WLAN fingerprinting based indoor positioning in the presence of censored and dropped data / Manh Kha Hoang ; Erster Gutachter: Prof. Dr.-Ing. Reinhold Häb-Umbach, zweiter Gutachter: Prof. Dr.-Ing. Peter A. Höher

WLAN Finge p in ing based Indoo Posi ioning in he P esence o Censo ed and D opped Da a Von de Fakul ä ü Elek o echnik, In o ma ik und Ma hema ik de Uni e si ä Pade bo n zu E langung des akademischen G ades Dok o de Ingenieu wissenscha en (D .-Ing.) genehmig e Disse a ion on M.Sc. Manh Kha Hoang E s e Gu ach e : P o . D .-Ing. Reinhold Häb-Umbach Zwei e Gu ach e : P o . D .-Ing. Pe e A. Höhe Tag de mündlichen P ü ung: 04. Mä z 2016 Pade bo n 2016 Diss. EIM-E/321 Acknowledgmen s Fi s o all, I would like o gi e a g ea hanks o P o . D .-Ing. Reinhold Häb-Umbach o being my esea ch supe iso . I would ha e ne e been able o comple e my esea ch wo k and my disse a ion wi hou his con inuous suppo , unde s anding and pa ience. He has kindly mo i a ed me o he new challenges and pa ien ly guided me o o e come he di i- cul ies. I eel e y lucky o ha e me P o . Häb-Umbach and wo ked wi h him. I would also like o hank P o . D .-Ing. Pe e A. Höhe who ha e e alua ed my disse a ion and gi en me he aluable esponses o imp o e i s quali y. Mo eo e , I would like o hank Vie namese go e nmen o g an ing he schola ship o my i s h ee yea s o s udying in Ge many. Wi hou i , I would ha e ne e had such a good chance o commence his PhD deg ee in Ge many. I also owe hanks o he Depa men o Communica ions Enginee ing, Uni e si y o Pade bo n, o p o iding me he li ing expense o my las wo yea s o my esea ch, in pa icula , once again hanks o P o . Häb-Umbach. I eally app ecia e all he suppo om all s a membe s o he Wo ld Uni e si y Se ice who ha e in oduced me o P o . Häb-Umbach, helped me o a ange he accomoda ion when I i s came o Ge many and con inuously encou aged me du ing my esea ch. Special hanks o my colleagues and s uden s a he Depa men o Communica ions En- ginee ing who ha e always suppo ed me o p oceed my esea ch and imp o e my disse a i- on. In pa icula , I wish o hank D .-Ing Jö g Schmalens öe o all o his suppo , aluable echnical ideas and commen s, wi hou ha i would be e y di icul o me o comple e my wo k. Las bu no leas , I wish o hank my wi e, Thi Hien T ang Vu, and my li le daugh e , T a My Hoang, o wi hou hei encou agemen , suppo and pa ience, his wo k would ha e no been comple ed. In addi ion, I would like o hank my pa en s who always beside, con inuously suppo and encou age me o no only hese yea s bu also whole my li e. 3 Con en s 1. In oduc ion 1 2. Fundamen als o Indoo Posi ioning and S a e o Resea ch 5 2.1. P oximi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2. La e a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. Angula ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4. Finge p in ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5. Dead Reckoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6. Compa ison o Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3. Scien i ic Objec i es 18 4. Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 20 4.1. Mo i a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2. Pa ame e Es ima ion using EM Algo i hm . . . . . . . . . . . . . . . . . 22 4.2.1. In oduc ion o he EM Algo i hm . . . . . . . . . . . . . . . . . . 22 4.2.2. EM algo i hm o Censo ed Gaussian Da a . . . . . . . . . . . . . 23 4.2.3. EM algo i hm o Censo ed and D opped Gaussian Da a . . . . . . 30 4.3. Op imal Classi ica ion Rule o Censo ed and D opped Gaussian Da a . . . 33 5. Sma phone Adap a ion wi hin he MLLR F amewo k 35 5.1. Mo i a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2. Model Adap a ion in he P esence o Censo ed and D opped Da a . . . . . 36 5.2.1. Mean Adap a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2.2. Va iance Adap a ion . