在python中通过不规则网格集成2D数据

Question

因此，我具有在一个域中不规则采样的2D函数，并且我想计算表面下方的体积。 数据以[x,y,z] ，举一个简单的例子：

def f(x,y):
    return np.cos(10*x*y) * np.exp(-x**2 - y**2)

datrange1 = np.linspace(-5,5,1000)
datrange2 = np.linspace(-0.5,0.5,1000)

ar = []
for x in datrange1:
    for y in datrange2:
        ar += [[x,y, f(x,y)]]


for x in xrange2:
    for y in yrange2:
        ar += [[x,y, f(x,y)]] 

val_arr1 = np.array(ar)

data = np.unique(val_arr1)


xlist, ylist, zlist = data.T 

其中np.unique对第一列中的数据进行排序，然后对第二列中的数据进行排序。 数据是以这种方式排列的，因为我需要围绕原点进行更多采样，因为必须解决一个尖锐的特征。

现在，我想知道是使用构建二维插值功能scipy.interpolate.interp2d ，然后用这个积分， dblquad 。 事实证明，这不仅笨拙而且缓慢，而且还会引发错误：

RuntimeWarning: No more knots can be added because the number of B-spline
coefficients already exceeds the number of data points m. 

是否有更好的方法来集成以这种方式排列或克服此错误的数据？

Answer 1

如果您可以围绕感兴趣的特征以足够高的分辨率对数据进行采样，然后在其他地方进行稀疏采样，那么问题定义就变成了如何定义每个采样下的区域。 对于常规的矩形样本，此操作很容易，并且可能以原点附近的分辨率为增量逐步进行。 我追求的方法是为每个样本生成2维Voronoi单元，以确定它们的面积。 我把大部分的代码从这个答案，因为它几乎都已经所需的组件。

import numpy as np
from scipy.spatial import Voronoi

#taken from: # https://stackoverflow.com/questions/28665491/getting-a-bounded-polygon-coordinates-from-voronoi-cells
#computes voronoi regions bounded by a bounding box
def square_voronoi(xy, bbox): #bbox: (min_x, max_x, min_y, max_y)
    # Select points inside the bounding box
    points_center = xy[np.where((bbox[0] <= xy[:,0]) * (xy[:,0] <= bbox[1]) * (bbox[2] <= xy[:,1]) * (bbox[2] <= bbox[3]))]
    # Mirror points
    points_left = np.copy(points_center)
    points_left[:, 0] = bbox[0] - (points_left[:, 0] - bbox[0])
    points_right = np.copy(points_center)
    points_right[:, 0] = bbox[1] + (bbox[1] - points_right[:, 0])
    points_down = np.copy(points_center)
    points_down[:, 1] = bbox[2] - (points_down[:, 1] - bbox[2])
    points_up = np.copy(points_center)
    points_up[:, 1] = bbox[3] + (bbox[3] - points_up[:, 1])
    points = np.concatenate((points_center, points_left, points_right, points_down, points_up,), axis=0)
    # Compute Voronoi
    vor = Voronoi(points)
    # Filter regions (center points should* be guaranteed to have a valid region)
    # center points should come first and not change in size
    regions = [vor.regions[vor.point_region[i]] for i in range(len(points_center))]
    vor.filtered_points = points_center
    vor.filtered_regions = regions
    return vor

#also stolen from: https://stackoverflow.com/questions/28665491/getting-a-bounded-polygon-coordinates-from-voronoi-cells
def area_region(vertices):
    # Polygon's signed area
    A = 0
    for i in range(0, len(vertices) - 1):
        s = (vertices[i, 0] * vertices[i + 1, 1] - vertices[i + 1, 0] * vertices[i, 1])
        A = A + s
    return np.abs(0.5 * A)

def f(x,y):
    return np.cos(10*x*y) * np.exp(-x**2 - y**2)

#sampling could easily be shaped to sample origin more heavily
sample_x = np.random.rand(1000) * 10 - 5 #same range as example linspace
sample_y = np.random.rand(1000) - .5
sample_xy = np.array([sample_x, sample_y]).T

vor = square_voronoi(sample_xy, (-5,5,-.5,.5)) #using bbox from samples
points = vor.filtered_points
sample_areas = np.array([area_region(vor.vertices[verts+[verts[0]],:]) for verts in vor.filtered_regions])
sample_z = np.array([f(p[0], p[1]) for p in points])

volume = np.sum(sample_z * sample_areas)

我还没有完全测试过，但是原理应该起作用，并且数学运算也可以完成。