Non-linear curve fitting in SciPy

Introduction

We use non-linear curve fitting to find a smooth curve that best matches data points when the relationship is not a straight line.

You want to model how a plant grows over time with a curve that bends.

You have data from a chemical reaction that follows a curve, not a line.

You want to predict sales that grow quickly then slow down, not just increase steadily.

You need to fit a curve to data points that form a wave or exponential shape.

Syntax

SciPy

from scipy.optimize import curve_fit

popt, pcov = curve_fit(function, xdata, ydata, p0=None)

function is the model you think fits your data, like an exponential or sine curve.

p0 is optional starting guesses for the curve parameters to help fitting.

Examples

Fit a straight line (linear) model to data.

SciPy

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

popt, pcov = curve_fit(model, xdata, ydata)

Fit an exponential curve with initial guesses for parameters.

SciPy

import numpy as np

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

popt, pcov = curve_fit(model, xdata, ydata, p0=[1, 0.1, 0])

Sample Program

This program fits an exponential curve to noisy data and shows the fitted parameters and plot.

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

# Create example data: y = 2 * exp(1.5 * x) + noise
xdata = np.linspace(0, 2, 50)
y = 2 * np.exp(1.5 * xdata)
noise = 0.2 * np.random.normal(size=xdata.size)
ydata = y + noise

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

# Fit curve
popt, pcov = curve_fit(model, xdata, ydata, p0=[1, 1])

# Print fitted parameters
a_fit, b_fit = popt
print(f"Fitted parameters: a = {a_fit:.2f}, b = {b_fit:.2f}")

# Plot data and fitted curve
plt.scatter(xdata, ydata, label='Data')
plt.plot(xdata, model(xdata, *popt), 'r-', label='Fitted curve')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.title('Non-linear curve fitting example')
plt.show()

Important Notes

Good initial guesses (p0) help the fitting find the best curve faster.

Curve fitting finds parameters that minimize the difference between your data and the curve.

Plotting the data and fitted curve helps you see how well the curve fits.

Summary

Non-linear curve fitting finds curves that best match data when relationships are not straight lines.

Use scipy.optimize.curve_fit with a model function and your data.

Check fitted parameters and plot results to understand the fit quality.

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