Matplotlib plot_surface用于二维多元线性回归

Question

我有很多三维数据点：x1、x2 和 y。我能够计算这些点的多元线性回归，并且能够在 3D 散点图上显示这些点，但我不知道如何绘制我为这些点计算的多元线性回归：就像您可以在 1D 线性回归中绘制最佳拟合线一样，我对绘制 2D 线性回归的"最佳拟合平面"感兴趣。

我的代码如下：

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
# collect data into numpy arrays
X = []
Y = []
for line in open('data_2d.csv'):
x1, x2, y = line.split(',')
X.append([1, float(x1), float(x2)]) # here X[i][0] represents x0 = 1
Y.append(float(y))
X = np.array(X)
Y = np.array(Y)
# calculate weights
w = np.linalg.solve(np.dot(X.T,X), np.dot(X.T, Y))
Yhat = np.dot(X, w) # results of linear regression for data points
# plot results
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:,1], X[:,2], Y)
ax.plot_surface(X[:,1], X[:,2], Yhat) # doesn't seem to work
plt.show()

Answer 1

事实证明，plot_surface要求其每个输入都是坐标矩阵，而不是像我之前使用的那样是值列表。我的解决方案如下：

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
# collect data into numpy arrays
X = []
Y = []
for line in open('data_2d.csv'): # contains 3 columns: x1, x2, and y
x1, x2, y = line.split(',')
X.append([1, float(x1), float(x2)]) # here X[i][0] represents x0 = 1
Y.append(float(y))
X = np.array(X)
Y = np.array(Y)
# calculate weights for computing solutions
w = np.linalg.solve(X.T.dot(X), X.T.dot(Y))
# calculate r-squared error given weights
Yhat = X.dot(w)
d1 = Y - Yhat
d2 = Y - Y.mean()
r2 = 1 - d1.dot(d1) / d2.dot(d2)
print("r-squared value of", r2)
# calculate plane of best fit
divs = 2 # number of divisions in surface: generates divs^2 points.
# The surface is a plane, so just 2^2 = 4 points can define it.
# plane spans all values of x1 and x2 from data
x1_range = np.linspace(min(X[:,1]),max(X[:,1]),divs)
x2_range = np.linspace(min(X[:,2]),max(X[:,2]),divs)
X_plane = []
for i in range(divs):
for j in range(divs):
X_plane.append([1, x1_range[i], x2_range[j]])
X_plane = np.array(X_plane)
# values of y are equal to the linear regression of points on the plane
Yhat2 = X_plane.dot(w)
# rearrange Yhat2 into a coordinate matrix for display as a surface
Yhat2_surface = []
for i in range(divs):
Yhat2_surface.append(Yhat2[ divs*i : divs*i+divs ])
Yhat2_surface = np.array(Yhat2_surface)
Yhat2 = Yhat2_surface
# generate coordinate matrices for x1 and x2 values
X2, X1 = np.meshgrid(x1_range, x2_range) # intentional ordering: X2, *then* X1
# plot results
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:,1], X[:,2], Y) # supplied data
ax.plot_surface(X1, X2, Yhat2, color='y', alpha=0.1) # plane of best fit
plt.show()

输出如下所示。点表示输入数据，黄色矩形表示其最佳拟合平面，用plot_surface绘制。

Answer 2

这是我拼凑的一个快速示例，演示了 3D 散点图、3D 曲面图和等值线图。

import numpy, scipy
import matplotlib
from mpl_toolkits.mplot3d import  Axes3D
from matplotlib import cm # to colormap 3D surfaces from blue to red
import matplotlib.pyplot as plt
graphWidth = 800 # units are pixels
graphHeight = 600 # units are pixels
# 3D contour plot lines
numberOfContourLines = 16

def SurfacePlot(equationFunc, data):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
matplotlib.pyplot.grid(True)
axes = Axes3D(f)
x_data = data[0]
y_data = data[1]
z_data = data[2]
xModel = numpy.linspace(min(x_data), max(x_data), 20)
yModel = numpy.linspace(min(y_data), max(y_data), 20)
X, Y = numpy.meshgrid(xModel, yModel)
Z = equationFunc(numpy.array([X, Y]))
axes.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=1, antialiased=True)
axes.scatter(x_data, y_data, z_data) # show data along with plotted surface
axes.set_title('Surface Plot (click-drag with mouse)') # add a title for surface plot
axes.set_xlabel('X Data') # X axis data label
axes.set_ylabel('Y Data') # Y axis data label
axes.set_zlabel('Z Data') # Z axis data label
plt.show()
plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems

def ContourPlot(equationFunc, data):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
axes = f.add_subplot(111)
x_data = data[0]
y_data = data[1]
z_data = data[2]
xModel = numpy.linspace(min(x_data), max(x_data), 20)
yModel = numpy.linspace(min(y_data), max(y_data), 20)
X, Y = numpy.meshgrid(xModel, yModel)
Z = equationFunc(numpy.array([X, Y]))
axes.plot(x_data, y_data, 'o')
axes.set_title('Contour Plot') # add a title for contour plot
axes.set_xlabel('X Data') # X axis data label
axes.set_ylabel('Y Data') # Y axis data label
CS = matplotlib.pyplot.contour(X, Y, Z, numberOfContourLines, colors='k')
matplotlib.pyplot.clabel(CS, inline=1, fontsize=10) # labels for contours
plt.show()
plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems

def ScatterPlot(data):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
matplotlib.pyplot.grid(True)
axes = Axes3D(f)
x_data = data[0]
y_data = data[1]
z_data = data[2]
axes.scatter(x_data, y_data, z_data)
axes.set_title('Scatter Plot (click-drag with mouse)')
axes.set_xlabel('X Data')
axes.set_ylabel('Y Data')
axes.set_zlabel('Z Data')
plt.show()
plt.close('all') # clean up after using pyplot or else thaere can be memory and process problems

def EquationFunc(data):
return 5.0 + numpy.sqrt(data[0]) + numpy.cos(data[1] / 5.0)

if __name__ == "__main__":
xData = numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0])
yData = numpy.array([11.0, 12.1, 13.0, 14.1, 15.0, 16.1, 17.0, 18.1, 90.0])
zData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.0, 9.9])
data = [xData, yData, zData]
ScatterPlot(data)
SurfacePlot(EquationFunc, data)
ContourPlot(EquationFunc, data)