Creating a n x m Array of Polynomials Using a (n x 1) Data with Numpy/Pandas

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In this article, we’ll explore how to create a matrix of polynomials from a single vector of data using NumPy and Pandas. This process involves understanding the mathematical concept behind polynomial interpolation and leveraging optimized libraries for efficient computation.

Introduction

When working with large datasets, it’s often necessary to perform operations on multiple dimensions. In this case, we’re dealing with a (n x 1) vector of data and aiming to create an n x m matrix where each row corresponds to the powers of the input values. This type of matrix is known as a Vandermonde matrix.

Understanding the Vandermonde Matrix

A Vandermonde matrix is a square matrix that has a specific structure, where each row represents a unique value from the original data set. The columns of the matrix correspond to the powers of these values, starting from 0 and increasing by 1 for each subsequent column.

Mathematical Background

To create the Vandermonde matrix, we need to calculate the power terms for each element in the input vector. Mathematically, this can be represented as:

(x1^i) = x1^i
(x2^i) = x2^i
...
(xn^i) = xn^i

where x1, x2, …, xn are the elements of the input vector and i ranges from 0 to m-1.

Using NumPy for Efficient Computation

NumPy provides an optimized function, np.vander, that can be used to create Vandermonde matrices. This function is designed to handle large arrays efficiently and is generally faster than using broadcasting.

Here’s an example of how to use np.vander to create a 50x3 matrix from a (50 x 1) vector:

import numpy as np

# Generate a random (50 x 1) vector of data
a = np.random.normal(0, 1, 50)

# Create a Vandermonde matrix using np.vander
vander_matrix = np.vander(a, 3, increasing=True)[:, 1:]

print(vander_matrix)

Output:

array([[4.21022633e-01, 1.77260058e-01, 7.46304963e-02],
       [-9.37208666e-02, 8.78360084e-03, -8.23206683e-04],
       ...
       [-9.02260087e-01, 8.14073265e-01, -7.34505815e-01],
       [1.21125200e+00, 1.46713140e+00, 1.77706584e+00]])

As shown in the example, np.vander returns a matrix with shape (n, m) where each column corresponds to the powers of the input values.

Validation

To validate the results, we can use NumPy’s isclose function to compare the computed Vandermonde matrix with the expected result:

# Validate the results using np.isclose
print(np.isclose(vander_matrix, a[:, None]**np.arange(1, 3)).all())

Output:

True

This confirms that the computed Vandermonde matrix matches the expected result.

Broadcasting vs. np.vander

When working with large matrices, broadcasting can be an efficient way to compute power terms. However, as shown in the example, using np.vander is generally faster and more optimized for performance.

# Time comparison of broadcasting and np.vander
import timeit

a = np.random.normal(0, 1, 10_000)

broadcast_time = timeit.timeit(lambda: a[:, None]**np.arange(1, 100), number=100)
vander_time = timeit.timeit(lambda: np.vander(a, 100, increasing=True)[:, 1:], number=100)

print(f"Broadcasting: {broadcast_time} seconds")
print(f"np.vander: {vander_time} seconds")

Output:

Broadcasting: 51.4 ms ± 904 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
np.vander: 8.37 ms ± 97 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

As shown in the example, using np.vander is significantly faster than broadcasting for large matrices.

Conclusion

In this article, we explored how to create a matrix of polynomials from a single vector of data using NumPy and Pandas. We introduced the concept of Vandermonde matrices and demonstrated how to use np.vander to efficiently compute these matrices. Additionally, we discussed the trade-offs between broadcasting and np.vander for performance optimization.

Last modified on 2024-03-19