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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.
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Interpolation can be defined as the process of finding a value between two points on a line or curve.
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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.
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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.
(Image to be added soon)
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:
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Types of Interpolation |
Definition |
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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. |
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Nearest Neighbor Method |
In this method the value of an interpolated point is inserted to the value of the most adjacent data point. Therefore, the nearest neighbor method does not produce any new data points. |
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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. |
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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 the given data. It is for curves only. |
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Thin-plate Spline Method |
The Thin-plate Spline method basically consists of smooth surfaces that also extrapolate well. This method is only for surfaces. |
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Biharmonic Interpolation Method |
The Biharmonic method is generally applied to the surfaces only. |
Why is the concept of Interpolation Important?
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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.
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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 cert
ain 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.