Efficiently create a density plot for high-density regions, points for sparse regions

Question

I need to make a plot that functions like a density plot for high-density regions on the plot, but below some threshold uses individual points. I couldn\'t find any existing

Answer 1

After a night to sleep on it and reading through Oz123's suggestions, I figured it out. The trick is to compute which bin each x,y point falls into (xi,yi), then test if H[xi,yi] (actually, in my case H[yi,xi]) is beneath the threshold. The code is below, and runs very fast for large numbers of points and is much cleaner:

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')