Fit curve to data, get analytical form, & check when curve crosses threshold

问题

I have 40 points for each curve and I would like to smooth the function and estimate when the curve crosses a threshold on the y axis. Is there a fitting function that I can easily apply this to, I can use interpolate to plot the new function but I can't figure out how to request the x value for which y = threshold.

Unfortunately the curves don't all have the same shape so I can't use scipy.optimize.curve_fit.

Thanks!

Update: Adding two curves:

Curve 1

[942.153,353.081,53.088,125.110,140.851,188.170,70.536,-122.473,-369.061,-407.945,88.734,484.334,267.762,65.831,74.010,-55.781,-260.024,-466.830,-524.511,-76.833,-36.779,-117.366,218.578,175.662,185.653,299.285,215.276,546.048,1210.132,3087.326,7052.849,13867.824,27156.939,51379.664,91908.266,148874.563,215825.031,290073.219,369567.781,437031.688]

Curve 2

[-39034.039,-34637.941,-24945.094,-16697.996,-9247.398,-2002.051,3409.047,3658.145,7542.242,11781.340,11227.688,10089.035,9155.883,8413.980,5289.578,3150.676,4590.023,6342.871,3294.719,580.567,-938.586,-3919.738,-5580.390,-3141.793,-2785.945,-2683.597,-4287.750,-4947.902,-7347.554,-8919.457,-6403.359,-6722.011,-8181.414,-6807.566,-7603.218,-6298.371,-6909.523,-5878.675,-5193.578,-7193.980]

x values are

[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]

回答1:

For fitting a smooth curve, you can fit Legendre polynomials using numpy.polynomial.legendre.Legendre's fit method.

# import packages we need later
import matplotlib.pyplot as plt
import numpy as np

Fitting Legendre Polynomials

Preparing data as numpy arrays:

curve1 = \
np.asarray([942.153,353.081,53.088,125.110,140.851,188.170,70.536,-122.473,-369.061,-407.945,88.734,484.334,267.762,65.831,74.010,-55.781,-260.024,-466.830,-524.511,-76.833,-36.779,-117.366,218.578,175.662,185.653,299.285,215.276,546.048,1210.132,3087.326,7052.849,13867.824,27156.939,51379.664,91908.266,148874.563,215825.031,290073.219,369567.781,437031.688])
curve2 = \
np.asarray([-39034.039,-34637.941,-24945.094,-16697.996,-9247.398,-2002.051,3409.047,3658.145,7542.242,11781.340,11227.688,10089.035,9155.883,8413.980,5289.578,3150.676,4590.023,6342.871,3294.719,580.567,-938.586,-3919.738,-5580.390,-3141.793,-2785.945,-2683.597,-4287.750,-4947.902,-7347.554,-8919.457,-6403.359,-6722.011,-8181.414,-6807.566,-7603.218,-6298.371,-6909.523,-5878.675,-5193.578,-7193.980])
xvals = \
np.asarray([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40])

Lets fit the Legendre polynomials, degree being the highest degree polynomial used, the first few is here for example.

degree=10
legendrefit_curve1 = np.polynomial.legendre.Legendre.fit(xvals, curve1, deg=degree)
legendrefit_curve2 = np.polynomial.legendre.Legendre.fit(xvals, curve2, deg=degree)

Calculate these fitted curves at evenly spaced points using the linspace method. n is the number of point pairs we want to have.

n=100
fitted_vals_curve1 = legendrefit_curve1.linspace(n=n)
fitted_vals_curve2 = legendrefit_curve2.linspace(n=n)

Let's plot the result, along with a threshold (using axvline):

plt.scatter(xvals, curve1)
plt.scatter(xvals, curve2)

plt.plot(fitted_vals_curve1[0],fitted_vals_curve1[1],c='r')
plt.plot(fitted_vals_curve2[0],fitted_vals_curve2[1],c='k')

threshold=100000
plt.axhline(y=threshold)

The curves fit beautifully.

When is the threshold crossed?

To check where the threshold is crossed in each series, you can do:

for x, y in zip(fitted_vals_curve1[0], fitted_vals_curve1[1]):
    if y > threshold:
        xcross_curve1 = x
        break

for x, y in zip(fitted_vals_curve2[0], fitted_vals_curve2[1]):
    if y > threshold:
        xcross_curve2 = x
        break

xcross_curve1 and xcross_curve2 will hold the x value where curve1 and curve2 crossed the threshold if they did cross the threshold; if they did not, they will be undefined.

Let's plot them to check if it works (link to axhline docs):

plt.scatter(xvals, curve1)
plt.scatter(xvals, curve2)

plt.plot(fitted_vals_curve1[0],fitted_vals_curve1[1],c='r')
plt.plot(fitted_vals_curve2[0],fitted_vals_curve2[1],c='k')

plt.axhline(y=threshold)

try: plt.axvline(x=xcross_curve1)
except NameError: print('curve1 is not passing the threshold',c='b')

try: plt.axvline(x=xcross_curve2)
except NameError: print('curve2 is not passing the threshold')

As expected, we get this plot:

(and a text output: curve2 is not passing the threshold.)

If you would like to increase accuracy of xcross_curve1 or xcross_curve2, you can increase degree and n defined above.

From Legendre to Polynomial form

We have fitted a curve, which roughly has the form:

where P_n is the nth Legendre polynomial, s(x) is some function which transforms x to the range P_n expects (some math stuff which we don't need to know now).

