(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Sphere function.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_sphere(X)
            array([14., 77.])

        """
        if fun_control is not None:
            self.fun_control = fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        X = np.atleast_2d(X)
        offset = np.ones(X.shape[1]) * self.offset
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, np.sum((X[i] - offset) ** 2))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_cubed(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Cubed function.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_cubed(X)
            array([ 0., 27.])
        """

        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        X = np.atleast_2d(X)
        offset = np.ones(X.shape[1]) * self.offset
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, np.sum((X[i] - offset) ** 3))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_forrester(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Forrester function. Function used by [Forr08a, p.83].
           f(x) = (6x- 2)^2 sin(12x-4) for x in [0,1].
           Starts with three sample points at x=0, x=0.5, and x=1.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_forrester(X)
            array([  0.        ,  11.99999999])
        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        X = np.atleast_2d(X)
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, (6.0 * X[i] - 2) ** 2 * np.sin(12 * X[i] - 4))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_branin(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        r"""Branin function. The 2-dim Branin function is defined as
            $$
            y = a (x_2 - b x_1^2 + c x_1 - r) ^2 + s (1 - t) \cos(x_1) + s,
            $$
            where values of $a, b, c, r, s$ and $t$ are:
            $a = 1$, $b = 5.1 / (4\pi^2)$, $c = 5 / \pi$, $r = 6$, $s = 10$ and $t = 1 / (8\pi)$.
            It has three global minima with $f(x) = 0.39788736$ at
            $$
            (-\pi, 12.275),
            $$
            $$
            (\pi, 2.275),
            $$
            and
            $$
            (9.42478, 2.475).
            $$
            Input domain: This function is usually evaluated on the square $x_1 \in [-5, 10] \times x_2 \in [0, 15]$.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
                pi = np.pi
                X = np.array([[0,0],
                    [-pi, 12.275],
                    [pi, 2.275],
                    [9.42478, 2.475]])
                fun = analytical()
                fun.fun_branin(X)
                array([55.60211264,  0.39788736,  0.39788736,  0.39788736])

        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        if X.shape[1] != 2:
            raise Exception
        x1 = X[:, 0]
        x2 = X[:, 1]
        a = 1
        b = 5.1 / (4 * np.pi**2)
        c = 5 / np.pi
        r = 6
        s = 10
        t = 1 / (8 * np.pi)
        y = a * (x2 - b * x1**2 + c * x1 - r) ** 2 + s * (1 - t) * np.cos(x1) + s
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_branin_modified(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Modified Branin function.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_branin_modified(X)
            array([  0.        ,  11.99999999])

        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        if X.shape[1] != 2:
            raise Exception
        x = X[:, 0]
        y = X[:, 1]
        X1 = 15 * x - 5
        X2 = 15 * y
        a = 1
        b = 5.1 / (4 * np.pi**2)
        c = 5 / np.pi
        d = 6
        e = 10
        ff = 1 / (8 * np.pi)
        y = (a * (X2 - b * X1**2 + c * X1 - d) ** 2 + e * (1 - ff) * np.cos(X1) + e) + 5 * x
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def branin_noise(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Branin function with noise.

        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            (np.ndarray): A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.branin_noise(X)
            array([  0.        ,  11.99999999])

        """
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        if X.shape[1] != 2:
            raise Exception
        x = X[:, 0]
        y = X[:, 1]
        X1 = 15 * x - 5
        X2 = 15 * y
        a = 1
        b = 5.1 / (4 * np.pi**2)
        c = 5 / np.pi
        d = 6
        e = 10
        ff = 1 / (8 * np.pi)
        noiseFree = (a * (X2 - b * X1**2 + c * X1 - d) ** 2 + e * (1 - ff) * np.cos(X1) + e) + 5 * x
        noise_y = []
        for i in noiseFree:
            noise_y.append(i + np.random.standard_normal() * 15)
        return np.array(noise_y)

    def fun_sin_cos(self, X, fun_control=None):
        """Sinusoidal function.
        Args:
            X (array):
                input
            fun_control (dict):
                dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            (np.ndarray): A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_sin_cos(X)
            array([-1.        , -0.41614684])
        """

        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        if X.shape[1] != 2:
            raise Exception
        x0 = X[:, 0]
        x1 = X[:, 1]
        y = 2.0 * np.sin(x0 + self.hz) + 0.5 * np.cos(x1 + self.hz)
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    # def fun_forrester_2(self, X):
    #     """
    #     Function used by [Forr08a, p.83].
    #     f(x) = (6x- 2)^2 sin(12x-4) for x in [0,1].
    #     Starts with three sample points at x=0, x=0.5, and x=1.

