Fitting custom models in SciPy - Step-by-Step Execution

Concept Flow - Fitting custom models

Define model function

↓

Prepare data (x, y)

↓

Choose initial parameters

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Call curve_fit with model, data, initial params

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curve_fit adjusts params to minimize error

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Obtain best-fit parameters and covariance

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Use fitted model for prediction or plotting

The flow shows defining a model, preparing data, choosing initial guesses, fitting with curve_fit, and getting best parameters.

Execution Sample

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

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

params, cov = curve_fit(model, xdata, ydata, p0=[1, 0])

This code fits a line y = a*x + b to data points using curve_fit.

Execution Table

Step	Action	Parameters (a, b)	Model Output	Error (ydata - model)
1	Initial guess	[1.0, 0.0]	[1, 2, 3, 4, 5]	[1.1, 2.1, 3.0, 4.1, 5.2]
2	curve_fit tries params [1.5, 0.5]	[1.5, 0.5]	[2.0, 3.5, 5.0, 6.5, 8.0]	[0.1, 0.6, 1.0, 1.6, 2.2]
3	curve_fit tries params [2.0, 0.1]	[2.0, 0.1]	[2.1, 4.1, 6.1, 8.1, 10.1]	[0.0, 0.0, -0.1, 0.0, 0.1]
4	curve_fit converges	[2.02, 0.04]	[2.06, 4.08, 6.10, 8.12, 10.14]	[0.04, 0.02, -0.10, -0.02, 0.06]

💡 curve_fit stops when parameter changes no longer reduce error significantly.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	Final
a	None	1.0	1.5	2.0	2.02
b	None	0.0	0.5	0.1	0.04
model output	None	[1, 2, 3, 4, 5]	[2, 3.5, 5, 6.5, 8]	[2.1, 4.1, 6.1, 8.1, 10.1]	[2.06, 4.08, 6.10, 8.12, 10.14]
error	None	[1.1, 2.1, 3.0, 4.1, 5.2]	[0.1, 0.6, 1.0, 1.6, 2.2]	[0.0, 0.0, -0.1, 0.0, 0.1]	[0.04, 0.02, -0.10, -0.02, 0.06]

Key Moments - 3 Insights

Why do we need to provide initial parameter guesses?

curve_fit uses initial guesses to start searching for best parameters. Without them, it may not find the best fit or may fail to converge, as shown in step 1 of the execution_table.

What does curve_fit actually minimize?

curve_fit minimizes the difference between the model output and actual data (error). This is seen in the error column of execution_table where errors get smaller each step.

Why do parameters change multiple times during fitting?

curve_fit iteratively adjusts parameters to reduce error. Each step in execution_table shows new parameters closer to the best fit.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3, what is the approximate value of parameter 'a'?

A 1.5

B 2.0

C 2.02

D 1.0

💡 Hint

Check the Parameters column at Step 3 in execution_table.

At which step does the error between model and data become smallest?

A Step 3

B Step 2

C Step 4

D Step 1

💡 Hint

Look at the Error column in execution_table and find the smallest values.

If initial guess for 'b' was 1.0 instead of 0.0, how would the first model output change?

A Model output would increase by 1.0 for all x

B Model output would decrease by 1.0 for all x

C Model output would stay the same

D Model output would be zero

💡 Hint

Recall model is a*x + b; changing b shifts output by b.

Concept Snapshot

Fitting custom models with scipy:
- Define model function: f(x, params)
- Prepare data arrays x and y
- Provide initial parameter guesses p0
- Use curve_fit(model, x, y, p0) to fit
- curve_fit returns best-fit parameters and covariance
- Use fitted params to predict or plot model

Full Transcript

This visual execution shows how to fit a custom model using scipy's curve_fit. First, you define a model function that takes input x and parameters. Then, you prepare your data points xdata and ydata. You provide initial guesses for parameters to help curve_fit start. The curve_fit function tries different parameters to minimize the difference between model output and actual data. The execution table traces parameter changes and error reduction step by step until convergence. The variable tracker shows how parameters and errors evolve. Key moments clarify why initial guesses matter, what curve_fit minimizes, and why parameters update multiple times. The quiz tests understanding of parameter values, error reduction, and effect of initial guesses. The snapshot summarizes the fitting steps for quick reference.