有效地为高密度区域创建密度图，为稀疏区域创建点

Question

我需要制作一个图表，其功能类似于图中高密度区域的密度图，但低于某个阈值则使用单个点。我找不到任何类似于我在matplotlib缩略图库或谷歌搜索中所需的代码。我有一个我自己编写的工作代码，但它有些棘手，而且（更重要的是）当点数/分档数量很大时需要花费不可思议的长时间。这是代码：

import numpy as np
import math
import matplotlib as mpl
import matplotlib.pyplot as plt
import pylab
import numpy.random

#Create the colormap:
halfpurples = {'blue': [(0.0,1.0,1.0),(0.000001, 0.78431373834609985, 0.78431373834609985),
(0.25, 0.729411780834198, 0.729411780834198), (0.5,
0.63921570777893066, 0.63921570777893066), (0.75,
0.56078433990478516, 0.56078433990478516), (1.0, 0.49019607901573181,
0.49019607901573181)],

    'green': [(0.0,1.0,1.0),(0.000001,
    0.60392159223556519, 0.60392159223556519), (0.25,
    0.49019607901573181, 0.49019607901573181), (0.5,
    0.31764706969261169, 0.31764706969261169), (0.75,
    0.15294118225574493, 0.15294118225574493), (1.0, 0.0, 0.0)],

    'red': [(0.0,1.0,1.0),(0.000001,
    0.61960786581039429, 0.61960786581039429), (0.25,
    0.50196081399917603, 0.50196081399917603), (0.5,
    0.41568627953529358, 0.41568627953529358), (0.75,
    0.32941177487373352, 0.32941177487373352), (1.0,
    0.24705882370471954, 0.24705882370471954)]} 

halfpurplecmap = mpl.colors.LinearSegmentedColormap('halfpurples',halfpurples,256)

#Create x,y arrays of normally distributed points
npts = 1000
x = numpy.random.standard_normal(npts)
y = numpy.random.standard_normal(npts)

#Set bin numbers in both axes
nxbins = 25
nybins = 25

#Set the cutoff for resolving the individual points
minperbin = 1

#Make the density histrogram
H, yedges, xedges = np.histogram2d(y,x,bins=(nybins,nxbins))
#Reorient the axes
H =  H[::-1]

extent = [xedges[0],xedges[-1],yedges[0],yedges[-1]]

#Compute all bins where the density plot value is below (or equal to) the threshold
lowxleftedges = [[xedges[i] for j in range(len(H[:,i])) if H[j,i] <= minperbin] for i in range(len(H[0,:]))] 
lowxrightedges = [[xedges[i+1] for j in range(len(H[:,i])) if H[j,i] <= minperbin] for i in range(len(H[0,:]))] 
lowyleftedges = [[yedges[-(j+2)] for j in range(len(H[:,i])) if H[j,i] <= minperbin] for i in range(len(H[0,:]))]
lowyrightedges = [[yedges[-(j+1)] for j in range(len(H[:,i])) if H[j,i] <= minperbin] for i in range(len(H[0,:]))]

#Flatten and convert to numpy array
lowxleftedges = np.asarray([item for sublist in lowxleftedges for item in sublist])
lowxrightedges = np.asarray([item for sublist in lowxrightedges for item in sublist])
lowyleftedges = np.asarray([item for sublist in lowyleftedges for item in sublist])
lowyrightedges = np.asarray([item for sublist in lowyrightedges for item in sublist])

#Find all points that lie in these regions
lowdatax = [[x[i] for j in range(len(lowxleftedges)) if lowxleftedges[j] <= x[i] and x[i] <= lowxrightedges[j] and lowyleftedges[j] <= y[i] and y[i] <= lowyrightedges[j]] for i in range(len(x))]
lowdatay = [[y[i] for j in range(len(lowyleftedges)) if lowxleftedges[j] <= x[i] and x[i] <= lowxrightedges[j] and lowyleftedges[j] <= y[i] and y[i] <= lowyrightedges[j]] for i in range(len(y))]

