在 Python 中使用回归函数的详尽优化算法

Question

我必须使用最小二乘法和留一交叉验证来估计 回归的均方误差： 回归函数

对于从 1 到 30 的 p，找出 p 的最佳数字，以便执行最佳回归，定义 p 的最佳值是什么，用于构建我的回归函数。

问题是我真的不知道该怎么做，根本没有。我理解背后的所有数学，我可以手工完成，我知道Python，但我有心理障碍。Scikit Learn有什么可以帮忙的吗？我知道他们有套索和 Ridge 用于特征选择，但这就像手动进行特征选择一样，我需要一些东西来计算 p 的每个值的函数权重，并计算它们的最小二乘误差。问题是我的数据只有一个特征 x，我正在申请 p 的几个值。

Answer 1

我发现这个问题足以编写此示例代码，它在我的一些测试数据上使用您的公式 - 您需要用您自己的数据替换测试数据。这段代码至少应该让你入门。此示例使用 scipy.optimize.differential_evolution 模块自动生成非线性求解器的初始参数估计值，并使用 matplotlib 绘制结果。

import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings
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, B0, B1, B2, B3, B4, B2p1, B2p, p):
returnVal = B0 # start with B0 and add the other terms
returnVal += B1 * numpy.sin(2.0 * numpy.pi * x)
returnVal += B2 * numpy.cos(2.0 * numpy.pi * x)
returnVal += B3 * numpy.sin(2.0 * numpy.pi * 2.0 * x)
returnVal += B4 * numpy.cos(2.0 * numpy.pi * 2.0 * x)
returnVal += B2p1 * numpy.sin(2.0 * numpy.pi * p * x)
returnVal += B2p * numpy.cos(2.0 * numpy.pi * p * x)
return  returnVal

# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
val = func(xData, *parameterTuple)
return numpy.sum((yData - val) ** 2.0)

def generate_Initial_Parameters():
parameterBounds = []
parameterBounds.append([-1.0, 1.0]) # seach bounds for B0
parameterBounds.append([-1.0, 1.0]) # seach bounds for B1
parameterBounds.append([-1.0, 1.0]) # seach bounds for B2
parameterBounds.append([-1.0, 1.0]) # seach bounds for B3
parameterBounds.append([-1.0, 1.0]) # seach bounds for B4
parameterBounds.append([-1.0, 1.0]) # seach bounds for B2p1
parameterBounds.append([-1.0, 1.0]) # seach bounds for B2p
parameterBounds.append([-1.0, 1.0]) # seach bounds for p
# "seed" the numpy random number generator for repeatable results
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
# generate initial parameter values
geneticParameters = generate_Initial_Parameters()
# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Parameters', 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('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)