Exploring 1D Interpolation with SciPy: Linear, Nearest, Cubic & More

Interpolation is an important method for approximating discrete functions, allowing estimation of function values at unsampled points based on known values.

Historically, Liu Zhuo in the 6th century used equally spaced quadratic interpolation for astronomical calculations; later Newton and Lagrange formulated general interpolation formulas for both equally and unequally spaced data. Today interpolation remains essential for data processing, function table creation, numerical integration, differentiation, root finding, and solving differential equations.

Given data:

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

Plot the data points:

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

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

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

To interpolate the x‑y relationship, SciPy's one‑dimensional interpolation function is called as:

<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', 'nearest', 'nearest-up', 'zero', 'quadratic', 'cubic', etc.

Define 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>

Generate dense x values and compute interpolated y values:

<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>

Plot and compare the interpolation results:

<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>