Мне не удалось сопоставить данные с уравнением пиков Гаусса, и если посмотреть на диаграмму рассеяния, то сами данные не имеют такой формы.Я смог согласовать его с пиковым уравнением Гамильтона «y = Gb * pow (x / mu, log (mu / x) / (B * B)) + (Vbmax * x) / (x + sigma_b)», здесьявляется графическим установщиком Python для этого уравнения с использованием ваших данных.
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([1612, 1710, 1755, 2692, 4082, 5988, 6672, 6579, 6506, 3865, 2244, 2042, 2057], dtype=float)
yData = numpy.array([37, 38, 39, 39.33, 39.66, 40, 40.33, 40.66, 41, 41.33, 41.66, 42, 43], dtype=float)
def func(x, Gb, mu, B, Vbmax, sigma_b): # Hamilton peak equation from zunzun.com
return Gb * numpy.power(x / mu, numpy.log(mu/x)/(B*B)) + (Vbmax * x) / (x + sigma_b)
# 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([0.0, maxY]) # search bounds for Gb
parameterBounds.append([minX, maxX]) # search bounds for mu
parameterBounds.append([0.0, 1.0]) # search bounds for B
parameterBounds.append([minY, maxY]) # search bounds for Vbmax
parameterBounds.append([0.0, minX]) # search bounds for sigma_b
# "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), 100)
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)
ОБНОВЛЕНИЕ с данными, транспонированными в соответствии с комментариями, и смещением в качестве подогнанного параметра дляУравнение пиков Гаусса, см. Начальную оценку параметров по значениям данных
import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
xData = numpy.array([37, 38, 39, 39.33, 39.66, 40, 40.33, 40.66, 41, 41.33, 41.66, 42, 43], dtype=float)
yData = numpy.array([1612, 1710, 1755, 2692, 4082, 5988, 6672, 6579, 6506, 3865, 2244, 2042, 2057], dtype=float)
def func(x, a, b, c, offset):
return a * numpy.exp(-0.5 * numpy.power((x-b) / c, 2.0)) + offset
# initial parameter estimates from the data
a = max(xData)
b = max((xData) - min(xData)) / 2.0 + min(xData)
c = 1.0 # my guess from the equation
offset = min(yData)
initialParameters = numpy.array([a, b, c, offset])
# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)
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('Parameters:', fittedParameters)
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)