使用 Python 进行反距离加权 (IDW) 插值

Question

问题： 在 Python 中计算点位置的反距离加权 (IDW) 插值的最佳方法是什么？

一些背景： 目前我正在使用 RPy2 与 R 及其 gstat 模块交互。不幸的是，gstat 模块与 arcgisscripting 冲突，我通过在单独的进程中运行基于 RPy2 的分析来解决这一问题。即使这个问题在最近/未来的版本中得到解决，并且效率可以提高，我仍然想消除对安装 R 的依赖。

gstat 网站确实提供了一个独立的可执行文件，它更容易与我的 python 打包脚本，但我仍然希望有一个不需要多次写入磁盘和启动外部进程的 Python 解决方案。在我正在执行的处理中，对独立点和值集的插值函数的调用次数可能接近 20,000 次。

我特别需要对点进行插值，因此在性能方面，使用 ArcGIS 中的 IDW 函数生成栅格听起来比使用 R 更糟糕......除非有一种方法可以有效地仅屏蔽我需要的点。即使进行了这样的修改，我也不认为性能会那么好。我将研究这个选项作为另一种选择。更新：这里的问题是您与所使用的单元格大小相关。如果减小像元大小以获得更好的精度，则处理需要很长时间。您还需要通过按点提取来跟进......如果您想要特定点的值，那么总的来说这是一个丑陋的方法。

我查看了 scipy 文档，但看起来没有计算 IDW 的直接方法。

我正在考虑推出自己的实现，可能使用一些 scipy 功能来定位最近的点并计算距离。

我错过了一些明显的东西吗？是否有一个我没见过的 python 模块完全符合我的要求？借助 scipy 创建自己的实现是明智的选择吗？

Answer 1

于 10 月 20 日更改：此类 Invdisttree 结合了反距离加权和
scipy.spatial.KDTree。
忘记最初的暴力答案；
恕我直言，这是离散数据插值的选择方法。

""" invdisttree.py: inverse-distance-weighted interpolation using KDTree
    fast, solid, local
"""
from __future__ import division
import numpy as np
from scipy.spatial import cKDTree as KDTree
    # http://docs.scipy.org/doc/scipy/reference/spatial.html

__date__ = "2010-11-09 Nov"  # weights, doc

#...............................................................................
class Invdisttree:
    """ inverse-distance-weighted interpolation using KDTree:
invdisttree = Invdisttree( X, z )  -- data points, values
interpol = invdisttree( q, nnear=3, eps=0, p=1, weights=None, stat=0 )
    interpolates z from the 3 points nearest each query point q;
    For example, interpol[ a query point q ]
    finds the 3 data points nearest q, at distances d1 d2 d3
    and returns the IDW average of the values z1 z2 z3
        (z1/d1 + z2/d2 + z3/d3)
        / (1/d1 + 1/d2 + 1/d3)
        = .55 z1 + .27 z2 + .18 z3  for distances 1 2 3

    q may be one point, or a batch of points.
    eps: approximate nearest, dist <= (1 + eps) * true nearest
    p: use 1 / distance**p
    weights: optional multipliers for 1 / distance**p, of the same shape as q
    stat: accumulate wsum, wn for average weights

How many nearest neighbors should one take ?
a) start with 8 11 14 .. 28 in 2d 3d 4d .. 10d; see Wendel's formula
b) make 3 runs with nnear= e.g. 6 8 10, and look at the results --
    |interpol 6 - interpol 8| etc., or |f - interpol*| if you have f(q).
    I find that runtimes don't increase much at all with nnear -- ymmv.

p=1, p=2 ?
    p=2 weights nearer points more, farther points less.
    In 2d, the circles around query points have areas ~ distance**2,
    so p=2 is inverse-area weighting. For example,
        (z1/area1 + z2/area2 + z3/area3)
        / (1/area1 + 1/area2 + 1/area3)
        = .74 z1 + .18 z2 + .08 z3  for distances 1 2 3
    Similarly, in 3d, p=3 is inverse-volume weighting.

Scaling:
    if different X coordinates measure different things, Euclidean distance
    can be way off.  For example, if X0 is in the range 0 to 1
    but X1 0 to 1000, the X1 distances will swamp X0;
    rescale the data, i.e. make X0.std() ~= X1.std() .

A nice property of IDW is that it's scale-free around query points:
if I have values z1 z2 z3 from 3 points at distances d1 d2 d3,
the IDW average
    (z1/d1 + z2/d2 + z3/d3)
    / (1/d1 + 1/d2 + 1/d3)
is the same for distances 1 2 3, or 10 20 30 -- only the ratios matter.
In contrast, the commonly-used Gaussian kernel exp( - (distance/h)**2 )
is exceedingly sensitive to distance and to h.

