Polynomial fitting in SciPy - Step-by-Step Execution

Concept Flow - Polynomial fitting

Start with data points

↓

Choose polynomial degree

↓

Use polyfit to find coefficients

↓

Create polynomial function

↓

Evaluate polynomial at x values

↓

Plot or analyze fitted curve

↓

End

We start with data points, pick the polynomial degree, find coefficients using polyfit, create a polynomial function, evaluate it, then analyze or plot the fit.

Execution Sample

SciPy

import numpy as np

x = np.array([0,1,2,3])
y = np.array([1,3,7,13])
coeffs = np.polyfit(x, y, 2)
y_fit = np.polyval(coeffs, x)

This code fits a 2nd degree polynomial to points (x,y) and calculates fitted y values.

Execution Table

Step	Action	Input	Output	Notes
1	Input data arrays	x=[0,1,2,3], y=[1,3,7,13]	Data ready	Data points prepared
2	Choose degree	degree=2	Degree set	Quadratic fit chosen
3	Call polyfit	x, y, degree=2	coeffs=[3.0, 1.0, 1.0]	Coefficients for x^2, x, constant
4	Create polynomial function	coeffs	poly(x) = 3*x^2 + 1*x + 1	Polynomial function formed
5	Evaluate polynomial	poly, x	y_fit=[1,3,7,13]	Fitted y values computed
6	Compare y and y_fit	y, y_fit	Match	Perfect fit for given points

💡 All steps completed, polynomial fit matches data points

Variable Tracker

Variable	Start	After Step 3	After Step 5	Final
x	[0,1,2,3]	[0,1,2,3]	[0,1,2,3]	[0,1,2,3]
y	[1,3,7,13]	[1,3,7,13]	[1,3,7,13]	[1,3,7,13]
degree	None	2	2	2
coeffs	None	[3.0,1.0,1.0]	[3.0,1.0,1.0]	[3.0,1.0,1.0]
y_fit	None	None	[1,3,7,13]	[1,3,7,13]

Key Moments - 3 Insights

Why does polyfit return coefficients in the order of highest degree first?

Polyfit returns coefficients starting with the highest power term (x^degree) to the constant term. This matches the standard polynomial form and is shown in step 3 of the execution table.

Why do the fitted y values exactly match the original y values?

Because the polynomial degree (2) is enough to perfectly fit the 4 data points given, as seen in step 5 and 6 of the execution table.

What happens if we choose a polynomial degree too low or too high?

Choosing too low degree may underfit (poor fit), too high may overfit (too wiggly). This affects coefficients and fitted values, visible by comparing y_fit to y in step 6.

Visual Quiz - 3 Questions

Look at the execution table, what are the polynomial coefficients after step 3?

A [0.0, 1.0, 2.0]

B [3.0, 1.0, 1.0]

C [1.0, 2.0, 3.0]

D [3.0, 2.0, 1.0]

💡 Hint

Check the 'Output' column in step 3 of the execution table.

At which step do we evaluate the polynomial at the x values?

A Step 2

B Step 4

C Step 5

D Step 6

💡 Hint

Look for 'Evaluate polynomial' action in the execution table.

If we change the degree to 1, how would the coefficients likely change?

A There would be 2 coefficients, fitting a line

B There would be 3 coefficients as before

C Coefficients would be all zeros

D Coefficients would be the same as degree 2

💡 Hint

Polynomial degree determines number of coefficients: degree + 1.

Concept Snapshot

Polynomial fitting uses data points and fits a polynomial curve.
Use np.polyfit(x, y, degree) to get coefficients.
Coefficients are ordered from highest degree to constant.
Evaluate fit with np.polyval(coeffs, x).
Degree controls curve flexibility and fit quality.

Full Transcript

Polynomial fitting means finding a curve that best matches given data points. We start with x and y data arrays. We pick a polynomial degree, like 2 for a quadratic. Using NumPy's polyfit, we get coefficients that define the polynomial. These coefficients start with the highest power term. Then we create a polynomial function and evaluate it at the x values to get fitted y values. If the degree is right, the fitted values closely match the original y values. This process helps us understand trends or predict new values.