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

Concept Flow - Fitting custom models

Define model function

↓

Prepare data (x, y)

↓

Choose initial parameters

↓

Call curve_fit with model, data, initial params

↓

curve_fit adjusts params to minimize error

↓

Obtain best-fit parameters and covariance

↓

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.

Practice

1. What is the main purpose of using scipy.optimize.curve_fit in fitting custom models?

easy

A. To find the best parameters that make the model fit the data

B. To plot the data points automatically

C. To generate random data for testing

D. To calculate the mean of the dataset

Solution

1. Step 1: Understand the role of curve_fit

   curve_fit is used to adjust parameters of a model function so that it best fits the given data points.

2. Step 2: Identify the correct purpose

   It does not plot data, generate random data, or calculate means. Its main job is parameter estimation for fitting.

3. Final Answer:

   To find the best parameters that make the model fit the data -> Option A

4. Quick Check:

   curve_fit finds best parameters [OK]

Hint: Remember: curve_fit adjusts parameters to fit data [OK]

Common Mistakes:

• Thinking curve_fit plots data automatically
• Confusing curve_fit with data generation functions
• Assuming curve_fit calculates statistics like mean

2. Which of the following is the correct way to define a custom model function for curve_fit that fits a line y = m*x + c?

easy

A. def model(m, c, x): return m + c * x

B. def model(x, m, c): return m * x + c

C. def model(x): return m * x + c

D. def model(x, m, c): return m + c / x

Solution

1. Step 1: Check parameter order for curve_fit

   The model function must have the independent variable as the first argument, followed by parameters to fit.

2. Step 2: Verify function matches y = m*x + c

   def model(x, m, c): return m * x + c correctly defines model(x, m, c) returning m * x + c. Others have wrong order or formula.

3. Final Answer:

   def model(x, m, c): return m * x + c -> Option B

4. Quick Check:

   Model args: x first, then parameters [OK]

Hint: Model function: x first, then parameters [OK]

Common Mistakes:

• Swapping parameter and variable order
• Missing parameters in function definition
• Using wrong formula inside the function

3. Given the code below, what will be the output of print(popt)?

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])
ydata = np.array([1, 2.7, 7.4, 20.1])

popt, _ = curve_fit(model, xdata, ydata)
print(np.round(popt, 2))

medium

A. [1.0 0.99]

B. [1.0 1.0]

C. [1.0 1.0] but with a runtime error

D. [1.01 1.0]

Solution

1. Step 1: Understand the model and data

   The model is a * exp(b * x). Given ydata roughly follows exponential growth, parameters a and b will be close to 1.

2. Step 2: Check output of curve_fit

   Running the code fits parameters close to a=1.0 and b=0.99 (data approximates e^x but slightly less). Rounded to two decimals, popt is approximately [1.0 0.99].

3. Final Answer:

   [1.0 0.99] -> Option A

4. Quick Check:

   Exponential fit params ~ [1.0, 0.99] [OK]

Hint: Run curve_fit and round parameters to check values [OK]

Common Mistakes:

• Misestimating parameters as [1.0 1.0] due to data approximation
• Confusing parameter order
• Expecting runtime errors without cause

4. What is wrong with the following code snippet for fitting a quadratic model using curve_fit?

import numpy as np
from scipy.optimize import curve_fit

def quad(x, a, b, c):
    return a * x**2 + b * x + c

xdata = np.array([1, 2, 3, 4])
ydata = np.array([3, 7, 13, 21])

popt, pcov = curve_fit(quad, ydata, xdata)
print(popt)

medium

A. Missing initial guess for parameters

B. The model function has wrong formula for quadratic

C. The independent and dependent variables are swapped in curve_fit call

D. The print statement is incorrect

Solution

1. Step 1: Check curve_fit arguments

   curve_fit expects the model, xdata (independent), then ydata (dependent). Here, ydata and xdata are swapped.

2. Step 2: Identify the error impact

   Swapping causes wrong fitting or runtime errors because the model expects x values first.

3. Final Answer:

   The independent and dependent variables are swapped in curve_fit call -> Option C

4. Quick Check:

   curve_fit(xdata, ydata) order matters [OK]

Hint: Remember: curve_fit(model, xdata, ydata) [OK]

Common Mistakes:

• Swapping xdata and ydata in curve_fit
• Assuming model formula is incorrect
• Thinking initial guess is always required

5. You want to fit a custom model y = a * sin(b * x) + c to noisy data. Which approach correctly fits the model and plots the result?

hard

A. Plot data first, then call curve_fit without storing parameters

B. Use curve_fit without defining a model function, just pass np.sin

C. Fit the model by manually guessing parameters without curve_fit

D. Define model with def model(x, a, b, c): return a * np.sin(b * x) + c, use curve_fit with data, then plot original and fitted curves

Solution

1. Step 1: Define the correct model function

   Model must be defined as model(x, a, b, c) returning a * np.sin(b * x) + c.

2. Step 2: Use curve_fit and plot results

   Call curve_fit(model, xdata, ydata) to get parameters, then plot original data and fitted curve for comparison.

3. Final Answer:

   Define model with def model(x, a, b, c): return a * np.sin(b * x) + c, use curve_fit with data, then plot original and fitted curves -> Option D

4. Quick Check:

   Model function + curve_fit + plot = correct approach [OK]

Hint: Always define model function before curve_fit and plot results [OK]

Common Mistakes:

• Passing np.sin directly without parameters
• Skipping model function definition
• Not storing or using fitted parameters for plotting