The NumPy polyfit() function fits a set of data points to a polynomial, even if the underlying function is not continuous:
- Continuing with the price data of BHP and VALE, look at the difference of their close prices and fit it to a polynomial of the third power:
bhp=np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True) vale=np.loadtxt('VALE.csv', delimiter=',', usecols=(6,), unpack=True) t = np.arange(len(bhp)) poly = np.polyfit(t, bhp - vale, 3) print("Polynomial fit", poly)The polynomial fit (in this example, a cubic polynomial was chosen) is as follows:
Polynomial fit [ 1.11655581e-03 -5.28581762e-02 5.80684638e-01 5.79791202e+01] - The numbers you see are the coefficients of the polynomial. Extrapolate to the next value with the
polyval()function and the polynomial object that we got from the fit:print("Next value", np.polyval(poly, t[-1] + 1))The next value we predict will be this:
Next value 57.9743076081 - Ideally, the difference between the close prices of BHP and VALE should be as small as possible. In an extreme case, it might be zero at some point. Find out when our polynomial fit reaches zero with the
roots()function:print( "Roots", np.roots(poly))
The roots of the polynomial are as follows:
Roots [ 35.48624287+30.62717062j 35.48624287-30.62717062j -23.63210575 +0.j ] - Another thing you may have learned in calculus class was to find extrema—these could be potential maxima or minima. Remember, from calculus, that these are the points where the derivative of our function is zero. Differentiate the polynomial fit with the
polyder()function:der = np.polyder(poly) print("Derivative", der)The coefficients of the derivative polynomial are as follows:
Derivative [ 0.00334967 -0.10571635 0.58068464] - Get the roots of the derivative:
print("Extremas", np.roots(der))The extremas that we get are as follows:
Extremas [ 24.47820054 7.08205278]Let's double-check and compute the values of the fit with the
polyval()function:vals = np.polyval(poly, t)
- Now, find the maximum and minimum values with the
argmax()and theargmin()function:vals = np.polyval(poly, t) print(np.argmax(vals)) print(np.argmin(vals))
This gives us the expected results shown in the following screenshot. OK, not quite the same results, but, if we backtrack to step 1, we can see that
twas defined with thearange()function:7 24
Plot the data and the fit it to get the following plot:
Obviously, the smooth line is the fit and the jagged line is the underlying data. But as it's not that good a fit, you might want to try a higher order polynomial.
We fit data to a polynomial with the polyfit() function. We learned about the polyval() function that computes the values of a polynomial, the roots() function that returns the roots of the polynomial, and the polyder() function that gives back the derivative of a polynomial (see polynomials.py):
from __future__ import print_function
import numpy as np
import sys
import matplotlib.pyplot as plt
bhp=np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True)
vale=np.loadtxt('VALE.csv', delimiter=',', usecols=(6,), unpack=True)
t = np.arange(len(bhp))
poly = np.polyfit(t, bhp - vale, 3)
print("Polynomial fit", poly)
print("Next value", np.polyval(poly, t[-1] + 1))
print("Roots", np.roots(poly))
der = np.polyder(poly)
print("Derivative", der)
print("Extremas", np.roots(der))
vals = np.polyval(poly, t)
print(np.argmax(vals))
print(np.argmin(vals))
plt.plot(t, bhp - vale, label='BHP - VALE')
plt.plot(t, vals, '-—', label='Fit')
plt.title('Polynomial fit')
plt.xlabel('Days')
plt.ylabel('Difference ($)')
plt.grid()
plt.legend()
plt.show()You could do a number of things to improve the fit. For example, try a different power as, in this section, a cubic polynomial was chosen. Consider smoothing the data before fitting it. One way you could smooth the data is with a moving average. You can find examples of simple and EMA calculations in the Chapter 3, Getting Familiar with Commonly Used Functions.