An Introduction
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:
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- y1= ((y2-y1)/ (x2- x1))* (x2- x1)
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
Types of Interpolation Methods
There are various different types of Interpolation Methods. Here they are:
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.
Drawbacks of Interpolation Method
While Interpolation is known to solve a lot of Mathematical and Statistical problems, it does have certain drawbacks and criticisms. One such drawback is that although the method of Interpolation is simple and has been known to Mathematicians and people in general, for a long time, it has been known to lack the necessary accuracy and precision.
In the ancient Greek and Babylonian civilizations, the method of Interpolation was crudely used for prediction purposes. They would determine various factors such as the right time for sowing seeds (in farming practices), calculate astronomical points in space and time to determine celestial events up in the sky, and plan strategies for monsoons, crop yield, growth and movement.
Today, the same methods are being used in the modern-day problems of the world. People use these methods of Interpolation for the fairly unpredictable stock markets, in solving data related to security analysis, for determining volatility of the highly unpredictable public-traded shares and bonds, and this overpowering mass of data makes the employment of Interpolation unreasonable as it can lead to many faulty predictions.
More often than not, the use of Interpolation in regression analysis, in this way leads to the yielding of an “error term”, that is obtaining a set of values that do not represent the factual relationship between the variables most crucial for successful prediction. Interpolation must be employed for simple predictions such as determining the interest rate or value of any variable for which the data point is missing.
FAQs on Interpolation
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, 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 Neighbor Method
Cubic Spline Interpolation Method
Shape-Preservation Method
Thin-plate Spline Method
Biharmonic Interpolation Method
3. What are the Various Types of Interpolation?
There are several types of Interpolations found in Mathematical models. Some of them are:
(i) Linear Interpolation (when linear polynomials are used to construct further points of data.
(ii) Polynomial Interpolation (which involves a given set of data from the polynomial of the lowest degree and lets it pass by the points in the data set for Interpolation data.
(iii) Bilinear Interpolation (Interpolation type using two distinct variables leading to the formation of a 2D rectilinear grid; Interpolation is done in two different directions, one after the other).
(iv) Spline Interpolation (when an interpolant is a special place-wise polynomial).
4. How is Interpolation useful in Statistics?
Interpolation is widely used in Statistical models for commercial and Mathematical research, as it helps predict the future probable points in data analysis. From the obtained sets, one can determine where the general consistent trend will lead the price (of a product or service), potential yield (and growth for a company) or gain insights into the stock market under a given market dynamic. Financial analysts have consistently exploited this method to reach logical conclusions in the bonds market and financial world.
5. Give an Example of Interpolation.
There can be various examples of Interpolation found in day to day life as well as in the Mathematical world. When one wants to predict the growth in the market price of fuel such as petrol or diesel and extend it along a straight line on the graph with respect to the global market trends such as currency value of a country or increment or decrement in share market values, one gets a data set of points that points towards an increment of fuel price as share market value falls or a currency value declines.
In the physical world, the mass and volume of a given substance can be determined by the similar plotting method of Interpolation to find how the volume of a substance changes with any increase or decrease in the mass of the substance.