Linear Least-Squares Problems Using Scipy

Linear least-squares problems are fundamental in many areas of science and engineering. These problems involve finding the best-fit solution to a system of linear equations by minimizing the sum of the squared residuals. In Python, the scipy library provides powerful tools to solve these problems efficiently. This article will explore linear least-squares problems using scipy, focusing on practical implementations and technical details.

What is Linear Least-Squares Problems?

Linear least-squares problems aim to find a vector x that minimizes the sum of the squared differences between the observed values and the values predicted by a linear model. Mathematically, given a matrix A and a vector b, the goal is to solve:

\text{minimize} \quad \| \mathbf{A} \mathbf{x} - \mathbf{b} \|_2^2

where,

\| \cdot \|_2 denotes the Euclidean norm. This problem arises in various applications, including data fitting, signal processing, and machine learning.

Solving Linear Least-Squares Problems with Scipy

Scipy's optimize module provides several functions to solve linear least-squares problems. The primary functions for least-squares problems are:

• scipy.optimize.lstsq: Solves the linear least-squares problem using a direct approach.
• scipy.optimize.least_squares: Provides more flexibility with non-linear least-squares problems and offers advanced options for solving optimization problems.

1. Linear Least-Squares Problems with scipy.optimize.lstsq

The scipy.optimize.lstsq function provides a straightforward way to solve linear least-squares problems. The function signature is:

Syntax:

scipy.optimize.lstsq(a, b, rcond='warn')

where,

• a: The input matrix A.
• b: The output vector b.
• rcond: Cutoff for small singular values. If rcond is None, the default value is used.

Example:

import numpy as np
from scipy.linalg import lstsq

# Example data
A = np.array([[1, 1], [1, 2], [1, 3]])
b = np.array([1, 2, 3])

# Solve least-squares problem
x, residuals, rank, s = lstsq(A, b)

print("Solution:", x)
print("Residuals:", residuals)

Output:

Solution: [1.80069003e-16 1.00000000e+00]
Residuals: 0.0

2. Linear Least-Squares Problemswith scipy.optimize.least_squares

For more complex scenarios, such as non-linear least-squares problems or when advanced optimization options are needed, scipy.optimize.least_squares is useful. Its function signature is:

Syntax:

scipy.optimize.least_squares(fun, x0, jac='2-point', method='trf', **kwargs)

• fun: The function that computes the residuals.
• x0: Initial guess for the solution.
• jac: The Jacobian matrix (optional).
• method: The optimization method ('trf', 'dogbox', or 'lm')

Example:

import numpy as np
from scipy.linalg import lstsq
from scipy.optimize import least_squares

# Define the data for least-squares problem
A = np.array([[1, 1], [1, 2], [1, 3]])
b = np.array([1, 2, 3])

# Solve least-squares problem using lstsq
x_lstsq, residuals, rank, s = lstsq(A, b)
print("Solution using lstsq:", x_lstsq)
print("Residuals using lstsq:", residuals)

# Define the residuals function for least_squares
def residuals_func(x):
    return np.dot(A, x) - b

# Initial guess for the solution
x0 = np.zeros(A.shape[1])

# Solve least-squares problem using least_squares
result = least_squares(residuals_func, x0)
print("Solution using least_squares:", result.x)
print("Residuals using least_squares:", result.fun)

Output:

Solution using lstsq: [1.80069003e-16 1.00000000e+00]
Residuals using lstsq: 0.0
Solution using least_squares: [5.84184113e-16 1.00000000e+00]
Residuals using least_squares: [ 4.4408921e-16  0.0000000e+00 -4.4408921e-16]

Comparison of Methods

Both lsq_linear and lstsq serve different purposes:

• lsq_linear: Suitable for bounded problems and large sparse matrices. It provides more flexibility with constraints.
• lstsq: Ideal for general-purpose least-squares solutions when no bounds are required.

Conclusion

Linear least-squares problems are essential in various scientific and engineering applications. Scipy's optimize module provides powerful tools for solving these problems, with lstsq offering a direct approach and least_squares providing more flexibility. Understanding the capabilities and limitations of these functions allows for effective problem-solving and accurate data analysis.