Polynomial fitting in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of polynomial fit coefficients

What is the output of the following code that fits a polynomial of degree 2 to the data points?

SciPy

import numpy as np
from numpy import polyfit
x = np.array([0, 1, 2, 3])
y = np.array([1, 3, 7, 13])
coeffs = polyfit(x, y, 2)
print(coeffs)

A [1. 1. 1.]

B [2. 1. 1.]

C [1. 2. 1.]

D [1. 1. 2.]

💡 Hint

Remember that polyfit returns coefficients starting from the highest degree term.

The points fit the polynomial y = 1*x^2 + 1*x + 1 exactly, so the coefficients are [1, 1, 1].

❓ data_output

intermediate

Number of coefficients from polynomial fit

How many coefficients does polyfit return when fitting a polynomial of degree 4 to 6 data points?

SciPy

import numpy as np
from numpy import polyfit
x = np.arange(6)
y = np.array([1, 4, 9, 16, 25, 36])
coeffs = polyfit(x, y, 4)
print(len(coeffs))

A 7

B 6

C 4

D 5

💡 Hint

The number of coefficients is degree + 1.

A polynomial of degree 4 has 5 coefficients, from x^4 down to the constant term.

❓ visualization

advanced

Plotting polynomial fit and data points

Which option produces a plot showing the original data points as red dots and the polynomial fit curve as a blue line?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from numpy import polyfit, polyval
x = np.array([0, 1, 2, 3, 4])
y = np.array([1, 3, 7, 13, 21])
coeffs = polyfit(x, y, 2)
x_fit = np.linspace(0, 4, 100)
y_fit = polyval(coeffs, x_fit)
plt.scatter(x, y, color='red')
plt.plot(x_fit, y_fit, color='blue')
plt.show()

A Scatter plot with red dots for data and blue line for fit

B Scatter plot with blue dots for data and red line for fit

C Line plot with red line for data and blue dots for fit

D Line plot with green line for data and orange dots for fit

💡 Hint

Check the colors and plot types used for data and fit.

The code uses plt.scatter with red color for data points and plt.plot with blue color for the polynomial curve.

🧠 Conceptual

advanced

Effect of polynomial degree on overfitting

What happens if you fit a polynomial of very high degree to a small noisy dataset?

A The polynomial degree does not affect the fit quality

B The polynomial ignores noise and fits the true trend perfectly

C The polynomial fits the noise and generalizes poorly to new data

D The polynomial always underfits the data

💡 Hint

Think about what happens when a model is too complex for the data.

High-degree polynomials can wiggle to fit noise, causing overfitting and poor predictions on new data.

🔧 Debug

expert

Identify error in polynomial fitting code

What error does the following code raise?

SciPy

import numpy as np
from numpy import polyfit
x = np.array([1, 2, 3])
y = np.array([2, 4])
coeffs = polyfit(x, y, 2)

A TypeError: polyfit() missing required positional argument

B ValueError: x and y must have the same length

C SyntaxError: invalid syntax

D No error, code runs successfully

💡 Hint

Check the lengths of x and y arrays.

polyfit requires x and y to have the same length; here y has fewer elements, causing a ValueError.