    #     Args:
    #         X (flooat): input values (1-dim)

    #     Returns:
    #         float: function value
    #     """
    #     try:
    #         X.shape[1]
    #     except ValueError:
    #         X = np.array(X)

    # X = np.atleast_2d(X)
    #     # y = X[:, 1]
    #     y = (6.0 * X - 2) ** 2 * np.sin(12 * X - 4)
    #     if self.sigma != 0:
    #         noise_y = np.array([], dtype=float)
    #         for i in y:
    #             noise_y = np.append(
    #                 noise_y, i + np.random.normal(loc=0, scale=self.sigma, size=1)
    #             )
    #         return noise_y
    #     else:
    #         return y

    def fun_runge(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Runge function. Formula: f(x) = 1/ (1 + sum(x_i) - offset)^2. Dim: k >= 1.
           Interval: -5 <= x <= 5

        Args:
            X (array): input
            fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3], [4, 5, 6]])
            >>> fun = analytical()
            >>> fun.fun_runge(X)
            array([0.0625    , 0.015625  , 0.00390625])

        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        if X.ndim == 1:
            X = X.reshape(-1, 1)
        offset = np.ones(X.shape[1]) * self.offset
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, (1 / (1 + np.sum((X[i] - offset) ** 2))))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_wingwt(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        r"""Wing weight function. Example from Forrester et al. to understand the weight
            of an unpainted light aircraft wing as a function of nine design and operational parameters:
            $W=0.036 S_W^{0.758}  Wfw^{0.0035} ( A / (\cos^2 \Lambda))^{0.6} q^{0.006}  \lambda^{0.04} ( (100 Rtc)/(\cos
              \Lambda) ))^{-0.3} (Nz Wdg)^{0.49}$

        | Symbol    | Parameter                              | Baseline | Minimum | Maximum |
        |-----------|----------------------------------------|----------|---------|---------|
        | $S_W$     | Wing area ($ft^2$)                     | 174      | 150     | 200     |
        | $W_{fw}$  | Weight of fuel in wing (lb)            | 252      | 220     | 300     |
        | $A$       | Aspect ratio                          | 7.52     | 6       | 10      |
        | $\Lambda$ | Quarter-chord sweep (deg)              | 0        | -10     | 10      |
        | $q$       | Dynamic pressure at cruise ($lb/ft^2$) | 34       | 16      | 45      |
        | $\lambda$ | Taper ratio                            | 0.672    | 0.5     | 1       |
        | $R_{tc}$  | Aerofoil thickness to chord ratio      | 0.12     | 0.08    | 0.18    |
        | $N_z$     | Ultimate load factor                   | 3.8      | 2.5     | 6       |
        | $W_{dg}$  | Flight design gross weight (lb)         | 2000     | 1700    | 2500    |
        | $W_p$     | paint weight (lb/ft^2)                   | 0.064 |   0.025  | 0.08    |

        Args:
            X (array): input
            fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
            >>> fun = analytical()
            >>> fun.fun_wingwt(X)
            array([0.0625    , 0.015625  , 0.00390625])

        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        #
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            Sw = X[i, 0] * (200 - 150) + 150
            Wfw = X[i, 1] * (300 - 220) + 220
            A = X[i, 2] * (10 - 6) + 6
            L = (X[i, 3] * (10 - (-10)) - 10) * np.pi / 180
            q = X[i, 4] * (45 - 16) + 16
            la = X[i, 5] * (1 - 0.5) + 0.5
            Rtc = X[i, 6] * (0.18 - 0.08) + 0.08
            Nz = X[i, 7] * (6 - 2.5) + 2.5
            Wdg = X[i, 8] * (2500 - 1700) + 1700
            Wp = X[i, 9] * (0.08 - 0.025) + 0.025
            # calculation on natural scale
            W = 0.036 * Sw**0.758 * Wfw**0.0035 * (A / np.cos(L) ** 2) ** 0.6 * q**0.006
            W = W * la**0.04 * (100 * Rtc / np.cos(L)) ** (-0.3) * (Nz * Wdg) ** (0.49) + Sw * Wp
            y = np.append(y, W)
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_xsin(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Example function.
        Args:
            X (array): input
            fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
            >>> fun = analytical()
            >>> fun.fun_xsin(X)
            array([0.84147098, 0.90929743, 0.14112001])

