Data Visualization With Multidimensional Scaling

Buja, Andreas; Swayne, Deborah F.; Littman, Michael L.; Dean, Nathaniel; Hofmann, Heike; Chen, Lisha

doi:10.1198/106186008x318440

Cited by 293 publications

(186 citation statements)

References 25 publications

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“…Calculation of the effective dimensionality of a dynamical system, and identification of order parameters describing the low-dimensional "intrinsic manifold" to which the system dynamics are effectively restrained, is a long-standing problem in as seemingly disparate fields as data visualization (10), speech recognition (11), semisupervised learning (12), and spectral clustering (13). The fraction of native contacts (Q) (8,14) and the folding probability (P fold ) (8,15) have been used as reaction coordinates for protein folding, but such coarse variables may lump together structurally and kinetically disparate conformations and can prove inadequate for larger proteins with frustrated folding funnels (5,8).…”

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confidence: 99%

Systematic determination of order parameters for chain dynamics using diffusion maps

Ferguson

Panagiotopoulos

Debenedetti

et al. 2010

Proc. Natl. Acad. Sci. U.S.A.

155

315

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We employ the diffusion map approach as a nonlinear dimensionality reduction technique to extract a dynamically relevant, low-dimensional description of n-alkane chains in the ideal-gas phase and in aqueous solution. In the case of C 8 we find the dynamics to be governed by torsional motions. For C 16 and C 24 we extract three global order parameters with which we characterize the fundamental dynamics, and determine that the low free-energy pathway of globular collapse proceeds by a "kink and slide" mechanism, whereby a bend near the end of the linear chain migrates toward the middle to form a hairpin and, ultimately, a coiled helix. The low-dimensional representation is subtly perturbed in the solvated phase relative to the ideal gas, and its geometric structure is conserved between C 16 and C 24 . The methodology is directly extensible to biomolecular self-assembly processes, such as protein folding.I t has long been suspected that cooperative couplings between degrees of freedom render the effective dimensionality of biophysical systems far less than the 3R-dimensional coordinate space of the R constituent atoms (1-5). This has been framed in the projection operator formalism (6) as a separation of time scales in which the important dynamics reside in a "slow subspace" (7) and is associated with a smooth underlying free energy surface (8). For example, two-dimensional descriptions have been formulated for dialanine (9) and a coarse-grained model of the src homology 3 domain (5).Calculation of the effective dimensionality of a dynamical system, and identification of order parameters describing the low-dimensional "intrinsic manifold" to which the system dynamics are effectively restrained, is a long-standing problem in as seemingly disparate fields as data visualization (10), speech recognition (11), semisupervised learning (12), and spectral clustering (13). The fraction of native contacts (Q) (8,14) and the folding probability (P fold ) (8, 15) have been used as reaction coordinates for protein folding, but such coarse variables may lump together structurally and kinetically disparate conformations and can prove inadequate for larger proteins with frustrated folding funnels (5, 8). Empirical order parameters also tend to perform poorly on landscapes exhibiting multiple local free-energy (FE) minima or lacking well-defined unfolded and folded basins. Principal components analysis (PCA) is a popular linear dimensionality reduction technique applied extensively to biophysical systems (1-4, 16) which seeks to describe the "essential subspace" (2) of the dynamics by a set of orthogonal vectors oriented along the directions of largest variance in the data. For the highly nonlinear intrinsic manifolds one expects for complex molecular systems (5), the linearity of this technique renders it appropriate in local regions, but results in a poor characterization of the global features (5, 17). This deficiency leads to poor PCA estimates of the effective dimensionality (17) far in excess of the dimensionality of the phas...

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mentioning

confidence: 99%

Systematic determination of order parameters for chain dynamics using diffusion maps

Ferguson

Panagiotopoulos

Debenedetti

et al. 2010

Proc. Natl. Acad. Sci. U.S.A.

155

315

View full text Add to dashboard Cite

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“…TPE is based on the framework of MDS (28). Given a real, symmetric, nonnegative, zero diagonal n × n dissimilarity matrix D for a set of n objects S ¼ f1;…;ng in a high-dimensional space, MDS finds a p-dimensional Euclidean embedding X ¼ fx 1 ;…;x n g ⊆ R p of the objects that minimizes a loss function such as stress σðXÞ ¼ ∑ starting with the n singleton clusters and ending with the trivial cluster.…”

Section: Methodsmentioning

confidence: 99%

Tree preserving embedding

Shieh

Hashimoto

Airoldi

2011

Proc. Natl. Acad. Sci. U.S.A.

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The goal of dimensionality reduction is to embed high-dimensional data in a low-dimensional space while preserving structure in the data relevant to exploratory data analysis such as clusters. However, existing dimensionality reduction methods often either fail to separate clusters due to the crowding problem or can only separate clusters at a single resolution. We develop a new approach to dimensionality reduction: tree preserving embedding. Our approach uses the topological notion of connectedness to separate clusters at all resolutions. We provide a formal guarantee of cluster separation for our approach that holds for finite samples. Our approach requires no parameters and can handle general types of data, making it easy to use in practice and suggesting new strategies for robust data visualization.hierarchical clustering | multidimensional scaling V isualization is an important first step in the analysis of highdimensional data (1). High-dimensional data often has low intrinsic dimensionality, making it possible to embed the data in a low-dimensional space while preserving much of its structure (2). However, it is rarely possible to preserve all types of structure in the embedding. Therefore, dimensionality reduction methods can only aim to preserve particular types of structure. Linear methods such as principal component an alysis (PCA) (3) and classical multidimensional scaling (MDS) (4-6) preserve global distances, while nonlinear methods such as manifold learning methods (7-9) preserve local distances defined by kernels or neighborhood graphs. However, most dimensionality reduction methods fail to preserve clusters (10), which are often of greatest interest.Clusters are difficult to preserve in embeddings due to the so-called crowding problem (11). When the intrinsic dimensionality of the data exceeds the embedding dimensionality, there is not enough space in the embedding to allow clusters to separate. Therefore, clusters are forced to collapse on top of each other in the embedding. As the embedding dimensionality increases, there is more space in the embedding for clusters to separate and the crowding problem disappears, making it possible to preserve clusters exactly (12). However, because the embedding dimensionality is at most two or three for visualization purposes, the crowding problem is prevalent in practice. When the clusters are known, they can be used to guide the embedding to avoid the crowding problem (13). However, the embedding is often used to help find the clusters in the first place. Therefore, it is important to solve the crowding problem without knowledge of the clusters.Force-based methods such as stochastic neighbor embedding (SNE) (14), variants of SNE (10,11,15,16), and local MDS (17) have been proposed to overcome the crowding problem. Force-based methods use attractive forces to pull together similar points and repulsive forces to push apart dissimilar points. SNE and its variants use forces based on kernels, while local MDS uses forces based on neighborhood graphs. Force-base...

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“…The well-established statistical technique of multidimensional scaling [38,39] offers the required visualization capability [15,40].…”

Section: A Multidimensional Scalingmentioning

confidence: 99%