Scipy's Multivariate Earth Mover Distance Not Working As Intended?

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Introduction

The multivariate Earth Mover's Distance (EMD) is a measure of the distance between two multivariate probability distributions. It is a generalization of the traditional Earth Mover's Distance, which is used to measure the distance between two univariate probability distributions. The multivariate EMD is a powerful tool for comparing and analyzing multivariate data, and it has many applications in fields such as statistics, machine learning, and data science.

Background

The multivariate EMD is based on the concept of optimal transport, which is a mathematical framework for measuring the distance between two probability distributions. The idea is to find the optimal way to transport one distribution to another, while minimizing the total cost of transportation. In the case of the multivariate EMD, the cost of transportation is measured by the distance between the corresponding points in the two distributions.

The Issue

You are using the wasserstein_distance_nd function from the SciPy library to compute the multivariate EMD between two Gaussian multivariate samples. However, you have noticed that the results are not as expected. Specifically, the EMD is not zero when the two samples are drawn from the same distribution.

Sanity Check

To verify that the issue is not with the wasserstein_distance_nd function, you have performed a quick sanity check. You have drawn two Gaussian multivariate samples from the same distribution, and then computed the EMD between the two samples using the wasserstein_distance_nd function. However, the results are not zero, which is unexpected.

Possible Causes

There are several possible causes for this issue:

  • Numerical instability: The wasserstein_distance_nd function uses numerical methods to compute the EMD, which can be prone to numerical instability. This can lead to small errors in the computation, which can accumulate and cause the EMD to be non-zero even when the two samples are drawn from the same distribution.
  • Sampling variability: When drawing samples from a multivariate distribution, there is always some degree of sampling variability. This can cause the two samples to be slightly different, even if they are drawn from the same distribution.
  • Implementation issues: There may be implementation issues with the wasserstein_distance_nd function that are causing the EMD to be non-zero even when the two samples are drawn from the same distribution.

Troubleshooting

To troubleshoot this issue, you can try the following:

  • Check the documentation: Make sure that you are using the wasserstein_distance_nd function correctly, and that you are passing the correct arguments to the function.
  • Verify the input data: Make sure that the input data is correct, and that the two samples are drawn from the same distribution.
  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.

Conclusion

The multivariate Earth Mover's Distance is a powerful tool for comparing and analyzing multivariate data. However, it can be prone to numerical instability and sampling variability, which can cause the EMD to be non-zero even when the two samples are drawn from the same distribution. By troubleshooting the issue and using a different implementation, you can ensure that the EMD is computed correctly.

Code

Here is an example of how to use the wasserstein_distance_nd function to compute the multivariate EMD between two Gaussian multivariate samples:

import numpy as np
from scipy.stats import multivariate_normal
from scipy.spatial.distance import wasserstein_distance_nd

mean = np.array([0, 0])
cov = np.array([[1, 0], [0, 1]])



n_samples = 1000



sample1 = multivariate_normal.rvs(mean, cov, n_samples)
sample2 = multivariate_normal.rvs(mean, cov, n_samples)



emd = wasserstein_distance_nd(sample1, sample2)


print(emd)

Discussion

This code computes the multivariate EMD between two Gaussian multivariate samples using the wasserstein_distance_nd function. However, the results are not as expected, and the EMD is not zero even when the two samples are drawn from the same distribution.

Possible Solutions

There are several possible solutions to this issue:

  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.
  • Use a different algorithm: Try using a different algorithm for computing the multivariate EMD, such as the sinkhorn algorithm.

Conclusion

Q&A

Q: What is the multivariate Earth Mover's Distance?

A: The multivariate Earth Mover's Distance (EMD) is a measure of the distance between two multivariate probability distributions. It is a generalization of the traditional Earth Mover's Distance, which is used to measure the distance between two univariate probability distributions.

Q: What is the purpose of the multivariate EMD?

A: The multivariate EMD is used to compare and analyze multivariate data. It can be used to measure the similarity or dissimilarity between two multivariate distributions, and it can be used as a metric for clustering, classification, and other machine learning tasks.

