dwMDS (distributed weighted multidimensional scaling)

Last updated on Mar 28, 2023

Distributed weighted multidimensional scaling (dwMDS) is a powerful data analysis technique that is widely used in many fields, including psychology, sociology, marketing, and ecology. dwMDS is a method for mapping objects or observations from a high-dimensional space into a lower-dimensional space, such as a 2D or 3D plot, while preserving the pairwise distances or similarities between the objects. dwMDS differs from traditional MDS in that it allows for the computation of the mapping on a distributed computing system, making it suitable for large-scale datasets.

In this article, we will explain the concept of dwMDS, how it works, its advantages and limitations, and how it can be applied in different fields.

The concept of MDS

MDS is a statistical technique that aims to represent the pairwise distances or similarities between objects in a lower-dimensional space, such as a 2D or 3D plot. MDS is based on the idea that objects that are similar to each other should be close to each other in the lower-dimensional space, while objects that are dissimilar should be far apart.

MDS can be used to visualize the structure of a dataset, identify clusters or groups of similar objects, or to identify the underlying dimensions that explain the similarity or dissimilarity between the objects.

There are two main types of MDS: classical MDS and non-metric MDS. Classical MDS assumes that the distances or similarities between the objects are interval-scaled and can be directly translated into Euclidean distances in the lower-dimensional space. Non-metric MDS, on the other hand, does not make any assumptions about the scaling of the distances or similarities and uses iterative algorithms to find a mapping that best approximates the distances or similarities.

The concept of dwMDS

dwMDS is a variant of non-metric MDS that is designed to work on large-scale datasets that cannot be analyzed using traditional MDS algorithms. Traditional MDS algorithms are often computationally intensive and can require a lot of memory and processing power, making them unsuitable for large datasets.

dwMDS is designed to overcome these limitations by distributing the computation across multiple computing nodes or processors. This allows the algorithm to scale to datasets that are too large to be analyzed on a single machine.

dwMDS works by dividing the dataset into smaller subsets, or "chunks", each of which is analyzed independently on a separate computing node. The results from each chunk are then combined to produce a final mapping of the dataset in the lower-dimensional space.

How dwMDS works

The basic steps of dwMDS are as follows:

Partition the dataset into smaller subsets or "chunks".
Compute the pairwise distances or similarities between the objects in each chunk.
Assign weights to the distances or similarities based on their reliability or importance.
Combine the results from each chunk to produce a final mapping of the dataset in the lower-dimensional space.

The first step in dwMDS is to partition the dataset into smaller subsets or "chunks". The size of the chunks depends on the available computing resources and the size of the dataset. Ideally, the chunks should be small enough to fit into the memory of a single computing node, but large enough to provide reliable estimates of the pairwise distances or similarities.

Once the dataset has been partitioned, the pairwise distances or similarities between the objects in each chunk are computed using a non-metric MDS algorithm, such as SMACOF (Scaling by Majorizing a Complicated Function). SMACOF is an iterative algorithm that tries to find a mapping of the objects in the lower-dimensional space that best approximates the pairwise distances or similarities.

The next step in dwMDS is to assign weights to the distances or similarities based on their reliability or importance. This is done using a technique called weighted MDS, which assigns higher weights to more reliable or important distances or similarities, and lower weights to less reliable or important ones. There are several ways to assign weights in weighted MDS, including:

Inverse distance weighting: Assign higher weights to distances or similarities between closer objects, and lower weights to distances or similarities between more distant objects.
Median polish: Assign weights based on the median value of the distances or similarities within each chunk.
Bootstrap aggregation: Assign weights based on the stability of the distances or similarities across multiple resampled datasets.

Once the weights have been assigned, the results from each chunk are combined to produce a final mapping of the dataset in the lower-dimensional space. This is done using a technique called consensus MDS, which combines the individual mappings from each chunk into a single consensus mapping. There are several ways to combine the individual mappings in consensus MDS, including:

Weighted averaging: Compute a weighted average of the individual mappings, with the weights determined by the reliability or importance of the individual distances or similarities.
Shepard-Kruskal scaling: Use an iterative algorithm to find a mapping that best approximates the individual mappings.

The final output of dwMDS is a set of coordinates for each object in the lower-dimensional space, which can be plotted and visualized.

Advantages of dwMDS

dwMDS has several advantages over traditional MDS:

Scalability: dwMDS can analyze large-scale datasets that cannot be analyzed using traditional MDS algorithms.
Distributed computing: dwMDS can use distributed computing systems to parallelize the computation and reduce the processing time.
Flexibility: dwMDS can be adapted to different types of data and different types of distances or similarities.
Robustness: dwMDS is less sensitive to outliers or noise in the data than traditional MDS algorithms.

Limitations of dwMDS

dwMDS also has some limitations:

Complexity: dwMDS requires a good understanding of distributed computing systems and parallel processing.
Data preprocessing: dwMDS requires careful data preprocessing to ensure that the distances or similarities are reliable and informative.
Parameter tuning: dwMDS requires tuning of several parameters, such as the size of the chunks and the weights assigned to the distances or similarities.
Interpretation: The interpretation of the lower-dimensional mapping can be subjective and depends on the context and the research question.

Applications of dwMDS

dwMDS can be applied in many fields, including:

Psychology: dwMDS can be used to analyze psychological data, such as personality traits or psychological disorders, and to identify clusters or groups of similar individuals.
Sociology: dwMDS can be used to analyze social network data and to visualize the structure of social relationships.
Marketing: dwMDS can be used to analyze consumer preferences and to identify segments of consumers with similar preferences.
Ecology: dwMDS can be used to analyze ecological data, such as species abundance or environmental variables, and to identify ecological patterns or gradients.

Conclusion

dwMDS is a powerful data analysis technique that can be used to map objects from a high-dimensional space into a lower-dimensional space while preserving the pairwise distances or similarities between the objects. dwMDS is designed to work on large-scale datasets and uses distributed computing systems to parallelize the computation and reduce the processing time. dwMDS has several advantages over traditional MDS, including scalability, distributed computing, flexibility, and robustness. However, dwMDS also has some limitations, including complexity, data preprocessing, parameter tuning, and interpretation. dwMDS can be applied in many fields, including psychology, sociology, marketing, and ecology, to analyze different types of data and to identify underlying patterns or structures.