How do I perform a curve fit with an array of points and touching a specific point in that array

Question

I need help with curve fitting a given set of points. The points form a parabola and I ought to find the peak point of the result. Issue is when I do a curve fit, it sometimes doesn't touch the max y-coordinate even if the actual point is given in the input array. Following is the code snippet. Here 1.88 is the actual peak y-coordinate (13.05,1.88). But the graph generated by the code does not touch the point due to curve fitting. So is there a way to fit the curve making sure that it touches the max point given in the input array?

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit, minimize_scalar
fig = plt.gcf()
#fig.set_size_inches(18.5, 10.5)

x = [4.59,9.02,13.05,18.47,20.3]
y = [1.7,1.84,1.88,1.7,1.64]

def f(x, p1, p2, p3):
    return p3*(p1/((x-p2)**2 + (p1/2)**2))   

plt.plot(x,y,"ro")
popt, pcov = curve_fit(f, x, y)

# find the peak
fm = lambda x: -f(x, *popt)
r = minimize_scalar(fm, bounds=(1, 5))
print( "maximum:", r["x"], f(r["x"], *popt) )  #maximum: 2.99846874275 18.3928199902
plt.text(1,1.9,'maximum '+str(round(r["x"],2))+'( @'+str(round(f(r["x"], *popt),2)) + ' )') 
x_curve = np.linspace(min(x), max(x), 50)
plt.plot(x_curve, f(x_curve, *popt))
plt.plot(r['x'], f(r['x'], *popt), 'ko')
plt.show()

Hi and welcome. This is the property of a least-square fit. If you want the parabola to get to the max of your data, I suggest that give the equation in the form f( x ) = yMax - a**2 * (x - x0 )**2 (a**2 to ensure positive prefactor) — mikuszefski
@mikuszefski please see my answer which does not require changing the equation. — James Phillips
@JamesPhillips Agreed, by weighting you can get arbitrarily close. For touching you either use constraints or you rewrite the equation such that the constraints are automatically fulfilled. A few comments on top for the OP. 1) The graph is a Cauchy profile and not a parabola. 2) If this is a fit to data with noise, what makes you think that the max-value is special. The true max could be, e.g., at x=11.7. Why this point should be touched by the fit but not any other point? Is the a technical restriction such that values cannot be higher than a specific value? — mikuszefski

James Phillips James Phillips · Accepted Answer · 2019-07-08T12:28:40

Here is a graphical code example using your equation with weighted fitting, where I have made the max point larger to more easily see the effect of the weighting. In non-weighted curve fitting, all weights are implicitly 1.0 as all data points have equal weight. Scipy's curve_fit routine uses weights in the form of uncertainties, so that giving a point a very small uncertainty (which I have done) is like giving the point a very large weight. This technique can be used to make a fit pass arbitrarily close to any single data point by any software that can perform weghted fitting.

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

x = [4.59,9.02,13.05,18.47,20.3]
y = [1.7,1.84,2.0,1.7,1.64]


# note the single very small uncertainty - try making this value 1.0
uncertainties = numpy.array([1.0, 1.0, 1.0E-6, 1.0, 1.0])


# rename data to use previous example
xData = numpy.array(x)
yData = numpy.array(y)


def func(x, p1, p2, p3):
    return p3*(p1/((x-p2)**2 + (p1/2)**2))   


# these are the same as the scipy defaults
initialParameters = numpy.array([1.0, 1.0, 1.0])

# curve fit the test data, first without uncertainties to
# get us closer to initial starting parameters
ssqParameters, pcov = curve_fit(func, xData, yData, p0 = initialParameters)

# now that we have better starting parameters, use uncertainties
fittedParameters, pcov = curve_fit(func, xData, yData, p0 = ssqParameters, sigma=uncertainties, absolute_sigma=True)

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)

How do I perform a curve fit with an array of points and touching a specific point in that array

1 Answers