使用 matplotlib 绘制缩放和旋转的双变量分布
Plot scaled and rotated bivariate distribution using matplotlib
我正在尝试 plot bivariate gaussian distribution 使用 matplotlib。我想使用两个 scatter 点（A 组）、（B 组）的 xy 坐标来执行此操作。
我想通过调整 COV matrix 来调整 distribution 以说明每个组 velocity 及其与附加 xy 坐标的距离用作参考点。
我计算了每组xy坐标到参考点坐标的距离。距离表示为radius，标记为[GrA_Rad]，[GrB_Rad]。
所以他们离参考点越远，radius就越大。我还计算了标记为 [GrA_Vel]、[GrB_Vel] 的 velocity。每组的direction表示为orientation。这被标记为 [GrA_Rotation]、[GrB_Rotation]
关于distribution如何针对velocity和距离(radius)进行调整的问题：
我希望使用 SVD。具体来说，如果我有每个 scatter 的 rotation angle，这将提供 direction。 velocity可以用来描述一个scalingmatrix[GrA_Scaling],[GrB_Scaling]。所以这个scalingmatrix可以用来扩大x-direction中的radius，收缩y-direction中的radius。这表示COVmatrix。
最后，distribution mean 值是通过将组 location (x,y) 平移 velocity.
的一半得到的
简单地说：radius应用于每个组的scatter点。 COV矩阵由radius和velocity调整。因此，使用 scaling matrix 扩展 x-direction 中的 radius 并收缩 y-direction 中的 radius。 direction 是从 rotation angle 测得的。然后通过将组位置 (x,y) 平移 velocity.
的一半来确定 distribution mean 值
下面是这些变量的df
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data = d)
我已经为每个 xy 坐标制作了一个 animated plot。
GrA_X = [10,12,17,16,16,14,12,8]
GrA_Y = [10,12,13,7,6,7,8,8]
GrB_X = [5,8,13,16,19,15,13,5]
GrB_Y = [6,15,12,10,8,9,10,8]
Item_X = [6,8,14,18,13,11,16,15]
Item_Y = [10,12,8,12,15,12,10,8]
scatter_GrA = ax.scatter(GrA_X, GrA_Y)
scatter_GrB = ax.scatter(GrB_X, GrB_Y)
scatter_Item = ax.scatter(Item_X, Item_Y)
def animate(i) :
scatter_GrA.set_offsets([[GrA_X[0+i], GrA_Y[0+i]]])
scatter_GrB.set_offsets([[GrB_X[0+i], GrB_Y[0+i]]])
scatter_Item.set_offsets([[Item_X[0+i], Item_Y[0+i]]])
ani = animation.FuncAnimation(fig, animate, np.arange(0,9),
interval = 1000, blit = False)
更新
问题已更新，并且变得更加清晰。我已经更新了我的代码以匹配。这是最新的输出：
除了样式，我认为这与 OP 描述的相符。
这是用于生成上述图的代码：
dfake = ({
'GrA_X' : [15,15],
'GrA_Y' : [15,15],
'Reference_X' : [15,3],
'Reference_Y' : [15,15],
'GrA_Rad' : [15,25],
'GrA_Vel' : [0,10],
'GrA_Scaling' : [0,0.5],
'GrA_Rotation' : [0,45]
})
dffake = pd.DataFrame(dfake)
fig,axs = plt.subplots(1, 2, figsize=(16,8))
fig.subplots_adjust(0,0,1,1)
plotone(dffake, 'A', 0, xlim=(0,30), ylim=(0,30), fig=fig, ax=axs[0])
plotone(dffake, 'A', 1, xlim=(0,30), ylim=(0,30), fig=fig, ax=axs[1])
plt.show()
我使用的 plotone 函数的完整实现在下面的代码块中。如果您只想了解用于生成和转换二维高斯 PDF 的数学，请查看 mvpdf 函数（以及它所依赖的 rot 和 getcov 函数）：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
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
def mvpdf(x, y, xlim, ylim, radius=1, velocity=0, scale=0, theta=0):
"""Creates a grid of data that represents the PDF of a multivariate gaussian.
x, y: The center of the returned PDF
(xy)lim: The extent of the returned PDF
radius: The PDF will be dilated by this factor
scale: The PDF be stretched by a factor of (scale + 1) in the x direction, and squashed by a factor of 1/(scale + 1) in the y direction
theta: The PDF will be rotated by this many degrees
returns: X, Y, PDF. X and Y hold the coordinates of the PDF.