We want our fitted line in the form:

We'll use legendre() of scipy.special:

from scipy.special import legendre

We'll also use use np.pad (docs, good SO post).

legendredict={}
for icoef, coef in enumerate(legendrefit_curve1.coef):
    legendredict[icoef]=coef*np.pad(legendre(icoef).coef,(10-icoef,0),mode='constant')

legendredict will hold keys from 0 to 10, and each value in the dict will be a list of floats. The key is refering to the degree of the Polynomial, and the list of floats are expressing what are the coefficients of x**n values within that constituent polynomial of our fit, in a backwards order.

For example:

P_4 is:

legendredict[4] is:

isarray([ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
        0.00000000e+00,  0.00000000e+00,  3.29634565e+05,  3.65967884e-11,
       -2.82543913e+05,  1.82983942e-11,  2.82543913e+04])

Meaning that in the sum of P_ns (f(x), above), we have q_4 lot of P_4, which is equivalent to having 2.82543913e+04 of 1s, 1.82983942e-11 of x, -2.82543913e+05 of x^2, etc, only from the P_4 component.

So if we want to know how much 1s, xs, x^2s, etc we need to form the polynomial sum, we need to add the need for 1s, xs, x^2s, etcs from all the different P_ns. This is what we do:

polycoeffs = np.sum(np.stack(list(legendredict.values())),axis=0)

Then let's form a polynomial sum:

for icoef, coef in enumerate(reversed(polycoeffs)):
    print(str(coef)+'*s(x)**'+str(icoef),end='\n +')

Giving the output:

-874.1456709637822*s(x)**0
 +2893.7228005540596*s(x)**1
 +50415.38472217957*s(x)**2
 +-6979.322584205707*s(x)**3
 +-453363.49985790614*s(x)**4
 +-250464.7549807652*s(x)**5
 +1250129.5521521813*s(x)**6
 +1267709.5031024509*s(x)**7
 +-493280.0177807359*s(x)**8
 +-795684.224334346*s(x)**9
 +-134370.1696946264*s(x)**10
 +

(We are going to ignore the last + sign, formatting is not the main point here.)

We need to calculate s(x) as well. If we are working in a Jupyter Notebook / Google Colab, only executing a cell with legendrefit_curve1 returns:

From where we can clearly see that s(x) is -1.0512820512820513+0.05128205128205128x. If we want to do it in a more programmatic way:

2/(legendrefit_curve1.domain[1]-legendrefit_curve1.domain[0]) is 0.05128205128205128 & -1-2/(legendrefit_curve1.domain[1]-legendrefit_curve1.domain[0]) is just -1.0512820512820513

Which is true for some mathamatical reasons not much relevant here (related Q).

So we can define:

def s(input):
    a=-1-2/(legendrefit_curve1.domain[1]-legendrefit_curve1.domain[0])
    b=2/(legendrefit_curve1.domain[1]-legendrefit_curve1.domain[0])
    return a+b*input

Also, lets define, based on the above obtained sum of polynomials of s(x):

def polyval(x):
    return -874.1456709637822*s(x)**0+2893.7228005540596*s(x)**1+50415.38472217957*s(x)**2+-6979.322584205707*s(x)**3+-453363.49985790614*s(x)**4+-250464.7549807652*s(x)**5+1250129.5521521813*s(x)**6+1267709.5031024509*s(x)**7+-493280.0177807359*s(x)**8+-795684.224334346*s(x)**9+-134370.1696946264*s(x)**10

In a more programmatic way:

def polyval(x):
    return sum([coef*s(x)**icoef for icoef, coef in enumerate(reversed(polycoeffs))])

Check that our polynomial indeed fits:

plt.scatter(fitted_vals_curve1[0],fitted_vals_curve1[1],c='r')
plt.plot(fitted_vals_curve1[0],[polyval(val) for val in fitted_vals_curve1[0]])

It does:

So let's print out our pure polynomial sum, with s(x) being replaced by an explicit function:

for icoef, coef in enumerate(reversed(polycoeffs)):
    print(str(coef)+'*(-1.0512820512820513+0512820512820513*x)**'+str(icoef),end='\n +')

Giving the output:

-874.1456709637822*(-1.0512820512820513+0512820512820513*x)**0
 +2893.7228005540596*(-1.0512820512820513+0512820512820513*x)**1
 +50415.38472217957*(-1.0512820512820513+0512820512820513*x)**2
 +-6979.322584205707*(-1.0512820512820513+0512820512820513*x)**3
 +-453363.49985790614*(-1.0512820512820513+0512820512820513*x)**4
 +-250464.7549807652*(-1.0512820512820513+0512820512820513*x)**5
 +1250129.5521521813*(-1.0512820512820513+0512820512820513*x)**6
 +1267709.5031024509*(-1.0512820512820513+0512820512820513*x)**7
 +-493280.0177807359*(-1.0512820512820513+0512820512820513*x)**8
 +-795684.224334346*(-1.0512820512820513+0512820512820513*x)**9
 +-134370.1696946264*(-1.0512820512820513+0512820512820513*x)**10
 +

Which can be simplified, as desired. (Ignore the last + sign.)

If want a higher (lower) degree polynomial fit, just fit higher (lower) degrees of Legendre polynomials.

来源：https://stackoverflow.com/questions/63037985/fit-curve-to-data-get-analytical-form-check-when-curve-crosses-threshold

标签

python

curve-fitting

spline