Introduction to Polyfit in Python

Overview

In scientific computing, linear regression is a common technique to fit a model to data assuming a linear relationship between the independent and dependent variables. However, in many cases, the relationship between the variables may be more complex than linear. Polynomial regression is a generalization of linear regression that allows us to fit a polynomial function to data. The polyfit function in Python is a convenient way to perform polynomial regression for a given set of data points.

Function Signature

The polyfit function in NumPy library has the following signature:

numpy.polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False)

where:

• x and y are the data points to be fitted. They must be of equal length and shape.
• deg is the degree of the polynomial function to be fitted.
• rcond is the cutoff ratio for small singular values of x. Default is 1E-10.
• full is a flag indicating whether to return additional information. Default is False.
• w is a vector of weights to apply to each data point. Default is None.
• cov is a flag indicating whether to return the covariance matrix in addition to the fitted coefficients. Default is False.

Usage

To demonstrate the usage of polyfit, let's generate some sample data points and fit them to a polynomial function of degree 2. We will plot the original data and the fitted function using matplotlib.

import numpy as np
import matplotlib.pyplot as plt

# Generate some sample data
x = np.array([0, 1, 2, 3, 4, 5])
y = np.array([2.1, 3.9, 6.2, 8.1, 11.1, 14.4])

# Fit a polynomial function of degree 2
coefficients = np.polyfit(x, y, 2)

# Evaluate the fitted function at 100 equally spaced points
xfit = np.linspace(0, 5, 100)
yfit = np.polyval(coefficients, xfit)

# Plot the data and the fitted function
plt.plot(x, y, 'o')
plt.plot(xfit, yfit, '-')
plt.legend(['data', 'fitted function'])
plt.show()

The output plot should show the original data points as circles and the fitted function as a continuous line.

Conclusion

The polyfit function in Python provides a convenient way to perform polynomial regression for a given set of data points. It is a powerful tool for curve fitting applications in scientific computing.