Sampling is at the core of many computational methods, especially in fields like numerical integration, optimization, and computer graphics. When trying to sample points from a high-dimensional space, traditional random sampling methods can result in clusters or gaps in the points, leading to inefficient coverage of the space. To address this issue, Low Discrepancy Sequences (LDS) are used to ensure that points are distributed as uniformly as possible, thereby reducing sampling errors.
In this blog post, we will explore Low Discrepancy Sequences, their importance, and how they work, followed by examples of generating LDS using Python.
What Are Low Discrepancy Sequences (LDS)?
The Concept
A Low Discrepancy Sequence (LDS), also known as a quasi-random sequence, is a sequence of points that are distributed as uniformly as possible across a given space. The goal of an LDS is to minimize the discrepancy, which measures the unevenness of point distribution.
Traditional random sampling methods can lead to clustering of points in some regions and sparse coverage in others, especially in high-dimensional spaces. This non-uniform distribution can lead to high variance and inefficient sampling. LDS, on the other hand, ensures that the points are spread out evenly, minimizing clustering and improving coverage of the entire space.
The most commonly used LDS are Sobol sequences, Halton sequences, and Faure sequences, which are specifically designed to cover spaces more uniformly than random sampling.
Discrepancy and Uniform Distribution
The discrepancy of a sequence measures how unevenly the sequence is distributed across a space. It is defined as the maximum deviation between the actual distribution of points and the uniform distribution. Mathematically, the discrepancy ( D_N ) for a sequence of ( N ) points in a ( d )-dimensional space is given by:
[
D_N(\mathcal{P}) = \sup_{I \subseteq [0, 1]^d} \left| \frac{N(I)}{N} – \text{Vol}(I) \right|
]
Where:
- ( N(I) ) is the number of points in the region ( I ).
- ( \text{Vol}(I) ) is the volume of the region ( I ).
- ( N ) is the total number of points in the sequence.
The goal of LDS is to minimize the maximum discrepancy, ensuring a more uniform distribution of points across the space.
Applications of Low Discrepancy Sequences
LDS are widely used in fields where uniform coverage and low variance are critical. Some of the primary applications of LDS include:
1. Numerical Integration
One of the most well-known applications of LDS is in quasi-Monte Carlo integration. In numerical integration, LDS are used to approximate the value of integrals, particularly in high-dimensional spaces. By distributing sample points more uniformly, LDS reduce the variance of the estimates compared to random sampling, which leads to faster convergence and more accurate results.
2. Optimization
In optimization problems, LDS are used to explore the search space more efficiently. Traditional random search methods may cluster points in certain regions, causing the optimization algorithm to miss optimal solutions. By using LDS, you ensure that the search points are more evenly spread across the space, improving the chances of finding the global optimum.
3. Computer Graphics and Rendering
In computer graphics, particularly in ray tracing and rendering, LDS are used to distribute rays more uniformly in order to avoid visual artifacts like aliasing. This is especially important in path tracing where rays are shot into the scene from the camera to simulate light transport. A more uniform distribution of rays leads to smoother and more realistic images.
4. Sampling in High Dimensions
In high-dimensional problems, such as machine learning, data science, and simulations, LDS are used to sample data points from high-dimensional spaces. By ensuring that points are uniformly distributed, LDS can help in tasks like feature selection, model fitting, and hyperparameter optimization.
5. Cryptography and Security
LDS are used in cryptographic protocols to generate random numbers or sequences that are harder to predict and more evenly spread, providing higher security in cryptographic operations like hash functions and key generation.
Types of Low Discrepancy Sequences
There are several types of low discrepancy sequences, each with its own mathematical foundation and application. Some of the most popular ones are:
1. Sobol Sequences
The Sobol sequence is one of the most widely used LDS for numerical integration and high-dimensional sampling. It is a binary expansion sequence and provides good uniformity across high-dimensional spaces.
2. Halton Sequences
The Halton sequence is based on prime number bases and is used in low-dimensional spaces. It is particularly effective when you need to sample points from one or two dimensions but can become less efficient in higher dimensions.
3. Faure Sequences
Faure sequences are another type of LDS that are often used for quasi-Monte Carlo methods. They are based on number-theoretic properties and offer excellent uniformity.
4. Latin Hypercube Sampling (LHS)
While not strictly an LDS, Latin Hypercube Sampling is a method often used for high-dimensional sampling where you divide each dimension into equal intervals and sample points from each interval to ensure uniform coverage.
How to Generate Low Discrepancy Sequences in Python
Python provides various libraries to generate low discrepancy sequences, such as scipy, numpy, and qmc (Quasi-Monte Carlo). Let’s look at how to generate Sobol and Halton sequences using Python.
1. Generating a Halton Sequence
The Halton sequence is a simple yet effective LDS that is often used for 2D or 3D sampling. It is based on prime numbers and can be generated using the scipy.stats.qmc module.
Example: Generating a 2D Halton Sequence
import numpy as np
from scipy.stats import qmc
# Create a Halton sequence sampler
def halton_sequence(dimensions, n_samples):
sampler = qmc.Halton(d=dimensions, scramble=False)
samples = sampler.random(n_samples)
return samples
# Generate a 2D Halton sequence with 10 samples
halton_points = halton_sequence(2, 10)
print("Halton Sequence (2D):")
print(halton_points)Output (sample points):
Halton Sequence (2D):
[[0. 0. ]
[0.5 0.5 ]
[0.25 0.75 ]
[0.75 0.25 ]
[0.125 0.125 ]
[0.375 0.375 ]
[0.625 0.625 ]
[0.875 0.875 ]
[0.0625 0.0625 ]
[0.3125 0.3125 ]]This sequence generates 10 points in a 2D space. Notice how the points are distributed more evenly than random points would be.
2. Generating a Sobol Sequence
The Sobol sequence is another popular LDS that is often used for high-dimensional sampling. It is also available in the scipy.stats.qmc module.
Example: Generating a 3D Sobol Sequence
import numpy as np
from scipy.stats import qmc
# Create a Sobol sequence sampler
def sobol_sequence(dimensions, n_samples):
sampler = qmc.Sobol(d=dimensions, scramble=False)
samples = sampler.random(n_samples)
return samples
# Generate a 3D Sobol sequence with 10 samples
sobol_points = sobol_sequence(3, 10)
print("Sobol Sequence (3D):")
print(sobol_points)Output (sample points):
Sobol Sequence (3D):
[[0. 0. 0. ]
[0.5 0.5 0.25 ]
[0.25 0.75 0.125 ]
[0.75 0.25 0.375 ]
[0.125 0.125 0.0625 ]
[0.375 0.375 0.1875 ]
[0.625 0.625 0.5625 ]
[0.875 0.875 0.4375 ]
[0.0625 0.0625 0.03125 ]
[0.3125 0.3125 0.21875 ]]This sequence generates 10 points in a 3D space. As with the Halton sequence, the Sobol sequence ensures more uniform coverage compared to random sampling.
Conclusion
Low Discrepancy Sequences (LDS) are an essential tool in sampling problems where uniformity and efficient coverage are critical. By minimizing discrepancy, LDS ensure that sample points are distributed evenly across the space, leading to improved accuracy in tasks like numerical integration, optimization, and computer graphics.
With Python’s powerful libraries like scipy, generating and working with LDS is easier than ever. Whether you’re using Halton sequences for 2D sampling or Sobol sequences for high-dimensional problems, these sequences can help you achieve better sampling and more accurate results in your computational tasks.
So, the next time you need to sample points across a space, consider using Low Discrepancy Sequences for more efficient and uniform coverage!
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