        """
        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        X = np.atleast_2d(X)
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            y = np.append(y, X[i] * np.sin(1.0 / X[i]))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_rosen(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Rosenbrock function.
        Args:
            X (array): input
            fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2,], [4, 5 ]])
            >>> fun = analytical()
            >>> fun.fun_rosen(X)
            array([24,  0])
        """

        if fun_control is None:
            fun_control = self.fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)
        if X.shape[1] != 2:
            raise Exception
        x0 = X[:, 0]
        x1 = X[:, 1]
        b = 10
        y = (x0 - 1) ** 2 + b * (x1 - x0**2) ** 2
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            return y

    def fun_random_error(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
        """Return errors for testing spot stability.
        Args:
            X (array): input
            fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

        Returns:
            np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

        Examples:
            >>> from spotpython.fun.objectivefunctions import analytical
            >>> import numpy as np
            >>> X = np.array([[1, 2,], [4, 5 ]])
            >>> fun = analytical()
            >>> fun.fun_random_error(X)
            array([24,  0])

        """
        if fun_control is not None:
            self.fun_control = fun_control
        if not isinstance(X, np.ndarray):
            X = np.array(X)

        if X.ndim == 1:
            X = X.reshape(-1, 1)
        y = np.array([], dtype=float)
        for i in range(X.shape[0]):
            # provoke error:
            if random() < 0.1:
                y = np.append(y, np.nan)
            else:
                y = np.append(y, np.sum(X[i]))
        if self.fun_control["sigma"] > 0:
            return self.add_noise(y)
        else:
            print(y)
            return y

add_noise(y)

Adds noise to the input data. This method takes in a list of float values y as input and adds noise to the data using a random number generator. The method returns a numpy array containing the noisy data.

Parameters:

Name	Type	Description	Default
self	analytical	analytical class object.	required
y	List[float]	Input data.	required

Returns:

Type	Description
ndarray	np.ndarray: Noisy data.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
    import numpy as np
    y = np.array([1, 2, 3, 4, 5])
    fun = analytical(sigma=1.0, seed=123)
    fun.add_noise(y)
array([0.01087865, 1.63221335, 4.28792526, 4.19397442, 5.9202309 ])

Source code in spotpython/fun/objectivefunctions.py

def add_noise(self, y: List[float]) -> np.ndarray:
    """
    Adds noise to the input data.
    This method takes in a list of float values y as input and adds noise to
    the data using a random number generator. The method returns a numpy array
    containing the noisy data.

    Args:
        self (analytical): analytical class object.
        y (List[float]): Input data.

    Returns:
        np.ndarray: Noisy data.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
            import numpy as np
            y = np.array([1, 2, 3, 4, 5])
            fun = analytical(sigma=1.0, seed=123)
            fun.add_noise(y)
        array([0.01087865, 1.63221335, 4.28792526, 4.19397442, 5.9202309 ])

    """
    # Use own rng:
    if self.fun_control["seed"] is not None:
        rng = default_rng(seed=self.fun_control["seed"])
    # Use class rng:
    else:
        rng = self.rng
    noise_y = np.array([], dtype=float)
    for y_i in y:
        noise_y = np.append(
            noise_y,
            y_i + rng.normal(loc=0, scale=self.fun_control["sigma"], size=1),
        )
    return noise_y

branin_noise(X, fun_control=None)

Branin function with noise.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.branin_noise(X)
array([  0.        ,  11.99999999])

Source code in spotpython/fun/objectivefunctions.py

def branin_noise(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Branin function with noise.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        (np.ndarray): A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.branin_noise(X)
        array([  0.        ,  11.99999999])

    """
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    if X.shape[1] != 2:
        raise Exception
    x = X[:, 0]
    y = X[:, 1]
    X1 = 15 * x - 5
    X2 = 15 * y
    a = 1
    b = 5.1 / (4 * np.pi**2)
    c = 5 / np.pi
    d = 6
    e = 10
    ff = 1 / (8 * np.pi)
    noiseFree = (a * (X2 - b * X1**2 + c * X1 - d) ** 2 + e * (1 - ff) * np.cos(X1) + e) + 5 * x
    noise_y = []
    for i in noiseFree:
        noise_y.append(i + np.random.standard_normal() * 15)
    return np.array(noise_y)

fun_branin(X, fun_control=None)