#Flatten and convert into numpy array
lowdatax = np.asarray([item for sublist in lowdatax for item in sublist])
lowdatay = np.asarray([item for sublist in lowdatay for item in sublist])

#Plot
fig1 = plt.figure()
ax1 = fig1.add_subplot(111)
ax1.plot(lowdatax,lowdatay,linestyle='.',marker='o',mfc='k',mec='k')
cp1 = ax1.imshow(H,interpolation='nearest',extent=extent,cmap=halfpurplecmap,vmin=minperbin)
fig1.colorbar(cp1)

fig1.savefig('contourtest.eps')

此代码生成如下所示的图像：

但是，当在较大的数据集上使用时，程序需要几秒到几分钟。有关如何提高速度的任何想法？谢谢！

Answer 1

这应该这样做：

import matplotlib.pyplot as plt, numpy as np, numpy.random, scipy

#histogram definition
xyrange = [[-5,5],[-5,5]] # data range
bins = [100,100] # number of bins
thresh = 3  #density threshold

#data definition
N = 1e5;
xdat, ydat = np.random.normal(size=N), np.random.normal(1, 0.6, size=N)

# histogram the data
hh, locx, locy = scipy.histogram2d(xdat, ydat, range=xyrange, bins=bins)
posx = np.digitize(xdat, locx)
posy = np.digitize(ydat, locy)

#select points within the histogram
ind = (posx > 0) & (posx <= bins[0]) & (posy > 0) & (posy <= bins[1])
hhsub = hh[posx[ind] - 1, posy[ind] - 1] # values of the histogram where the points are
xdat1 = xdat[ind][hhsub < thresh] # low density points
ydat1 = ydat[ind][hhsub < thresh]
hh[hh < thresh] = np.nan # fill the areas with low density by NaNs

plt.imshow(np.flipud(hh.T),cmap='jet',extent=np.array(xyrange).flatten(), interpolation='none', origin='upper')
plt.colorbar()   
plt.plot(xdat1, ydat1, '.',color='darkblue')
plt.show()

Answer 2

对于记录，这是使用scipy.stats.gaussian_kde而不是2D直方图的新尝试的结果。 根据目的，可以设想颜色网格和轮廓的不同组合。

import numpy as np
from matplotlib import pyplot as plt
from scipy.stats import gaussian_kde

# parameters
npts = 5000         # number of sample points
bins = 100          # number of bins in density maps
threshold = 0.01    # density threshold for scatter plot

# initialize figure
fig, ax = plt.subplots()

# create a random dataset
x1, y1 = np.random.multivariate_normal([0, 0], [[1, 0], [0, 1]], npts/2).T
x2, y2 = np.random.multivariate_normal([4, 4], [[4, 0], [0, 1]], npts/2).T
x = np.hstack((x1, x2))
y = np.hstack((y1, y2))
points = np.vstack([x, y])

# perform kernel density estimate
kde = gaussian_kde(points)
z = kde(points)

# mask points above density threshold
x = np.ma.masked_where(z > threshold, x)
y = np.ma.masked_where(z > threshold, y)

# plot unmasked points
ax.scatter(x, y, c='black', marker='.')

# get bounds from axes
xmin, xmax = ax.get_xlim()
ymin, ymax = ax.get_ylim()

# prepare grid for density map
xedges = np.linspace(xmin, xmax, bins)
yedges = np.linspace(ymin, ymax, bins)
xx, yy = np.meshgrid(xedges, yedges)
gridpoints = np.array([xx.ravel(), yy.ravel()])

# compute density map
zz = np.reshape(kde(gridpoints), xx.shape)

# plot density map
im = ax.imshow(zz, cmap='CMRmap_r', interpolation='nearest',
               origin='lower', extent=[xmin, xmax, ymin, ymax])

# plot threshold contour
cs = ax.contour(xx, yy, zz, levels=[threshold], colors='black')

# show
fig.colorbar(im)
plt.show()

Answer 3

你的问题是二次的 - 对于npts = 1000，你有一个数组大小达到10 ^ 6点，而不是你用列表推导迭代这些列表。
现在，这当然是一个品味问题，但我发现列表理解可以产生一个难以理解的完全代码，并且它们有时仅稍微快一些......但这不是我的观点。
我的观点是，对于大型数组操作，你有像numpy函数：

np.where, np.choose etc.

