Curve fitting (curve_fit) in SciPy - Mini Project: Build & Apply

Curve fitting with scipy curve_fit

📖 Scenario: You have collected some data points from an experiment measuring how a plant grows over days. You want to find a simple mathematical formula that fits this data well. This helps you understand the growth pattern and predict future growth.

🎯 Goal: Build a Python program that uses scipy.optimize.curve_fit to find the best fitting curve for the plant growth data. You will define a function for the curve, fit it to the data, and print the fitted parameters.

📋 What You'll Learn

Create arrays for days and plant heights

Define a simple linear function for growth

Use curve_fit to find the best parameters

Print the fitted parameters

💡 Why This Matters

🌍 Real World

Scientists and engineers often collect data and want to find a simple formula that explains it. Curve fitting helps find that formula.

💼 Career

Data scientists and analysts use curve fitting to model trends and make predictions in fields like biology, finance, and engineering.

1

Create data arrays for days and plant heights

Create two variables: days and heights. Set days to the list [1, 2, 3, 4, 5, 6, 7] and heights to the list [2.3, 2.9, 3.8, 4.5, 5.1, 5.8, 6.3].

SciPy

# Create the data lists for days and heights
# Your code here

Hint

Use Python lists to store the numbers exactly as given.

2

Define a linear function for plant growth

Define a function called linear_growth that takes x, a, and b as inputs and returns a * x + b. This function models a straight line.

SciPy

days = [1, 2, 3, 4, 5, 6, 7]
heights = [2.3, 2.9, 3.8, 4.5, 5.1, 5.8, 6.3]

# Define the linear_growth function
# Your code here

Hint

Use def to create the function and return the formula a * x + b.

3

Use curve_fit to find the best fitting parameters

Import curve_fit from scipy.optimize. Use curve_fit with linear_growth, days, and heights to get the fitted parameters. Save the first output as params.

SciPy

days = [1, 2, 3, 4, 5, 6, 7]
heights = [2.3, 2.9, 3.8, 4.5, 5.1, 5.8, 6.3]

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

# Import curve_fit and find params
# Your code here

Hint

Use from scipy.optimize import curve_fit and call curve_fit with the function and data.

4

Print the fitted parameters

Write a print statement to display the text "Fitted parameters:" followed by the params variable.

SciPy

from scipy.optimize import curve_fit

days = [1, 2, 3, 4, 5, 6, 7]
heights = [2.3, 2.9, 3.8, 4.5, 5.1, 5.8, 6.3]

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

params, _ = curve_fit(linear_growth, days, heights)

# Print the fitted parameters
# Your code here

Hint

Use print("Fitted parameters:", params) to show the result.

Practice

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

easy

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

B. To sort data points in ascending order

C. To calculate the average of data points

D. To remove outliers from the dataset

Solution

1. Step 1: Understand the role of curve fitting

   Curve fitting is used to find parameters of a model that best match the given data points.

2. Step 2: Identify what curve_fit does

   The curve_fit function from scipy optimizes parameters to minimize the difference between model predictions and data.

3. Final Answer:

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

4. Quick Check:

   Curve fitting = best parameters for model [OK]

Hint: Remember: curve_fit finds model parameters to fit data [OK]

Common Mistakes:

• Confusing curve fitting with data sorting
• Thinking curve_fit calculates averages
• Assuming curve_fit removes outliers

2. Which of the following is the correct way to call curve_fit with a model function f, data arrays xdata and ydata?

easy

A. popt = curve_fit(f, ydata, xdata)

B. popt, pcov = curve_fit(xdata, ydata, f)

C. curve_fit(f, ydata, xdata)

D. popt, pcov = curve_fit(f, xdata, ydata)

Solution

1. Step 1: Recall the function signature of curve_fit

   The correct call is curve_fit(function, xdata, ydata), where the model function is first, then x and y data.

2. Step 2: Match the options with the correct signature

   popt, pcov = curve_fit(f, xdata, ydata) matches the correct order and unpacks both parameters and covariance matrix.

3. Final Answer:

   popt, pcov = curve_fit(f, xdata, ydata) -> Option D

4. Quick Check:

   curve_fit(function, xdata, ydata) [OK]

Hint: Remember order: function, xdata, ydata in curve_fit [OK]

Common Mistakes:

• Swapping xdata and ydata
• Not unpacking both outputs
• Passing arguments in wrong order

3. Given the code:

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])

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

What will be printed?

medium

A. [2.00 0.00]

B. [2.02 0.04]

C. [1.00 2.00]

D. [0.06 2.04]

Solution

1. Step 1: Understand the model and data

   The model is linear: y = a*x + b. The data roughly follows y = 2*x + 0.

2. Step 2: Use curve_fit to find parameters

   Running curve_fit finds parameters close to a=2.02 and b=0.04, fitting the data closely.

3. Final Answer:

   [2.02 0.04] -> Option B

4. Quick Check:

   Fitted slope ~2.02, intercept ~0.04 [OK]

Hint: Check model form and data trend to guess parameters [OK]

Common Mistakes:

• Confusing parameter order
• Ignoring rounding
• Assuming exact integers

4. Consider this code snippet:

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])
ydata = np.array([2, 4, 6])

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

What is the main issue here?

medium

A. The model function is incorrect

B. The data arrays have different lengths

C. xdata and ydata are swapped in curve_fit call

D. curve_fit cannot handle linear models

Solution

1. Step 1: Check the order of arguments in curve_fit

   The correct order is curve_fit(model, xdata, ydata). Here, ydata and xdata are swapped.

2. Step 2: Understand the impact of swapping

   Swapping causes the model to fit x as a function of y, which is not intended and leads to wrong parameters.

3. Final Answer:

   xdata and ydata are swapped in curve_fit call -> Option C

4. Quick Check:

   Order is model, xdata, ydata [OK]

Hint: Always put xdata before ydata in curve_fit [OK]

Common Mistakes:

• Swapping xdata and ydata
• Assuming model function is wrong
• Ignoring argument order importance

5. You have noisy data from a quadratic process: y = a*x**2 + b*x + c. You want to fit this model using curve_fit. Which of the following code snippets correctly defines the model and fits the data xdata, ydata?

hard

A. def model(x, a, b, c):
return a * x**2 + b * x + c
popt, pcov = curve_fit(model, xdata, ydata)

B. def model(a, b, c, x):
return a * x**2 + b * x + c
popt, pcov = curve_fit(model, xdata, ydata)

C. def model(x, a, b):
return a * x**2 + b * x
popt, pcov = curve_fit(model, xdata, ydata)

D. def model(x, a, b, c):
return a * x + b * x**2 + c
popt, pcov = curve_fit(model, ydata, xdata)

Solution

1. Step 1: Define the model function correctly

   The model must have x as the first argument, followed by parameters a, b, c. The formula is a*x**2 + b*x + c.

2. Step 2: Call curve_fit with correct argument order

   Use curve_fit(model, xdata, ydata) to fit the model to data.

3. Final Answer:

   def model(x, a, b, c): return a * x**2 + b * x + c popt, pcov = curve_fit(model, xdata, ydata) -> Option A

4. Quick Check:

   Model args: x first, then parameters; curve_fit(model, xdata, ydata) [OK]

Hint: Model function: x first, parameters next; call curve_fit accordingly [OK]

Common Mistakes:

• Putting parameters before x in model
• Missing one parameter in model
• Swapping xdata and ydata in curve_fit