Linear Interpolation is the technique of determining the values of the functions of any intermediate points when the values of two adjacent points are known. Linear interpolation is basically the estimation of an unknown value that falls within two known values. Linear Interpolation is used in various disciplines like statistical, economics, price determination, etc. It is used to fill the gaps in the statistical data for the sake of continuity of information.
By using the following formula we can Linearly interpolate the given data point
Here (x1, y1) are the coordinates of the first data point. And (x2,y2) are coordinates of the second data point, where x is the point on which we perform interpolation and y is the interpolated value.
Example Problem:
Let’s take an example for better understanding. We have the following data values where x denotes the number and y is the function of the square root of x. Our task is to find the square root of 5.5 (x).
|
x |
1 |
2 |
3 |
4 |
5 |
6 |
|---|---|---|---|---|---|---|
|
y ( f(x) = √x ) |
1 |
1.4142 |
1.7320 |
2 |
2.2360 |
2.4494 |
We can use the Linear Interpolation method here.
1. Find the two adjacent (x1, y1) ,(x2,y2) from the x. i.e. (5,2.2360) and (6,2.4494).
Where x1 = 5, x2= 6, y1 = 2.2360, y2 = 2.4494, and we interpolate at point x = 5.5.
2. Using the formula y(x) = y1 + (x – x1) \frac{(y2 – y1) }{ (x2 – x1)}
3. After putting the values in the above equation.
y = 2.3427
y = 2.3427At x = 5.5 the value of Y will be 2.3427. So by using linear interpolation we can easily determine the value of a function between two intervals.
Approach 1:
Using the formula
Example: Suppose we have a dataset of the population of a city and the year.
|
X(Year) |
2016 |
2017 |
2018 |
2019 |
2021 |
|---|---|---|---|---|---|
|
Y(Population) |
10001 |
12345 |
74851 |
12124 |
5700 |
Here, X is the year and Y is the population in any city. Our task to find the population of the city in the year 2020.
We choose our (x1, y1) ,(x2,y2) as x1=2019 , y1=12124, x2=2021, y2=5700, x = 2020, y = ?
Here (x1, y1) and (x2, y2) are two adjacent points and x is the year for which we want to predict the value of the y population.
Python3
# Python3 code# Implementing Linear interpolation# Creating Function to calculate the# linear interpolationdef interpolation(d, x): output = d[0][1] + (x - d[0][0]) * ((d[1][1] - d[0][1])/(d[1][0] - d[0][0])) return output# Driver Codedata=[[2019, 12124],[2021, 5700]]year_x=2020# Finding the interpolationprint("Population on year {} is".format(year_x), interpolation(data, year_x)) |
Population on year 2020 is 8912.0
Approach 2:
Using scipy.interpolate.interp1d
Similarly, we can achieve linear interpolation using a scipy library function called interpolate.interp1d.
Syntax : scipy.interpolate.interp1d(x, y, kind=’linear’, axis=- 1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)
|
Sr. no. |
Parameters |
Description |
|---|---|---|
|
1. |
x |
A 1-D array of real values. |
|
2. |
y |
A N-D array of real values. |
|
3. |
kind |
i.e. kind of interpolation do you want it can be ‘linear’, ‘nearest’, ‘nearest-up’, ‘zero’, ‘slinear’, ‘quadratic’, ‘cubic’, ‘previous’, or ‘next’. ‘zero’, ‘slinear’, ‘quadratic’ and ‘cubic’, by defaults it is linear. |
|
4. |
axis |
Specifies the axis of y along which we interpolate. |
|
5. |
copy |
It holds boolean values if True, the class makes internal copies of x and y . |
|
6. |
bounds_error |
It holds boolean values If True, a ValueError is raised when interpolation is attempted on a value outside the range of x. |
Example:
Let’s have a random dataset :
X = [1,2,3,4,5], Y = [11,2.2,3.5,-88,1], and we want to find the value of Y at point 2.5.
Python3
# Implementation of Linear Interpolation using Python3 code# Importing libraryfrom scipy.interpolate import interp1dX = [1,2,3,4,5] # random x valuesY = [11,2.2,3.5,-88,1] # random y values# test valueinterpolate_x = 2.5# Finding the interpolationy_interp = interp1d(X, Y)print("Value of Y at x = {} is".format(interpolate_x), y_interp(interpolate_x)) |
Output
Value of y at x = 2.5 is 2.85
