{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:11:43Z","timestamp":1760213503157,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2016,10,2]],"date-time":"2016-10-02T00:00:00Z","timestamp":1475366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007660","name":"University of Antwerp","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007660","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Localization systems are increasingly valuable, but their location estimates are only useful when the uncertainty of the estimate is known. This uncertainty is currently calculated as the location error given a ground truth, which is then used as a static measure in sometimes very different environments. In contrast, we propose the use of the conditional entropy of a posterior probability distribution as a complementary measure of uncertainty. This measure has the advantage of being dynamic, i.e., it can be calculated during localization based on individual sensor measurements, does not require a ground truth, and can be applied to discrete localization algorithms. Furthermore, for every consistent location estimation algorithm, both the location error and the conditional entropy measures must be related, i.e., a low entropy should always correspond with a small location error, while a high entropy can correspond with either a small or large location error. We validate this relationship experimentally by calculating both measures of uncertainty in three publicly available datasets using probabilistic Wi-Fi fingerprinting with eight different implementations of the sensor model. We show that the discrepancy between these measures, i.e., many location estimates having a high location error while simultaneously having a low conditional entropy, is largest for the least realistic implementations of the probabilistic sensor model. Based on the results presented in this paper, we conclude that conditional entropy, being dynamic, complementary to location error, and applicable to both continuous and discrete localization, provides an important extra means of characterizing a localization method.<\/jats:p>","DOI":"10.3390\/s16101636","type":"journal-article","created":{"date-parts":[[2016,10,3]],"date-time":"2016-10-03T10:17:01Z","timestamp":1475489821000},"page":"1636","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Conditional Entropy and Location Error in Indoor Localization Using Probabilistic Wi-Fi Fingerprinting"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0064-5020","authenticated-orcid":false,"given":"Rafael","family":"Berkvens","sequence":"first","affiliation":[{"name":"iMinds, MOSAIC, University of Antwerp, Faculty of Applied Engineering, Antwerp 2020, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Herbert","family":"Peremans","sequence":"additional","affiliation":[{"name":"Engineering Management, University of Antwerp, Faculty of Applied Economics, Antwerp 2000, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1152-6617","authenticated-orcid":false,"given":"Maarten","family":"Weyn","sequence":"additional","affiliation":[{"name":"iMinds, MOSAIC, University of Antwerp, Faculty of Applied Engineering, Antwerp 2020, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,10,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.compeleceng.2015.01.019","article-title":"Raspberry Pi as a Sensor Web node for home automation","volume":"44","year":"2015","journal-title":"Comput. Electri. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3058","DOI":"10.1109\/ACCESS.2015.2508648","article-title":"Localization challenges for the emergence of the smart world","volume":"3","author":"Pahlavan","year":"2015","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1007\/s11390-015-1597-z","article-title":"Infrastructure-free floor localization through crowdsourcing","volume":"30","author":"Ye","year":"2015","journal-title":"J. Comput. Sci. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1016\/j.adhoc.2013.10.007","article-title":"Location error estimation in wireless ad hoc networks","volume":"13","author":"Gribben","year":"2014","journal-title":"Ad Hoc Netw."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Berkvens, R., Weyn, M., and Peremans, H. (2015, January 1\u20134). Asynchronous, electromagnetic sensor fusion in RatSLAM. Proceedings of the IEEE Sensors 2015, Busan, Korea.","DOI":"10.1109\/ICSENS.2015.7370552"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Inaba, M., and Corke, P. (2016). Robotics Research: The 16th International Symposium ISRR, Springer.","DOI":"10.1007\/978-3-319-28872-7"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Steckel, J., and Peremans, H. (2013). BatSLAM: Simultaneous localization and mapping using biomimetic sonar. PloS ONE, 8.","DOI":"10.1371\/journal.pone.0054076"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Berkvens, R., Weyn, M., and Peremans, H. (2015, January 13\u201316). Localization performance quantification by conditional entropy. Proceedings of the 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Banff, AB, Canada.","DOI":"10.1109\/IPIN.2015.7346969"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1109\/MCOM.2015.7263356","article-title":"Platform for benchmarking of RF-based indoor localization solutions","volume":"53","author":"Lemic","year":"2015","journal-title":"IEEE Commun. Mag."},{"key":"ref_10","unstructured":"Torres-Sospedra, J., Mendoza, G., Montoliu, R., and Rambla, D. IndoorLoc Database Repository. Available online: http:\/\/indoorloc.uji.es\/."