Interpolation vs Extrapolation: Common Methods with Python Code

In this interesting post, we will explore two terms we often hear from data scientists i.e. Extrapolation & Interpolation. When we get access to data points, we try to present all the information that can be useful.

Let’s start exploring this bit from the AI universe.

Interpolation vs Extrapolation

The prefix inter- means between, and extra- means beyond.

Interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points.

Extrapolation is a type of estimation, beyond the original observation range, the value of a variable on the basis of its relationship with another variable.

Interpolation vs Extrapolation: Graphical Representation

Let’s consider a few points.

Fit a straight line(we can also fit fancy mathematical functions, just for simplicity we have considered fitting a straight line) in a way that generalizes all the points.

The line lets you both interpolate (generate expected values in between your data points) and extrapolate (generate expected values outside the range of your data points).

Interpolation is guessing data points that fall within the range of the data you have, i.e. between your existing data points. Extrapolation is guessing data points from beyond the range of your known data set.

An Interesting Example of Interpolation and Extrapolation

If I tell you that I had 7 dollars the day before yesterday, and 3 dollars left today, you can interpolate that I had  5 dollars yesterday. You could also extrapolate that I will have 1 dollar tomorrow. You could extrapolate that I will have -1 dollar the day after tomorrow, illustrating one of the many pitfalls of extrapolation.

Think over this pitfall.

It points towards a risk that extrapolation is subjected to greater uncertainty and a higher risk of producing meaningless results.

Common interpolation methods: Linear Interpolation, Polynomial Interpolation & Spline Interpolation with Python Code

Interpolation is a technique used to estimate values between two known values. It is commonly used in data analysis, numerical analysis, and computer graphics. There are several interpolation methods available, but here are three of the most popular ones:

Linear Interpolation with Python Code

What is Linear Interpolation?

Linear Interpolation is a simple method that assumes a linear relationship between two known data points. It connects the two points with a straight line and estimates values between them. It is a good choice for data that changes linearly.

Python Code for Linear Interpolation

To implement linear interpolation in Python, you can use the interp1d function from the scipy.interpolate module:

import numpy as np
from scipy.interpolate import interp1d

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 2, 4, 6])

# Linear interpolation function
f = interp1d(x, y)

# Estimate values
x_new = np.linspace(0, 3, num=10)
y_new = f(x_new)

# Plot original data and estimated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='linear')
plt.legend()
plt.show()

Polynomial Interpolation with Python Code

What is Polynomial Interpolation?

Polynomial interpolation is a method that fits a polynomial curve to the data points. It is a good choice for data that has a non-linear relationship between two points.

Python Code for Polynomial Interpolation

To implement polynomial interpolation in Python, you can use the polyfit function from the numpy module:

import numpy as np

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 1, 4, 9])

# Polynomial interpolation function
p = np.polyfit(x, y, deg=2)
f = np.poly1d(p)

# Estimate values
x_new = np.linspace(0, 3, num=10)
y_new = f(x_new)

# Plot original data and estimated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='polynomial')
plt.legend()
plt.show()

Spline Interpolation with Python Code

What is Spline Interpolation?

Spline interpolation is a method that fits a piecewise polynomial curve to the data points. It is a good choice for data that has a complex or erratic relationship between two points.

Python Code for Spline Interpolation

To implement spline interpolation in Python, you can use the splrep and splev functions from the scipy.interpolate module:

import numpy as np
from scipy.interpolate import splrep, splev

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 2, 4, 6])

# Spline interpolation function
spl = splrep(x, y)
f = splev(x, spl)

# Estimate values
x_new = np.linspace(0, 3, num=10)
y_new = splev(x_new, spl)

# Plot original data and estimated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='spline')
plt.legend()
plt.show()

Common extrapolation methods: Linear Extrapolation, Polynomial Extrapolation & Spline Extrapolation with Python Code

Extrapolation is the technique of estimating values outside the range of known data points. Extrapolation can be useful when trying to predict trends in data that extends beyond the available data. However, extrapolation can also be unreliable and can lead to inaccurate results if the data is not well-behaved. Here are three popular extrapolation methods:

Linear Extrapolation with Python Code

What is Linear Extrapolation?

Linear extrapolation is a simple method that extends a straight line beyond the range of known data points. It assumes that the relationship between the known data points is linear and that this relationship continues beyond the range of the data. Linear extrapolation is the easiest method to use, but it can be highly unreliable for data that doesn’t follow a straight line.

Python Code for Linear Extrapolation

Here is an example of linear extrapolation in Python:

import numpy as np

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 2, 4, 6])

# Linear extrapolation function
slope = (y[-1] - y[-2]) / (x[-1] - x[-2])
intercept = y[-1] - slope * x[-1]
x_new = np.array([4, 5])
y_new = slope * x_new + intercept

# Plot original data and extrapolated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='linear extrapolation')
plt.legend()
plt.show()

Polynomial Extrapolation with Python Code

What is Polynomial Extrapolation?

Polynomial extrapolation is a method that fits a polynomial curve to the data points and extends the curve beyond the range of the data. It is more accurate than linear extrapolation for data that doesn’t follow a straight line, but it can still be unreliable for highly complex data.

Python Code for Polynomial Extrapolation

Here is an example of polynomial extrapolation in Python:

import numpy as np

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 1, 4, 9])

# Polynomial extrapolation function
p = np.polyfit(x, y, deg=2)
x_new = np.array([3, 4])
y_new = np.polyval(p, x_new)

# Plot original data and extrapolated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='polynomial extrapolation')
plt.legend()
plt.show()

Spline Extrapolation with Python Code

What is Spline Extrapolation?

Spline extrapolation is a method that fits a piecewise polynomial curve to the data points and extends the curve beyond the range of the data. It is the most accurate method for extrapolation and can handle highly complex data, but it can also be the most computationally expensive.

Python Code for Spline Extrapolation

Here is an example of spline extrapolation in Python:

import numpy as np
from scipy.interpolate import splrep, splev

# Example data
x = np.array([0, 1, 2, 3])
y = np.array([0, 2, 4, 6])

# Spline extrapolation function
spl = splrep(x, y, k=1)
x_new = np.array([3, 4])
y_new = splev(x_new, spl, der=0)

# Plot original data and extrapolated values
import matplotlib.pyplot as plt

plt.plot(x, y, 'o', label='data')
plt.plot(x_new, y_new, '-', label='spline extrapolation')
plt.legend()
plt.show()

Frequently Asked Questions

Conclusion

Interpolation and extrapolation both try to extend what we have observed—to what we have not observed but do so in different directions or modes. Interpolation extends to what must have happened between observations, while extrapolation extends to what happens before, after, or beyond observations.

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