Branin function. The 2-dim Branin function is defined as $$ y = a (x_2 - b x_1^2 + c x_1 - r) ^2 + s (1 - t) \cos(x_1) + s, $$ where values of \(a, b, c, r, s\) and \(t\) are: \(a = 1\), \(b = 5.1 / (4\pi^2)\), \(c = 5 / \pi\), \(r = 6\), \(s = 10\) and \(t = 1 / (8\pi)\). It has three global minima with \(f(x) = 0.39788736\) at $$ (-\pi, 12.275), $$ $$ (\pi, 2.275), $$ and $$ (9.42478, 2.475). $$ Input domain: This function is usually evaluated on the square \(x_1 \in [-5, 10] \times x_2 \in [0, 15]\).

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
    pi = np.pi
    X = np.array([[0,0],
        [-pi, 12.275],
        [pi, 2.275],
        [9.42478, 2.475]])
    fun = analytical()
    fun.fun_branin(X)
    array([55.60211264,  0.39788736,  0.39788736,  0.39788736])

Source code in spotpython/fun/objectivefunctions.py

def fun_branin(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    r"""Branin function. The 2-dim Branin function is defined as
        $$
        y = a (x_2 - b x_1^2 + c x_1 - r) ^2 + s (1 - t) \cos(x_1) + s,
        $$
        where values of $a, b, c, r, s$ and $t$ are:
        $a = 1$, $b = 5.1 / (4\pi^2)$, $c = 5 / \pi$, $r = 6$, $s = 10$ and $t = 1 / (8\pi)$.
        It has three global minima with $f(x) = 0.39788736$ at
        $$
        (-\pi, 12.275),
        $$
        $$
        (\pi, 2.275),
        $$
        and
        $$
        (9.42478, 2.475).
        $$
        Input domain: This function is usually evaluated on the square $x_1 \in [-5, 10] \times x_2 \in [0, 15]$.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
            pi = np.pi
            X = np.array([[0,0],
                [-pi, 12.275],
                [pi, 2.275],
                [9.42478, 2.475]])
            fun = analytical()
            fun.fun_branin(X)
            array([55.60211264,  0.39788736,  0.39788736,  0.39788736])

    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    if X.shape[1] != 2:
        raise Exception
    x1 = X[:, 0]
    x2 = X[:, 1]
    a = 1
    b = 5.1 / (4 * np.pi**2)
    c = 5 / np.pi
    r = 6
    s = 10
    t = 1 / (8 * np.pi)
    y = a * (x2 - b * x1**2 + c * x1 - r) ** 2 + s * (1 - t) * np.cos(x1) + s
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_branin_factor(X, fun_control=None)

Calculates the Branin function of (x1, x2) with an additional factor based on the value of x3. If x3 = 1, the value of the Branin function is increased by 10. If x3 = 2, the value of the Branin function is decreased by 10. Otherwise, the value of the Branin function is not changed.

Parameters:

Name	Type	Description	Default
X	ndarray	A 2D numpy array with shape (n, 3) where n is the number of samples.	required
fun_control	Optional[Dict]	A dictionary containing control parameters for the function. If None, self.fun_control is used. Defaults to None.	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
    import numpy as np
    X = np.array([[0, 0, 0], [0, 0, 1], [0, 0, 2]])
    fun = analytical()
    fun.fun_branin_factor(X)
    array([55.60211264, 65.60211264, 45.60211264])

Source code in spotpython/fun/objectivefunctions.py

def fun_branin_factor(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """
    Calculates the Branin function of (x1, x2) with an additional factor based on the value of x3.
    If x3 = 1, the value of the Branin function is increased by 10.
    If x3 = 2, the value of the Branin function is decreased by 10.
    Otherwise, the value of the Branin function is not changed.

    Args:
        X (np.ndarray):
            A 2D numpy array with shape (n, 3) where n is the number of samples.
        fun_control (Optional[Dict]):
            A dictionary containing control parameters for the function.
            If None, self.fun_control is used. Defaults to None.