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3. Reg ession Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6. Hidden Ma ko Model o Indoo Use T acking 43 6.1. Mo i a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2. Hidden Ma ko Model o Indoo Use T acking . . . . . . . . . . . . . . 43 6.3. Fo wa d Algo i hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.4. Mo emen Vec o Es ima ion . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4.1. S ep De ec ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.4.2. Mo emen Heading Es ima ion . . . . . . . . . . . . . . . . . . . 50 6.4.3. Mo emen Vec o Calcula ion . . . . . . . . . . . . . . . . . . . . 51 6.5. In oduc ion o Pseudo S a es . . . . . . . . . . . . . . . . . . . . . . . . . 52 i Con en s ii 7. Expe imen al Resul s on Indoo Posi ioning 55 7.1. Pa ame e Es ima ion and Classi ica ion . . . . . . . . . . . . . . . . . . . 55 7.1.1. EM algo i hm o censo ed da a . . . . . . . . . . . . . . . . . . . 55 7.1.2. EM algo i hm o censo ed and d opped da a . . . . . . . . . . . . 58 7.2. Sma phone Adap a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2.1. Classi ica ion on A i icial Da a . . . . . . . . . . . . . . . . . . . 59 7.2.2. Classi ica ion on Field Da a . . . . . . . . . . . . . . . . . . . . . 60 7.3. HMM o Indoo Use T acking . . . . . . . . . . . . . . . . . . . . . . . 62 7.3.1. A i icial Da a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.3.2. Field Da a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8. Se e Based Indoo Na iga ion Sys em 65 8.1. O e iew o Se e A chi ec u e . . . . . . . . . . . . . . . . . . . . . . . 68 8.2. Da abase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2.1. Map Tile Da a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2.2. RSSI da a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3. Sha ed Memo y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.4. O e iew o Sma phone Applica ion . . . . . . . . . . . . . . . . . . . . 74 8.5. Sys em Ope a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.5.1. Da a Ga he ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.5.2. Localiza ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.5.3. Na iga ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.5.4. Fea u es Unde De elopmen . . . . . . . . . . . . . . . . . . . . . 83 8.6. Communica ion Se cu i y . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9. Conclusions 88 A. Appendix 91 A.1. De i a ion o EM Algo i hm . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.1.1. Compu a ion o I0. . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.1.2. Compu a ion o I1. . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.1.3. Compu a ion o I2. . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.1.4. Compu a ion o W En ies . . . . . . . . . . . . . . . . . . . . . . 94 A.1.5. Compu a ion o In o ma ion Ma ix I. . . . . . . . . . . . . . . . 95 A.2. Pa ame e s o Kalman Fil e . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3. And oid Sma phone Applica ion . . . . . . . . . . . . . . . . . . . . . . . 99 A.3.1. Mani es File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.4. Reques s and Responses o he Communica ion in Indoo Na iga ion Sys em 100 A.4.1. Se F inge P in Reques and Response . . . . . . . . . . . . . . . 100 Lis o abb e ia ions 102 No a ions and Symbols 104 Lis o igu es 108 Lis o ables 111 Con en s iii Re e ences 112 Lis o own publica ions 119 “In he age o au oma ion he abili y o na iga e pe sons and de ices in indoo en i onmen s has become inc easingly impo an o a ising numbe o applica ions.” Raine Mau z [1] “Despi e i s cu en limi a ions, indoo na iga ion’s huge po en ial economic and sociologi- cal capabili ies a e pushing o wa d he esea ch, de elopmen , implemen a ion, and sale o low-cos sys ems. In he nea u u e, his will change he way we in e ac wi h ou su oun- dings, wi h many ad an ages o he a ious s akeholde s in ol ed in any indoo business.” And ea Bo ino, Gio anni Malna i and