我正在尝试使用Python scikitlearn LinearRegression() function
实现多元线性回归。回归方程为:
y= c+ a1x1 + a2x2
附加条件是,c始终取(1,10)之间的值,`a1取(0,1)之间的值。
如何使用Python解决这类方程式?
[这里是使用您的方程式的示例有界图形3D表面贴合器示例,并使用3D散点图,3D表面图和轮廓图在c上限制了边界。您应该能够单击并拖动鼠标,并在3空间中旋转3D图以进行检查。当然,您可以根据需要更改或添加范围。
[请注意,本示例使用的是scipy的curve_fit(),该曲线允许在拟合参数上有界限,并且估计的初始参数必须在界限之内,以便curve_fit()可以开始。在此示例中,a1,a2和c的拟合参数值为:
合适的参数[9.71206053e-01 3.57603742e-02 1.63260453e-16]
参数“ c”有效地位于零的下限。如果从对curve_fit()的调用中删除了边界,则在此示例中,参数“ c”将为负。
import numpy, scipy, scipy.optimize
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(func, data, fittedParameters):
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 = func(numpy.array([X, Y]), *fittedParameters)
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 there can be memory and process problems
def ContourPlot(func, data, fittedParameters):
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 = func(numpy.array([X, Y]), *fittedParameters)
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 there 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 there can be memory and process problems
def func(data, a1, a2, c):
x1 = data[0]
x2 = data[1]
return c + (a1 * x1) + (x2 * a2)
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, 12.0])
data = [xData, yData, zData]
# initial parameter estimates must be within bounds
initialParameters = [1.0, 1.0, 0.5]
# bounds on parameters - initial parameters must be within these
# note that +/- infinity means "no bound" on that parameter
lowerBounds = (-numpy.Inf, -numpy.Inf, 0.0)
upperBounds = (numpy.Inf, numpy.Inf, 1.0)
parameterBounds = [lowerBounds, upperBounds]
# now call curve_fit passing in parameter bounds
fittedParameters, pcov = scipy.optimize.curve_fit(func, [xData, yData], zData, p0 = initialParameters, bounds = parameterBounds)
ScatterPlot(data)
SurfacePlot(func, data, fittedParameters)
ContourPlot(func, data, fittedParameters)
print('fitted parameters', fittedParameters)
modelPredictions = func(data, *fittedParameters)
absError = modelPredictions - zData
SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(zData))
print('RMSE:', RMSE)
print('R-squared:', Rsquared)