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
            import numpy as np
            X = np.array([[0, 0, 0], [0, 0, 1], [0, 0, 2]])
            fun = analytical()
            fun.fun_branin_factor(X)
            array([55.60211264, 65.60211264, 45.60211264])
    """
    if fun_control is None:
        fun_control = self.fun_control
    if len(X.shape) == 1:
        X = np.array([X])
    if X.shape[1] != 3:
        raise Exception("X must have shape (n, 3)")
    x1 = X[:, 0]
    x2 = X[:, 1]
    x3 = X[:, 2]
    a = 1
    b = 5.1 / (4 * np.pi**2)
    c = 5 / np.pi
    r = 6
    s = 10
    t = 1 / (8 * np.pi)
    y = a * (x2 - b * x1**2 + c * x1 - r) ** 2 + s * (1 - t) * np.cos(x1) + s
    for j in range(X.shape[0]):
        if x3[j] == 1:
            y[j] = y[j] + 10
        elif x3[j] == 2:
            y[j] = y[j] - 10
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_branin_modified(X, fun_control=None)

Modified Branin function.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_branin_modified(X)
array([  0.        ,  11.99999999])

Source code in spotpython/fun/objectivefunctions.py

def fun_branin_modified(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Modified Branin function.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_branin_modified(X)
        array([  0.        ,  11.99999999])

    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    if X.shape[1] != 2:
        raise Exception
    x = X[:, 0]
    y = X[:, 1]
    X1 = 15 * x - 5
    X2 = 15 * y
    a = 1
    b = 5.1 / (4 * np.pi**2)
    c = 5 / np.pi
    d = 6
    e = 10
    ff = 1 / (8 * np.pi)
    y = (a * (X2 - b * X1**2 + c * X1 - d) ** 2 + e * (1 - ff) * np.cos(X1) + e) + 5 * x
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_cubed(X, fun_control=None)

Cubed function.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_cubed(X)
array([ 0., 27.])

Source code in spotpython/fun/objectivefunctions.py

def fun_cubed(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Cubed function.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_cubed(X)
        array([ 0., 27.])
    """

    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    X = np.atleast_2d(X)
    offset = np.ones(X.shape[1]) * self.offset
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, np.sum((X[i] - offset) ** 3))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_forrester(X, fun_control=None)

Forrester function. Function used by [Forr08a, p.83]. f(x) = (6x- 2)^2 sin(12x-4) for x in [0,1]. Starts with three sample points at x=0, x=0.5, and x=1.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_forrester(X)
array([  0.        ,  11.99999999])

Source code in spotpython/fun/objectivefunctions.py

def fun_forrester(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Forrester function. Function used by [Forr08a, p.83].
       f(x) = (6x- 2)^2 sin(12x-4) for x in [0,1].
       Starts with three sample points at x=0, x=0.5, and x=1.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_forrester(X)
        array([  0.        ,  11.99999999])
    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    X = np.atleast_2d(X)
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, (6.0 * X[i] - 2) ** 2 * np.sin(12 * X[i] - 4))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_linear(X, fun_control=None)

Linear function.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_linear(X)
array([ 6., 15.])

Source code in spotpython/fun/objectivefunctions.py

def fun_linear(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Linear function.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_linear(X)
        array([ 6., 15.])

    """
    if fun_control is not None:
        self.fun_control = fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    X = np.atleast_2d(X)
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, np.sum(X[i]))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_random_error(X, fun_control=None)

Return errors for testing spot stability. Args: X (array): input fun_control (dict): dict with entries sigma (noise level) and seed (random seed).

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2,], [4, 5 ]])
>>> fun = analytical()
>>> fun.fun_random_error(X)
array([24,  0])

Source code in spotpython/fun/objectivefunctions.py

def fun_random_error(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Return errors for testing spot stability.
    Args:
        X (array): input
        fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2,], [4, 5 ]])
        >>> fun = analytical()
        >>> fun.fun_random_error(X)
        array([24,  0])

    """
    if fun_control is not None:
        self.fun_control = fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    if X.ndim == 1:
        X = X.reshape(-1, 1)
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        # provoke error:
        if random() < 0.1:
            y = np.append(y, np.nan)
        else:
            y = np.append(y, np.sum(X[i]))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        print(y)
        return y

fun_rosen(X, fun_control=None)

Rosenbrock function. Args: X (array): input fun_control (dict): dict with entries sigma (noise level) and seed (random seed).