"""
# create the coordinate grids
X,Y = np.meshgrid(np.linspace(*xlim), np.linspace(*ylim))
# stack them into the format expected by the multivariate pdf
XY = np.stack([X, Y], 2)
# displace xy by half the velocity
x,y = rot(theta) @ (velocity/2, 0) + (x, y)
# get the covariance matrix with the appropriate transforms
cov = getcov(radius=radius, scale=scale, theta=theta)
# generate the data grid that represents the PDF
PDF = sts.multivariate_normal([x, y], cov).pdf(XY)
return X, Y, PDF
def plotmv(x, y, xlim=None, ylim=None, radius=1, velocity=0, scale=0, theta=0, xref=None, yref=None, fig=None, ax=None):
"""Plot an xy point with an appropriately tranformed 2D gaussian around it.
Also plots other related data like the reference point.
"""
if xlim is None: xlim = (x - 5, x + 5)
if ylim is None: ylim = (y - 5, y + 5)
if fig is None:
fig = plt.figure(figsize=(8,8))
ax = fig.gca()
elif ax is None:
ax = fig.gca()
# plot the xy point
ax.plot(x, y, '.', c='C0', ms=20)
if not (xref is None or yref is None):
# plot the reference point, if supplied
ax.plot(xref, yref, '.', c='w', ms=12)
# plot the arrow leading from the xy point
if velocity > 0:
ax.arrow(x, y, *rot(theta) @ (velocity, 0),
width=.4, length_includes_head=True, ec='C0', fc='C0')
# fetch the PDF of the 2D gaussian
X, Y, PDF = mvpdf(x, y, xlim=xlim, ylim=ylim, radius=radius, velocity=velocity, scale=scale, theta=theta)
# normalize PDF by shifting and scaling, so that the smallest value is 0 and the largest is 1
normPDF = PDF - PDF.min()
normPDF = normPDF/normPDF.max()
# plot and label the contour lines of the 2D gaussian
cs = ax.contour(X, Y, normPDF, levels=6, colors='w', alpha=.5)
ax.clabel(cs, fmt='%.3f', fontsize=12)
# plot the filled contours of the 2D gaussian. Set levels high for smooth contours
cfs = ax.contourf(X, Y, normPDF, levels=50, cmap='viridis', vmin=-.9, vmax=1)
# create the colorbar and ensure that it goes from 0 -> 1
cbar = fig.colorbar(cfs, ax=ax)
cbar.set_ticks([0, .2, .4, .6, .8, 1])
# add some labels
ax.grid()
ax.set_xlabel('X distance (M)')
ax.set_ylabel('Y distance (M)')
# ensure that x vs y scaling doesn't disrupt the transforms applied to the 2D gaussian
ax.set_aspect('equal', 'box')
return fig, ax
def fetchone(df, l, i, **kwargs):
"""Fetch all the needed data for one xy point
"""
keytups = (
('x', 'Gr%s_X'%l),
('y', 'Gr%s_Y'%l),
('radius', 'Gr%s_Rad'%l),
('velocity', 'Gr%s_Vel'%l),
('scale', 'Gr%s_Scaling'%l),
('theta', 'Gr%s_Rotation'%l),
('xref', 'Reference_X'),
('yref', 'Reference_Y')
)
ret = {k:df.loc[i, l] for k,l in keytups}
# add in any overrides
ret.update(kwargs)
return ret
def plotone(df, l, i, xlim=None, ylim=None, fig=None, ax=None, **kwargs):
"""Plot exactly one point from the dataset
"""
# look up all the data to plot one datapoint
xydata = fetchone(df, l, i, **kwargs)
# do the plot
return plotmv(xlim=xlim, ylim=ylim, fig=fig, ax=ax, **xydata)
旧答案-2
我已经调整了我的答案以匹配 OP 发布的示例：
生成上图的代码如下：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
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
def datalimits(*data, pad=.15):
dmin,dmax = min(d.min() for d in data), max(d.max() for d in data)
spad = pad*(dmax - dmin)
return dmin - spad, dmax + spad
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data=d)
