<p>我从散点图中提取数据进行分析,发现多项式+正弦似乎不是一个最佳模型,因为低阶多项式没有很好地遵循数据的形状,高阶多项式在数据极值处表现出Runge的高曲率现象。(“0.1)可能是(a)和(b)方程中的一个很好的(a)和(0.1)的峰值(b)的搜索结果。在</p>
<p>这是一个图形化的Python曲线拟合器,在文件的顶部我加载了我提取的数据,所以你需要用实际的数据替换它。该算法使用scipy的差分进化遗传算法模块来估计非线性fitter的初始参数值,该算法使用拉丁超立方体算法确保参数空间的彻底搜索,并要求搜索范围。在这里,这些界限是从数据的最大值和最小值获取的。在</p>
<p>从拟合曲线中减去模型预测值后,您将只剩下要建模的正弦分量。我注意到在x=275附近似乎还有一个额外的窄的低振幅峰值。在</p>
<p><a href="https://i.stack.imgur.com/DHEW9.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DHEW9.png" alt="plot"/></a></p>
<pre><code>import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings
##########################################################
# load data section
f = open('/home/zunzun/temp/temp.dat')
textData = f.read()
f.close()
xData = []
yData = []
for line in textData.split('\n'):
if line: # ignore blank lines
spl = line.split()
xData.append(float(spl[0]))
yData.append(float(spl[1]))
xData = numpy.array(xData)
yData = numpy.array(yData)
##########################################################
# model to be fitted
def func(x, a, b, c, offset): # Extreme Valye Peak equation from zunzun.com
return a * numpy.exp(-1.0 * numpy.exp(-1.0 * ((x-b)/c))-((x-b)/c) + 1.0) + offset
##########################################################
# fitting section
# 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)
minData = min(minX, minY)
maxData = max(maxX, maxY)
parameterBounds = []
parameterBounds.append([minData, maxData]) # search bounds for a
parameterBounds.append([minData, maxData]) # search bounds for b
parameterBounds.append([minData, maxData]) # search bounds for c
parameterBounds.append([minY, maxY]) # search bounds for offset
# "seed" the numpy random number generator for repeatable results
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()
# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted 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()
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)
</code></pre>
<p>更新-
如果高频正弦分量是常数(我不知道),那么仅用几个周期对一小部分数据进行建模就足以确定拟合模型正弦波部分的方程和初始参数估计。在这里,我做了这个,结果如下:</p>
<p><a href="https://i.stack.imgur.com/8O9Is.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8O9Is.png" alt="sine"/></a></p>
<p>根据以下方程式:</p>
^{pr2}$
<p>结合这两个模型使用实际数据,而不是我的散点图提取的数据,应该接近您需要的。在</p>