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2,], [4, 5 ]])
>>> fun = analytical()
>>> fun.fun_rosen(X)
array([24,  0])

Source code in spotpython/fun/objectivefunctions.py

def fun_rosen(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Rosenbrock function.
    Args:
        X (array): input
        fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2,], [4, 5 ]])
        >>> fun = analytical()
        >>> fun.fun_rosen(X)
        array([24,  0])
    """

    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    if X.shape[1] != 2:
        raise Exception
    x0 = X[:, 0]
    x1 = X[:, 1]
    b = 10
    y = (x0 - 1) ** 2 + b * (x1 - x0**2) ** 2
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_runge(X, fun_control=None)

Runge function. Formula: f(x) = 1/ (1 + sum(x_i) - offset)^2. Dim: k >= 1. Interval: -5 <= x <= 5

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_runge(X)
array([0.0625    , 0.015625  , 0.00390625])

Source code in spotpython/fun/objectivefunctions.py

def fun_runge(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Runge function. Formula: f(x) = 1/ (1 + sum(x_i) - offset)^2. Dim: k >= 1.
       Interval: -5 <= x <= 5

    Args:
        X (array): input
        fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_runge(X)
        array([0.0625    , 0.015625  , 0.00390625])

    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)

    if X.ndim == 1:
        X = X.reshape(-1, 1)
    offset = np.ones(X.shape[1]) * self.offset
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, (1 / (1 + np.sum((X[i] - offset) ** 2))))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_sin_cos(X, fun_control=None)

Sinusoidal function. Args: X (array): input fun_control (dict): dict with entries sigma (noise level) and seed (random seed).

Returns:

Type	Description
ndarray	A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_sin_cos(X)
array([-1.        , -0.41614684])

Source code in spotpython/fun/objectivefunctions.py

def fun_sin_cos(self, X, fun_control=None):
    """Sinusoidal function.
    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        (np.ndarray): A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_sin_cos(X)
        array([-1.        , -0.41614684])
    """

    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    if X.shape[1] != 2:
        raise Exception
    x0 = X[:, 0]
    x1 = X[:, 1]
    y = 2.0 * np.sin(x0 + self.hz) + 0.5 * np.cos(x1 + self.hz)
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_sphere(X, fun_control=None)

Sphere function.

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3], [4, 5, 6]])
>>> fun = analytical()
>>> fun.fun_sphere(X)
array([14., 77.])

Source code in spotpython/fun/objectivefunctions.py

def fun_sphere(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Sphere function.

    Args:
        X (array):
            input
        fun_control (dict):
            dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3], [4, 5, 6]])
        >>> fun = analytical()
        >>> fun.fun_sphere(X)
        array([14., 77.])

    """
    if fun_control is not None:
        self.fun_control = fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    X = np.atleast_2d(X)
    offset = np.ones(X.shape[1]) * self.offset
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, np.sum((X[i] - offset) ** 2))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_wingwt(X, fun_control=None)

Wing weight function. Example from Forrester et al. to understand the weight of an unpainted light aircraft wing as a function of nine design and operational parameters: \(W=0.036 S_W^{0.758} Wfw^{0.0035} ( A / (\cos^2 \Lambda))^{0.6} q^{0.006} \lambda^{0.04} ( (100 Rtc)/(\cos \Lambda) ))^{-0.3} (Nz Wdg)^{0.49}\)

Symbol	Parameter	Baseline	Minimum	Maximum
\(S_W\)	Wing area (\(ft^2\))	174	150	200
\(W_{fw}\)	Weight of fuel in wing (lb)	252	220	300
\(A\)	Aspect ratio	7.52	6	10
\(\Lambda\)	Quarter-chord sweep (deg)	0	-10	10
\(q\)	Dynamic pressure at cruise (\(lb/ft^2\))	34	16	45
\(\lambda\)	Taper ratio	0.672	0.5	1
\(R_{tc}\)	Aerofoil thickness to chord ratio	0.12	0.08	0.18
\(N_z\)	Ultimate load factor	3.8	2.5	6
\(W_{dg}\)	Flight design gross weight (lb)	2000	1700	2500
\(W_p\)	paint weight (lb/ft^2)	0.064	0.025	0.08

Parameters:

Name	Type	Description	Default
X	array	input	required
fun_control	dict	dict with entries sigma (noise level) and seed (random seed).	None

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
>>> fun = analytical()
>>> fun.fun_wingwt(X)
array([0.0625    , 0.015625  , 0.00390625])

