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.