Non-linear curve fitting in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of curve fitting with exponential decay

What is the output of the following code snippet that fits an exponential decay model to data?

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, 5])
ydata = np.array([5, 3, 2, 1.2, 0.7, 0.4])

params, _ = curve_fit(model, xdata, ydata, p0=[5, 0.5])
print(np.round(params, 2))

A [5.00 0.50]

B [4.98 0.48]

C [5.10 0.60]

D [4.00 0.30]

💡 Hint

Look at how curve_fit estimates parameters close to initial guesses but adjusted to data.

The curve_fit function finds parameters close to the initial guess but adjusted to best fit the data. The output parameters are approximately [4.98, 0.48].

❓ data_output

intermediate

Number of iterations in curve fitting

After fitting a non-linear model using scipy.optimize.curve_fit, how can you find the number of iterations the optimizer took? What is the value of the 'nfev' key in the returned info dictionary?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 4, 50)
ydata = model(xdata, 3, 1.5) + 0.1 * np.random.normal(size=xdata.size)

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

# The number of function evaluations is stored in the 'nfev' attribute of the OptimizeResult object
# But curve_fit returns only params and cov, so we need to use full_output=True to get info
params, cov, info, mesg, ier = curve_fit(model, xdata, ydata, full_output=True)
print(info['nfev'])

A 50

B 100

C 500

D 200

💡 Hint

The 'nfev' key counts how many times the model function was evaluated during fitting.

The 'nfev' key in the info dictionary shows the number of function evaluations. For this data and model, it is typically around 200.

🔧 Debug

advanced

Identify the error in curve fitting code

What error does the following code raise when trying to fit a quadratic model to data?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([1, 2, 3, 4, 5])
ydata = np.array([2, 5, 10, 17, 26])

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

A RuntimeError: Optimal parameters not found

B TypeError: curve_fit() got an unexpected keyword argument 'p0'

C ValueError: Expected 3 initial parameters but got 2

D No error, outputs parameters

💡 Hint

Check the number of parameters in the model and the length of p0.

The model has 3 parameters (a, b, c) but p0 provides only 2 initial guesses. This mismatch causes a ValueError.

❓ visualization

advanced

Plotting fitted curve and data points

Which code snippet correctly plots the original data points and the fitted non-linear curve using matplotlib?

SciPy

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

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

xdata = np.linspace(0, 5, 50)
ydata = model(xdata, 3, 1.2) + 0.2 * np.random.normal(size=xdata.size)

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

# Plotting code here

A plt.scatter(xdata, ydata, label='Data')\nplt.plot(xdata, model(xdata, *params), 'r-', label='Fit')\nplt.legend()\nplt.show()

B plt.plot(xdata, ydata, 'ro', label='Data')\nplt.scatter(xdata, model(xdata, *params), label='Fit')\nplt.legend()\nplt.show()

C plt.plot(xdata, ydata, label='Data')\nplt.plot(xdata, model(xdata, params[0], params[1]), 'g--', label='Fit')\nplt.legend()\nplt.show()

D plt.scatter(xdata, ydata)\nplt.plot(xdata, model(xdata, params), 'b-', label='Fit')\nplt.legend()\nplt.show()

💡 Hint

Use scatter for data points and plot for the fitted curve. Unpack params with *params.

Option A correctly uses scatter for data points and plots the fitted curve by unpacking params. Others misuse plot/scatter or params.

🧠 Conceptual

expert

Understanding residuals in non-linear curve fitting

In non-linear curve fitting, what does the residual sum of squares (RSS) represent and why is minimizing it important?

A RSS measures the total squared difference between observed and predicted values; minimizing it finds the best fit parameters.

B RSS counts the number of data points; minimizing it reduces dataset size.

C RSS is the sum of absolute differences between parameters; minimizing it ensures parameters are small.

D RSS is the product of residuals; minimizing it maximizes the error.

💡 Hint

Think about how fitting tries to reduce the difference between model and data.

RSS sums squared differences between observed and predicted values. Minimizing RSS finds parameters that best fit the data.

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