Mastering 1D Interpolation in Python with SciPy: Linear, Nearest, Quadratic, Cubic

Interpolation is an important method for approximating discrete functions; it estimates function values at unsampled points using known values at a finite set of points.

Historically, as early as the 6th century Chinese scholar Liu Zhuo applied equally spaced quadratic interpolation to astronomical calculations. In the 17th century, Newton and Lagrange discussed general formulas for both equally and unequally spaced data. Today, interpolation remains a fundamental tool for data processing, function table creation, numerical integration, differentiation, root finding, and solving differential equations.

Given the data:

<code>x = [1, 2, 3, 4, 5, 6]
y = [300, 500, 800, 1300, 3000, 5000]
</code>

Plotting the raw data points:

<code>import matplotlib.pyplot as plt
%matplotlib inline

plt.scatter(x, y)
</code>

To interpolate the relationship between x and y , SciPy provides the one‑dimensional interpolation function interp1d :

<code>interp1d(x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)
</code>

Parameters:

x : independent variable data

y : dependent variable data

kind : type of interpolation; options include 'linear' – linear interpolation 'nearest' – nearest‑point interpolation 'nearest-up' – nearest‑point interpolation (upper) 'zero' – zero‑order interpolation 'quadratic' – quadratic spline interpolation 'cubic' – cubic spline interpolation

Creating interpolation functions for each method:

<code>from scipy.interpolate import interp1d
func_linear = interp1d(x, y, 'linear')
func_nearest = interp1d(x, y, 'nearest')
func_nearest_up = interp1d(x, y, 'nearest-up')
func_zero = interp1d(x, y, 'zero')
func_quadratic = interp1d(x, y, 'quadratic')
func_cubic = interp1d(x, y, 'cubic')
</code>

Generating dense points and evaluating each interpolation:

<code>import numpy as np
x1 = np.linspace(1, 6, 100)
y_linear = func_linear(x1)
y_nearest = func_nearest(x1)
y_nearest_up = func_nearest_up(x1)
y_zero = func_zero(x1)
y_quadratic = func_quadratic(x1)
y_cubic = func_cubic(x1)
</code>

Plotting the original data together with each interpolated curve for comparison:

<code>plt.figure(figsize=(20, 10))

plt.subplot(2, 3, 1)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_linear, label='linear', color='red')
plt.legend()

plt.subplot(2, 3, 2)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_nearest, label='nearest', color='red')
plt.legend()

plt.subplot(2, 3, 3)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_nearest_up, label='nearest_up', color='red')
plt.legend()

plt.subplot(2, 3, 4)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_zero, label='zero', color='red')
plt.legend()

plt.subplot(2, 3, 5)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_quadratic, label='quadratic', color='red')
plt.legend()

plt.subplot(2, 3, 6)
plt.scatter(x, y, label='raw data')
plt.plot(x1, y_cubic, label='cubic', color='red')
plt.legend()

plt.savefig('images/sci0202.png')
</code>