Spline interpolation in SciPy - Step-by-Step Execution

Concept Flow - Spline interpolation

Input data points (x, y)

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Choose spline type (e.g., cubic)

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Fit spline to data

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Evaluate spline at new points

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Output smooth interpolated values

Spline interpolation fits a smooth curve through given points by choosing a spline type, fitting it, then evaluating at new points.

Execution Sample

SciPy

import numpy as np
from scipy.interpolate import CubicSpline

x = np.array([0, 1, 2, 3, 4])
y = np.array([0, 1, 0, 1, 0])
cs = CubicSpline(x, y)
x_new = np.linspace(0, 4, 9)
y_new = cs(x_new)

This code fits a cubic spline to 5 points and evaluates it at 9 new points between 0 and 4.

Execution Table

Step	Action	Input/Variable	Result/Output
1	Define original x points	x = [0,1,2,3,4]	Array x created
2	Define original y points	y = [0,1,0,1,0]	Array y created
3	Create CubicSpline object	cs = CubicSpline(x, y)	Spline fitted to points
4	Create new x points for evaluation	x_new = np.linspace(0,4,9)	x_new = [0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]
5	Evaluate spline at x_new	y_new = cs(x_new)	y_new = [0.0,0.6875,1.0,0.6875,0.0,0.3125,1.0,0.6875,0.0]

💡 All new points evaluated, spline interpolation complete

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5
x	undefined	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]
y	undefined	undefined	[0,1,0,1,0]	[0,1,0,1,0]	[0,1,0,1,0]	[0,1,0,1,0]
cs	undefined	undefined	undefined	CubicSpline object	CubicSpline object	CubicSpline object
x_new	undefined	undefined	undefined	undefined	[0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]	[0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0]
y_new	undefined	undefined	undefined	undefined	undefined	[0.0,0.6875,1.0,0.6875,0.0,0.3125,1.0,0.6875,0.0]

Key Moments - 3 Insights

Why does the spline output y_new have values between the original y points?

Because spline interpolation fits a smooth curve, it estimates values between points, not just the original y values. See execution_table step 5 where y_new has intermediate values.

What happens if we evaluate the spline outside the original x range?

By default, CubicSpline extrapolates outside the original x range, which can give unexpected values. This is not shown here but important to know.

Why do we use np.linspace for x_new instead of just original x?

np.linspace creates evenly spaced points to see the smooth curve between original points, as shown in execution_table step 4.

Visual Quiz - 3 Questions

Look at the execution_table step 5, what is the interpolated y value at x_new = 2.0?

A 0.6875

B 1.0

C 0.0

D 0.3125

💡 Hint

Check the y_new array in step 5 of execution_table; the value at index corresponding to x_new=2.0 is 0.0

At which step is the spline function created and fitted to the data?

A Step 3

B Step 2

C Step 4

D Step 5

💡 Hint

Look at the Action column in execution_table; step 3 shows creation of CubicSpline object

If we change x_new to have fewer points, how does y_new change?

A y_new will have more values

B y_new will have fewer values

C y_new values will be unchanged

D y_new will be empty

💡 Hint

Refer to variable_tracker for x_new and y_new sizes; y_new matches length of x_new

Concept Snapshot

Spline interpolation fits a smooth curve through data points.
Use scipy.interpolate.CubicSpline(x, y) to create spline.
Evaluate spline at new points with cs(x_new).
Output is smooth y values between original points.
Useful for smooth data estimation and plotting.

Full Transcript

Spline interpolation is a way to draw a smooth curve through given points. We start with original x and y points. Then we create a cubic spline object using scipy's CubicSpline. This fits a smooth curve through the points. Next, we create new x points where we want to estimate y values. Finally, we evaluate the spline at these new points to get smooth y values. This helps us see smooth trends between data points instead of just dots.