高效地创建高密度区域的密度图，稀疏区域的点

Question

我需要绘制一个图表，用于绘图上高密度区域的密度图，但低于某个阈值时会使用单个点。我找不到任何现有的代码，看起来类似于我在matplotlib缩略图库或谷歌搜索中需要的代码。我有一个自己写的工作代码，但是当点数/点数很大时，它有点棘手，并且（更重要的是）需要很长的时间，这是无法接受的。下面是代码：

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') 

此代码生成看起来像这样的图像：

然而，在较大的数据使用时，将设置程序需要几秒钟到几分钟。任何想法如何加快这一点？谢谢！

Answer 1

这应做到：

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

您的问题是二次的 - 对于npts = 1000，您的数组大小达到10^6个点，并且您使用列表解析遍历这些列表。
现在，这是一个当然的味道问题，但我发现列表理解可以产生一个难以遵循的完全代码，并且它们有时只是稍微快一点......但这不是我的观点。
我的观点是，对于大数组操作你有一个像numpy的功能：

np.where, np.choose etc. 

见，你可以实现与NumPy的列表内涵的这个功能，你的代码应该跑得更快。

我理解正确吗？您的评论？

#Find all points that lie in these regions 

你在测试多边形内的点吗？如果是这样，请考虑在matplotlib中使用point in polygon。

Answer 3

经过一个晚上睡觉，并通过Oz123的建议阅读，我想出了。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，在我的情况下是H [yi，xi]）是否低于阈值。代码如下，并且运行速度非常快的大数目的点，是更清洁：

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') 

Answer 4

为了记录在案，这里是使用scipy.stats.gaussian_kde而不是2D直方图的新尝试的结果。 根据目的，可以设想不同的颜色啮合和轮廓组合。

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()