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

70

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.

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[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.