1-D Interpolation

The interp1d class in the scipy.interpolate is a convenient method to create a function based on fixed data points, which can be evaluated anywhere within the domain defined by the given data using linear interpolation.

By using the above data, let us create a interpolate function and draw a new interpolated graph.

f1 = interp1d(x, y,kind = 'linear')

f2 = interp1d(x, y, kind = 'cubic')

Using the interp1d function, we created two functions f1 and f2. These functions, for a given input x returns y. The third variable kind represents the type of the interpolation technique. 'Linear', 'Nearest', 'Zero', 'Slinear', 'Quadratic', 'Cubic' are a few techniques of interpolation.

Now, let us create a new input of more length to see the clear difference of interpolation. We will use the same function of the old data on the new data.

xnew = np.linspace(0, 4,30)
plt.plot(x, y, 'o', xnew, f(xnew), '-', xnew, f2(xnew), '--')
plt.legend(['data', 'linear', 'cubic','nearest'], loc = 'best')
plt.show()

The above program will generate the following output.

Splines

To draw smooth curves through data points, drafters once used thin flexible strips of wood, hard rubber, metal or plastic called mechanical splines. To use a mechanical spline, pins were placed at a judicious selection of points along a curve in a design, and then the spline was bent, so that it touched each of these pins.

Clearly, with this construction, the spline interpolates the curve at these pins. It can be used to reproduce the curve in other drawings. The points where the pins are located is called knots. We can change the shape of the curve defined by the spline by adjusting the location of the knots.

Univariate Spline

One-dimensional smoothing spline fits a given set of data points. The UnivariateSpline class in scipy.interpolate is a convenient method to create a function, based on fixed data points class %u2013 scipy.interpolate.UnivariateSpline(x, y, w = None, bbox = [None, None], k = 3, s = None, ext = 0, check_finite = False).

Parameters %u2212 Following are the parameters of a Univariate Spline.

• This fits a spline y = spl(x) of degree k to the provided x, y data.

• %u2018w%u2019 %u2212 Specifies the weights for spline fitting. Must be positive. If none (default), weights are all equal.

• %u2018s%u2019 %u2212 Specifies the number of knots by specifying a smoothing condition.

• %u2018k%u2019 %u2212 Degree of the smoothing spline. Must be <= 5. Default is k = 3, a cubic spline.

• Ext %u2212 Controls the extrapolation mode for elements not in the interval defined by the knot sequence.

• if ext = 0 or %u2018extrapolate%u2019, returns the extrapolated value.

• if ext = 1 or %u2018zero%u2019, returns 0

• if ext = 2 or %u2018raise%u2019, raises a ValueError

• if ext = 3 of %u2018const%u2019, returns the boundary value.

• check_finite %u2013 Whether to check that the input arrays contain only finite numbers.

Let us consider the following example.

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.plot(x, y, 'ro', ms = 5)
plt.show()

Use the default value for the smoothing parameter.

spl = UnivariateSpline(x, y)
xs = np.linspace(-3, 3, 1000)
plt.plot(xs, spl(xs), 'g', lw = 3)
plt.show() 

Manually change the amount of smoothing.

spl.set_smoothing_factor(0.5)
plt.plot(xs, spl(xs), 'b', lw = 3)
plt.show()

 

As listed below, this sub-package contains spline functions and classes, one-dimensional and multi-dimensional (univariate and multivariate) interpolation classes, Lagrange and Taylor polynomial interpolators, and wrappers for FITPACK and DFITPACK functions.

2-D Splines

For data on a grid:

For unstructured data: