Fitting custom models in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of custom model fitting with scipy.optimize.curve_fit

What is the output of the following code snippet that fits a custom 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([0, 1, 2, 3, 4])
ydata = np.array([1, 3, 7, 13, 21])

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

A [1.0, 2.0, 1.0]

B [0.0, 1.0, 1.0]

C [1.0, 1.0, 1.0]

D [1.0, 1.0, 2.0]

💡 Hint

Recall the quadratic formula: y = a*x² + b*x + c. Check the data points to guess coefficients.

The data fits y = x² + 2x + 1 exactly. The curve_fit finds parameters [1.0, 2.0, 1.0].

❓ data_output

intermediate

Number of parameters fitted in a custom exponential model

Given the custom exponential model y = a * exp(b * x) + c, how many parameters does scipy.optimize.curve_fit estimate?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 4, 5)
ydata = exp_model(xdata, 2, 0.5, 1) + 0.1 * np.random.normal(size=xdata.size)
params, _ = curve_fit(exp_model, xdata, ydata)
print(len(params))

A 4

B 2

C 3

D 1

💡 Hint

Count the number of parameters in the function definition after x.

The model has three parameters: a, b, and c. curve_fit estimates all three.

🔧 Debug

advanced

Identify the error in custom model fitting code

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

SciPy

import numpy as np
from scipy.optimize import curve_fit

def linear_model(x, m, c):
    return m * x + c

xdata = np.array([0, 1, 2, 3])
ydata = np.array([1, 3, 5, 7])

params, _ = curve_fit(linear_model, xdata, ydata, p0=[1])

A ValueError: x and y must have same length

B ValueError: 'p0' must be a list of length 2

C RuntimeError: Optimal parameters not found

D TypeError: linear_model() missing 1 required positional argument

💡 Hint

Check the initial guess p0 length compared to model parameters.

The model has two parameters but p0 provides only one initial guess, causing a ValueError.

❓ visualization

advanced

Plot result of fitting a sinusoidal custom model

Which plot correctly shows the fitted sinusoidal model y = a * sin(b * x + c) to noisy data?

SciPy

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

def sin_model(x, a, b, c):
    return a * np.sin(b * x + c)

xdata = np.linspace(0, 2 * np.pi, 50)
ydata = 3 * np.sin(2 * xdata + 0.5) + 0.3 * np.random.normal(size=xdata.size)
params, _ = curve_fit(sin_model, xdata, ydata, p0=[2, 2, 0])

plt.scatter(xdata, ydata, label='Data')
plt.plot(xdata, sin_model(xdata, *params), color='red', label='Fitted model')
plt.legend()
plt.show()

A Scatter points with a red exponential curve

B Scatter points with a flat red line

C Scatter points with a red linear line

D Scatter points with a red smooth sine curve closely following data

💡 Hint

Think about the shape of a sine wave and how fitting works.

The fitted model is a sine curve that closely matches the noisy data points, shown as scatter.

🧠 Conceptual

expert

Effect of initial parameter guess on fitting convergence

Which statement best describes the effect of the initial parameter guess p0 in scipy.optimize.curve_fit when fitting complex custom models?

A A poor initial guess can cause the algorithm to converge to a local minimum, not the global best fit

B Initial guesses have no effect; curve_fit always finds the global minimum

C Initial guesses only affect the speed but never the final result

D curve_fit ignores initial guesses if the model is nonlinear

💡 Hint

Consider how optimization algorithms work with complex landscapes.

Nonlinear fitting can get stuck in local minima if initial guesses are poor, affecting final fit quality.