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Torres-Sospedra, J., Montoliu, R., Mart\u0131nez-Us\u00f3, A., Avariento, J.P., Arnau, T.J., Benedito-Bordonau, M., and Huerta, J. (2014, January 27\u201330). UJIIndoorLoc: A new multi-building and multi-floor database for WLAN fingerprint-based indoor localization problems. Proceedings of the 2014 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Busan, Korea.","DOI":"10.1109\/IPIN.2014.7275492"},{"key":"ref_12","unstructured":"Kay, S.M. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1049\/ip-rsn:20000554","article-title":"GPS GDOP metric","volume":"147","author":"Yarlagadda","year":"2000","journal-title":"IEE Proc. Radar Sonar Navig."},{"key":"ref_14","unstructured":"MacKay, D.J.C. (2003). Information Theory, Inference, and Learning Algorithms, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1109\/COMST.2015.2464084","article-title":"Wi-Fi fingerprint-based indoor positioning: Recent advances and comparisons","volume":"18","author":"He","year":"2016","journal-title":"IEEE Commun. Surv. Tutorials"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Potorti, F., Barsocchi, P., Girolami, M., Torres-Sospedra, J., and Montoliu, R. (2015, January 13\u201316). Evaluating indoor localization solutions in large environments through competitive benchmarking: The EvAAL-ETRI competition. Proceedings of the 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Banff, AB, Canada.","DOI":"10.1109\/IPIN.2015.7346970"},{"key":"ref_17","unstructured":"Weyn, M. (2011). Opportunistic Seamless Localization. [Ph.D. Thesis, University of Antwerp]."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lemic, F., Behboodi, A., Handziski, V., and Wolisz, A. (2014, January 27\u201330). Experimental decomposition of the performance of fingerprinting-based localization algorithms. Proceedigns of the 2014 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Busan, Korea.","DOI":"10.1109\/IPIN.2014.7275503"},{"key":"ref_19","first-page":"300","article-title":"Comprehensive aurvey on distance\/similarity measures between probability Density Functions","volume":"1","author":"Cha","year":"2007","journal-title":"Int. J. Math. Models Methods Appl. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9263","DOI":"10.1016\/j.eswa.2015.08.013","article-title":"Comprehensive analysis of distance and similarity measures for Wi-Fi fingerprinting indoor positioning systems","volume":"42","author":"Montoliu","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Moreira, A., Nicolau, M.J., Meneses, F., and Costa, A. (2015, January 13\u201316). Wi-Fi fingerprinting in the real world\u2014RTLS@UM at the EvAAL competition. Proceedings of the 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN), Banff, AB, Canada.","DOI":"10.1109\/IPIN.2015.7346967"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.adhoc.2011.12.006","article-title":"Low-dimensional signal-strength fingerprint-based positioning in wireless LANs","volume":"12","author":"Milioris","year":"2014","journal-title":"Ad Hoc Netw."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1109\/TCOMM.2015.2442989","article-title":"A novel fused positioning feature for handling heterogeneous hardware Problem","volume":"63","author":"Fang","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.pmcj.2015.10.005","article-title":"Wi-Fi fingerprinting based on collaborative confidence level training","volume":"30","author":"Jing","year":"2015","journal-title":"Pervasive Mob. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Husen, M.N., and Lee, S. (2016, January 4\u20136). High performance indoor location Wi-Fi fingerprinting using invariant received signal strength. Proceedings of the 10th International Conference on Ubiquitous Information Management and Communication, Danang, Vietnam.","DOI":"10.1145\/2857546.2857589"},{"key":"ref_26","unstructured":"Papoulis, A., and Pillai, S.U. (2002). Probability, Random Variables and Stochastic Processes, McGraw-Hill. [4th ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1109\/SURV.2012.022412.00172","article-title":"A survey of wireless path loss prediction and coverage mapping methods","volume":"15","author":"Phillips","year":"2013","journal-title":"IEEE Commun. Surv. Tutorials"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liang, C.J.M., Priyantha, N.B., Liu, J., and Terzis, A. (2010, January 3\u20135). Surviving Wi-Fi interference in low power ZigBee networks. Proceedings of the 8th ACM Conference on Embedded Networked Sensor Systems, Zurich, Switzerland.","DOI":"10.1145\/1869983.1870014"},{"key":"ref_29","first-page":"60","article-title":"Interference-aware self-optimizing Wi-Fi for high efficiency internet of things in dense networks","volume":"89","author":"Lee","year":"2016","journal-title":"Comput. Commun."},{"key":"ref_30","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, John Wiley & Sons. [2nd ed.]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/16\/10\/1636\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:32:22Z","timestamp":1760211142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/16\/10\/1636"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,2]]},"references-count":30,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2016,10]]}},"alternative-id":["s16101636"],"URL":"https:\/\/doi.org\/10.3390\/s16101636","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2016,10,2]]}}}