Paolo Mon uschi [2] 1. In oduc ion Posi ioning and na iga ion ha e played impo an oles in many aspec s o human ci iliza- ion o housands o yea s. Demand o his se ice is inc easing s eadily in many aspec s called loca ion-based se ices (LBS) o loca ion-awa e sys ems [3, 4] such as anspo a i- on, secu i y, social ne wo king, ma ke ing, and so on. While posi ioning o localiza ion is he p ocess o de e mine he coo dina es o he a ge objec s, na iga ion is he p ocess o es ima e he ou e om one loca ion o ano he . A he beginning, posi ioning and na iga ion echniques we e in es iga ed o ou doo en- i onmen o suppo he anspo a ion. Ou doo posi ioning and na iga ion we e done by a combina ion o celes ial measu emen s and land ma ks, which helped people o con ol he essel o e a e y la ge dis ance. Wi h he de elopmen o echnology, sa elli e was in en- ed and subsequen ly, sa elli e based na iga ion sys ems such as Global Posi ioning Sys em (GPS) by Uni ed S a es and Global Na iga ion Sa elli e Sys em (GLONASS) by Russia we e de eloped. A he momen , he e a e wo o he sys ems being de eloped namely Galileo by Eu opean Union and he BeiDou Na iga ion Sa elli e Sys em by China. Among hem, GPS [5, 6] is he mos popula and success ul na iga ion sys em. GPS was i s de eloped o mili a y pu pose by he U.S. A my and i was made a ailable o ci ilian use in he 1990s. Wi h GPS, posi ion es ima e in h ee dimen ions is ob ained by applying a ci cula la e a ion echnique, which elies on ange measu emen s. Nowadays, GPS is used o suppo he ans- po a ion o mos o he ehicles such as ai planes, ca s and ships. Mo eo e , GPS can now be used on mos sma de ices (sma phones and able compu e s) o suppo he daily ac- i i ies wi h lowe posi ioning accu acy han he adi ional GPS de ices since lowe -quali y GPS chipse s a e used on po able de ices due o p ice limi a ion. In he ideal case o line-o - sigh condi ion, i.e., no o almos no obs acles be ween he de ice and he sa elli es, GPS can loca e a ecei e wi h an accu acy o a ew me e s. Howe e , in an u ban a ea, he accu acy deg ades d ama ically due o non-line-o -sigh p oblems. Beside he success ul GPS, in o ma ion om some o he sou ces, e.g., Global Sys em o Mobile Communica ions (GSM), a e also used o he ou doo posi ioning and na iga ion pu poses. GSM based posi ioning was de eloped o sa is y he Enhanced 911 (E-911) man- da e om he US Fede al Communica ions Commission which equi es cellula p o ide s o ack he loca ion o hei subsc ibe s o wi hin 50 m o o e 67% o he ime. The GSM posi ioning sys em may ei he solely employ he GSM in o ma ion [7, 8, 9] o combine i wi h o he sou ces o in o ma ion, e.g., GPS, ine ial senso s and WiFi, esul ing in hyb id sys ems [10, 11] in o de o p oduce a mo e accu a e posi ion es ima ion. In an indoo en i onmen , GPS signals a e no mally blocked o un eliable due o he a - enua ion o signal h ough oo s o walls. As a esul , he posi ioning accu acy is poo . The e o e, de eloping a eliable indoo posi ioning and na iga ion sys em is o a pa icula in e es . This opic has been a ac ed he conside a ion o many esea che s o e he las de- 1 Fundamen als o Indoo Posi ioning and S a e o Resea ch 8 whe e x=x y(2.4) A=   x2−x1y2−y1 . . .. . . xN−x1yN−y1   (2.5) b=1 2   (x2 2+y2 2)−(x2 1+y2 1)−( 2 2− 2 1) . . . (x2 N+y2 N)−(x2 1+y2 1)−( 2 N− 2 1)   (2.6) The LS solu ion o he abo e sys em o equa ions can be ob ained as ollows: x= (ATA)−1ATb.(2.7) BS1 BS2 BS3 M 1 2 3 Figu e 2.2.: Cicula la e a ion based posi ioning: he es ima ed posi ion o he mobile use Mis de- e mined based on he es ima ed dis ances be ween he mobile use and he e e ence poin s. A leas h ee e e ence poin s a e needed o calcula e he use posi ion. •Hype bolic la e a ion: These me hods use ange di e ences be ween he mobile objec and any pai o e e- ence poin s o o m a sys em o equa ions o hype bolas (Eq. (2.8)) which a e hen used o compu e he posi ion o he ecei e : dij = i− j =p(x−xi)2+ (y−yi)2−q(x−xj)2+ (y−yj)2,(2.8) he e, dij deno es he ange di e ence be ween i- h and j- h e e ence poin s, ∀i, j;i6= j. The solu ion can be easily ob ained by applying he LS me hod in a simila ashion as discussed in he ci cula la e a ion based me hod. The abo e sys em o equa ions can Fundamen als o Indoo Posi ioning and S a e o Resea ch 9 be sol ed as ollows [3]: Since di1= i− 1,(2.9) we ha e ( 1+di1)2= 2 i ⇔ 2 i− 2 1=d2 i1+ 2 1di1(2.10) hen (x−xi)2+ (y−yi)2−(x−x1)2−(y−y1)2= 2 i− 2 1 =d2 i1+ 2 1di1 ⇔x2 i+y2 i−(x2 1+y2 1)−2x(xi−x1)−2y(yi−y1) = d2 i1+ 2 1di1 x(xi−x1) + y(yi−y1) + di1 1=1 2(x2 i+y2 i)−(x2 1+y2 1)−d2 i1; (2.11) whe e i= 1,···, N. The abo e sys em o equa ions in Eq. 2.11 can be w i en in ma ix o m: Ax =b,(2.12) whe e x=  x y 1 (2.13) A=     x2−x1y2−y1d21 . . .. . . . . . xN−x1yN−y1dN1     (2.14) b=1 2   (x2 2+y2 2)−(x2 1+y2 1)−d2 21 . . . (x2 N+y2 N)−(x2 1+y2 1)−d2 N1   (2.15) The LS solu ion o he abo e sys em o equa ions is hen ob ained as: x= (ATA)−1ATb.(2.16) Many indoo localiza ion me hods employing la e a ion echniques ha e been p oposed, see he desc ip ions in [46, 47, 48] and he e e ences he ein. The posi ioning accu acy depends on how accu a e he es ima e o he dis ance be ween ansmi e and ecei e is. La e a ion based echniques can be di ided in o 2ca ego ies: ime based la e a ion and signal s eng h based la e a ion. Fundamen als o Indoo Posi ioning and S a e o Resea ch 10 Time based la e a ion echniques employ he TOA o TDOA in o ma ion o compu e he dis ances. In [49], a ime based la e a ion posi ioning sys em has been de eloped. The objec o be loca ed is he mobile ag ansmi ing ul asonic pulses in a oughly hemisphe ical pa e n a ound he op o i , once eques ed om a mas e s a ion ia a adio signal. A cen al posi ioning uni polls he ne wo k o he ul asonic ecei e s moun ed on he ceiling o he oom o e ie e he ime o ligh (TOF) o he ul asound emi ed by he mobile ag (i de ec ed a he ecei e ). The TOF in o ma ion is measu ed as he ime di e ence be ween he momen he mas e sends ou he eques o mobile ags o emi ing ul asonic pulse and he i s signal peak de ec ed on he ecei e . Fo ime synch oniza ion, he mas e s a ion sends he ese signal o all ecei e s simul aneously wi h he ul asound emi ing eques . Dis ances be ween mobile ag and ecei e s a e hen compu ed and he posi ion o he mobile ag is de e mined by applying he LS app oach. The epo ed expe imen al esul s a e e y imp essi e wi h a posi ioning e o in he o de o cen ime e s. In addi ion o ime based la e a ion, signal s eng h based la e a ion echniques a e also employed in indoo posi ioning. These echniques employ he dependence o signal s eng h on p opaga ion dis ance. The e o e, he accu acy o hese echniques mos ly depends on how accu a e he es ima ed pa h loss model is. In [18], a modi ied pa h loss model o indoo en i onmen was de eloped, whe e he a enua ion o signal pene a ing h ough walls is conside ed. The pa ame e s o he pa h loss model a e ained empi ically. Once he dis ances be ween ansmi e s and ecei e a e ob ained, he inal es ima e o ecei e loca ion is ob ained by a ious me hods, e.g., by sol ing he sys em o ci cula o hypecbolic equa ions using he LS app oach as summa ized in [46]. In [47], me hods o imp o ing posi ioning esul s using RSSI based la e a ion echniques we e explo ed. Ins ead o using he heo e ical pa h loss model, he au ho s p oposed wo app oaches, eg ession based and co ela ion based. In he eg ession based me hod, polyno- mial eg ession was used o model he ela ionship be ween he RSSI and dis ance, and he coe icien s o he polynomial a e es ima ed by employing LS app oxima ion. The co ela- ion based me hod u ilizes he ac ha he signal p opaga ion om close-by loca ions o an AP is highly co ela ed as hey ace he simila p opaga ion en i onmen . This me hod ecu - si ely pe o ms localiza ion in g adually educed local a eas o app oach he ue loca ion. In each i e a ion, a ce ain numbe o RSSI eadings which