下面的代码是一个双变量高斯分布。通过调整COV matrix以考虑特定变量来产生分布。具体来说，每个XY坐标都应用一个半径（[_Rad]）。然后通过COV因子（matrix）调整scaling [_Scaling]，以扩展x-direction中的半径并收缩y-direction中的半径。其方向由rotation angle（[_Rotation]）度量。输出表示为probability函数。
问题。半径应仅覆盖设定区域。当我尝试将脚本转换为一组坐标（位于底部的链接）时，概率会扩展到整个框架。您会看到颜色的闪烁，它表示交替出现的概率。但是坐标的半径范围是8-25。该区域应固定或固定。它不应该扩展整个框架
我试图将这些区域修复为0.5，但这不是问题。我希望更改代码，以使概率仅受所提供半径的影响。
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
from matplotlib.animation import FuncAnimation
#Create data limits used for the animation frame. Bit rough
DATA_LIMITS = [-100, 100]
def datalimits(*data):
return DATA_LIMITS # dmin - spad, dmax + spad
#This is the function used for the rotation matrix
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
#Used for the covariance matrix
def getcov(radius=1, scale=1, theta=0):
cov = np.array([
[radius*(scale + 1), 0],
[0, radius/(scale + 1)]
])
r = rot(theta)
return r @ cov @ r.T
#This is the multivariate probability distribution function
def mvpdf(x, y, xlim, ylim, radius=1, velocity=0, scale=0, theta=0):
X,Y = np.meshgrid(np.linspace(*xlim), np.linspace(*ylim))
XY = np.stack([X, Y], 2)
x,y = rot(theta) @ (velocity/2, 0) + (x, y)
cov = getcov(radius=radius, scale=scale, theta=theta)
PDF = sts.multivariate_normal([x, y], cov).pdf(XY)
return X, Y, PDF
#Used for the animation function
def mvpdfs(xs, ys, xlim, ylim, radius=None, velocity=None, scale=None, theta=None):
PDFs = []
for i,(x,y) in enumerate(zip(xs,ys)):
kwargs = {
'radius': radius[i] if radius is not None else 1,
'velocity': velocity[i] if velocity is not None else 0,
'scale': scale[i] if scale is not None else 0,
'theta': theta[i] if theta is not None else 0,
'xlim': xlim,
'ylim': ylim
}
X, Y, PDF = mvpdf(x, y,**kwargs)
PDFs.append(PDF)
return X, Y, np.sum(PDFs, axis=0)
fig, ax = plt.subplots(figsize = (10,4))
ax.set_xlim(DATA_LIMITS)
ax.set_ylim(DATA_LIMITS)
#animate the scatter points
line_a, = ax.plot([], [], '.', c='red', alpha = 0.5, markersize=5, animated=True)
line_b, = ax.plot([], [], '.', c='blue', alpha = 0.5, markersize=5, animated=True)
cfs = None
def plotmvs(tdf, xlim=None, ylim=None, fig=fig, ax=ax):
global cfs
if cfs:
for tp in cfs.collections:
tp.remove()
df = tdf[1]
if xlim is None: xlim = datalimits(df['X'])
if ylim is None: ylim = datalimits(df['Y'])
PDFs = []
for (group, gdf), group_line in zip(df.groupby('group'), (line_a, line_b)):
# Update the scatter line data
group_line.set_data(*gdf[['X','Y']].values.T)
kwargs = {
'radius': gdf['Radius'].values if 'Radius' in gdf else None,
'velocity': gdf['Velocity'].values if 'Velocity' in gdf else None,
'scale': gdf['Scaling'].values if 'Scaling' in gdf else None,
