Non-linear curve fitting in SciPy - Step-by-Step Execution

Concept Flow - Non-linear curve fitting

Start with data points

↓

Choose model function

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Initial guess for parameters

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Use curve_fit to optimize parameters

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Check fit quality

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Use fitted curve for prediction or plotting

We start with data and a model, guess parameters, optimize them to fit data, then use the fit.

Execution Sample

SciPy

import numpy as np
from scipy.optimize import curve_fit

def model(x, a, b):
    return a * np.exp(b * x)

xdata = np.array([0,1,2,3,4])
ydata = np.array([1,2.7,7.4,20.1,54.6])

params, _ = curve_fit(model, xdata, ydata, p0=[1, 0.5])

Fit an exponential model y = a * exp(b * x) to given data points.

Execution Table

Step	Action	Parameter Guess	Function Evaluation	Error Reduction
1	Initial guess	[1, 0.5]	[1.0, 1.6487, 2.7183, 4.4817, 7.3891]	High error
2	Optimize parameters	[1.5, 1.0]	[1.5, 4.07, 11.0, 29.6, 79.4]	Error decreases
3	Optimize parameters	[1.2, 0.9]	[1.2, 2.95, 7.25, 17.8, 43.7]	Error decreases
4	Optimize parameters	[1.0, 0.9]	[1.0, 2.46, 6.05, 14.9, 36.7]	Error decreases
5	Optimize parameters	[1.0, 0.8]	[1.0, 2.22, 4.93, 11.0, 24.5]	Error decreases
6	Optimize parameters	[1.0, 0.7]	[1.0, 2.01, 4.05, 8.1, 16.3]	Error decreases
7	Optimize parameters	[1.0, 0.6]	[1.0, 1.82, 3.32, 6.05, 11.0]	Error decreases
8	Optimize parameters	[1.0, 0.55]	[1.0, 1.73, 2.85, 4.7, 8.0]	Error decreases
9	Final parameters found	[1.0, 0.54]	[1.0, 1.72, 2.79, 4.5, 7.7]	Minimal error
10	Exit	-	-	Converged to best fit

💡 Optimization converged when error stopped decreasing significantly.

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	After 6	After 7	After 8	After 9	Final
params	[1, 0.5]	[1.5, 1.0]	[1.2, 0.9]	[1.0, 0.9]	[1.0, 0.8]	[1.0, 0.7]	[1.0, 0.6]	[1.0, 0.55]	[1.0, 0.54]	[1.0, 0.54]	[1.0, 0.54]

Key Moments - 3 Insights

Why do we need an initial guess for parameters?

curve_fit uses the initial guess to start optimization. Without it, the function may not find the best fit. See execution_table step 1 where the initial guess is set.

What does 'error decreases' mean in the optimization steps?

It means the difference between the model predictions and actual data gets smaller, so the fit improves. This is shown in execution_table rows 2 to 9.

Why does the optimization stop at step 10?

Because the error no longer decreases significantly, meaning the best fit parameters are found. This is the exit condition in execution_table.

Visual Quiz - 3 Questions

Look at the variable_tracker table, what are the final fitted parameters?

A [1.0, 0.5]

B [1.0, 0.54]

C [1.5, 1.0]

D [1.2, 0.9]

💡 Hint

Check the 'Final' column in variable_tracker for 'params'.

At which step in the execution_table does the optimization first show 'Minimal error'?

A Step 9

B Step 7

C Step 5

D Step 10

💡 Hint

Look for the row where 'Error Reduction' says 'Minimal error'.

If the initial guess was very far from the true parameters, what would likely happen in the execution_table?

A The initial guess row would show minimal error.

B The optimization would stop immediately.

C Optimization steps would show larger error reductions over more steps.

D Parameters would not change from the initial guess.

💡 Hint

Consider how optimization improves parameters step by step as shown in execution_table.

Concept Snapshot

Non-linear curve fitting uses a model function and data.
Start with an initial guess for parameters.
Use scipy.optimize.curve_fit to find best parameters.
Optimization iteratively reduces error.
Stop when error no longer improves.
Use fitted parameters for predictions or plotting.

Full Transcript

Non-linear curve fitting means finding parameters of a model function that best match given data points. We start with data and a chosen model, like an exponential function. We guess initial parameters to start the process. Then, using scipy's curve_fit, the parameters are adjusted step by step to reduce the difference between the model's output and the actual data. This process repeats until the error stops decreasing significantly, meaning the best fit is found. The final parameters can then be used to predict new values or visualize the fitted curve.