Neighbourhood relation preservation (NRP) A rank-based data visualisation quality assessment criterion Jigang Sun PhD studies finished in July 2011 PhD.

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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Problem: loose constraints

Rank based quality measures|},:{}:{|),( jkDDkDDkjiR ijikijikdata

|},:{}:{|),( jkLLkLLkjiR ijikijikoutput

• 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?