Source code in spotpython/fun/objectivefunctions.py

def fun_wingwt(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    r"""Wing weight function. Example from Forrester et al. to understand the weight
        of an unpainted light aircraft wing as a function of nine design and operational parameters:
        $W=0.036 S_W^{0.758}  Wfw^{0.0035} ( A / (\cos^2 \Lambda))^{0.6} q^{0.006}  \lambda^{0.04} ( (100 Rtc)/(\cos
          \Lambda) ))^{-0.3} (Nz Wdg)^{0.49}$

    | Symbol    | Parameter                              | Baseline | Minimum | Maximum |
    |-----------|----------------------------------------|----------|---------|---------|
    | $S_W$     | Wing area ($ft^2$)                     | 174      | 150     | 200     |
    | $W_{fw}$  | Weight of fuel in wing (lb)            | 252      | 220     | 300     |
    | $A$       | Aspect ratio                          | 7.52     | 6       | 10      |
    | $\Lambda$ | Quarter-chord sweep (deg)              | 0        | -10     | 10      |
    | $q$       | Dynamic pressure at cruise ($lb/ft^2$) | 34       | 16      | 45      |
    | $\lambda$ | Taper ratio                            | 0.672    | 0.5     | 1       |
    | $R_{tc}$  | Aerofoil thickness to chord ratio      | 0.12     | 0.08    | 0.18    |
    | $N_z$     | Ultimate load factor                   | 3.8      | 2.5     | 6       |
    | $W_{dg}$  | Flight design gross weight (lb)         | 2000     | 1700    | 2500    |
    | $W_p$     | paint weight (lb/ft^2)                   | 0.064 |   0.025  | 0.08    |

    Args:
        X (array): input
        fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
        >>> fun = analytical()
        >>> fun.fun_wingwt(X)
        array([0.0625    , 0.015625  , 0.00390625])

    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    #
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        Sw = X[i, 0] * (200 - 150) + 150
        Wfw = X[i, 1] * (300 - 220) + 220
        A = X[i, 2] * (10 - 6) + 6
        L = (X[i, 3] * (10 - (-10)) - 10) * np.pi / 180
        q = X[i, 4] * (45 - 16) + 16
        la = X[i, 5] * (1 - 0.5) + 0.5
        Rtc = X[i, 6] * (0.18 - 0.08) + 0.08
        Nz = X[i, 7] * (6 - 2.5) + 2.5
        Wdg = X[i, 8] * (2500 - 1700) + 1700
        Wp = X[i, 9] * (0.08 - 0.025) + 0.025
        # calculation on natural scale
        W = 0.036 * Sw**0.758 * Wfw**0.0035 * (A / np.cos(L) ** 2) ** 0.6 * q**0.006
        W = W * la**0.04 * (100 * Rtc / np.cos(L)) ** (-0.3) * (Nz * Wdg) ** (0.49) + Sw * Wp
        y = np.append(y, W)
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y

fun_xsin(X, fun_control=None)

Example function. Args: X (array): input fun_control (dict): dict with entries sigma (noise level) and seed (random seed).

Returns:

Type	Description
ndarray	np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

Examples:

>>> from spotpython.fun.objectivefunctions import analytical
>>> import numpy as np
>>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
>>> fun = analytical()
>>> fun.fun_xsin(X)
array([0.84147098, 0.90929743, 0.14112001])

Source code in spotpython/fun/objectivefunctions.py

def fun_xsin(self, X: np.ndarray, fun_control: Optional[Dict] = None) -> np.ndarray:
    """Example function.
    Args:
        X (array): input
        fun_control (dict): dict with entries `sigma` (noise level) and `seed` (random seed).

    Returns:
        np.ndarray: A 1D numpy array with shape (n,) containing the calculated values.

    Examples:
        >>> from spotpython.fun.objectivefunctions import analytical
        >>> import numpy as np
        >>> X = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9, 10, 11, 12]])
        >>> fun = analytical()
        >>> fun.fun_xsin(X)
        array([0.84147098, 0.90929743, 0.14112001])

    """
    if fun_control is None:
        fun_control = self.fun_control
    if not isinstance(X, np.ndarray):
        X = np.array(X)
    X = np.atleast_2d(X)
    y = np.array([], dtype=float)
    for i in range(X.shape[0]):
        y = np.append(y, X[i] * np.sin(1.0 / X[i]))
    if self.fun_control["sigma"] > 0:
        return self.add_noise(y)
    else:
        return y