The NumPy polyfit function can fit 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, let's 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, int(sys.argv[1])) print "Polynomial fit", polyThe polynomial fit (in this example, a cubic polynomial was chosen):
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
polyvalfunction and the polynomial object we got from the fit:print "Next value", np.polyval(poly, t[-1] + 1)
The next value we predict will be:
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
rootsfunction: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 we 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
polyderfunction:der = np.polyder(poly) print "Derivative", der
The coefficients of the derivative polynomial are as follows:
Derivative [ 0.00334967 -0.10571635 0.58068464]The numbers you see are the coefficients of the derivative polynomial.
- Get the roots of the derivative and find the extrema:
print "Extremas", np.roots(der)
The extrema that we get are:
Extremas [ 24.47820054 7.08205278]Let's double check; compute the values of the fit with
polyval:vals = np.polyval(poly, t)
- Now, find the maximum and minimum values with
argmaxandargmin:vals = np.polyval(poly, t) print np.argmax(vals) print np.argmin(vals)
This gives us the following expected results. Ok, not quite the same results, but, if we backtrack to step 1, we can see that
twas defined with thearangefunction:7 24
- Plot the data and the fit it as follows:
plot(t, bhp - vale) plot(t, vals) show()
It results in this plot:
Obviously, the smooth line is the fit and the jagged line is the underlying data. It's not that good a fit, so 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):
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
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, int(sys.argv[1]))
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)
plot(t, bhp - vale)
plot(t, vals)
show()There are a number of things you could do to improve the fit. Try a different power as, in this tutorial, a cubic polynomial was chosen. Consider smoothing the data before fitting it. One way you could smooth is with a moving average. Examples of simple and exponential moving average calculations can be found in the previous chapter.