What is Interpolation? Everything You Need To Know

What Is Interpolation?

Interpolation is estimating or measuring an unknown quantity between two known quantities. This is often used in defined piecewise functions in mathematics, meaning they have discontinuous graphs.

Interpolation Formula

The process of interpolation involves creating a smooth curve between two data points. The curve is created by plotting the point on the graph at which the distance between two points is equal to half of their difference in y-coordinates. It is important because it ensures that your data points are evenly spaced along your line.

The interpolation formula is as follows:

Types of Interpolation

Following major types of interpolation is used to create a smooth transition between two points.

• Cubic Spline Interpolation: This type of interpolation creates a curved line to connect two points in a graph. It's also called "quadratic spline interpolation" or "quadratic smoothing."
• Lagrange Basis Interpolation: This type of interpolation is similar to cubic spline interpolation in that it creates a curved line between two points on a graph but differs in how it chooses different possible curves. Lagrange basis interpolation considers all potential curves and then selects the one that produces the best fit for the data set.
• Linear Interpolation: This method requires a straight line or curve between two points. It is used when there is an exact relationship between the values, but no data points are available.
• Nearest Neighbor Interpolation: This method uses the closest known value to predict the value between two known values. It is useful when there are limited data points available.
• Spline Interpolation: This method uses a curve that passes through as many data points as possible, eliminating any gaps in your data set and making it more robust against errors in measurement or sampling.

Uses of Interpolation

Interpolation is a mathematical function that takes the values of nearby points and uses them to predict the value of the unknown end. It can be used in a variety of industries, including

Geodesy: Interpolation is used to map out features on Earth's surface, such as mountains or ocean currents, using satellite imagery.

Engineering: Interpolation predicts how materials behave in extreme conditions, such as high temperatures or pressure.

Statistical analysis: Interpolation can be used to smooth out data sets so that they become more evenly distributed. For example, if you have a spike in sales one day, you can use interpolation to smooth out the rest of your sales data for that month so that the overall trend looks smooth instead of erratic.

Interpolation vs. Extrapolation

Interpolation is estimating the value of an unknown number between known values. It is a handy tool for predicting trends and other patterns in data.

Extrapolation is the method of estimating a value beyond what has been observed. It can be helpful when you want to predict future events or trends, but it's important to remember that extrapolation is only sometimes accurate.

These are two different ways to arrive at an answer. While interpolation takes into historical account data and makes an educated guess as to what might happen next, extrapolation doesn't consider any information beyond what has already happened.

Examples of Interpolation

from scipy.interpolate import interp1d

import numpy as np

import matplotlib.pyplot as plotlib

# Define parameters.

x_sample_start = 0

x_sample_end = 15

x_sample_number = 16

x_sample_endpoint = True

x_new_sample_number = 60

y_exponent = 2

y_divisor = 15.0

# Create an array of x sample data points.

x = np.linspace(

    x_sample_start,

    x_sample_end,

    num=x_sample_number,

    endpoint=x_sample_endpoint)

print("x array values:")

print(x)

# Create an array of y sample data points as a sine function of x.

y = np.sin(-x**y_exponent/y_divisor)

print("y array values:")

print(y)

# Create interpolation functions to generate new y data points.

linear_interpolation_function = interp1d(x, y, kind='linear')

cubic_interpolation_function = interp1d(x, y, kind='cubic')

quadratic_interpolation_function = interp1d(x, y, kind='quadratic')

# Define an array of new x data points.

x_new = np.linspace(

    x_sample_start,

    x_sample_end,

    num=x_new_sample_number,

    endpoint=x_sample_endpoint)

# Plot the functions.

plotlib.plot(

    x, y, 'o',

    x_new, linear_interpolation_function(x_new), '-',

    x_new, cubic_interpolation_function(x_new), '--',

    x_new, quadratic_interpolation_function(x_new), ':'

)

# Plot the legend.

plotlib.legend(['data', 'linear', 'cubic', 'quadratic'], loc='best')

# Display the plot.

plotlib.show()

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FAQs

1. Why is interpolation needed?

Interpolation is needed because it helps us understand the difference between two points on a graph. The more data we have, the more accurate our interpolation will be.

2. What do you mean by interpolation?

Interpolation is a method of calculating the value of a function or data between two known points. This can be done by fitting a polynomial to the data, or by guessing and checking.

3. What is interpolation with an example?

Interpolation is the process of finding the area under a curve. It is used to find missing data in a graph or table, and can also be used to find the slope of a curve at a certain point.

For example, if you know that the average height of adults in your town is 5'9", but you don't know how tall each person is, you could use interpolation to determine the average weight of all adults in your town by looking at each person's height and weight and knowing that it follows a linear pattern.

4. What is interpolation in math simple?

Interpolation is a mathematical process that calculates the value of an unknown number between two known numbers.

For example, if you have a list of numbers and the second number is missing, interpolation can calculate the missing number.

5. What are interpolation and extrapolation?

Interpolation and extrapolation are two methods for predicting future values.

Interpolation refers to taking known values and extrapolating them into the future, while extrapolation refers to taking known values, then extending them beyond their normal range of values.

6. What are interpolations and their types?

• Cubic Spline Interpolation
• Lagrange Basis Interpolation.
• Linear Interpolation
• Nearest Neighbor Interpolation
• Spline Interpolation