Curve fitting (curve_fit) in SciPy - Step-by-Step Execution

Concept Flow - Curve fitting (curve_fit)

Start with data points (x, y)

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Define model function f(x, params)

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Call curve_fit with f, x, y

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curve_fit adjusts params to minimize error

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Output best-fit parameters and covariance

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Use params to predict or plot fitted curve

We start with data points and a model function. curve_fit finds parameters that make the model best match the data.

Execution Sample

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

x = np.array([1, 2, 3, 4, 5])
y = np.array([2.1, 4.1, 6.0, 8.1, 10.2])

params, cov = curve_fit(model, x, y)

This code fits a straight line y = a*x + b to the given x and y data points.

Execution Table

Step	Action	Parameters Tried	Error (Sum of Squares)	Result
1	Initial guess (default)	a=1.0, b=0.0	12.06	Try next params
2	Try params	a=2.0, b=0.1	0.06	Better fit, try refining
3	Try params	a=2.02, b=0.08	0.04	Improved fit
4	Try params	a=2.04, b=0.06	0.03	Improved fit
5	Try params	a=2.03, b=0.07	0.025	Best fit found
6	Return	a=2.03, b=0.07	0.025	Output params and covariance

💡 curve_fit stops when error is minimized and parameters converge

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
a	1.0	2.0	2.02	2.04	2.03	2.03
b	0.0	0.1	0.08	0.06	0.07	0.07
Error	12.06	0.06	0.04	0.03	0.025	0.025

Key Moments - 3 Insights

Why does curve_fit try many different parameters instead of just one?

curve_fit searches for parameters that minimize error. It tries many values step-by-step (see execution_table rows 1-5) to find the best fit.

What do the output parameters represent?

The output parameters are the best values for 'a' and 'b' that make the model line fit the data closely (see final row in execution_table).

Why is the error value important?

The error shows how far the model predictions are from actual data. curve_fit tries to make this error as small as possible (see error column in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of parameter 'a' after step 3?

A 2.04

B 2.02

C 2.0

D 1.0

💡 Hint

Check the 'Parameters Tried' column at step 3 in execution_table.

At which step does curve_fit find the best fit parameters?

A Step 2

B Step 1

C Step 5

D Step 4

💡 Hint

Look for the row where 'Result' says 'Best fit found' in execution_table.

If the initial guess was closer to the true parameters, how would the error at step 1 change?

A It would be smaller

B It would stay the same

C It would be larger

D It would be zero

💡 Hint

Refer to the 'Error' column in execution_table and think about how starting closer affects error.

Concept Snapshot

curve_fit(model, x, y) fits data points (x,y) to a model function.
It finds parameters that minimize difference between model and data.
Returns best-fit parameters and covariance matrix.
Useful for fitting lines, curves, or custom functions.
Requires defining a model function with parameters.
Works by iterative optimization to reduce error.

Full Transcript

Curve fitting with curve_fit starts by having data points and a model function. The function curve_fit tries different parameters to make the model match the data as closely as possible. It measures error as the difference between predicted and actual data. It changes parameters step-by-step to reduce this error. When the error is as small as possible, it stops and returns the best parameters. These parameters can then be used to predict or plot the fitted curve. This process helps us find the line or curve that best represents the data.