Curve fitting (curve_fit) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of curve fitting with linear model

What is the output of the following code snippet that fits a linear model to data using curve_fit?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([0, 1, 2, 3, 4])
ydata = np.array([1, 3, 5, 7, 9])

params, covariance = curve_fit(linear_model, xdata, ydata)
print(params)

A [0.5 1.5]

B [1. 2.]

C [2. 1.]

D [3. 0.]

💡 Hint

Think about the equation of a line y = a*x + b and the data points given.

The data points follow the line y = 2*x + 1 exactly, so the fitted parameters are a=2 and b=1.

❓ data_output

intermediate

Number of parameters returned by curve_fit

Given the following model and data, how many parameters does curve_fit return?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 5, 6)
ydata = np.array([1, 3, 7, 13, 21, 31])

params, covariance = curve_fit(quadratic_model, xdata, ydata)
print(len(params))

A 3

B 2

C 6

D 1

💡 Hint

Count the number of parameters in the quadratic model function.

The quadratic model has three parameters: a, b, and c, so curve_fit returns three values.

🔧 Debug

advanced

Identify the error in curve_fit usage

What error does the following code raise when run?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([0, 1, 2, 3])
ydata = np.array([0, 0.84, 0.91, 0.14])

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

A ValueError: initial guess p0 must have length 2

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

C TypeError: model() missing 1 required positional argument

D RuntimeError: Optimal parameters not found

💡 Hint

Check the length of the initial guess array compared to the number of parameters in the model.

The model has two parameters (a and b), but p0 is given as a list with only one element, causing a ValueError.

❓ visualization

advanced

Plotting fitted curve with noisy data

Which option correctly plots the noisy data points and the fitted curve from curve_fit?

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, 4, 50)
y = model(xdata, 2.5, -1.3)
np.random.seed(0)
noise = 0.2 * np.random.normal(size=xdata.size)
ydata = y + noise

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

plt.scatter(xdata, ydata, label='Noisy data')
plt.plot(xdata, model(xdata, *params), 'r-', label='Fitted curve')
plt.legend()
plt.show()

A Line plot of noisy data points with blue dots showing the fitted curve

B Scatter plot of noisy data points with a red line showing the fitted exponential curve

C Scatter plot of noisy data points with no fitted curve shown

D Only the fitted curve plotted as a blue dashed line, no data points

💡 Hint

Look for a scatter plot for data and a red line for the fitted curve.

The code plots noisy data as scattered points and overlays the fitted curve as a red line.

🧠 Conceptual

expert

Understanding covariance matrix from curve_fit

What does the covariance matrix returned by curve_fit represent?

A It is the matrix used to transform the input data before fitting

B It contains the residuals of the fitted curve compared to the data points

C It shows the correlation between the input x values and the output y values

D It estimates the uncertainty and correlation between the fitted parameters

💡 Hint

Think about what uncertainty means in parameter estimation.

The covariance matrix estimates how much the fitted parameters vary and how they relate to each other.