Multidimensional Scaling, Second Edition

Cox, Trevor F.; Cox, Michael

doi:10.1201/9781420036121

Cited by 936 publications

(714 citation statements)

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“…We study two important scenarios: (i) average-case landscape of the variety of clustering algorithms over a number of realworld data sets, and (ii) a landscape over artificial data sets generated by mixtures of Gaussian components. In both cases multidimensional scaling [14] is employed to visualize the landscape. In the case of controlled artificial data sets, we also obtain a dynamic trace of the changes in the landscape caused by varying the density and isolation of clusters.…”

Section: Introductionmentioning

confidence: 99%

Landscape of clustering algorithms

Jain

Topchy

Law

et al. 2004

Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004.

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Section: Introductionmentioning

confidence: 99%

Landscape of clustering algorithms

Jain

Topchy

Law

et al. 2004

Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004.

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“…Combinations of these measures are also of interest, for example a 11 + a 12 + a 21 = |X ∪ Y|. Based on these four counts, a 11 , a 22 , a 12 , a 21 , one can derive a number of different similarity coefficients as shown in table 1, (Cox and Cox, 2000).…”

Section: Similarity Measuresmentioning

confidence: 99%

“…We refer the reader to the books by Borg and Groenen (1997) and Cox and Cox (2000) for a thorough treatment of the subject. Formally, let Δ = {δ ij } be an r × r matrix representing the pair-wise dissimilarity (or distance) between r points.…”

Section: Multidimensional Scalingmentioning

confidence: 99%

“…Three methods form the basis of this framework: Williams' Index a technique we recently introduced (MartinFernandez et al, 2005); STAPLE's algorithm ; and a novel visualization based on Multidimensional Scaling (MDS), a statistical tool to explore (dis)similarity data (Borg and Groenen, 1997;Cox and Cox, 2000). Second, we discuss the validity of our results by comparing our framework (purely based on common agreement) with an evaluation against a set of manual segmentations (used as ground truth).…”

Section: Introductionmentioning

confidence: 99%

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On evaluating brain tissue classifiers without a ground truth

et al. 2007

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In this paper, we present a set of techniques for the evaluation of brain tissue classifiers on a large data set of MR images of the head. Due to the difficulty of establishing a gold standard for this type of data, we focus our attention on methods which do not require a ground truth, but instead rely on a common agreement principle. Three different techniques are presented: the Williams' index, a measure of common agreement; STAPLE, an Expectation Maximization algorithm which simultaneously estimates performance parameters and constructs an estimated reference standard; and Multidimensional Scaling, a visualization technique to explore similarity data. We apply these different evaluation methodologies to a set eleven different segmentation algorithms on forty MR images. We then validate our evaluation pipeline by building a ground truth based on human expert tracings. The evaluations with and without a ground truth are compared. Our findings show that comparing classifiers without a gold standard can provide a lot of interesting information. In particular, outliers can be easily detected, strongly consistent or highly variable techniques can be readily discriminated, and the overall similarity between different techniques can be assessed. On the other hand, we also find that some information present in the expert segmentations is not captured by the automatic classifiers, suggesting that common agreement alone may not be sufficient for a precise performance evaluation of brain tissue classifiers.

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“…The classical example is Principal Component Analysis (PCA) [1] but in recent years methods based on manifold learning such as Multidimensional Scaling [2], Local Linear Embedding [3], Isomap [4], Laplacian Eigenmaps [5] or Hessian Eigenmaps [6] have received a great deal of attention. The common assumption in these methods is that sample data lie in a low dimensional manifold and their goal is to identify the metric on the underlying manifolds from which a suitable low dimensional representation is derived that allows to adequately approximate the original manifold metric with the natural one in the low dimensional representation.…”

Section: Introductionmentioning

confidence: 99%

Diffusion Maps for dimensionality reduction and visualization of meteorological data

et al. 2015

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Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in:Neurocomputing 163 (2015) AbstractThe growing interest in big data problems implies the need for unsupervised methods for data visualization and dimensionality reduction. Diffusion Maps (DM) is a recent technique that can capture the lower dimensional geometric structure underlying the sample patterns in a way which can be made to be independent of the sampling distribution. Moreover, DM allows to define an embedding whose Euclidean metric relates to the sample's intrinsic one which, in turn, enables a principled application of k-means clustering. In this work we give a self-contained review of DM and discuss two methods to compute the DM embedding coordinates to new out-of-sample data. Then, we will apply them on two meteorological data problems that involve respectively time and spatial compression of numerical weather forecasts and show how DM is capable to, first, greatly reduce the initial dimension while still capturing relevant information in the original data and, also, how the sample-derived DM embedding coordinates can be extended to new patterns.

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Multidimensional Scaling, Second Edition

Cited by 936 publications

References 0 publications

Landscape of clustering algorithms

Landscape of clustering algorithms

On evaluating brain tissue classifiers without a ground truth

Diffusion Maps for dimensionality reduction and visualization of meteorological data

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