## Pages

### Spline Interpolation Example in Python

Interpolation is a method of estimating unknown data points in a given dataset range. Discovering new values between two data points makes the curve smoother. Spline interpolation is a type of piecewise polynomial interpolation method.
The SciPy API provides several functions to implement the interpolation method for a given dataset.

In this tutorial, you'll learn how to apply interpolation for a given dataset by using SciPy API functions in Python. The tutorial covers;

1. Preparing test data
2. Direct spline interpolation
3. Spline interpolation with InterpolatedUnivariateSpline
4. Source code listing

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```from scipy import interpolate
import matplotlib.pyplot as plt
import numpy as np ```
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Preparing test data

We'll create a simple dataset for this tutorial. The target data contains x and y components of curve data. You may apply the same method for your target data.

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```y = [1,3,4,3,5,7,5,6,8,7,8,9,8,7]
n = len(y)
x = range(0, n)```
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Next, we'll visualize the data on a graph and apply linear interpolation. Linear interpolation iin a graph is simply connecting data points linearly.

` `
```plt.plot(x,y,'ro')
plt.plot(x,y, 'b')
plt.title("Data set and linear interpolation")
plt.show() ```

Direct Spline Interpolation

To represent spline interpolation smoothing coefficients can be taken parametrically or directly. The 'splrep' function helps us to define the curve with direct method. It provides t, c, k tuple containing the vector of knots, the B-spline coefficients, and the degree of the spline. After taking the coefficients we'll use 'splev' function to evaluate the spline.

The 'splev' evaluates the value of the smoothing polynomial based on given coefficients. Finally, we'll visualize the curve on a graph.

` `
```tck = interpolate.splrep(x, y, s=0)
xfit = np.arange(0, n-1, np.pi/50)
yfit = interpolate.splev(xfit, tck, der=0)

plt.plot(x, y, 'ro')
plt.plot(xfit, yfit,'b')
plt.plot(xfit, yfit)
plt.title("Direct spline interpolantion")
plt.show() ```

Spline interpolation with InterpolatedUnivariateSpline

We can also do spline smoothing by using the InterpolatedUnivariateSpline class. Based on UnivariateSpline class, InterpolatedUnivarateSpline fits the curve based on x and y data. Below, we apply the method for out target test data and visualize it on a graph.

` `
```s = interpolate.InterpolatedUnivariateSpline(x, y)
xfit = np.arange(0, n-1, np.pi/50)
yfit = s(xfit)

plt.plot(x, y, 'ro')
plt.plot(xfit, yfit,'green')
plt.title("InterpolatedUnivariateSpline interpolation")
plt.show()```
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In this tutorial, we've briefly learned how to implement spline interpolation by using SciPy API's interpolation functions in Python. The full source code is listed below.

Source code listing

` `
```from scipy import interpolate
import matplotlib.pyplot as plt
import numpy as np

y = [1,3,4,3,5,7,5,6,8,7,8,9,8,7]
n = len(y)
x = range(0, n)```
` `
```plt.plot(x,y,'ro')
plt.plot(x,y, 'b')
plt.title("Data set and linear interpolation")
plt.show()

tck = interpolate.splrep(x, y, s=0)
xfit = np.arange(0, n-1, np.pi/50)
yfit = interpolate.splev(xfit, tck, der=0)

plt.plot(x, y, 'ro')
plt.plot(xfit, yfit,'b')
plt.plot(xfit, yfit)
plt.title("Direct spline interpolantion")
plt.show()

s = interpolate.InterpolatedUnivariateSpline(x, y)
xfit = np.arange(0, n-1, np.pi/50)
yfit = s(xfit)

plt.plot(x, y, 'ro')
plt.plot(xfit, yfit,'green')
plt.title("InterpolatedUnivariateSpline interpolation")
plt.show() ```
`  `

References: