Polynomial fitting in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the polynomial fitting function from numpy.

SciPy

from numpy import [1]

Drag options to blanks, or click blank then click option'

Apolyfit

Binterpolate

Cstats

Doptimize

Attempts:

3 left

💡 Hint

Common Mistakes

Trying to import from scipy submodules like optimize or stats.

Using interpolate which is for interpolation, not fitting.

The polynomial fitting function polyfit can be imported directly from numpy.

2fill in blank

medium

Complete the code to fit a polynomial of degree 2 to data points x and y.

SciPy

coefficients = [1](x, y, 2)

Drag options to blanks, or click blank then click option'

Acurve_fit

Bpolyfit

Cminimize

Dlinregress

Attempts:

3 left

💡 Hint

Common Mistakes

Using curve_fit which is for general curve fitting, not specifically polynomials.

Using linregress which fits only linear regression.

The polyfit function fits a polynomial of specified degree to data points.

3fill in blank

hard

Fix the error in the code to evaluate the polynomial at points x_new using the coefficients. Assume polyval is imported from numpy.

SciPy

y_new = [1](coefficients, x_new)

Drag options to blanks, or click blank then click option'

Apolyval1d

Bpolyfit

Cpolyval2d

Dpolyval

Attempts:

3 left

💡 Hint

Common Mistakes

Using polyfit instead of polyval to evaluate values.

Using polyval2d which is for 2D polynomials.

The polyval function evaluates a polynomial at given points using coefficients.

4fill in blank

hard

Fill both blanks to create a dictionary of polynomial coefficients for degrees 1 and 3 from data x and y.

SciPy

coeffs = {1: [1](x, y, 1), 3: [2](x, y, 3)}

Drag options to blanks, or click blank then click option'

Apolyfit

Bcurve_fit

Dminimize

Attempts:

3 left

💡 Hint

Common Mistakes

Using different functions for each degree.

Using curve_fit which requires a function definition.

Use polyfit to fit polynomials of different degrees.

5fill in blank

hard

Fill all three blanks to create a dictionary of polynomial fits for degrees 1, 2, and 4, then evaluate degree 2 polynomial at x_new. Assume polyfit and polyval imported from numpy.

SciPy

coeffs = {1: [1](x, y, 1), 2: [2](x, y, 2), 4: [3](x, y, 4)}
y_new = polyval(coeffs[2], x_new)

Drag options to blanks, or click blank then click option'

Apolyfit

Dcurve_fit

Attempts:

3 left

💡 Hint

Common Mistakes

Using curve_fit for polynomial fitting.

Mixing different fitting functions.

Use polyfit for all polynomial fits. Then use polyval to evaluate the polynomial.

Practice

1. What does the scipy.polyfit function do in polynomial fitting?

easy

A. It calculates the coefficients of the polynomial that best fits the data.

B. It plots the data points on a graph.

C. It predicts future data points without fitting.

D. It normalizes the data before fitting.

Solution

1. Step 1: Understand the purpose of polyfit

   polyfit takes data points and finds polynomial coefficients that best fit those points.

2. Step 2: Differentiate from other functions

   Plotting or normalization are not done by polyfit; it only calculates coefficients.

3. Final Answer:

   It calculates the coefficients of the polynomial that best fits the data. -> Option A

4. Quick Check:

   polyfit = coefficients [OK]

Hint: Remember: polyfit finds coefficients, not plots or predictions [OK]

Common Mistakes:

• Confusing polyfit with plotting functions
• Thinking polyfit predicts future points directly
• Assuming polyfit normalizes data automatically

2. Which of the following is the correct syntax to fit a 3rd degree polynomial to data arrays x and y using SciPy?

easy

A. coeffs = scipy.polyfit(y, x, 3)

B. coeffs = scipy.polyfit(x, y, 3)

C. coeffs = scipy.polyfit(x, y)

D. coeffs = scipy.polyfit(x, y, degree=3)

Solution

1. Step 1: Check the order of arguments in polyfit

   The correct order is polyfit(x, y, degree).

2. Step 2: Confirm the degree argument is positional, not keyword

   polyfit expects degree as the third positional argument, not as a keyword.

3. Final Answer:

   coeffs = scipy.polyfit(x, y, 3) -> Option B

4. Quick Check:

   Correct syntax = coeffs = scipy.polyfit(x, y, 3) [OK]

Hint: Remember: polyfit(x, y, degree) with degree as positional [OK]

Common Mistakes:

• Swapping x and y arguments
• Omitting the degree argument
• Using degree as a keyword argument

3. Given the code:

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

What is the output printed?

medium

A. [ 1. 4. 9. 16.]

B. [ 1. 2. 4. 8.]

C. [ 0. 1. 4. 9.]

D. [ 1. 3. 7. 13.]

Solution

1. Step 1: Fit a 2nd degree polynomial to points

   The points (x, y) fit exactly to y = 1 + 2x + x^2, so polyfit finds coefficients close to [1, 2, 1].

2. Step 2: Use polyval to compute fitted values at x

   Evaluating the polynomial at x gives the original y values: [1, 3, 7, 13].

3. Final Answer:

   [ 1. 3. 7. 13.] -> Option D

4. Quick Check:

   polyval(coeffs, x) = original y [OK]

Hint: polyval with polyfit coeffs returns fitted y values [OK]

Common Mistakes:

• Confusing input arrays order
• Expecting different output than original y
• Misunderstanding polynomial degree effect

4. What is wrong with this code snippet for polynomial fitting?

import numpy as np
from scipy import polyfit, polyval
x = np.array([1, 2, 3])
y = np.array([2, 4, 6])
coeffs = polyfit(x, y, 2)
fitted = polyval(coeffs, x)
print(fitted)

medium

A. The code is correct and will run without errors.

B. The arrays x and y must be lists, not numpy arrays.

C. The degree 2 polynomial is too high for 3 points; use degree 1 instead.

D. polyval cannot be used with coefficients from polyfit.

Solution

1. Step 1: Check polynomial degree vs data points

   Fitting a degree 2 polynomial to 3 points is mathematically valid and will produce a polynomial that fits all points exactly.

2. Step 2: Validate data types and function usage

   Using numpy arrays is correct; polyval works with polyfit coefficients; no syntax errors present.

3. Final Answer:

   The code is correct and will run without errors. -> Option A

4. Quick Check:

   Degree 2 polynomial with 3 points = code runs fine [OK]

Hint: Degree equal to number of points minus one fits exactly [OK]

Common Mistakes:

• Using too high polynomial degree for few points
• Thinking numpy arrays are invalid input
• Believing polyval can't use polyfit output

5. You have noisy data points and want to fit a polynomial that smooths the noise but avoids overfitting. Which approach is best?

hard

A. Use polyfit with degree zero to get a constant fit.

B. Fit a high-degree polynomial to capture all fluctuations.

C. Fit a low-degree polynomial and check the fit visually.

D. Fit multiple polynomials of different degrees and average coefficients.

Solution

1. Step 1: Understand overfitting and noise smoothing

   High-degree polynomials fit noise too closely, causing overfitting; low-degree polynomials smooth data better.

2. Step 2: Use visual check to confirm fit quality

   Plotting fitted curve helps decide if degree is appropriate and avoids overfitting.

3. Final Answer:

   Fit a low-degree polynomial and check the fit visually. -> Option C

4. Quick Check:

   Low degree + visual check = smooth fit [OK]

Hint: Low degree + visual check avoids overfitting [OK]

Common Mistakes:

• Choosing too high degree polynomial
• Using degree zero which ignores trends
• Averaging coefficients from different fits