Python中某些特征的正系数线性回归
Posted
技术标签:
【中文标题】Python中某些特征的正系数线性回归【英文标题】：Linear regression with positive coefficients for SOME of the features in Python 【发布时间】：2019-05-22 04:34:55 【问题描述】：我正在尝试找到一种方法来拟合线性回归。但是，我想强制 一些 驱动因素的系数为正。
据我了解，scipy.optimize.nnls 可以进行非负最小二乘，但适用于所有驱动程序。
有没有办法自动完成？
非常感谢。
【问题讨论】：
【参考方案1】：这是一个图形拟合器，它在拟合函数中具有“砖墙”，强制拟合参数之一为正。请注意，在此示例中，拟合度非常差 - 如果您移除“砖墙”，示例拟合度会大大提高。本例使用默认的 scipy curve_fit() 初始参数估计全部为 1.0，并且不使用 scipy 的遗传算法来帮助寻找初始参数估计。使用这种技术时，初始参数估计必须在“砖墙”条件之外，以便非线性拟合器可以正常开始。
import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
xData = numpy.array([19.1647, 18.0189, 16.9550, 15.7683, 14.7044, 13.6269, 12.6040, 11.4309, 10.2987, 9.23465, 8.18440, 7.89789, 7.62498, 7.36571, 7.01106, 6.71094, 6.46548, 6.27436, 6.16543, 6.05569, 5.91904, 5.78247, 5.53661, 4.85425, 4.29468, 3.74888, 3.16206, 2.58882, 1.93371, 1.52426, 1.14211, 0.719035, 0.377708, 0.0226971, -0.223181, -0.537231, -0.878491, -1.27484, -1.45266, -1.57583, -1.61717])
yData = numpy.array([0.644557, 0.641059, 0.637555, 0.634059, 0.634135, 0.631825, 0.631899, 0.627209, 0.622516, 0.617818, 0.616103, 0.613736, 0.610175, 0.606613, 0.605445, 0.603676, 0.604887, 0.600127, 0.604909, 0.588207, 0.581056, 0.576292, 0.566761, 0.555472, 0.545367, 0.538842, 0.529336, 0.518635, 0.506747, 0.499018, 0.491885, 0.484754, 0.475230, 0.464514, 0.454387, 0.444861, 0.437128, 0.415076, 0.401363, 0.390034, 0.378698])
def func(x, a, b, offset): #exponential curve fitting function
# force a to be positive by using "brick wall" that
# returns a large value, and therefore a large error,
# if parameter a is not positive
if a <= 0.0:
return 1.0E10
return a * numpy.exp(-b*x) + offset
fittedParameters, pcov = curve_fit(func, xData, yData)
print(fittedParameters)
print()
modelPredictions = func(xData, *fittedParameters)
absError = modelPredictions - yData
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(yData))
print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)
print()
##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
axes = f.add_subplot(111)
# first the raw data as a scatter plot
axes.plot(xData, yData, 'D')
# create data for the fitted equation plot
xModel = numpy.linspace(min(xData), max(xData))
yModel = func(xModel, *fittedParameters)
# now the model as a line plot
axes.plot(xModel, yModel)
axes.set_xlabel('X Data') # X axis data label
axes.set_ylabel('Y Data') # Y axis data label
plt.show()
plt.close('all') # clean up after using pyplot
graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)
【讨论】：
嗨@JamesPhillips，新年快乐！据我在您的示例中了解，仅使用一个驱动程序来预测目标。但是在多元回归的情况下，你如何应用它，其中一些变量被限制为正而不是其他？【参考方案2】：根据有关约束多元回归的评论，这是一个图形 3D 表面拟合器，它也有一个“砖墙”来强制拟合参数之一为正。可以使用受约束或不受约束的函数版本调用 curve_fit 以进行比较。
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 thaere 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 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 func(data, a, b, c):
# extract the individual data arrays used in the equation
x = data[0]
y = data[1]
return a*x + b*y + c
def constrainedFunction(data, a, b, c):
# use a "brick wall" to ensure parameter c is positive
# return a large value and therefor large error
if c <= 0.0:
return 1.0E10
else:
return func(data, a, b, c) # call the unconstrained function
if __name__ == "__main__":
xData = numpy.array([-10.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0])
yData = numpy.array([-10.0, 11.0, 12.1, 13.0, 14.1, 15.0, 16.1, 17.0, 18.1, 19.0])
zData = numpy.array([-30.0, 1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.0, 9.9])
data = [xData, yData, zData]
initialParameters = [1.0, 1.0, 1.0] # these are the same as scipy default values in this example
# here a non-linear surface fit is made with scipy's curve_fit()
fittedParameters, pcov = scipy.optimize.curve_fit(constrainedFunction, [xData, yData], zData, p0 = initialParameters)
ScatterPlot(data)
SurfacePlot(func, data, fittedParameters)
ContourPlot(func, data, fittedParameters)
print('fitted prameters', fittedParameters)
【讨论】：
以上是关于Python中某些特征的正系数线性回归的主要内容，如果未能解决你的问题，请参考以下文章