    """
# anykernel( dj / av dj ) is also scale-free
# error analysis, |f(x) - idw(x)| ? todo: regular grid, nnear ndim+1, 2*ndim

    def __init__( self, X, z, leafsize=10, stat=0 ):
        assert len(X) == len(z), "len(X) %d != len(z) %d" % (len(X), len(z))
        self.tree = KDTree( X, leafsize=leafsize )  # build the tree
        self.z = z
        self.stat = stat
        self.wn = 0
        self.wsum = None;

    def __call__( self, q, nnear=6, eps=0, p=1, weights=None ):
            # nnear nearest neighbours of each query point --
        q = np.asarray(q)
        qdim = q.ndim
        if qdim == 1:
            q = np.array([q])
        if self.wsum is None:
            self.wsum = np.zeros(nnear)

        self.distances, self.ix = self.tree.query( q, k=nnear, eps=eps )
        interpol = np.zeros( (len(self.distances),) + np.shape(self.z[0]) )
        jinterpol = 0
        for dist, ix in zip( self.distances, self.ix ):
            if nnear == 1:
                wz = self.z[ix]
            elif dist[0] < 1e-10:
                wz = self.z[ix[0]]
            else:  # weight z s by 1/dist --
                w = 1 / dist**p
                if weights is not None:
                    w *= weights[ix]  # >= 0
                w /= np.sum(w)
                wz = np.dot( w, self.z[ix] )
                if self.stat:
                    self.wn += 1
                    self.wsum += w
            interpol[jinterpol] = wz
            jinterpol += 1
        return interpol if qdim > 1  else interpol[0]

#...............................................................................
if __name__ == "__main__":
    import sys

    N = 10000
    Ndim = 2
    Nask = N  # N Nask 1e5: 24 sec 2d, 27 sec 3d on mac g4 ppc
    Nnear = 8  # 8 2d, 11 3d => 5 % chance one-sided -- Wendel, mathoverflow.com
    leafsize = 10
    eps = .1  # approximate nearest, dist <= (1 + eps) * true nearest
    p = 1  # weights ~ 1 / distance**p
    cycle = .25
    seed = 1

    exec "\n".join( sys.argv[1:] )  # python this.py N= ...
    np.random.seed(seed )
    np.set_printoptions( 3, threshold=100, suppress=True )  # .3f

    print "\nInvdisttree:  N %d  Ndim %d  Nask %d  Nnear %d  leafsize %d  eps %.2g  p %.2g" % (
        N, Ndim, Nask, Nnear, leafsize, eps, p)

    def terrain(x):
        """ ~ rolling hills """
        return np.sin( (2*np.pi / cycle) * np.mean( x, axis=-1 ))

    known = np.random.uniform( size=(N,Ndim) ) ** .5  # 1/(p+1): density x^p
    z = terrain( known )
    ask = np.random.uniform( size=(Nask,Ndim) )

#...............................................................................
    invdisttree = Invdisttree( known, z, leafsize=leafsize, stat=1 )
    interpol = invdisttree( ask, nnear=Nnear, eps=eps, p=p )

    print "average distances to nearest points: %s" % \
        np.mean( invdisttree.distances, axis=0 )
    print "average weights: %s" % (invdisttree.wsum / invdisttree.wn)
        # see Wikipedia Zipf's law
    err = np.abs( terrain(ask) - interpol )
    print "average |terrain() - interpolated|: %.2g" % np.mean(err)

    # print "interpolate a single point: %.2g" % \
    #     invdisttree( known[0], nnear=Nnear, eps=eps )

Answer 2

编辑：@Denis是对的，线性Rbf（例如 scipy .interpolate.Rbf 与“function ='线性'”）与 IDW 不同...

（注意，如果您使用大量点，所有这些都会使用过多的内存！ ）

这是 IDW 的一个简单示例：

def simple_idw(x, y, z, xi, yi):
    dist = distance_matrix(x,y, xi,yi)

    # In IDW, weights are 1 / distance
    weights = 1.0 / dist

    # Make weights sum to one
    weights /= weights.sum(axis=0)

    # Multiply the weights for each interpolated point by all observed Z-values
    zi = np.dot(weights.T, z)
    return zi

而线性 Rbf 则如下：（

def linear_rbf(x, y, z, xi, yi):
    dist = distance_matrix(x,y, xi,yi)

    # Mutual pariwise distances between observations
    internal_dist = distance_matrix(x,y, x,y)

    # Now solve for the weights such that mistfit at the observations is minimized
    weights = np.linalg.solve(internal_dist, z)

    # Multiply the weights for each interpolated point by the distances
    zi =  np.dot(dist.T, weights)
    return zi

此处使用 distance_matrix 函数：）

def distance_matrix(x0, y0, x1, y1):
    obs = np.vstack((x0, y0)).T
    interp = np.vstack((x1, y1)).T

    # Make a distance matrix between pairwise observations
    # Note: from <http://stackoverflow.com/questions/1871536>
    # (Yay for ufuncs!)
    d0 = np.subtract.outer(obs[:,0], interp[:,0])
    d1 = np.subtract.outer(obs[:,1], interp[:,1])

    return np.hypot(d0, d1)