'theta': gdf['Rotation'].values if 'Rotation' in gdf else None,
'xlim': xlim,
'ylim': ylim
}
X, Y, PDF = mvpdfs(gdf['X'].values, gdf['Y'].values, **kwargs)
PDFs.append(PDF)
#I've played around with these functions a bit. This is
#where I think the probability subtraction from both teams results
#in the uneven or _flickering_ background probability
PDF = PDFs[0] - PDFs[1]
normPDF = PDF - PDF.min()
normPDF = normPDF / normPDF.max()
#This function attempted to _fix_ the background
#to 0.5 or neutral but you can still see the areas
#not covered by scatter points is still a different
#probability
#normPDF = PDF * .5/max(PDF.max(), -PDF.min()) + .5
#create the contour
cfs = ax.contourf(X, Y, normPDF, cmap='viridis', alpha = 0.8,
levels=10)
return cfs.collections + [line_a, line_b]
#This big data frame houses the XY coordinates, Radius, Scaling factor
#Rotation for each respective frame
n = 10
time = range(n)
d = ({
'A1_X' : [13.3,13.16,12.99,12.9,12.79,12.56,12.32,12.15,11.93,11.72],
'A1_Y' : [26.12,26.44,26.81,27.18,27.48,27.82,28.13,28.37,28.63,28.93],
'A2_X' : [6.97,6.96,7.03,6.98,6.86,6.76,6.55,6.26,6.09,5.9],
'A2_Y' : [10.92,10.83,10.71,10.52,10.22,10.02,9.86,9.7,9.54,9.37],
'A3_X' : [-31.72,-31.93,-32.18,-32.43,-32.7,-32.89,-33.15,-33.51,-33.84,-34.17],
'A3_Y' : [21.25,21.52,21.7,21.98,22.25,22.47,22.7,22.95,23.2,23.4],
'A4_X' : [37.54,37.42,37.3,37.14,36.97,36.77,36.56,36.37,36.13,35.89],
'A4_Y' : [7.31,7.35,7.38,7.43,7.5,7.58,7.65,7.68,7.69,7.69],
'A5_X' : [-5.37,-5.31,-5.28,-5.34,-5.41,-5.42,-5.68,-5.84,-6.1,-6.31],
'A5_Y' : [-5.42,-5.7,-6,-6.15,-6.41,-6.67,-6.88,-7.11,-7.33,-7.49],
'A6_X' : [-3.33,-3.15,-2.97,-2.94,-2.88,-2.79,-2.69,-2.66,-2.54,-2.67],
'A6_Y' : [13.69,13.86,14.09,14.34,14.73,15.01,15.38,15.83,16.15,16.73],
'A7_X' : [-4.4,-4.56,-4.83,-5.02,-5.18,-5.51,-5.81,-6.03,-6.31,-6.7],
'A7_Y' : [21.34,21.53,21.69,21.89,22.03,22.35,22.63,22.91,23.14,23.34],
'A8_X' : [-14.89,-15.12,-15.26,-15.52,-15.96,-16.37,-16.7,-17.08,-17.55,-17.95],
'A8_Y' : [3.7,3.41,3.14,2.84,2.58,2.26,2.07,1.78,1.45,1.23],
'A9_X' : [-51.92,-52.04,-52.15,-52.26,-52.36,-52.54,-52.76,-52.98,-53.17,-53.4],
'A9_Y' : [16.45,16.44,16.5,16.61,16.59,16.52,16.52,16.43,16.45,16.49],
'A10_X' : [-15.18,-15.18,-15.18,-15.18,-15.18,-15.18,-15.18,-15.18,-15.18,-15.18],
'A10_Y' : [26.02,26.02,26.02,26.02,26.02,26.02,26.02,26.02,26.02,26.02],
'A11_X' : [15.5,15.22,14.9,14.59,14.36,14.08,13.74,13.43,13.13,12.82],
'A11_Y' : [7.25,7.36,7.51,7.61,7.72,7.88,8.05,8.18,8.5,8.8],
'A12_X' : [-5.36,-5.35,-5.33,-5.28,-5.18,-5.12,-4.99,-4.83,-4.8,-4.71],
'A12_Y' : [19.02,18.77,18.56,18.41,18.22,18.03,17.9,17.72,17.69,17.58],