limitpad = .5
clevels = 5
cflevels = 50
xmin,xmax = datalimits(df['GrA_X'], df['GrB_X'], pad=limitpad)
ymin,ymax = datalimits(df['GrA_Y'], df['GrB_Y'], pad=limitpad)
X,Y = np.meshgrid(np.linspace(xmin, xmax), np.linspace(ymin, ymax))
fig = plt.figure(figsize=(10,6))
ax = plt.gca()
Zs = []
for l,color in zip('AB', ('red', 'yellow')):
# plot all of the points from a single group
ax.plot(df['Gr%s_X'%l], df['Gr%s_Y'%l], '.', c=color, ms=15, label=l)
Zrows = []
for _,row in df.iterrows():
x,y = row['Gr%s_X'%l], row['Gr%s_Y'%l]
cov = getcov(radius=row['Gr%s_Rad'%l], scale=row['Gr%s_Scaling'%l], theta=row['Gr%s_Rotation'%l])
mnorm = sts.multivariate_normal([x, y], cov)
Z = mnorm.pdf(np.stack([X, Y], 2))
Zrows.append(Z)
Zs.append(np.sum(Zrows, axis=0))
# plot the reference points
# create Z from the difference of the sums of the 2D Gaussians from group A and group B
Z = Zs[0] - Zs[1]
# normalize Z by shifting and scaling, so that the smallest value is 0 and the largest is 1
normZ = Z - Z.min()
normZ = normZ/normZ.max()
# plot and label the contour lines
cs = ax.contour(X, Y, normZ, levels=clevels, colors='w', alpha=.5)
ax.clabel(cs, fmt='%2.1f', colors='w')#, fontsize=14)
# plot the filled contours. Set levels high for smooth contours
cfs = ax.contourf(X, Y, normZ, levels=cflevels, cmap='viridis', vmin=0, vmax=1)
# create the colorbar and ensure that it goes from 0 -> 1
cbar = fig.colorbar(cfs, ax=ax)
cbar.set_ticks([0, .2, .4, .6, .8, 1])
ax.set_aspect('equal', 'box')
旧答案-1
很难确切地说出您在寻找什么。可以通过协方差矩阵缩放和旋转多元高斯分布。以下是如何根据您的数据执行此操作的示例：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
def getcov(scale, theta):
cov = np.array([
[1*(scale + 1), 0],
[0, 1/(scale + 1)]
])
r = rot(theta)
return r @ cov @ r.T
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data=d)
xmin,xmax = min(df['GrA_X'].min(), df['GrB_X'].min()), max(df['GrA_X'].max(), df['GrB_X'].max())
ymin,ymax = min(df['GrA_Y'].min(), df['GrB_Y'].min()), max(df['GrA_Y'].max(), df['GrB_Y'].max())
X,Y = np.meshgrid(
np.linspace(xmin - (xmax - xmin)*.1, xmax + (xmax - xmin)*.1),
np.linspace(ymin - (ymax - ymin)*.1, ymax + (ymax - ymin)*.1)
)
fig,axs = plt.subplots(df.shape[0], sharex=True, figsize=(4, 4*df.shape[0]))
fig.subplots_adjust(0,0,1,1,0,-.82)
for (_,row),ax in zip(df.iterrows(), axs):
for c in 'AB':
x,y = row['Gr%s_X'%c], row['Gr%s_Y'%c]
cov = getcov(scale=row['Gr%s_Scaling'%c], theta=row['Gr%s_Rotation'%c])
mnorm = sts.multivariate_normal([x, y], cov)
Z = mnorm.pdf(np.stack([X, Y], 2))
ax.contour(X, Y, Z)
ax.plot(row['Gr%s_X'%c], row['Gr%s_Y'%c], 'x')
ax.set_aspect('equal', 'box')
这输出：
我正在尝试 plot bivariate gaussian distribution 使用 matplotlib。我想使用两个 scatter 点（A 组）、（B 组）的 xy 坐标来执行此操作。
我想通过调整 COV matrix 来调整 distribution 以说明每个组 velocity 及其与附加 xy 坐标的距离用作参考点。
我计算了每组xy坐标到参考点坐标的距离。距离表示为radius，标记为[GrA_Rad]，[GrB_Rad]。
所以他们离参考点越远，radius就越大。我还计算了标记为 [GrA_Vel]、[GrB_Vel] 的 velocity。每组的direction表示为orientation。这被标记为 [GrA_Rotation]、[GrB_Rotation]
关于distribution如何针对velocity和距离(radius)进行调整的问题：
我希望使用 SVD。具体来说，如果我有每个 scatter 的 rotation angle，这将提供 direction。 velocity可以用来描述一个scalingmatrix[GrA_Scaling],[GrB_Scaling]。所以这个scalingmatrix可以用来扩大x-direction中的radius，收缩y-direction中的radius。这表示COVmatrix。
最后，distribution mean 值是通过将组 location (x,y) 平移 velocity.
