Interpolation

Interpolation is a useful mathematical and statistical tool that is used to estimate values between any two given points. In this article, you will learn about this tool, the formula for interpolation and how to use it.

  • Interpolation can be defined as the process of finding a value between two points on a line or curve.

  •  Now to help us remember what it means, we should think of the first part of the word, which is 'inter,' and which means 'enter,' and that  reminds us to look 'inside' the data we originally had. 

  • Interpolation is a tool which is not only useful in statistics, but is a tool that is also useful in the field of science, business or any time whenever there is a need to predict values that fall within any two existing data points.


Examples of Interpolation-

Here's an example which will illustrate the concept of interpolation and give you a better understanding of the concept of interpolation. Let’s suppose a gardener planted a tomato plant and she measured and kept track of the growth of the tomato plant every other day. This gardener is a very curious person, and she would like to estimate how tall her plant was on the fourth day.

Her table of observations basically looked like the table given below:

Day

Height (mm)

1

0

3

4

5

8

7

12

9

16


Based on the given chart, it's not too difficult to figure out whether the plant was probably 6 mm tall on the fourth day and this is because this disciplined tomato plant grew in a linear pattern; that is there was a linear relationship between the number of days measured and the plant's growth. Linear pattern basically means that the points created a straight line. We could estimate it by plotting the given data on a graph.

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But what if the plant does not grow with a convenient linear pattern? What if its growth looked more like that in the picture given below?

(Image to be added soon)

What do you think the gardener will do in order to make an estimation based on the above curve? Well, that is where the interpolation formula comes into picture.


Formula of Interpolation

The interpolation formula can be written as -

y - \[y_{1}\] = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\]

Now , if we go back to the tomato plant example, the first set of values for day three are given as (3,4), the second set of values for day five are given as (5,8), and the value for x is 4 since we want to find the height of the tomato plant, y, on the fourth day. After substituting these given values into the formula, we can easily calculate the estimated height of the plant on the fourth day.

y – y1 =( (y2- y1) / ( x2- x1))* (x- x1)

Putting the values we have been given,

y – 4 = ((8- 4) / ( 5- 3))* (x- 3)

y – 4 =4/2 (x-3)

y – 4 = 2(x-3)

y – 4 = 2(4-3)

y= 2(1) +4

y = 6

There are various different types of interpolation methods. Here they are:

Types of Interpolation Methods

Types of Interpolation

Definition

Linear Interpolation Method 

The Linear Interpolation method applies a distinct linear polynomial between each pair of the given data points for the curves, or within the sets of three points for surfaces.

Nearest Neighbour Method 

In this method the value of an interpolated point is inserted to the value of the most adjacent data point. Therefore, the nearest neighbour method does not produce any new data points.

Cubic Spline Interpolation Method 

The Cube Spline method fits a different cubic polynomial between each pair of the given data points for the curves, or between sets of three points for surfaces.

Shape-Preservation Method

The Shape-preservation method is also known as Piecewise cubic Hermite interpolation (PCHIP). This method preserves the monotonicity and the shape of thegiven data. It is for curves only.

Thin-plate Spline Method 

The Thin-plate Spline method basically consists of smooth surfaces that also extrapolate well. This method is only for surfaces.

Biharmonic Interpolation Method

The Biharmonic method is generally applied to the surfaces only.


Why is the Concept of Interpolation Important?

  • The concept of Interpolation is used to simplify complicated functions by sampling any given data points and interpolating these data points using a simpler function. 

  • Commonly Polynomials are used for the process of interpolation because they are much easier to evaluate, differentiate, and integrate and are known as polynomial interpolation.

FAQ (Frequently Asked Questions)

1. What Do You Mean By Interpolation and What is the Difference Between Interpolation and Extrapolation?

Interpolation can basically be described as guessing data points that fall within the range of the data you are already provided with that is between your existing data points. Extrapolation can be defined as guessing data points from beyond the range of your data set (beyond the data what you have been provided you with)

Interpolation can be defined as an estimation of a value within two known values in a given sequence of values.  When graphical data contains a gap, but the data is available on either side of the gap or at a few specific points within the gap, interpolation is a method that allows us to estimate the values within the gap.

2. What is the Interpolation Formula and What is Interpolation and its Types?

So, it can be understood that the formula for interpolation is a method of curve fitting using the linear polynomials and hence to construct new data points within the given range of a discrete set of known data points(the data points). Linear interpolation can be used since very early antiquity for filling the unknown values in any table.

As we know that Interpolation can be defined as a  process of using the points with known values or  the given sample points to estimate values at other unknown points. Interpolation methods can be used to predict unknown values for any geographic point data, for example elevation, rainfall, chemical concentrations, noise levels, and so on. Here are the types of interpolation methods -

  • Linear Interpolation Method  

  • Nearest Neighbour Method  

  • Cubic Spline Interpolation Method 

  • Shape-Preservation Method 

  • Thin-plate Spline Method 

  • Biharmonic Interpolation Method