'A13_X' : [-45.76,-45.91,-46.13,-46.41,-46.62,-46.82,-47.07,-47.35,-47.61,-47.87],
'A13_Y' : [18.9,18.96,19.03,19.12,19.12,19.18,19.31,19.42,19.45,19.53],
'A14_X' : [-10.28,-10.3,-10.23,-10.36,-10.53,-10.69,-10.84,-10.95,-11.17,-11.37],
'A14_Y' : [18.25,18.42,18.56,18.73,18.86,18.98,19.02,19.19,19.3,19.46],
'A15_X' : [29.77,29.6,29.45,29.24,28.9,28.68,28.42,28.06,27.75,27.49],
'A15_Y' : [11.59,11.38,11.19,11.02,10.85,10.71,10.58,10.39,10.18,9.98],
'B1_X' : [38.35,38.1,37.78,37.55,37.36,37.02,36.78,36.46,36.21,35.79],
'B1_Y' : [12.55,12.58,12.58,12.55,12.5,12.47,12.43,12.48,12.44,12.44],
'B2_X' : [14.6,14.38,14.16,13.8,13.45,13.11,12.71,12.3,12.06,11.61],
'B2_Y' : [4.66,4.44,4.24,4.1,4.01,3.84,3.67,3.56,3.44,3.47],
'B3_X' : [-12.16,-12.35,-12.53,-12.73,-12.91,-13.01,-13.24,-13.44,-13.68,-13.93],
'B3_Y' : [20.07,20.26,20.34,20.5,20.62,20.69,20.72,20.73,20.63,20.58],
'B4_X' : [-3.27,-3.1,-2.83,-2.49,-2.34,-2.13,-1.97,-1.8,-1.67,-1.59],
'B4_Y' : [-6.25,-6.37,-6.52,-6.61,-6.76,-6.89,-7.01,-7.1,-7.13,-7.33],
'B5_X' : [-21.47,-21.63,-21.84,-22.03,-22.28,-22.53,-22.77,-22.99,-23.27,-23.52],
'B5_Y' : [8.94,8.87,8.79,8.68,8.61,8.56,8.48,8.35,8.22,8.12],
'B6_X' : [-13.81,-13.83,-13.91,-14.02,-14.15,-14.31,-14.54,-14.77,-14.96,-15.24],
'B6_Y' : [25.45,25.81,25.94,26.26,26.56,26.75,26.92,27.07,27.22,27.25],
'B7_X' : [-6.28,-6.33,-6.43,-6.44,-6.61,-6.8,-7.02,-7.22,-7.46,-7.7],
'B7_Y' : [13.82,13.6,13.43,13.26,13.12,13.09,13.07,13.14,13.19,13.32],
'B8_X' : [28.39,28.09,27.91,27.76,27.4,27.14,26.91,26.69,26.34,26.1],
'B8_Y' : [8.36,8.2,8.13,8.1,8.01,7.94,7.84,7.76,7.8,7.84],
'B9_X' : [-7.55,-7.54,-7.57,-7.65,-7.77,-7.87,-8.01,-8.06,-8.06,-8.06],
'B9_Y' : [17.98,17.94,17.97,18.02,18.05,18.09,18.07,18.02,17.97,17.92],
'B10_X' : [-32.36,-32.63,-32.92,-33.25,-33.54,-33.78,-34.13,-34.37,-34.69,-35.01],
'B10_Y' : [13.27,13.48,13.67,13.9,14.14,14.48,14.76,15.05,15.31,15.62],
'B11_X' : [-44.08,-44.19,-44.33,-44.47,-44.64,-44.78,-44.92,-45.16,-45.36,-45.56],
'B11_Y' : [15.9,16.09,16.22,16.38,16.49,16.63,16.7,16.79,16.85,16.94],
'B12_X' : [-16.47,-16.67,-16.76,-16.86,-16.99,-17.24,-17.48,-17.76,-17.98,-18.29],
'B12_Y' : [29.76,29.96,30.07,30.3,30.45,30.59,30.61,30.67,30.62,30.66],
'B13_X' : [-50.27,-50.38,-50.55,-50.74,-50.92,-51.02,-51.13,-51.3,-51.46,-51.65],
'B13_Y' : [16.31,16.3,16.31,16.33,16.36,16.28,16.25,16.22,16.21,16.27],
'B14_X' : [-15.55,-15.81,-16.05,-16.35,-16.67,-16.96,-17.35,-17.76,-18.09,-18.6],
'B14_Y' : [8.56,8.53,8.54,8.57,8.62,8.6,8.58,8.49,8.44,8.55],
'B15_X' : [9.79,9.47,9.2,8.77,8.41,8.07,7.65,7.19,6.76,6.42],
'B15_Y' : [27.61,27.79,27.99,28.16,28.37,28.53,28.68,28.82,28.9,29.06],
