In this article, we will learn Interpolation using the SciPy module in Python. First, we will discuss interpolation and its types with implementation.
Interpolation and Its Types
Interpolation is a technique of constructing data points between given data points. The scipy.interpolate is a module in Python SciPy consisting of classes, spline functions, and univariate and multivariate interpolation classes. Interpolation is done in many ways some of them are :
- 1-D Interpolation
- Spline Interpolation
- Univariate Spline Interpolation
- RBF Interpolation
Let’s discuss all the methods one by one and visualize the results.
1-D Interpolation
To create a function based on fixed data points, scipy.interpolate.interp1d is used. It takes data points x and y and returns a function that can be called with new x and returns the corresponding y point.
Syntax: scipy.interpolate.interp1d(x , y , kind , axis , copy , bounds_error , fill_value , assume_sorted)
Python
# Import the required Python libraries import matplotlib.pyplot as plt from scipy import interpolate import numpy as np # Initialize input values x and y x = np.arange(0, 10) y = x**2 # Interpolation temp = interpolate.interp1d(x, y) xnew = np.arange(0, 9, 0.2) ynew = temp(xnew) plt.title("1-D Interpolation") plt.plot(x, y, '*', xnew, ynew, '-', color="green") plt.show() |
Output:
Spline Interpolation
In spline interpolation, a spline representation of the curve is computed, and then the spline is computed at the desired points. The function splrep is used to find the spline representation of a curve in a two-dimensional plane.
- To find the B-spline representation of a 1-D curve, scipy.interpolate.splrep is used.
Syntax: scipy.interpolate.splrep(x, y, w, xb, xe, k, task, s, t, full_output, per, quiet)
- To compute a B-spline or its derivatives, scipy.interpolate.splev is used.
Syntax: scipy.interpolate.splev(x , tck , der , ext)
Python
# Import the required Python libraries import numpy as np import matplotlib.pyplot as plt from scipy import interpolate # Initialize the input values x = np.arange(0, 10) y = np.cos(x**3) # Interpolation # To find the spline representation of a # curve in a 2-D plane using the function # splrep temp = interpolate.splrep(x, y, s=0) xnew = np.arange(0, np.pi**2, np.pi/100) ynew = interpolate.splev(xnew, temp, der=0) plt.figure() plt.plot(x, y, '*', xnew, ynew, xnew, np.cos(xnew), x, y, 'b', color="green") plt.legend(['Linear', 'Cubic Spline', 'True']) plt.axis([-0.1, 6.5, -1.1, 1.1]) plt.title('Cubic-spline Interpolation in Python') plt.show() |
Output:
Univariate Spline
It is a 1-D smoothing spline that fits a given group of data points. The scipy.interpolate.UnivariateSpline is used to fit a spline y = spl(x) of degree k to the provided x, y data. s specifies the number of knots by specifying a smoothing condition. The scipy.interpolate.UnivariateSpline. set_smoothing_factor: Spline computation with the given smoothing factor s and with the knots found at the last call.
Syntax: scipy.interpolate.UnivariateSpline( x, y, w, bbox, k, s, ext)
Python
# Import the required libraries import matplotlib.pyplot as plt from scipy.interpolate import UnivariateSpline x = np.linspace(-3, 3, 50) y = np.exp(-x**2) + 0.1 * np.random.randn(50) plt.title("Univariate Spline") plt.plot(x, y, 'g.', ms=8) # Using the default values for the # smoothing parameter spl = UnivariateSpline(x, y) xs = np.linspace(-3, 3, 1000) plt.plot(xs, spl(xs), 'green', lw=3) # Manually change the amount of smoothing spl.set_smoothing_factor(0.5) plt.plot(xs, spl(xs), color='black', lw=3) plt.show() |
Output:
Radial basis function for Interpolation
The scipy.interpolate.Rbf is used for interpolating scattered data in n-dimensions. The radial basis function is defined as corresponding to a fixed reference data point. The scipy.interpolate.Rbf is a class for radial basis function interpolation of functions from N-D scattered data to an M-D domain.
Syntax: scipy.interpolate.Rbf(*args)
Python
# Import the required libraries import numpy as np from scipy.interpolate import Rbf import matplotlib.pyplot as plt # setup the data values x = np.linspace(0, 10, 9) y = np.cos(x/2) xi = np.linspace(0, 10, 110) # Interpolation using RBF rbf = Rbf(x, y) fi = rbf(xi) plt.subplot(2, 1, 2) plt.plot(x, y, '*', color="green") plt.plot(xi, fi, 'green') plt.plot(xi, np.sin(xi), 'black') plt.title('Radial basis function Interpolation') plt.show() |
Output:
