Inverse Distance Weighted (IDW) Interpolation with Python

Question

The Question: What is the best way to calculate inverse distance weighted (IDW) interpolation in Python, for point locations?

Some Background: Currently I'm using RPy2 to interface with R and it's gstat module. Unfortunately, the gstat module conflicts with arcgisscripting which I got around by running RPy2 based analysis in a separate process. Even if this issue is resolved in a recent/future release, and efficiency can be improved, I'd still like to remove my dependency on installing R.

The gstat website does provide a stand alone executable, which is easier to package with my python script, but I still hope for a Python solution which doesn't require multiple writes to disk and launching external processes. The number of calls to the interpolation function, of separate sets of points and values, can approach 20,000 in the processing I'm performing.

I specifically need to interpolate for points, so using the IDW function in ArcGIS to generate rasters sounds even worse than using R, in terms of performance.....unless there is a way to efficiently mask out only the points I need. Even with this modification, I wouldn't expect performance to be all that great. I will look into this option as another alternative. UPDATE: The problem here is you are tied to the cell size you are using. If you reduce the cell-size to get better accuracy, processing takes a long time. You also need to follow up by extracting by points.....over all an ugly method if you want values for specific points.

I have looked at the scipy documentation, but it doesn't look like there is a straight forward way to calculate IDW.

I'm thinking of rolling my own implementation, possibly using some of the scipy functionality to locate the closest points and calculate distances.

Am I missing something obvious? Is there a python module I haven't seen that does exactly what I want? Is creating my own implementation with the aid of scipy a wise choice?

Answer 1

Edit: @Denis is right, a linear Rbf (e.g. scipy.interpolate.Rbf with "function='linear'") isn't the same as IDW...

(Note, all of these will use excessive amounts of memory if you're using a large number of points!)

Here's a simple exampe of IDW:

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

Whereas, here's what a linear Rbf would be:

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

(Using the distance_matrix function here:)

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&gt;
    # (Yay for ufuncs!)
    d0 = np.subtract.outer(obs[:,0], interp[:,0])
    d1 = np.subtract.outer(obs[:,1], interp[:,1])

    return np.hypot(d0, d1)

Putting it all together into a nice copy-paste example yields some quick comparison plots:

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&gt;
    # (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 2

changed 20 Oct: this class Invdisttree combines inverse-distance weighting and scipy.spatial.KDTree.
Forget the original brute-force answer; this is imho the method of choice for scattered-data interpolation.

""" 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__ = "2010-10-20 Oct"  # weights

#...............................................................................
class Invdisttree:
    """ inverse-distance-weighted interpolation using KDTree:
invdisttree = Invdisttree( X, z )  -- points, values
interpol = invdisttree( q, nnear=6, eps=0, p=1, weights=None, stat=0 )
    interpolates z from the 6 points nearest each query point q;
    q may be one point, or a batch of points
    eps: approximate nearest, dist <= (1 + eps) * true nearest
    p: weights ~ 1 / distance**p; p=2 falls off faster
    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.

If different X coordinates measure different things, Euclidean distance
can be way off, unreasonable.  For example, if X0 is in the range 0 to 1
but X1 0 to 1000, the X1 distances will swamp X0; rescale.

A nice property of IDW is that it's scale-free around query points:
if I have values f1 f2 f3 from 3 people at distances d1 d2 d3,
the IDW average
    (f1/d1 + f2/d2 + f3/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 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, so p=2 falls off faster
    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.sum( x, axis=-1 ) / Ndim )

    known = np.random.uniform( size=(N,Ndim) ) ** 2  # a bit clumpy
    z = terrain( known )
    ask = np.random.uniform( size=(Nask,Ndim) )

#...............................................................................
    invdisttree = Invdisttree( known, z, leafsize=leafsize, stat=1 )
    weights = None
    for jweights in (0, 1):
        print "" if jweights == 0  else "\nwith weights:"

        interpol = invdisttree( ask, nnear=Nnear, eps=eps, p=p, weights=weights )

        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)
        weights = err

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