'A1_Radius' : [10.33,10.34,10.34,10.37,10.38,10.37,10.36,10.36,10.35,10.35],
'A2_Radius' : [9.05,9.06,9.07,9.08,9.09,9.09,9.08,9.06,9.05,9.04],
'A3_Radius' : [13.04,13.15,13.29,13.44,13.6,13.72,13.88,14.1,14.31,14.52],
'A4_Radius' : [25,25,25,25,25,25,25,25,25,24.81],
'A5_Radius' : [11.24,11.33,11.44,11.49,11.59,11.68,11.77,11.86,11.95,12.02],
'A6_Radius' : [8.19,8.19,8.18,8.17,8.15,8.14,8.13,8.11,8.11,8.09],
'A7_Radius' : [8.18,8.19,8.2,8.21,8.22,8.25,8.27,8.29,8.31,8.33],
'A8_Radius' : [9.71,9.79,9.85,9.94,10.05,10.17,10.26,10.38,10.53,10.65],
'A9_Radius' : [25,25,25,25,25,25,25,25,25,25],
'A10_Radius' : [9.08,9.08,9.08,9.08,9.08,9.08,9.08,9.08,9.08,9.08],
'A11_Radius' : [11.12,11.03,10.91,10.81,10.74,10.64,10.54,10.45,10.34,10.23],
'A12_Radius' : [8.07,8.06,8.05,8.05,8.04,8.03,8.02,8.01,8.01,8.01],
'A13_Radius' : [24.16,24.36,24.62,24.98,25,25,25,25,25,25],
'A14_Radius' : [8.3,8.3,8.3,8.31,8.32,8.33,8.34,8.35,8.37,8.39],
'A15_Radius' : [17.86,17.75,17.66,17.52,17.28,17.14,16.96,16.73,16.53,16.38],
'B1_Radius' : [25,25,25,25,25,25,24.84,24.45,24.15,23.65],
'B2_Radius' : [11.3,11.28,11.26,11.19,11.11,11.06,10.99,10.91,10.88,10.77],
'B3_Radius' : [8.46,8.48,8.5,8.52,8.54,8.55,8.57,8.59,8.61,8.63],
'B4_Radius' : [11.54,11.59,11.65,11.69,11.75,11.81,11.86,11.9,11.92,12],
'B5_Radius' : [10.08,10.13,10.18,10.24,10.3,10.37,10.43,10.5,10.59,10.66],
'B6_Radius' : [8.89,8.92,8.94,8.98,9.02,9.06,9.1,9.14,9.18,9.21],
'B7_Radius' : [8.19,8.2,8.22,8.23,8.24,8.24,8.25,8.25,8.26,8.26],
'B8_Radius' : [17.34,17.15,17.03,16.93,16.69,16.52,16.38,16.25,16.01,15.84],
'B9_Radius' : [8.13,8.13,8.14,8.14,8.15,8.15,8.16,8.16,8.16,8.16],
'B10_Radius' : [13.38,13.5,13.64,13.8,13.94,14.05,14.23,14.35,14.53,14.7],
'B11_Radius' : [22.24,22.35,22.5,22.65,22.84,23,23.16,23.44,23.67,23.9],
'B12_Radius' : [9.68,9.74,9.77,9.82,9.86,9.92,9.96,10.02,10.05,10.1],
'B13_Radius' : [25,25,25,25,25,25,25,25,25,25],
'B14_Radius' : [9.17,9.2,9.23,9.26,9.3,9.34,9.4,9.47,9.52,9.59],
'B15_Radius' : [9.8,9.76,9.74,9.7,9.67,9.63,9.59,9.55,9.5,9.48],
'A1_Scaling' : [0,0.07,0.1,0.09,0.06,0.1,0.09,0.05,0.07,0.08],
'A2_Scaling' : [0,0.01,0.01,0.02,0.06,0.03,0.04,0.07,0.03,0.04],
'A3_Scaling' : [0,0.07,0.06,0.08,0.09,0.05,0.07,0.11,0.1,0.09],
'A4_Scaling' : [0,0.01,0.01,0.02,0.02,0.03,0.03,0.02,0.03,0.04],
'A5_Scaling' : [0,0.05,0.05,0.02,0.04,0.04,0.07,0.05,0.07,0.04],
'A6_Scaling' : [0,0.04,0.05,0.04,0.09,0.05,0.09,0.12,0.07,0.21],
'A7_Scaling' : [0,0.04,0.06,0.05,0.03,0.13,0.1,0.07,0.08,0.11],
'A8_Scaling' : [0,0.08,0.06,0.09,0.16,0.16,0.08,0.14,0.19,0.12],
'A9_Scaling' : [0,0.01,0.01,0.02,0.01,0.02,0.03,0.03,0.02,0.03],
'A10_Scaling' : [0,0,0,0,0,0,0,0,0,0],
