extrapolating data with numpy/python

After discussing with you in the Python chat - you're fitting your data to an exponential. This should give a relatively good indicator since you're not looking for long term extrapolation.

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

def exponential_fit(x, a, b, c):
    return a*np.exp(-b*x) + c

if __name__ == "__main__":
    x = np.array([0, 1, 2, 3, 4, 5])
    y = np.array([30, 50, 80, 160, 300, 580])
    fitting_parameters, covariance = curve_fit(exponential_fit, x, y)
    a, b, c = fitting_parameters
    
    next_x = 6
    next_y = exponential_fit(next_x, a, b, c)
    
    plt.plot(y)
    plt.plot(np.append(y, next_y), 'ro')
    plt.show()

The red dot in the on far right axis shows the next "predicted" point.

As pointed out by this answer to a related question, as of version 0.17.0 of scipy, there is an option in scipy.interpolate.interp1d that allows linear extrapolation. In your case, you could do:

>>> import numpy as np
>>> from scipy import interpolate

>>> x = [1, 2, 3, 4, 5]
>>> y = [5, 10, 15, 20, 25]
>>> f = interpolate.interp1d(x, y, fill_value = "extrapolate")
>>> print(f(6))
30.0

Since your data is approximately linear you can do a linear regression, and then use the results from that regression to calculate the next point, using y = w[0]*x + w[1] (keeping the notation from the linked example for y = mx + b).

If your data is not approximately linear and you don't have some other theoretical form for a regression, then general extrapolations (using say polynomials or splines) are much less reliable as they can go a bit crazy beyond the known data points. For example, see the accepted answer here.

You can also use numpy's polyfit:

data = np.array([[1,5], [2,10], [3,15], [4,20], [5,25]])
fit = np.polyfit(data[:,0], data[:,1] ,1) #The use of 1 signifies a linear fit.

fit
[  5.00000000e+00   1.58882186e-15]  #y = 5x + 0

line = np.poly1d(fit)
new_points = np.arange(5)+6

new_points
[ 6, 7, 8, 9, 10]

line(new_points)
[ 30.  35.  40.  45.  50.]

This allows you to alter the degree of the polynomial fit quite easily as the function polyfit take thes following arguments np.polyfit(x data, y data, degree). Shown is a linear fit where the returned array looks like fit[0]*x^n + fit[1]*x^(n-1) + ... + fit[n-1]*x^0 for any degree n. The poly1d function allows you turn this array into a function that returns the value of the polynomial at any given value x.

In general extrapolation without a well understood model will have sporadic results at best.

---

Exponential curve fitting.

from scipy.optimize import curve_fit

def func(x, a, b, c):
    return a * np.exp(-b * x) + c

x = np.linspace(0,4,5)
y = func(x, 2.5, 1.3, 0.5)
yn = y + 0.2*np.random.normal(size=len(x))

fit ,cov = curve_fit(func, x, yn)
fit
[ 2.67217435  1.21470107  0.52942728]         #Variables

y
[ 3.          1.18132948  0.68568395  0.55060478  0.51379141]  #Original data

func(x,*fit)
[ 3.20160163  1.32252521  0.76481773  0.59929086  0.5501627 ]  #Fit to original + noise