Addendum for “The TBM global distance measure for the association of uncertain combat ID...

Information Fusion 18 (2014) 197–198

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Information Fusion

journal homepage: www.elsevier .com/locate / inf fus

Addendum for ‘‘The TBM global distance measure for the associationof uncertain combat ID declarations’’

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TBM−distJ−distE−distT−distB−dist

Fig. 1. Multi-object classification performance using non-specific CID dec(over 1000 Monte Carlo runs).

http://dx.doi.org/10.1016/j.inffus.2013.10.009

DOI of original article: http://dx.doi.org/10.1016/j.inffus.2005.04.004⇑ Corresponding author. Tel.: +61 3 9626 8370.

E-mail addresses: Branko.Ristic@dsto.defence.gov.au (B. Ristic), mihai.florea@-ca.thalesgroup.com (M.C. Florea), ebosse@gel.ulaval.ca (É. Bossé).

Branko Ristic a,⇑, Mihai Cristian Florea b, Éloi Bossé c

a Land Division, Defence Science and Technology Organisation, 506 Lorimer Street, Melbourne VIC 3207, Australiab Thales Research & Technology (TRT) Canada, 1405 Boul. du Parc Technologique, Quebec, G1P 4P5, Canadac Computer sciences and software engineering department, 1065, av. de la Médecine Université Laval Québec (QC), G1V 0A6, Canada

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larations

Several authors, e.g. [2–6], have blindly re-used (without scru-tiny) the expressions for distance measures listed in [1] in the con-text of Dempster-Shafer theory. Unfortunately, [1] contains twotypographical errors: the first in Eq. (34), the second in Eq. (36),as originally noted in [7]. The purpose of this addendum is to pro-vide the clarification on the correct formulations in order to stoppropagating those errors in future related publications.

Eq. (34) should be expressed as:

hmi;mji ¼ 2XA # H

miðAÞ �mjðAÞ ð1Þ

Upon substitution of Eq. (1) into Eq. (32) of [1], the correct expres-sion for the Euclidean distance readily follows as:

dij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðhmi;mii þ hmj;mji � 2hm;mjiÞ

r

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

A # Hm2

i ðAÞ þX

A # Hm2

j ðAÞ � 2X

A # HmiðAÞ mjðAÞ

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

A # Hm2

i ðAÞ � 2miðAÞmjðAÞ þm2j ðAÞ

h ir

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

A # H½miðAÞ �mjðAÞ�2

qð2Þ

The correct expression for Bhattacharya distance, introduced byEq. (36) of [1], should state:

dij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

XA # H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimiðAÞ �mjðAÞ

qrð3Þ

In simulations presented in Section 4 of [1], however, the cor-rect expressions, i.e. (2) for Euclidian distance and (3) for Bhattach-arya distance, have been implemented. Hence we emphasize thatthe results shown in Figs. 3 and 5 of [1] are valid, as well as theconclusions of the study drawn in Section 5.

The Tessem’s distance was defined in Eq. (35) of [1] on single-ton, i.e

dij ¼maxh2HjBetPiðhÞ � BetPjðhÞj

In order to measure the effect of computing the Tessem distanceas the maximum over all subsets A from H, i.e.

dij ¼maxA # HjBetPiðAÞ � BetPjðAÞj

we have re-run the simulations described in Section 4 of [1] fornon-specific CID declarations (results shown in Fig. 3 of [1]). Thenew results are plotted in Fig. 1. For completeness we have includedall five distance measures (only Tessem’s distance is implementeddifferently, as the maximum over all subsets A from H). ComparingFig. 1 with Fig. 3 of [1] we conclude that there is almost no differ-ence in performance.

In summary, the TMB based dissimilarity measure is indeedsuperior to the other four considered distance measures, in thecontext of association of uncertain combat ID declarations.

References

[1] B. Ristic, P. Smets, The TBM global distance measure for the association ofuncertain combat ID declaration, Information Fusion 7 (2006) 276–284.

[2] V. Khatibi, G. Montazer, A new evidential distance measure based on beliefintervals, Scientia Iranica 17 (2D) (2010) 119–132.

198 B. Ristic et al. / Information Fusion 18 (2014) 197–198

[3] C. Shi, Y. Cheng, Q. Pan, Y. Lu, A new method to determine evidence distance, in:2010 International Conference on Computational Intelligence and SoftwareEngineering, CiSE 2010, 2010.

[4] X. Li, J. Dezert, F. Smarandache, X. Huang, Evidence supporting measure ofsimilarity for reducing the complexity in information fusion, InformationSciences 181 (10) (2011) 1818–1835.

[5] Z. Liu, J. Dezert, Q. Pan, G. Mercier, Combination of sources of evidence withdifferent discounting factors based on a new dissimilarity measure, DecisionSupport Systems 52 (1) (2011) 133–141.

[6] A. Samet, E. Lefevre, S.B. Yahia, Reliability estimation with extrinsic and intrinsicmeasure in belief function theory, in: 2013 5th International Conference onModeling, Simulation and Applied Optimization, ICMSAO 2013, 2013.

[7] M.C. Florea, É. Bossé, Crisis management using dempster-shafer theory: usingdissimilarity measures to characterize sources’ reliability, in: NATO RTO 086 –C3I in Crisis, Emergency and Consequence Management, Bucharest, Romania,11–12 May, 2009.