Polynomial fitting in SciPy - Mini Project: Build & Apply

Polynomial fitting

📖 Scenario: You have collected some data points from a small experiment measuring temperature over time. You want to find a smooth curve that fits these points well. This will help you understand the trend and make predictions.

🎯 Goal: Build a polynomial model that fits the given data points using numpy. You will create the data, set the polynomial degree, fit the polynomial, and then print the polynomial coefficients.

📋 What You'll Learn

Create a dictionary called data_points with exact keys time and temperature and their respective lists of values

Create a variable called degree and set it to the exact integer 2

Use numpy to fit a polynomial of degree degree to the data points

Store the polynomial coefficients in a variable called coefficients

Print the coefficients variable

💡 Why This Matters

🌍 Real World

Polynomial fitting is used in science and engineering to find smooth curves that describe data trends, such as temperature changes, stock prices, or sensor readings.

💼 Career

Data scientists and analysts use polynomial fitting to model relationships in data, make predictions, and communicate insights clearly.

1

DATA SETUP: Create the data points dictionary

Create a dictionary called data_points with two keys: 'time' and 'temperature'. Set data_points['time'] to the list [0, 1, 2, 3, 4] and data_points['temperature'] to the list [1, 3, 7, 13, 21].

SciPy

# Create the data_points dictionary with time and temperature lists
# Your code here

Hint

Use a dictionary with keys 'time' and 'temperature'. Each key should have a list of numbers as its value.

2

CONFIGURATION: Set the polynomial degree

Create a variable called degree and set it to the integer 2.

SciPy

data_points = {'time': [0, 1, 2, 3, 4], 'temperature': [1, 3, 7, 13, 21]}
# Set the polynomial degree to 2
# Your code here

Hint

Just assign the number 2 to a variable named degree.

3

CORE LOGIC: Fit the polynomial to the data

Import numpy as np and polyfit from numpy.polynomial.polynomial. Use polyfit with data_points['time'], data_points['temperature'], and degree to fit the polynomial. Store the result in a variable called coefficients.

SciPy

data_points = {'time': [0, 1, 2, 3, 4], 'temperature': [1, 3, 7, 13, 21]}
degree = 2
# Import numpy and polyfit, then fit the polynomial
# Your code here

Hint

Use polyfit from numpy.polynomial.polynomial to fit the polynomial. The order of arguments is x values, y values, then degree.

4

OUTPUT: Print the polynomial coefficients

Print the variable coefficients to display the polynomial coefficients.

SciPy

import numpy as np
from numpy.polynomial.polynomial import polyfit

data_points = {'time': [0, 1, 2, 3, 4], 'temperature': [1, 3, 7, 13, 21]}
degree = 2
coefficients = polyfit(data_points['time'], data_points['temperature'], degree)
# Print the coefficients
# Your code here

Hint

Use print(coefficients) to show the polynomial coefficients.

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