将它们全部放在一个漂亮的复制粘贴示例中会产生一些快速比较图：

<sub>（来源：jkington 位于 www.geology.wisc.edu）

（来源：jkington 位于 www.geology.wisc.edu）

（来源：jkington 位于 www.geology.wisc.edu）</sub>

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import Rbf

def main():
    # Setup: Generate data...
    n = 10
    nx, ny = 50, 50
    x, y, z = map(np.random.random, [n, n, n])
    xi = np.linspace(x.min(), x.max(), nx)
    yi = np.linspace(y.min(), y.max(), ny)
    xi, yi = np.meshgrid(xi, yi)
    xi, yi = xi.flatten(), yi.flatten()

    # Calculate IDW
    grid1 = simple_idw(x,y,z,xi,yi)
    grid1 = grid1.reshape((ny, nx))

    # Calculate scipy's RBF
    grid2 = scipy_idw(x,y,z,xi,yi)
    grid2 = grid2.reshape((ny, nx))

    grid3 = linear_rbf(x,y,z,xi,yi)
    print grid3.shape
    grid3 = grid3.reshape((ny, nx))


    # Comparisons...
    plot(x,y,z,grid1)
    plt.title('Homemade IDW')

    plot(x,y,z,grid2)
    plt.title("Scipy's Rbf with function=linear")

    plot(x,y,z,grid3)
    plt.title('Homemade linear Rbf')

    plt.show()

def simple_idw(x, y, z, xi, yi):
    dist = distance_matrix(x,y, xi,yi)

    # In IDW, weights are 1 / distance
    weights = 1.0 / dist

    # Make weights sum to one
    weights /= weights.sum(axis=0)

    # Multiply the weights for each interpolated point by all observed Z-values
    zi = np.dot(weights.T, z)
    return zi

def linear_rbf(x, y, z, xi, yi):
    dist = distance_matrix(x,y, xi,yi)

    # Mutual pariwise distances between observations
    internal_dist = distance_matrix(x,y, x,y)

    # Now solve for the weights such that mistfit at the observations is minimized
    weights = np.linalg.solve(internal_dist, z)

    # Multiply the weights for each interpolated point by the distances
    zi =  np.dot(dist.T, weights)
    return zi


def scipy_idw(x, y, z, xi, yi):
    interp = Rbf(x, y, z, function='linear')
    return interp(xi, yi)

def distance_matrix(x0, y0, x1, y1):
    obs = np.vstack((x0, y0)).T
    interp = np.vstack((x1, y1)).T

    # Make a distance matrix between pairwise observations
    # Note: from <http://stackoverflow.com/questions/1871536>
    # (Yay for ufuncs!)
    d0 = np.subtract.outer(obs[:,0], interp[:,0])
    d1 = np.subtract.outer(obs[:,1], interp[:,1])

    return np.hypot(d0, d1)


def plot(x,y,z,grid):
    plt.figure()
    plt.imshow(grid, extent=(x.min(), x.max(), y.max(), y.min()))
    plt.hold(True)
    plt.scatter(x,y,c=z)
    plt.colorbar()

if __name__ == '__main__':
    main()

Answer 3

我还需要一些快速的东西，我从 @joerington 解决方案开始，最终在 numba 结束，

我总是在 scipy、numpy 和 numba 之间进行实验，并选择最好的一个。对于这个问题，我使用 numba，因为额外的 tmp 内存可以忽略不计，但速度却非常快。

使用 numpy 需要在内存和速度之间进行权衡。例如，在 16GB 内存上，如果您想在其他 50000 个点上计算 50000 个点的插值，无论如何，它都会耗尽内存或速度非常慢。

因此，为了节省内存，我们需要使用 for 循环，以获得最小的临时内存分配。但在 numpy 中编写 for 循环意味着失去可能的矢量化。为此，我们有 numba。您可以为在 numpy 上接受 for 循环的函数添加 numba jit，它将在硬件上有效地矢量化 + 在多核上实现额外的并行性。对于巨大的数组情况，它会提供更好的速度，并且您可以在 GPU 上运行它，而无需编写 cuda

一个非常简单的片段是计算距离矩阵，在 IDW 情况下，我们需要逆距离矩阵。但即使对于 IDW 以外的方法，您也可以执行类似的操作。

另外，对于计算斜边的自定义方法，我有几个注意点此处

@nb.njit((nb.float64[:, :], nb.float64[:, :]), parallel=True)
def f2(d0, d1):
    print('Numba with parallel')
    res = np.empty((d0.shape[0], d1.shape[0]), dtype=d0.dtype)
    for i in nb.prange(d0.shape[0]):
        for j in range(d1.shape[0]):
            res[i, j] = np.sqrt((d0[i, 0] - d1[j, 0])**2 + (d0[i, 1] - d1[j, 1])**2)
    return res

最近 numba 也与 scikit 兼容，所以这是+1

请参阅：

为什么 np.hypot 和 np.subtract.outer 与普通广播相比非常快？使用 Numba 并行加速 numpy 进行距离矩阵计算

numpy 中的自定义 dtype 用于纬度、经度，以实现更快的距离矩阵/克里格/IDW 插值计算