使用python进行非线性回归-有什么简单的方法可以更好地拟合此数据？ [英] Nonlinear regression with python - what's a simple method to fit this data better?
问题描述
我有一些要拟合的数据，因此可以在给定温度下对物理参数的值进行一些估算。
我使用了numpy.polyfit对于二次模型，但拟合度不如我希望的那样好，并且我对回归没有太多经验。
我包括了numpy提供的散点图和模型：
import numpy，scipy，matplotlib
导入matplotlib.pyplot as plt
来自scipy.optimize导入curve_fit
来自scipy.optimize导入different_evolution
进口警告
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。 3.74888、3.16206、2.58882、1.93371、1.52426、1.11421、0.719035、0.377708、0.0226971，-0.223181，-0.537231，-0.8 78491，-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， 0.415076、0.401363、0.390034、0.378698]）
def func（x，a，b，Offset）：＃Sigmoid A具有zunzun.com的偏移量
返回1.0 /（ 1.0 + numpy.exp（-a *（xb）））+偏移
＃遗传算法的函数以最小化（平方误差之和）
def sumOfSquaredError（parameterTuple） ：
warnings.filterwarnings（ ignore）＃不通过遗传算法打印警告
val = func（xData，* parameterTuple）
return numpy.sum（（yData-val）** 2.0）
def generate_Initial_Parameters（）：
＃用于边界的最小和最大值
maxX = max（xData）
minX = min（xData）
maxY = max（yData）
minY = min（yData）
parameterBounds = []
parameterBounds.append（[minX，maxX]）＃
的搜索范围parameterBounds.append（[minX，maxX]）＃b $ b $的搜索范围b parameterBounds.append（[0.0，maxY]）＃偏移量的搜索范围
＃为可重复的结果种子 numpy随机数生成器
result = differential_evolution（sumOfSquaredError，parameterBounds，seed = 3）
返回结果。x
＃生成初始参数值
GeneticParameters = generate_Initial_Parameters（）
＃曲线拟合测试数据
fitParameters，pcov = curve_fit（func，xData，yData，GeneticParameters）
print（'Parameters'，fittedParameters）
modelPredictions = func（xData，* fittedParameters）
absError = modelPredictions-yData
SE = numpy.square（absError）＃平方e rrors
MSE = numpy.mean（SE）＃均方误差
RMSE = numpy.sqrt（MSE）＃均方根误差，RMSE
Rsquared = 1.0-（numpy.var（absError ）/ numpy.var（yData））
print（'RMSE：'，RMSE）
print（'R-squared：'，Rsquared）
############################################# ###########
＃图形输出部分
def ModelAndScatterPlot（graphWidth，graphHeight）：
f = plt.figure（figsize =（graphWidth / 100.0，graphHeight / 100.0 ），dpi = 100）
轴= f.add_subplot（111）
＃首先将原始数据作为散点图
轴.plot（xData，yData，'D' ）
＃创建拟合方程图的数据
xModel = numpy.linspace（min（xData），max（xData））
yModel = func（xModel，* fittedParameters）
＃现在将模型作为线图
axes.plot（xModel，yModel）
axes.set_xlabel（'X Data'）＃X轴数据标签
axes.set_ylabel（'Y Data'）＃Y轴数据标签
plt.show （）
plt.close（'all'）＃使用pyplot后清理
graphWidth = 800
graphHeight = 600
ModelAndScatterPlot（graphWidth，graphHeight）
I have some data that I want to fit so I can make some estimations for the value of a physical parameter given a certain temperature.
I used numpy.polyfit for a quadratic model, but the fit isn't quite as nice as I'd like it to be and I don't have much experience with regression.
I have included the scatter plot and the model provided by numpy: S vs Temperature; blue dots are experimental data, black line is the model
The x axis is temperature (in C) and the y axis is the parameter, which we'll call S. This is experimental data, but in theory S should tends towards 0 as temperature increases and reach 1 as temperature decreases.
My question is: How can I fit this data better? What libraries should I use, what kind of function might approximate this data better than a polynomial, etc?
I can provide code, coefficients of the polynomial, etc, if it's helpful.
Here is a Dropbox link to my data. (Somewhat important note to avoid confusion, although it won't change the actual regression, the temperature column in this data set is Tc - T, where Tc is the transition temperature (40C). I converted this using pandas into T by calculating 40 - x).
This example code uses an equation that has two shape parameters, a and b, and an offset term (that does not affect curvature). The equation is "y = 1.0 / (1.0 + exp(-a(x-b))) + Offset" with parameter values a = 2.1540318329369712E-01, b = -6.6744890642157646E+00, and Offset = -3.5241299859669645E-01 which gives an R-squared of 0.988 and an RMSE of 0.0085.
The example contains your posted data with Python code for fitting and graphing, with automatic initial parameter estimation using the scipy.optimize.differential_evolution genetic algorithm. The scipy implementation of Differential Evolution uses the Latin Hypercube algorithm to ensure a thorough search of parameter space, and this requires bounds within which to search - in this example code, these bounds are based on the maximum and minimum data values.
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, a, b, Offset): # Sigmoid A With Offset from zunzun.com
return 1.0 / (1.0 + numpy.exp(-a * (x-b))) + Offset
# 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():
# min and max used for bounds
maxX = max(xData)
minX = min(xData)
maxY = max(yData)
minY = min(yData)
parameterBounds = []
parameterBounds.append([minX, maxX]) # search bounds for a
parameterBounds.append([minX, maxX]) # search bounds for b
parameterBounds.append([0.0, maxY]) # search bounds for Offset
# "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)
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)
##########################################################
# 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)
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