# A Well-defined Guide on Interpolation Vs. Extrapolation

Interpolation means finding unknown data that lies within the range of given values while extrapolation means projecting known data to obtain unknown values. But this is not the only fact that sets them apart.. Join ScienceStruck as we explore the meaning, methods, and applications of each of these two techniques of numerical analysis that are very similar yet have distinct differences.

Padmini Krishna

Last Updated: Mar 26, 2018

Did You Know?

**Curve Fitting**, a numerical method of statistical analysis is a very good example of both

*interpolation*as well as

*extrapolation*.In it, a few measured data points are used to plot a mathematical function, and then, a known curve that fits best to that function is constructed. Since the shape of the fitted curve is known, it can even be extended beyond the range of given values.

**Interpolation**and

**Extrapolation**, sound like complex scientific hypotheses, they are fundamental techniques that are used in mathematics and statistics.

In fact, they have significant applications in real world scenarios, for example: data modeling which can be used to forecast weather, stock market trends, and much more. Methods like these also form the building blocks of elaborate fields of study like Digital Signal Processing. Have you always been gripped by the detail, perfection, and captivating beauty of computer-generated design patterns? Did you know that the software used to generate these patterns also use mathematical techniques like interpolation and extrapolation to create them?

If we have succeeded in arousing your curiosity, then allow us to resolve every doubt you have ever had about interpolation and extrapolation.

Interpolation and Extrapolation

What It Really Is

Interpolation

**(def.)**The numerical method of interpolation refers to the calculation of values that lie somewhere in the middle of the given discrete set of data points.

✏ Drawing straight or curved lines that seem to be a good fit between each set of given points would naturally help give the graph a definitive shape resembling that of a known function.

Extrapolation

**(def.)**The numerical method of extrapolation is used to calculate points that are outside the range of the given set of discrete data points by using relevant methods of assumption.

✏ Just like in the previous analogy, a larger set of values can be obtained by interpolating the function, either linearly or by taking help of the shape of a predictable nonlinear function.

✏ Since the shape of the graph has been defined based on a known function, the corresponding values for any point that has not only been included in the given discrete set of data points but also is beyond the scope or range of the given data, can be predicted.

Various Methods Used

Interpolation

Interpolation is generally done on mathematical functions by making use of curve fitting or regression techniques (the analysis of the relationship between variables).

Some methods of interpolation that are generally used are:

♦ Linear Interpolation

The interpolation which results from joining the points on the graph with a straight line is called Linear Interpolation.

♦ Nonlinear Interpolation

•

If the values plotted on the graph represent a polynomial function (an equation whose degree is more than 1), interpolation to obtain unknown values is done by substituting the coordinates of the required point in the equation of the polynomial function. This is called Polynomial Interpolation.

**Polynomial Interpolation**If the values plotted on the graph represent a polynomial function (an equation whose degree is more than 1), interpolation to obtain unknown values is done by substituting the coordinates of the required point in the equation of the polynomial function. This is called Polynomial Interpolation.

**Spline Interpolation**

Polynomial Interpolation has its limitations, and can yield unreliable results for higher degrees unless certain special sets of polynomials are used in calculations. Spline Interpolation is an alternative technique that is more efficient in which a combination of different low-degree polynomials for different segments of the graph are used for interpolation.

♦ Multivariate Interpolation

Multivariate Interpolation simply means that interpolation is done on functions having more than one variable, and hence, span over more than one spatial dimension. Interpolation using Gaussian processes is a good example of Multivariate Interpolation.

Extrapolation

It is impossible to extrapolate a set of finite data without using some method of interpolation to figure out which mathematical function can be applied as the basis to predict additional data. Interpolation using

Commonly used extrapolation techniques are:

*Lagrange Polynomials*, or by trying to find the*Newton's Series*for the data, are two common methods of interpolation that generally precede extrapolation.Commonly used extrapolation techniques are:

♦ Linear Extrapolation

Just like with Linear Interpolation, this method is built on the fact that the given data points have a linear relationship with each other.

♦ Polynomial Extrapolation

Using any method of interpolation will not yield accurate results upon extrapolation; iterative numerical methods wherein finite distances are calculated are generally used in this technique. A befitting polynomial function that can be used for extrapolation needs to be deduced.

♦ Conic Extrapolation

This method, wherein conic section templates are used for extrapolation, is mostly implemented using software.

♦ French Curve Extrapolation

The historic set of curves that were standardized by renowned mathematician, Euler, can also be used in the extrapolation of certain kinds of data.

♦ Extrapolation using Fast Fourier Transform

Fast Fourier Transform is a mathematical technique where the data used is convoluted. Using Fast Fourier Transform for extrapolation saves a lot of computation time.

Examples of Real World Applications

Interpolation

The various techniques of interpolation see their widest application in the field of engineering. Here are a few examples of real scenarios where interpolation is truly a godsend.

✏ At the recipient end, once the sampled version of the signal is obtained, the original signal, obviously, needs to be reconstructed.

✏ How does one obtain the entire original signal from the bits and pieces that have been transmitted? Answer: Interpolation.

Airplane, or Aerospace Engineering

✏ For a rocket that is sent into outer space, the value of each of its parameters, like speed, acceleration, temperature, and so on, can be represented in the form of mathematical functions. Readings of these values are taken at regular intervals.

✏ To obtain the values for those instants where readings were not explicitly taken, interpolation can be used.

Thermodynamics and Fluid Mechanics

✏ In various industries pertaining to chemicals, mechanical engineering, or material sciences, knowing the thermal and physical properties of various fluid substances is a must.

✏ Although the calculation and computation of many of these values is done with the help of software and databases, on a very high scale, the sheer volume of numbers to be crunched makes it a tedious, time-consuming process.

✏ Although the calculation and computation of many of these values is done with the help of software and databases, on a very high scale, the sheer volume of numbers to be crunched makes it a tedious, time-consuming process.

Extrapolation

Extrapolation and prediction go hand in hand, and hence, in almost every industry whenever anything needs to be planned for the future, or with reference to projected values, extrapolation is always used.

Forecasting

Here are a few real examples where extrapolation is used to forecast future trends.

• It is possible to predict what the population of the world will be in any future year by extrapolating the data from the past few decades that we currently have.

• It is possible to predict what the population of the world will be in any future year by extrapolating the data from the past few decades that we currently have.

Engineering and Mathematical Applications

1. When audio signals are transmitted over long distances, they are susceptible to disturbance and noise. Using extrapolation techniques to predict the signal during its reconstruction, at the recipient end, can help obtain a much clearer signal.

2. The field of image processing has benefited greatly because of extrapolation. As a result of nonlinear extrapolation of data, HDTVs have the simplicity of computation.

2. The field of image processing has benefited greatly because of extrapolation. As a result of nonlinear extrapolation of data, HDTVs have the simplicity of computation.

*Interpolation*and

*Extrapolation*are parallel concepts. However, if one were to decide which method is "better", then, in terms of reliability,

**interpolation will always have the upper hand**because it uses a sort of approximation of actual measured values in the computation of unknown values; whereas extrapolation is done on the basis of assumptions and comes with varying degrees of uncertainty. In fact, scientific papers have declared a warning to the world about the dangers of extrapolating large groups of critical data.

We hope that we were able to simplify these primary techniques of statistical analysis for you.