SciPy Interpolation

Interpolation estimates unknown values that lie between values you already know.

What is Interpolation?

Suppose you measured the temperature in Karachi at 6 am, 9 am, 12 pm and 3 pm. What was it at 10:30 am? You did not measure it, but you can estimate it from the nearby readings. That is interpolation.

Linear Interpolation With NumPy

For a quick straight-line estimate, NumPy's interp() is enough:

import numpy as np

hours = [6, 9, 12, 15]
temps = [27, 31, 35, 34]
print(np.interp(10.5, hours, temps))

33.0

At 10:30 (10.5 hours), halfway between 31 and 35 degrees, the estimate is 33.

Interpolation With SciPy

scipy.interpolate.interp1d builds a function from your points. You can then call it with any value in the range:

import numpy as np
from scipy.interpolate import interp1d

hours = np.array([6, 9, 12, 15])
temps = np.array([27, 31, 35, 34])

f = interp1d(hours, temps)
print(f(10.5))
print(f([7, 8, 13]))

33.0
[28.33333333 29.66666667 34.66666667]

Smooth Curves With Cubic Splines

Straight lines between points make sharp corners. A cubic spline draws a smooth curve through every point, which is often more realistic for things like temperature:

import numpy as np
from scipy.interpolate import CubicSpline

hours = np.array([6, 9, 12, 15])
temps = np.array([27, 31, 35, 34])

cs = CubicSpline(hours, temps)
print(np.round(cs([7, 10.5, 13]), 2))

[28.02 33.31 35.47]

Compare with the straight-line answers above. The spline estimates that the temperature kept rising a little after 12 pm before falling, which a straight line cannot show.

Filling Gaps in a Data Series

A sensor missed some readings. Fill them in with interpolation:

import numpy as np

days = np.arange(1, 8)
readings = np.array([20.0, 22.0, np.nan, 25.0, np.nan, np.nan, 31.0])

known = ~np.isnan(readings)
filled = readings.copy()
filled[~known] = np.interp(days[~known], days[known], readings[known])
print(filled)

[20.  22.  23.5 25.  27.  29.  31. ]

The empty days now have values that fit smoothly between their neighbours.

Extrapolation Is Risky

By default, interp1d refuses values outside the range of your data:

from scipy.interpolate import interp1d

f = interp1d([6, 9, 12, 15], [27, 31, 35, 34])
f(18)

ValueError: A value (18.0) in x_new is above the interpolation range's maximum value (15).

This protects you. Guessing far outside the known data can give very wrong answers.

Exercise