简单地说：radius应用于每个组的scatter点。 COV矩阵由radius和velocity调整。因此，使用 scaling matrix 扩展 x-direction 中的 radius 并收缩 y-direction 中的 radius。 direction 是从 rotation angle 测得的。然后通过将组位置 (x,y) 平移 velocity.
distribution mean 值
下面是这些变量的df
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data = d)
我已经为每个 xy 坐标制作了一个 animated plot。
GrA_X = [10,12,17,16,16,14,12,8]
GrA_Y = [10,12,13,7,6,7,8,8]
GrB_X = [5,8,13,16,19,15,13,5]
GrB_Y = [6,15,12,10,8,9,10,8]
Item_X = [6,8,14,18,13,11,16,15]
Item_Y = [10,12,8,12,15,12,10,8]
scatter_GrA = ax.scatter(GrA_X, GrA_Y)
scatter_GrB = ax.scatter(GrB_X, GrB_Y)
scatter_Item = ax.scatter(Item_X, Item_Y)
def animate(i) :
scatter_GrA.set_offsets([[GrA_X[0+i], GrA_Y[0+i]]])
scatter_GrB.set_offsets([[GrB_X[0+i], GrB_Y[0+i]]])
scatter_Item.set_offsets([[Item_X[0+i], Item_Y[0+i]]])
ani = animation.FuncAnimation(fig, animate, np.arange(0,9),
interval = 1000, blit = False)
更新
问题已更新，并且变得更加清晰。我已经更新了我的代码以匹配。这是最新的输出：
除了样式，我认为这与 OP 描述的相符。
这是用于生成上述图的代码：
dfake = ({
'GrA_X' : [15,15],
'GrA_Y' : [15,15],
'Reference_X' : [15,3],
'Reference_Y' : [15,15],
'GrA_Rad' : [15,25],
'GrA_Vel' : [0,10],
'GrA_Scaling' : [0,0.5],
'GrA_Rotation' : [0,45]
})
dffake = pd.DataFrame(dfake)
fig,axs = plt.subplots(1, 2, figsize=(16,8))
fig.subplots_adjust(0,0,1,1)
plotone(dffake, 'A', 0, xlim=(0,30), ylim=(0,30), fig=fig, ax=axs[0])
plotone(dffake, 'A', 1, xlim=(0,30), ylim=(0,30), fig=fig, ax=axs[1])
plt.show()
我使用的 plotone 函数的完整实现在下面的代码块中。如果您只想了解用于生成和转换二维高斯 PDF 的数学，请查看 mvpdf 函数（以及它所依赖的 rot 和 getcov 函数）：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
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
def mvpdf(x, y, xlim, ylim, radius=1, velocity=0, scale=0, theta=0):
"""Creates a grid of data that represents the PDF of a multivariate gaussian.
x, y: The center of the returned PDF
(xy)lim: The extent of the returned PDF
radius: The PDF will be dilated by this factor
scale: The PDF be stretched by a factor of (scale + 1) in the x direction, and squashed by a factor of 1/(scale + 1) in the y direction
theta: The PDF will be rotated by this many degrees
returns: X, Y, PDF. X and Y hold the coordinates of the PDF.
"""
# create the coordinate grids
X,Y = np.meshgrid(np.linspace(*xlim), np.linspace(*ylim))
# stack them into the format expected by the multivariate pdf
XY = np.stack([X, Y], 2)
# displace xy by half the velocity
x,y = rot(theta) @ (velocity/2, 0) + (x, y)
# get the covariance matrix with the appropriate transforms
cov = getcov(radius=radius, scale=scale, theta=theta)
# generate the data grid that represents the PDF
PDF = sts.multivariate_normal([x, y], cov).pdf(XY)
return X, Y, PDF
def plotmv(x, y, xlim=None, ylim=None, radius=1, velocity=0, scale=0, theta=0, xref=None, yref=None, fig=None, ax=None):
"""Plot an xy point with an appropriately tranformed 2D gaussian around it.
