如何在非线性网格上插值不规则分布的数据

Question

我得到了一个大约80.000个点(x,y,z)的数据集，而这些点不规则地分布在(x,y) in [0,a]x[0,b]平面上，在每个点(x,y)上，物理量z取一定的值。为了进一步评估数据，我想将其插值到网格上。

之前，我使用scipy.interpolate.griddata成功地将它们插值到一个规则的，二次的，二维网格上。然而，这种规则网格有一个缺点，即它不能适当地对z变化剧烈的区域进行建模，而在z变化很小的区域中有太多的数据点。

我希望有一个非线性(最好仍然是二次的，但网格大小可变)网格，在z变化剧烈的区域有更多的网格点，在z变化轻微的区域有更少的数据点。

Answer 1

我认为你把它弄反了:你的网格可以尽可能规则，但每个网格点应该使用相同数量的样本点进行评估，从而允许在高样本密度的区域进行强梯度变化，并在数据稀疏区域施加平滑。

我使用了逆距离加权树。我在python中浮动的实现:导入numpy为np从scipy。空间导入cKDTree

class invdisttree(object):
    """
    Compute the score of query points based on the scores of their k-nearest neighbours,
    weighted by the inverse of their distances.
    @reference:
    https://en.wikipedia.org/wiki/Inverse_distance_weighting
    Example:
    --------
    import numpy as np
    import matplotlib.pyplot as plt
    from invdisttree import invdisttree
    import matplotlib.pyplot as plt
    # create sample points with structured scores
    X1 = 10 * np.random.rand(1000, 2) -5
    def func(x, y):
        return np.sin(x**2 + y**2) / (x**2 + y**2)
    z1 = func(X1[:,0], X1[:,1])
    # 'train'
    tree = invdisttree(X1, z1)
    # 'test'
    spacing = np.linspace(-5., 5., 100)
    X2 = np.meshgrid(spacing, spacing)
    grid_shape = X2[0].shape
    X2 = np.reshape(X2, (2, -1)).T
    z2 = tree(X2)
    fig, (ax1, ax2, ax3) = plt.subplots(1,3, sharex=True, sharey=True, figsize=(10,3))
    ax1.contourf(spacing, spacing, func(*np.meshgrid(spacing, spacing)))
    ax1.set_title('Ground truth')
    ax2.scatter(X1[:,0], X1[:,1], c=z1, linewidths=0)
    ax2.set_title('Samples')
    ax3.contourf(spacing, spacing, z2.reshape(grid_shape))
    ax3.set_title('Reconstruction')
    plt.show()
    """
    def __init__(self, X=None, z=None, leafsize=10):
        if not X is None:
            self.tree = cKDTree(X, leafsize=leafsize )
        if not z is None:
            self.z = z
    def fit(self, X=None, z=None, leafsize=10):
        """
        Arguments:
        ----------
            X: (N, d) ndarray
                Coordinates of N sample points in a d-dimensional space.
            z: (N,) ndarray
                Corresponding scores.
            leafsize: int (default 10)
                Leafsize of KD-tree data structure;
                should be less than 20.
        Returns:
        --------
            invdisttree instance: object
        """
        return self.__init__(X, z, leafsize)
    def __call__(self, X, k=6, eps=1e-6, p=2, regularize_by=1e-9):
        self.distances, self.idx = self.tree.query(X, k, eps=eps, p=p)
        self.distances += regularize_by
        weights = self.z[self.idx.ravel()].reshape(self.idx.shape)
        mw = np.sum(weights/self.distances, axis=1) / np.sum(1./self.distances, axis=1)
        return mw
    def transform(self, X, k=6, p=2, eps=1e-6, regularize_by=1e-9):
        """
        Arguments:
        ----------
            X: (N, d) ndarray
                Coordinates of N query points in a d-dimensional space.
            k: int (default 6)
                Number of nearest neighbours to use.
            p: int or inf
                Which Minkowski p-norm to use.
                1 is the sum-of-absolute-values "Manhattan" distance
                2 is the usual Euclidean distance
                infinity is the maximum-coordinate-difference distance
            eps: float (default 1e-6)
                Return approximate nearest neighbors; the k-th returned value
                is guaranteed to be no further than (1+eps) times the
                distance to the real k-th nearest neighbor.
            regularise_by: float (default 1e-9)
                Regularise distances to prevent division by zero
                for sample points with the same location as query points.
        Returns:
        --------
            z: (N,) ndarray
                Corresponding scores.
        """
        return self.__call__(X, k, eps, p, regularize_by)