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pythonnumpysplinederivative

Discontinuity in numerical derivative of an interpolating cubic spline

I am trying to to calculate and plot the numerical derivative (dy/dx) from two lists x and y. I am using the scipy.interpolate.UnivariateSpline and scipy.interpolate.UnivariateSpline.derivative to compute the slope. The plot of y vs x seems to be C1 continuous and I was expecting the slope dy/dx to be smooth as well when plotted against x. But then what is causing the little bump in the plot here? Also any suggestion on how I can massage the code to make it C1 continuous?

import numpy as np
from matplotlib import pyplot as plt
from scipy.interpolate import UnivariateSpline
x=[20.14141131550861, 20.29161104293003, 20.458574567775457, 20.653802880772922, 20.910446090013004, 21.404599384233677, 21.427939384233678, 21.451279384233676, 21.474619384233677, 21.497959384233678, 21.52129938423368, 21.52130038423368, 21.54463938423368, 21.56797938423368, 21.59131938423368, 21.61465938423368, 21.63799938423368, 22.132152678454354, 22.388795887694435, 22.5840242006919]
y=[-1.6629252348586834, -1.7625046339166028, -1.875358801338162, -2.01040013818419, -2.193327440415778, -2.5538174545988306, -2.571799827167608, -2.5896274995868005, -2.607298426787476, -2.624811539182082, -2.642165776735291, -2.642165776735291, -2.659360089028171, -2.6763934353217587, -2.693264784620056, -2.7099731157324367, -2.7265165368570314, -3.0965791078676754, -3.290845721407758, -3.440799238587583]
spl1 = UnivariateSpline(x,y,s=0)
dydx = spl1.derivative(n=1)
T = dydx(x)
plt.plot(x,y,'-x')
plt.plot(x,T,'-')
plt.show()

enter image description here

Solution

  • The given data points look like they define a nice C1-smooth curve, but they do not. Plotting the slopes (difference of y over difference of x) shows this:

    plt.plot(np.diff(y)/np.diff(x))

    slopes

    There are some duplicate values of y in the array, which look like they don't belong, also some near-duplicate (but not duplicate) values of x.

    The easiest way to fix the spline is to allow a tiny bit of smoothing:

    spl1 = UnivariateSpline(x, y, s=1e-5)

    makes the derivative what you expected:

    smoothed

    Removing the "bad apple" also helps, though not as much.

    spl1 = UnivariateSpline(x[:10] + x[11:], y[:10] + y[11:], s=0)