Also plots other related data like the reference point.
"""
if xlim is None: xlim = (x - 5, x + 5)
if ylim is None: ylim = (y - 5, y + 5)
if fig is None:
fig = plt.figure(figsize=(8,8))
ax = fig.gca()
elif ax is None:
ax = fig.gca()
# plot the xy point
ax.plot(x, y, '.', c='C0', ms=20)
if not (xref is None or yref is None):
# plot the reference point, if supplied
ax.plot(xref, yref, '.', c='w', ms=12)
# plot the arrow leading from the xy point
if velocity > 0:
ax.arrow(x, y, *rot(theta) @ (velocity, 0),
width=.4, length_includes_head=True, ec='C0', fc='C0')
# fetch the PDF of the 2D gaussian
X, Y, PDF = mvpdf(x, y, xlim=xlim, ylim=ylim, radius=radius, velocity=velocity, scale=scale, theta=theta)
# normalize PDF by shifting and scaling, so that the smallest value is 0 and the largest is 1
normPDF = PDF - PDF.min()
normPDF = normPDF/normPDF.max()
# plot and label the contour lines of the 2D gaussian
cs = ax.contour(X, Y, normPDF, levels=6, colors='w', alpha=.5)
ax.clabel(cs, fmt='%.3f', fontsize=12)
# plot the filled contours of the 2D gaussian. Set levels high for smooth contours
cfs = ax.contourf(X, Y, normPDF, levels=50, cmap='viridis', vmin=-.9, vmax=1)
# create the colorbar and ensure that it goes from 0 -> 1
cbar = fig.colorbar(cfs, ax=ax)
cbar.set_ticks([0, .2, .4, .6, .8, 1])
# add some labels
ax.grid()
ax.set_xlabel('X distance (M)')
ax.set_ylabel('Y distance (M)')
# ensure that x vs y scaling doesn't disrupt the transforms applied to the 2D gaussian
ax.set_aspect('equal', 'box')
return fig, ax
def fetchone(df, l, i, **kwargs):
"""Fetch all the needed data for one xy point
"""
keytups = (
('x', 'Gr%s_X'%l),
('y', 'Gr%s_Y'%l),
('radius', 'Gr%s_Rad'%l),
('velocity', 'Gr%s_Vel'%l),
('scale', 'Gr%s_Scaling'%l),
('theta', 'Gr%s_Rotation'%l),
('xref', 'Reference_X'),
('yref', 'Reference_Y')
)
ret = {k:df.loc[i, l] for k,l in keytups}
# add in any overrides
ret.update(kwargs)
return ret
def plotone(df, l, i, xlim=None, ylim=None, fig=None, ax=None, **kwargs):
"""Plot exactly one point from the dataset
"""
# look up all the data to plot one datapoint
xydata = fetchone(df, l, i, **kwargs)
# do the plot
return plotmv(xlim=xlim, ylim=ylim, fig=fig, ax=ax, **xydata)
旧答案-2
我已经调整了我的答案以匹配 OP 发布的示例：
生成上图的代码如下：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
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
def datalimits(*data, pad=.15):
dmin,dmax = min(d.min() for d in data), max(d.max() for d in data)
spad = pad*(dmax - dmin)
return dmin - spad, dmax + spad
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data=d)
limitpad = .5
clevels = 5
cflevels = 50
xmin,xmax = datalimits(df['GrA_X'], df['GrB_X'], pad=limitpad)
ymin,ymax = datalimits(df['GrA_Y'], df['GrB_Y'], pad=limitpad)
X,Y = np.meshgrid(np.linspace(xmin, xmax), np.linspace(ymin, ymax))
fig = plt.figure(figsize=(10,6))
ax = plt.gca()
Zs = []
for l,color in zip('AB', ('red', 'yellow')):
# plot all of the points from a single group
ax.plot(df['Gr%s_X'%l], df['Gr%s_Y'%l], '.', c=color, ms=15, label=l)
