Understanding the Difference: sklearn.manifold.MDS vs. skbio.pcoa
Introduction
Both sklearn.manifold.MDS and skbio.pcoa are dimensionality reduction techniques used for visualizing high-dimensional data in a lower-dimensional space. However, they differ in their core implementation, leading to different outcomes. In particular, sklearn.manifold.MDS can produce seemingly random results, while skbio.pcoa consistently returns stable visualizations. This article delves into the reasons behind this discrepancy.
The Core of the Issue: Random Initialization
The fundamental difference lies in the way these methods handle initialization. sklearn.manifold.MDS employs a random initialization strategy for its iterative optimization algorithm. This means the initial configuration of the points in the low-dimensional space is chosen randomly.
The Impact of Random Initialization
- Potential for Local Minima: The random initialization can lead
sklearn.manifold.MDSto converge to different local minima of the stress function, which measures the discrepancy between the original distances and the distances in the low-dimensional space. This results in seemingly random outputs for different runs. - Lack of Reproducibility: Without a fixed starting point, the results of
sklearn.manifold.MDSare not reproducible. Running the same code multiple times can produce different visualizations.
skbio.pcoa: Deterministic and Stable
In contrast, skbio.pcoa utilizes a deterministic approach. It performs Principal Coordinates Analysis (PCoA), which is a specific method that guarantees a consistent output based on the input data. PCoA is directly linked to the eigenvalues and eigenvectors of the distance matrix, ensuring stability and reproducibility.
Illustrative Example
Let’s consider an example using a sample distance matrix:
| A | B | C | D | |
|---|---|---|---|---|
| A | 0 | 1 | 2 | 3 |
| B | 1 | 0 | 1 | 2 |
| C | 2 | 1 | 0 | 1 |
| D | 3 | 2 | 1 | 0 |
Running sklearn.manifold.MDS with the same distance matrix multiple times might produce different layouts, while skbio.pcoa consistently generates the same visualization.
Code Example
sklearn.manifold.MDS
from sklearn.manifold import MDS from sklearn.metrics.pairwise import euclidean_distances import numpy as np # Sample distance matrix dist_matrix = np.array([[0, 1, 2, 3], [1, 0, 1, 2], [2, 1, 0, 1], [3, 2, 1, 0]]) # MDS with random initialization mds = MDS(n_components=2, random_state=None) mds.fit(dist_matrix) print(mds.embedding_) # Repeat the process to see different results mds = MDS(n_components=2, random_state=None) mds.fit(dist_matrix) print(mds.embedding_)
skbio.pcoa
from skbio.stats.ordination import pcoa # Perform PCoA pcoa_results = pcoa(dist_matrix) print(pcoa_results.samples.values)
Conclusion
While both sklearn.manifold.MDS and skbio.pcoa aim to reduce dimensionality, their core implementations lead to different behaviors. sklearn.manifold.MDS‘s random initialization results in potentially varying outputs, while skbio.pcoa‘s deterministic PCoA guarantees consistent and reproducible visualizations. The choice between these methods depends on the specific requirements of your analysis, particularly the need for reproducibility and the understanding of the underlying assumptions and limitations.