看到您可以使用NumPy实现列表推导的功能，并且您的代码应该运行得更快。

我的理解是正确的吗？

#Find all points that lie in these regions

你在测试多边形内的一个点吗？如果是这样，请考虑matplotlib中的point in polygon。

Answer 4

经过一夜的睡眠并阅读Oz123的建议，我明白了。诀窍是计算每个x，y点落入（xi，yi）的哪个bin，然后测试H [xi，yi]（实际上，在我的情况下H [yi，xi]）是否低于阈值。代码如下，对于大量的点运行速度非常快，而且更清晰：

import numpy as np
import math
import matplotlib as mpl
import matplotlib.pyplot as plt
import pylab
import numpy.random

#Create the colormap:
halfpurples = {'blue': [(0.0,1.0,1.0),(0.000001, 0.78431373834609985, 0.78431373834609985),
0.25, 0.729411780834198, 0.729411780834198), (0.5,
0.63921570777893066, 0.63921570777893066), (0.75,
0.56078433990478516, 0.56078433990478516), (1.0, 0.49019607901573181,
0.49019607901573181)],

    'green': [(0.0,1.0,1.0),(0.000001,
    0.60392159223556519, 0.60392159223556519), (0.25,
    0.49019607901573181, 0.49019607901573181), (0.5,
    0.31764706969261169, 0.31764706969261169), (0.75,
    0.15294118225574493, 0.15294118225574493), (1.0, 0.0, 0.0)],

    'red': [(0.0,1.0,1.0),(0.000001,
    0.61960786581039429, 0.61960786581039429), (0.25,
    0.50196081399917603, 0.50196081399917603), (0.5,
    0.41568627953529358, 0.41568627953529358), (0.75,
    0.32941177487373352, 0.32941177487373352), (1.0,
    0.24705882370471954, 0.24705882370471954)]} 

halfpurplecmap = mpl.colors.LinearSegmentedColormap('halfpurples',halfpurples,256)

#Create x,y arrays of normally distributed points
npts = 100000
x = numpy.random.standard_normal(npts)
y = numpy.random.standard_normal(npts)

#Set bin numbers in both axes
nxbins = 100
nybins = 100

#Set the cutoff for resolving the individual points
minperbin = 1

#Make the density histrogram
H, yedges, xedges = np.histogram2d(y,x,bins=(nybins,nxbins))
#Reorient the axes
H =  H[::-1]

extent = [xedges[0],xedges[-1],yedges[0],yedges[-1]]

#Figure out which bin each x,y point is in
xbinsize = xedges[1]-xedges[0]
ybinsize = yedges[1]-yedges[0]
xi = ((x-xedges[0])/xbinsize).astype(np.integer)
yi = nybins-1-((y-yedges[0])/ybinsize).astype(np.integer)

#Subtract one from any points exactly on the right and upper edges of the region
xim1 = xi-1
yim1 = yi-1
xi = np.where(xi < nxbins,xi,xim1)
yi = np.where(yi < nybins,yi,yim1)

#Get all points with density below the threshold
lowdensityx = x[H[yi,xi] <= minperbin]
lowdensityy = y[H[yi,xi] <= minperbin]

#Plot
fig1 = plt.figure()
ax1 = fig1.add_subplot(111)
ax1.plot(lowdensityx,lowdensityy,linestyle='.',marker='o',mfc='k',mec='k',ms=3)
cp1 = ax1.imshow(H,interpolation='nearest',extent=extent,cmap=halfpurplecmap,vmin=minperbin)
fig1.colorbar(cp1)

fig1.savefig('contourtest.eps')