Neighbourhood relation preservation (NRP)
A rank-based data visualisation quality assessment criterion
Jigang SunPhD studies finished in July 2011
PhD Supervisor: Colin Fyfe, Malcolm CroweUniversity of the West of Scotland
outline
• Multidimensional Scaling (MDS);• The need for a common quality measure for data
visualisation;• Local continuity meta-criterion (LCMC);• Definition of neighbourhood relation preservation (NRP);• Illustration of LCMC and NRP on mappings of data sets
created by different methods;
Multidimensional Scaling (MDS)
A group of information visualisation methods that projects data points from high dimensional data space to low, typically two dimensional, latent space in which the structure of the original data set can be identified by eye.
For example…
By LeftSammon, using graph distances, k=20
Samples of high dimensional data(each image is 28*28=784 dimensions)
2 dimensional projection
Various methods• The classical MDS, the stress function to be minimised is defined to be
spacelatent in and i points databetween distanceEuclidean the
space datain and i points databetween distanceEuclidean the
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Various methods
• Each method has its own criterion.
Mappings of open box
By Sammon’s mapping By LeftSammon mapping
Mappings can be assessed by eye
By CMDS
By LeftExp By RightExp
By Isomap
Mappings of open box
by Sammon's mapping by LeftSammon mapping
Sammon vs LeftSammon mapping
• Assessing mapping quality by eye is usually difficult
Local continuity meta-criterion (LCMC)
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Rank based quality measures|},:{}:{|),( jkDDkDDkjiR ijikijikdata
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• Traditional rank is used in trustworthiness and continuity (T&C )
• Problem 1: change of intermediate points is not considered
• p is mapped perfectly since rank of p does not change
• Rank is discrete; distance is continuous
Rank based quality measures
Problem 2: angle constraint is not considered
Neighbourhood relation preservation (NRP)
• Given and the difference in angle piq in data space and output space is less than ), we say that a neighbourhood relation of p over q with respect to i, , is preserved. We denote this as )=1; otherwise )=0;
• Φ(i,k)=, t=1.3• NRP(k)=1/N
Assessment to mappings of open box
Mappings of MNIST digits
By CMDS
By Isomap
By LeftExpBy RightExp
Assessment of mappings of digits
Conclusions• Multidimensional Scaling (MDS);• List of objective function of some MDS methods;• The need for a common quality measure for data visualisation;• Local continuity meta-criterion (LCMC);• Definition of Neighbourhood relation preservation (NRP);• Comparison of LCMC and NRP on mappings of data sets created by
different methods;
Thank you! Any questions?
