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

Concept Flow - Non-linear curve fitting

Start with data points

↓

Choose model function

↓

Initial guess for parameters

↓

Use curve_fit to optimize parameters

↓

Check fit quality

↓

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.

Practice

1. What is the main purpose of using scipy.optimize.curve_fit in data analysis?

easy

A. To sort data points in ascending order

B. To find the best-fitting curve for data when the relationship is non-linear

C. To calculate the mean of a dataset

D. To generate random numbers for simulations

Solution

1. Step 1: Understand the function's purpose

   scipy.optimize.curve_fit is designed to fit a curve to data points, especially when the relationship is not a straight line.

2. Step 2: Compare options with the function's goal

   Options B, C, and D describe unrelated tasks like sorting, averaging, or random number generation, which are not the purpose of curve fitting.

3. Final Answer:

   To find the best-fitting curve for data when the relationship is non-linear -> Option B

4. Quick Check:

   Curve fitting = best-fitting curve [OK]

Hint: Curve fitting finds best curve, not sorting or averaging [OK]

Common Mistakes:

• Confusing curve fitting with data sorting
• Thinking curve_fit calculates averages
• Assuming curve_fit generates random data

2. Which of the following is the correct way to import the curve_fit function from SciPy?

easy

A. import curve_fit from scipy.optimize

B. import scipy.curve_fit

C. from scipy import curve_fit

D. from scipy.optimize import curve_fit

Solution

1. Step 1: Recall correct import syntax in Python

   To import a specific function from a module, use from module import function syntax.

2. Step 2: Match syntax with options

   from scipy.optimize import curve_fit matches the correct syntax: from scipy.optimize import curve_fit. Options B, C, and D use incorrect syntax or wrong module paths.

3. Final Answer:

   from scipy.optimize import curve_fit -> Option D

4. Quick Check:

   Correct import = from module import function [OK]

Hint: Use 'from module import function' to import specific functions [OK]

Common Mistakes:

• Using 'import scipy.curve_fit' which is invalid
• Trying 'from scipy import curve_fit' when it's in optimize submodule
• Incorrect order like 'import curve_fit from ...'

3. What will be the output of the following code snippet?

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])

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

medium

A. [1.00 1.00]

B. [1.02 1.00]

C. [1.00 0.99]

D. [0.99 1.00]

Solution

1. Step 1: Understand the model and data

   The model is an exponential function: a * exp(b * x). The ydata roughly follows this pattern with a near 1 for a and about 1 for b.

2. Step 2: Run curve_fit and round parameters

   Using curve_fit on given data returns parameters close to [1.00, 0.99]. Rounding to two decimals gives [1.00 0.99].

3. Final Answer:

   [1.00 0.99] -> Option C

4. Quick Check:

   Fitted params ≈ [1.00, 0.99] [OK]

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

Common Mistakes:

• Assuming parameters are exactly 1.00 and 1.00
• Confusing parameter order or values
• Ignoring rounding effects

4. Identify the error in the following code snippet for non-linear curve fitting:

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])

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

medium

A. Missing unpacking of the tuple returned by curve_fit

B. Model function has wrong parameters

C. xdata and ydata have different lengths

D. curve_fit is not imported correctly

Solution

1. Step 1: Check the return value of curve_fit

   curve_fit returns a tuple: (parameters, covariance). The code assigns this tuple to a single variable without unpacking.

2. Step 2: Identify the correct usage

   Correct usage unpacks the tuple: params, _ = curve_fit(...). Without unpacking, printing params shows the tuple, not just parameters.

3. Final Answer:

   Missing unpacking of the tuple returned by curve_fit -> Option A

4. Quick Check:

   curve_fit returns tuple, unpack it [OK]

Hint: Always unpack curve_fit output: params, _ = curve_fit(...) [OK]

Common Mistakes:

• Assigning curve_fit output to one variable without unpacking
• Assuming curve_fit returns only parameters
• Ignoring the covariance matrix returned

5. You want to fit a non-linear model y = a * x / (b + x) to data using curve_fit. Which of the following code snippets correctly defines the model and fits the data?

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([1, 2, 3, 4, 5])
ydata = np.array([0.5, 1.2, 1.8, 2.4, 2.9])

params, covariance = curve_fit(model, xdata, ydata)
print(np.round(params, 2))

hard

A. Correctly defines model and fits data using curve_fit

B. Model function should use addition instead of division

C. curve_fit requires initial guess parameters to work

D. xdata and ydata lengths must be different for curve_fit

Solution

1. Step 1: Check model function correctness

   The model y = a * x / (b + x) is correctly implemented as return a * x / (b + x).

2. Step 2: Verify curve_fit usage

   The code calls curve_fit(model, xdata, ydata) and unpacks parameters and covariance correctly. Initial guesses are optional here.

3. Final Answer:

   Correctly defines model and fits data using curve_fit -> Option A

4. Quick Check:

   Model and curve_fit usage correct [OK]

Hint: Define model exactly, call curve_fit with data and unpack results [OK]

Common Mistakes:

• Changing division to addition in model
• Thinking initial guesses are always required
• Using different lengths for xdata and ydata