scipy克里金插值

### 使用 SciPy 进行克里金插值

尽管 `SciPy` 并不直接提供专门用于克里金插值的功能模块，但可以借助其优化和其他辅助功能来构建克里金插值器。通常情况下，更推荐使用专门为克里金设计的库如 `PyKrige`[^3] 来完成这项工作。然而，如果确实希望通过 `SciPy` 实现简单的克里金插值，则可以通过编写自定义代码并结合 `scipy.optimize` 和其他子模块来进行。

下面是一个基于 `SciPy` 的简单一维普通克里金插值示例：

import numpy as np
from scipy.spatial.distance import pdist, squareform
from scipy.optimize import minimize


def variogram_model(distance, params):
    """球状变差函数模型"""
    sill, range_, nugget = params
    gamma = ((distance / range_) ** 2 * (1 - distance / range_)) * \
            (distance <= range_)
    return sill * gamma + nugget


def kriging_system(x_known, z_known, x_unknown, model_params):
    distances = squareform(pdist(np.vstack((x_known, x_unknown)).reshape(-1, 1)))
    C = variogram_model(distances[:len(x_known), :len(x_known)], model_params)
    c = variogram_model(distances[-1:, :-1], model_params).ravel()
    
    A = np.zeros((C.shape[0]+1, C.shape[1]+1))
    A[:-1, :] = 1
    A[:, -1] = 1
    b = np.hstack([c, [1]])
    
    weights = np.linalg.solve(A, b)[:-1]
    estimate = np.dot(weights.T, z_known)

    return estimate


if __name__ == "__main__":
    # 已知样本点坐标及其对应的观测值
    known_points_x = np.array([0., 1., 2., 3., 4.])
    observed_values_z = np.random.rand(len(known_points_x))

    # 待预测的新位置
    new_point_x = 2.5
    
    initial_guesses = [np.var(observed_values_z)*0.9, max(abs(np.diff(known_points_x))), 0.]  # 初始猜测参数[sill, range, nugget]

    result = minimize(lambda p: sum(((kriging_system(
        known_points_x,
        observed_values_z,
        xi,
        p)-zi)**2 for xi, zi in zip(new_point_x.ravel(), observed_values_z))),
                      initial_guesses,
                      method='L-BFGS-B',
                      bounds=((None, None),
                              (min(np.diff(known_points_x)), max(known_points_x)),
                              (0, None)))

    estimated_value_at_new_location = kriging_system(
        known_points_x,
        observed_values_z,
        new_point_x,
        result.x)

    print(f"The estimated value at location {new_point_x} is approximately "
          f"{estimated_value_at_new_location:.6f}.")

此脚本展示了如何创建一个基本的一维普通克里金估计器，并对其进行测试以估算给定新地点处的数据值。需要注意的是，这段代码仅适用于教学目的；对于实际应用而言，建议采用成熟的第三方库如 `PyKrige` 或者探索更多专业的地理信息系统软件中的相应工具。