Q: What are the possible causes of the issue with the multivariate EMD?

A: There are several possible causes of the issue with the multivariate EMD, including:

  • Numerical instability: The wasserstein_distance_nd function uses numerical methods to compute the EMD, which can be prone to numerical instability. This can lead to small errors in the computation, which can accumulate and cause the EMD to be non-zero even when the two samples are drawn from the same distribution.
  • Sampling variability: When drawing samples from a multivariate distribution, there is always some degree of sampling variability. This can cause the two samples to be slightly different, even if they are drawn from the same distribution.
  • Implementation issues: There may be implementation issues with the wasserstein_distance_nd function that are causing the EMD to be non-zero even when the two samples are drawn from the same distribution.

Q: How can I troubleshoot the issue with the multivariate EMD?

A: To troubleshoot the issue with the multivariate EMD, you can try the following:

  • Check the documentation: Make sure that you are using the wasserstein_distance_nd function correctly, and that you are passing the correct arguments to the function.
  • Verify the input data: Make sure that the input data is correct, and that the two samples are drawn from the same distribution.
  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.

Q: What are some possible solutions to the issue with the multivariate EMD?

A: There are several possible solutions to the issue with the multivariate EMD, including:

  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.
  • Use a different algorithm: Try using a different algorithm for computing the multivariate EMD, such as the sinkhorn algorithm.

Q: How can I ensure that the multivariate EMD is computed correctly?

A: To ensure that the multivariate EMD is computed correctly, you can try the following:

  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.
  • Use a different algorithm: Try using a different algorithm for computing the multivariate EMD, such as the sinkhorn algorithm.

Q: What are some common pitfalls to avoid when using the multivariate EMD?

A: There are several common pitfalls to avoid when using the multivariate EMD, including:

  • Numerical instability: The wasserstein_distance_nd function uses numerical methods to compute the EMD, which can be prone to numerical instability. This can lead to small errors in the computation, which can accumulate and cause the EMD to be non-zero even when the two samples are drawn from the same distribution.
  • Sampling variability: When drawing samples from a multivariate distribution, there is always some degree of sampling variability. This can cause the two samples to be slightly different, even if they are drawn from the same distribution.
  • Implementation issues: There may be implementation issues with the wasserstein_distance_nd function that are causing the EMD to be non-zero even when the two samples are drawn from the same distribution.

Conclusion

The multivariate Earth Mover's Distance is a powerful tool for comparing and analyzing multivariate data. However, it can be prone to numerical instability and sampling variability, which can cause the EMD to be non-zero even when the two samples are drawn from the same distribution. By troubleshooting the issue and using a different implementation, you can ensure that the EMD is computed correctly.

Code

Here is an example of how to use the wasserstein_distance_nd function to compute the multivariate EMD between two Gaussian multivariate samples:

import numpy as np
from scipy.stats import multivariate_normal
from scipy.spatial.distance import wasserstein_distance_nd


mean = np.array([0, 0])
cov = np.array([[1, 0], [0, 1]])



n_samples = 1000



sample1 = multivariate_normal.rvs(mean, cov, n_samples)
sample2 = multivariate_normal.rvs(mean, cov, n_samples)



emd = wasserstein_distance_nd(sample1, sample2)


print(emd)

Discussion

This code computes the multivariate EMD between two Gaussian multivariate samples using the wasserstein_distance_nd function. However, the results are not as expected, and the EMD is not zero even when the two samples are drawn from the same distribution.

Possible Solutions

There are several possible solutions to this issue:

  • Use a different implementation: Try using a different implementation of the multivariate EMD, such as the emd function from the scipy.spatial.distance module.
  • Increase the precision: Try increasing the precision of the numerical computation by using a higher-precision arithmetic library, such as the mpmath library.
  • Use a different algorithm: Try using a different algorithm for computing the multivariate EMD, such as the sinkhorn algorithm.