Fit sigmoid curve in python

Question

Thanks ahead! I am trying to fit a sigmoid curve over some data, below is my code

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

====== some code in between =======

plt.scatter(drag[0].w,drag[0].s, s = 10, label = 'drag%d'%0)
def sigmoid(x,x0,k):
    y = 1.0/(1.0+np.exp(-x0*(x-k)))
    return y
popt,pcov = curve_fit(sigmoid, drag[0].w, drag[0].s)
xx = np.linspace(10,1000,10)
yy = sigmoid(xx, *popt)
plt.plot(xx,yy,'r-', label='fit')
plt.legend(loc='upper left')
plt.xlabel('weight(kg)', fontsize=12)
plt.ylabel('wing span(m)', fontsize=12)
plt.show()

this is now showing the graph below which is not very rightfitting curve is the red one at bottom

What are the possible solutions?

Also I am open to other methods of fitting logistic curves on this set of data

Thanks again!

Answer 1

Here is an example graphical fitter using your equation with an amplitude scaling factor for my test data. This code uses scipy's Differential Evolution genetic algorithm to provide initial parameter estimates for curve_fit(), as the scipy default initial parameter estimates of all 1.0 are not always optimal. 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 those bounds are taken from the example data I provide, when using your own data please check that the bounds seem reasonable. Note that ranges on the parameters are much easier to provide than specific values for the initial parameter estimates.

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 sigmoid(x, amplitude, x0, k):
    return amplitude * 1.0/(1.0+numpy.exp(-x0*(x-k)))


# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
    warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
    val = sigmoid(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([minY, maxY]) # search bounds for amplitude
    parameterBounds.append([minX, maxX]) # search bounds for x0
    parameterBounds.append([minX, maxX]) # search bounds for k

    # "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(sigmoid, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()

modelPredictions = sigmoid(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 = sigmoid(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)