Zrows = []
for _,row in df.iterrows():
x,y = row['Gr%s_X'%l], row['Gr%s_Y'%l]
cov = getcov(radius=row['Gr%s_Rad'%l], scale=row['Gr%s_Scaling'%l], theta=row['Gr%s_Rotation'%l])
mnorm = sts.multivariate_normal([x, y], cov)
Z = mnorm.pdf(np.stack([X, Y], 2))
Zrows.append(Z)
Zs.append(np.sum(Zrows, axis=0))
# plot the reference points
# create Z from the difference of the sums of the 2D Gaussians from group A and group B
Z = Zs[0] - Zs[1]
# normalize Z by shifting and scaling, so that the smallest value is 0 and the largest is 1
normZ = Z - Z.min()
normZ = normZ/normZ.max()
# plot and label the contour lines
cs = ax.contour(X, Y, normZ, levels=clevels, colors='w', alpha=.5)
ax.clabel(cs, fmt='%2.1f', colors='w')#, fontsize=14)
# plot the filled contours. Set levels high for smooth contours
cfs = ax.contourf(X, Y, normZ, levels=cflevels, cmap='viridis', vmin=0, vmax=1)
# create the colorbar and ensure that it goes from 0 -> 1
cbar = fig.colorbar(cfs, ax=ax)
cbar.set_ticks([0, .2, .4, .6, .8, 1])
ax.set_aspect('equal', 'box')
旧答案-1
很难确切地说出您在寻找什么。可以通过协方差矩阵缩放和旋转多元高斯分布。以下是如何根据您的数据执行此操作的示例：
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import scipy.stats as sts
def rot(theta):
theta = np.deg2rad(theta)
return np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]
])
def getcov(scale, theta):
cov = np.array([
[1*(scale + 1), 0],
[0, 1/(scale + 1)]
])
r = rot(theta)
return r @ cov @ r.T
d = ({
'Time' : [1,2,3,4,5,6,7,8],
'GrA_X' : [10,12,17,16,16,14,12,8],
'GrA_Y' : [10,12,13,7,6,7,8,8],
'GrB_X' : [5,8,13,16,19,15,13,5],
'GrB_Y' : [6,15,12,7,8,9,10,8],
'Reference_X' : [6,8,14,18,13,11,16,15],
'Reference_Y' : [10,12,8,12,15,12,10,8],
'GrA_Rad' : [8.3,8.25,8.2,8,8.15,8.15,8.2,8.3],
'GrB_Rad' : [8.3,8.25,8.3,8.4,8.6,8.4,8.3,8.65],
'GrA_Vel' : [0,2.8,5.1,6.1,1.0,2.2,2.2,4.0],
'GrB_Vel' : [0,9.5,5.8,5.8,3.16,4.12,2.2,8.2],
'GrA_Scaling' : [0,0.22,0.39,0.47,0.07,0.17,0.17,0.31],
'GrB_Scaling' : [0,0.53,0.2,0.2,0.06,0.1,0.03,0.4],
'GrA_Rotation' : [0,45,23.2,-26.56,-33.69,-36.86,-45,-135],
'GrB_Rotation' : [0,71.6,36.87,5.2,8.13,16.70,26.57,90],
})
df = pd.DataFrame(data=d)
xmin,xmax = min(df['GrA_X'].min(), df['GrB_X'].min()), max(df['GrA_X'].max(), df['GrB_X'].max())
ymin,ymax = min(df['GrA_Y'].min(), df['GrB_Y'].min()), max(df['GrA_Y'].max(), df['GrB_Y'].max())
X,Y = np.meshgrid(
np.linspace(xmin - (xmax - xmin)*.1, xmax + (xmax - xmin)*.1),
np.linspace(ymin - (ymax - ymin)*.1, ymax + (ymax - ymin)*.1)
)
fig,axs = plt.subplots(df.shape[0], sharex=True, figsize=(4, 4*df.shape[0]))
fig.subplots_adjust(0,0,1,1,0,-.82)
for (_,row),ax in zip(df.iterrows(), axs):
for c in 'AB':
x,y = row['Gr%s_X'%c], row['Gr%s_Y'%c]
cov = getcov(scale=row['Gr%s_Scaling'%c], theta=row['Gr%s_Rotation'%c])
mnorm = sts.multivariate_normal([x, y], cov)
Z = mnorm.pdf(np.stack([X, Y], 2))
ax.contour(X, Y, Z)
ax.plot(row['Gr%s_X'%c], row['Gr%s_Y'%c], 'x')
ax.set_aspect('equal', 'box')
这输出：