'A11_Scaling' : [0,0.05,0.07,0.06,0.04,0.06,0.09,0.06,0.11,0.11],
'A12_Scaling' : [0,0.04,0.03,0.02,0.02,0.02,0.02,0.03,0,0.01],
'A13_Scaling' : [0,0.02,0.03,0.05,0.03,0.03,0.05,0.05,0.04,0.04],
'A14_Scaling' : [0,0.02,0.02,0.03,0.03,0.02,0.01,0.03,0.03,0.04],
'A15_Scaling' : [0,0.04,0.03,0.04,0.08,0.04,0.05,0.1,0.08,0.06],
'B1_Scaling' : [0,0.04,0.06,0.03,0.02,0.07,0.04,0.06,0.04,0.11],
'B2_Scaling' : [0,0.06,0.05,0.09,0.08,0.08,0.11,0.1,0.04,0.12],
'B3_Scaling' : [0,0.04,0.02,0.04,0.03,0.01,0.03,0.02,0.04,0.04],
'B4_Scaling' : [0,0.03,0.06,0.07,0.03,0.03,0.02,0.02,0.01,0.03],
'B5_Scaling' : [0,0.02,0.03,0.03,0.04,0.04,0.04,0.04,0.06,0.04],
'B6_Scaling' : [0,0.08,0.01,0.07,0.06,0.04,0.05,0.05,0.04,0.05],
'B7_Scaling' : [0,0.03,0.02,0.02,0.03,0.02,0.03,0.02,0.04,0.04],
'B8_Scaling' : [0,0.07,0.02,0.01,0.08,0.04,0.04,0.03,0.07,0.04],
'B9_Scaling' : [0,0,0,0.01,0.01,0.01,0.01,0,0,0],
'B10_Scaling' : [0,0.07,0.07,0.09,0.08,0.11,0.12,0.09,0.1,0.11],
'B11_Scaling' : [0,0.03,0.02,0.03,0.03,0.02,0.02,0.04,0.03,0.03],
'B12_Scaling' : [0,0.05,0.01,0.04,0.02,0.05,0.04,0.05,0.03,0.05],
'B13_Scaling' : [0,0.01,0.02,0.02,0.02,0.01,0.01,0.02,0.01,0.02],
'B14_Scaling' : [0,0.04,0.04,0.05,0.06,0.05,0.09,0.1,0.07,0.16],
'B15_Scaling' : [0,0.08,0.06,0.13,0.1,0.08,0.11,0.14,0.12,0.08],
'A1_Rotation' : [0,112.81,114.01,110.56,110.6,113.37,116.02,116.99,118.62,119.38],
'A2_Rotation' : [0,-94.27,-73.02,-87.73,-98.57,-102.94,-111.2,-119.95,-122.41,-124.66],
'A3_Rotation' : [0,128.47,135.84,134.06,134.75,133.94,134.69,136.59,137.5,138.85],
'A4_Rotation' : [0,164.64,165.45,163.48,161.42,160.63,161.31,162.54,164.99,167.17],
'A5_Rotation' : [0,-78.78,-81.19,-87.73,-92.23,-92.33,-101.96,-105.48,-111.09,-114.38],
'A6_Rotation' : [0,43.84,48.03,59.34,66.72,67.83,69.48,72.62,72.2,77.81],
'A7_Rotation' : [0,131.01,141.7,138.59,138.53,137.8,137.64,136.12,136.75,139.03],
'A8_Rotation' : [0,-127.32,-123.16,-125.78,-133.66,-135.66,-137.83,-138.76,-139.64,-141.01],
'A9_Rotation' : [0,-173.6,166.75,154.39,162.16,173.26,175.63,-178.69,-179.76,178.52],
'A10_Rotation' : [0,0,0,0,0,0,0,0,0,0],
'A11_Rotation' : [0,159.85,156.67,158.48,157.77,156.11,155.54,155.77,152.28,149.97],
'A12_Rotation' : [0,-87.73,-86.34,-82.59,-77.52,-76.38,-71.91,-67.87,-66.98,-65.81],
'A13_Rotation' : [0,157.86,160.03,161.3,165.63,165.1,162.84,162.02,163.42,163.4],
'A14_Rotation' : [0,96.06,80.08,98.93,112.07,119.3,125.62,125.42,130.18,131.8],
'A15_Rotation' : [0,-128.97,-128.41,-133.2,-139.65,-141,-143.05,-144.79,-145.08,-144.84],
'B1_Rotation' : [0,172.04,176.94,179.74,-177.22,-176.59,-175.81,-178,-177.26,-177.53],
