Curve fitting (curve_fit) in SciPy - Time & Space Complexity

Time Complexity: Curve fitting (curve_fit)

O(n)

Understanding Time Complexity

When using curve fitting, we want to know how the time needed grows as we give more data points.

We ask: How does the fitting process slow down when the input data gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.optimize import curve_fit

def model(x, a, b):
    return a * x + b

xdata = np.linspace(0, 10, 100)
ydata = 3.5 * xdata + 2 + np.random.normal(size=100)

params, covariance = curve_fit(model, xdata, ydata)

This code fits a straight line to 100 data points using curve_fit from scipy.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Repeated evaluation of the model function and calculation of residuals during optimization.
• How many times: The optimizer calls the model function multiple times, roughly proportional to the number of data points times the number of iterations.

How Execution Grows With Input

As the number of data points grows, the fitting process takes longer because it must check more points each iteration.

Input Size (n)	Approx. Operations
10	Low number of function calls, fast fitting
100	About 10 times more operations than 10 points
1000	About 10 times more operations than 100 points

Pattern observation: The time grows roughly linearly with the number of data points.

Final Time Complexity

Time Complexity: O(n)

This means the time to fit grows roughly in direct proportion to the number of data points.

Common Mistake

[X] Wrong: "Curve fitting time stays the same no matter how many points I have."

[OK] Correct: More points mean more calculations each iteration, so fitting takes longer as data grows.

Interview Connect

Understanding how curve fitting time grows helps you explain performance in real data tasks, showing you know how data size affects analysis speed.

Self-Check

"What if the model function is more complex and takes longer to compute? How would the time complexity change?"