Python quintic interpolation

Question

This article shows how to do interpolation in Python and looks at different 2d implementation methods. We will discuss useful functions for bivariate interpolation such as scipy.interpolate.interp2d, numpy.meshgrid, and Radial Basis Function for smoothing/interpolation (RBF) used in Python.

Table of Contents Show

Use scipy.interpolate.interp2d to Create 2D Interpolation in Python
Use numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy') 2 to Create Radial Basis Function for Interpolation in Python
How do you do interpolation in Python?
How to do cubic interpolation in Python?
How do you interpolate 2d data in Python?
How do you interpolate missing values in Python?

We will implement interpolation using the SciPy and Numpy libraries, making it easy.

Use scipy.interpolate.interp2d to Create 2D Interpolation in Python

First of all, let’s understand interpolation, a technique of constructing data points between given data points. Let’s assume two points, such as 1 and 2.

In this example, we can interpolate and find points 1.22 and 1.44, and many more.

Interpolation is often used in Machine Learning to fill in missing data in a dataset, called imputation. Interpolation is frequently used to make a dataset’s points more uniform.

Let’s see working with examples of interpolation in Python using the scipy.interpolate module.

The interp2d is a straightforward generalization of the

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

0 function. This function takes the x and y coordinates of the available data points as separate one-dimensional arrays and a two-dimensional array of values for each pair of x and y coordinates.

The data points are assumed to be on a regular and uniform x and y coordinate grid. The general function form is below.

class scipy.interpolate.interp2d(x, y, z, kind='linear', copy=True, bounds_error=False, fill_value=None)

Where

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

1,

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

2, and

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

3 are arrays, the kind could be

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

4 or may be left as optional. See also scipy.interpolate.interp2d detailed documentation.

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

Output:

scipy interpolate interp2d

Note that we have used numpy.meshgrid to make the grid; you can make a rectangular grid out of two one-dimensional arrays representing Cartesian or Matrix indexing. See numpy.meshgrid documentation.

The general function form is below.

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

The

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

8 represents one-dimensional coordinate arrays

#import libraries
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt

# x,y arrays
x = np.arange(-5.01, 5.01, 0.25)
y = np.arange(-5.01, 5.01, 0.25)
xx, yy = np.meshgrid(x, y)  #see details below for 'np.meshgrid'

#approximate function which is z:= f(x,y)
z = np.sin(xx**2+yy**2)
fun = interpolate.interp2d(x, y, z, kind='linear') # kind could be {'linear', 'cubic', 'quintic'}
xnew = np.arange(-5.01, 5.01, 1e-2)
ynew = np.arange(-5.01, 5.01, 1e-2)
znew = fun(xnew, ynew)

plt.plot(x, z[0, :], 'go-', xnew, znew[0, :], 'b-')
plt.show()

9,

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

0,…,

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

1.

Use numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy') 2 to Create Radial Basis Function for Interpolation in Python

This class of interpolation is used in the case of n-dimensional scattered data; for this, we use

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

2.

import matplotlib.pyplot as plt
import numpy as np
from scipy.interpolate.rbf import Rbf  # radial basis functions

# x y arrays
x = [1, 1, 2 ,3, 4, 4, 2, 6, 7]
y = [0, 2, 5, 6, 2, 4, 1, 5, 2]
z = [1]*len(x)

#RBF Func
rbf_fun = Rbf(x, y, z, function='gaussian')

x_new = np.linspace(0, 8, 81)
y_new = np.linspace(0, 8, 82)

x_grid, y_grid = np.meshgrid(x_new, y_new)
z_new = rbf_fun(x_grid.ravel(), y_grid.ravel()).reshape(x_grid.shape)

plt.pcolor(x_new, y_new, z_new);
plt.plot(x, y, 'o');
plt.xlabel('x'); plt.ylabel('y');
plt.title('RBF Gaussian interpolation');

Output:

Radial Basis Function

This class of interpolating functions converts

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

4 scattered data to

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

5 with

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

6. Smoothing and interpolating scattered data in n-dimensions can be accomplished using RBF interpolation.

Still, as there is a chance of extrapolation, like getting values outside the data range, this should be done carefully.

How do you do interpolation in Python?

interpolate package..

import numpy as np from scipy import interpolate import matplotlib. pyplot as plt x = np. linspace(0, 4, 12) y = np. ... .

xnew = np. linspace(0, 4,30) plt. plot(x, y, 'o', xnew, f(xnew), '-', xnew, f2(xnew), '--') plt. ... .

import matplotlib. pyplot as plt from scipy..

How to do cubic interpolation in Python?

S′i(xi+1)=S′i+1(xi+1),i=1,…,n−2,S″i(xi+1)=S″i+1(xi+1),i=1,…,n−2, which gives us 2(n−2) equations. Two more equations are required to compute the coefficients of Si(x). These last two constraints are arbitrary, and they can be chosen to fit the circumstances of the interpolation being performed.

How do you interpolate 2d data in Python?

Interpolate over a 2-D grid. x, y and z are arrays of values used to approximate some function f: z = f(x, y) which returns a scalar value z. This class returns a function whose call method uses spline interpolation to find the value of new points.

How do you interpolate missing values in Python?

You can interpolate missing values ( NaN ) in pandas. DataFrame and Series with interpolate() . This article describes the following contents. Use dropna() and fillna() to remove missing values NaN or to fill them with a specific value.