'B2_Rotation' : [0,-135.52,-136.05,-144.97,-150.41,-151.25,-152.35,-154.52,-154.34,-158.27],
'B3_Rotation' : [0,136.51,144.7,143.29,144.27,144.21,149.19,152.95,159.91,164.19],
'B4_Rotation' : [0,-34.26,-30.8,-24.57,-28.69,-29.15,-30.36,-29.83,-28.55,-32.58],
'B5_Rotation' : [0,-157.4,-157.25,-155.34,-157.7,-160.03,-160.33,-158.82,-158.14,-158.21],
'B6_Rotation' : [0,92.27,101.22,104.5,106.63,110.85,116.25,120.35,122.84,128.35],
'B7_Rotation' : [0,-105.22,-111.46,-106.51,-115.8,-125.96,-134.82,-144.07,-152.07,-160.76],
'B8_Rotation' : [0,-151.19,-154.33,-157.13,-160.12,-161.43,-160.74,-160.39,-164.53,-167.14],
'B9_Rotation' : [0,-73.61,-155.92,158.51,161.67,160.64,169.2,175.38,-178.71,-172.8],
'B10_Rotation' : [0,142.69,144.41,144.93,143.57,139.61,139.91,138.54,138.86,138.47],
'B11_Rotation' : [0,119.54,127.13,128.87,133.1,133.59,136.35,140.45,143.34,144.72],
'B12_Rotation' : [0,134.03,132.68,125.29,126.67,132.59,139.67,144.84,150.2,153.44],
'B13_Rotation' : [0,-177.72,178.64,176.87,175.25,-177.73,-175.96,-175.29,-175.61,-178.46],
'B14_Rotation' : [0,-173.78,-177.74,179.11,176.83,178.31,179.19,-178.19,-177.33,-179.89],
'B15_Rotation' : [0,152.39,147.79,151.7,151.38,151.96,153.65,155.22,157.01,156.86],
})
tuples = [((t, k.split('_')[0][0], int(k.split('_')[0][1:]), k.split('_')[1]), v[i])
for k,v in d.items() for i,t in enumerate(time)]
df = pd.Series(dict(tuples)).unstack(-1)
df.index.names = ['time', 'group', 'id']
interval_ms = 200
delay_ms = 1000
ani = FuncAnimation(fig, plotmvs, frames=df.groupby('time'),
blit=True, interval=interval_ms, repeat_delay=delay_ms)
plt.show()
答案 0 :(得分：1)
您的问题实际上并不在于概率分布函数的空间范围，而在于您用来显示合成函数的归一化。特别是，零值意味着零概率，并且如果该值与每一帧中的相同颜色相关联，那当然是有利的。您在代码中使用的规范化，即
normPDF = PDF - PDF.min()
normPDF = normPDF / normPDF.max()
不保证此属性。例如，请考虑PDF的最小值和最大值
分别为-0.8和0.9。转换后，normPDF的最小值和最大值将为0和1，即您进行转换将执行-0.8 --> 0和0.9 --> 1。同时，原始的零概率也正在转换：0 --> 0.8/(0.9+0.8) = 0.47058823529411764。由于分布函数在每一帧中都会发生变化，因此最大值也会发生变化，因此零概率将在每一帧中转换为不同的值。
为避免这种情况，最好使用保留零概率的规范化。我想到的最简单的变换是将概率归一化为间隔[-1,1]（因为相互减去概率分布，因此-1），这可以通过将分布除以最大值得到。概率：
normPDF = (PDFs[0]-PDFs[1])/max(PDFs[0].max(),PDFs[1].max())
通过这种归一化，normPDF中的任何值都不会小于-1或大于1。但是请注意，normPDF不一定涵盖从-1到1的所有值，而是局限于这些值。此外，根据定义，零概率始终保持零。现在，分布已经归一化了，我们仍然需要注意，零概率始终被分配为相同的颜色。这可以通过告诉contourf等高线图的级别来完成。这可以使用keyword级别来完成，该级别可以是整数（然后仅轮廓的数量是固定的；在此不够）或包含所有所需级别的可迭代项。最简单的方法是等距排列的级别，np.linspace(-1,1,n)（您要使用的级别是n）将仅在归一化限制之间绘制级别：
cfs = ax.contourf(X, Y, normPDF, cmap='viridis', alpha = 1, levels=np.linspace(-1,1,10))
警告：如果您确实要对每个帧分别进行标准化，就像现在在代码中那样，分配给某些颜色的值（0除外）很可能会发生变化从一帧到另一帧。这是因为PDF的最小值和最大值会发生变化。如果您还打算在图旁边显示一个颜色条，则还必须更新每个帧的颜色条。但是，这将使动画很难遵循。相反，我建议首先计算 all 帧的PDF，计算所有帧的最大值，然后使用此“全局”最大值立即归一化所有帧。这样，您只需要绘制一次颜色条。