a e he closes aining poin s o he p e ious es ima ed posi ion a e kep and used o i he p opaga ion model. Expe imen al esul s on bo h a i icial da a and eal da a showed he conside able imp o emen s compa ed o he o iginal RSSI based la e a ion me hods. An o e iew o la e a ion based echniques o Ul a-Wideband signals is gi en in [48] whe e posi ioning schemes o ime based la e a ion and RSSI based la e a ion a e discussed. In gene al, ime based la e a ion posi ioning echniques a e mo e p ecise han RSSI based la e a ion echniques, howe e , hey equi e addi ional ha dwa e. 2.3. Angula ion Angula ion based posi ioning echniques employ he angle o a i al o a wi eless signal o de e mine he posi ion o a mobile objec . Theo e ically, hese echniques a e able o compu e he posi ion o he ecei e when he e a e a leas wo e e ence poin s, and i he posi ion Fundamen als o Indoo Posi ioning and S a e o Resea ch 11 o he mobile objec does no lie on he line connec ing he e e ence poin s. BS1BS2 M α1α2 Figu e 2.3.: Angula ion based posi ioning: he es ima ed posi ion o he mobile use Mis de e mined based on he es ima ed angle αibe ween he mobile use and he e e ence poin s. A leas wo e e ence poin s a e needed o calcula e he use posi ion wi h he condi ion ha he use posi ion does no lie on he line connec ing he e e ence poin s. Fig. 2.3 illus a es he angula ion based posi ioning echniques. In his sys em, he connec- ed line be ween wo e e ences can be conside ed as an in e nal e e ence. The angle be - ween he ansmi e and he ecei e αican be de e mined by: an αi=y−yi x−xi ⇔xsin αi−ycos αi=xisin αi−yicos αi;i= 1,···, N. (2.17) Howe e , hese echniques also su e om he e o s caused by he NLOS p oblem. I he e a e mo e han wo e e ence poin s, he sys em o equa ions migh no p oduce a unique solu ion. To sol e he sys em o equa ions, an app oxima e solu ion is needed. Again, he LS me hod is he mos common solu ion o sol ing he sys em o equa ions which is de ined by Eq. (2.17). Eq. (2.17) can be w i en in ma ix o m as ollows Ax =b,(2.18) whe e x=x y(2.19) A=  −sin α1cos α1 . . .. . . −sin αNcos αN   (2.20) b=   y1cos α1−x1sin α1 . . . yNcos αN−xNsin αN   (2.21) Fundamen als o Indoo Posi ioning and S a e o Resea ch 12 The LS solu ion o he abo e sys em o equa ions can be ob ained as ollows: x= (ATA)−1ATb.(2.22) An indoo sa elli e posi ioning sys em has been p oposed in [38], howe e , di e en om he sa elli e based GPS. This sys em employs he in a ed signals ins ead o ele omagne ic wa e and angula ion echnique ins ead o la e a ion echnique. The sys em consis s o 3 in a ed ligh sou ces, called indoo sa elli es, moun ed a ixed posi ions as emi e s, and he in a ed inciden angle senso s a he mobile objec s measu ing he inciden angle om each emi e . The posi ion o he mobile objec s a e hen ob ained by applying he LS app oach. In [50], angula ion based echniques ha e been employed o es ima e he cu en posi ion o an objec . ML, LS, To al Leas Squa es (TLS) and Weigh ed LS (WLS) algo i hms we e applied o sol e he sys em o equa ions. To imp o e he pe o mance o he posi ioning sys em, a me hod based on Weigh ed TLS (WTLS) was de i ed. The op imis ic esul s wi h WTLS based app oach we e p esen ed wi h simula ion da a which app oach he C ame -Rao Lowe Bound (CRLB). Ano he example o an AOA based localiza ion sys em can be ound in [51]. The sys em consis s o a se o passi e he mal in a ed senso s ins alled in he oom edges o de ec he he mal adia ion o he human skin. The he mal senso s a e he mopile-a ays, whe e each con ains a numbe o pixels and each pixel has a ield o iew. The hea sou ce posi ion is de e mined ia he p inciple o AOA by compu ing he in e sec ion poin c ea ed by he di ec ions o he pixels wi h he highes ou pu s. 2.4. Finge p in ing Finge p in ing based posi ioning echniques a e he me hods o es ima e he posi ion o an objec which ely on aining da a om a se o e e ence poin s (ancho poin s) wi h known loca ions. Fig. 2.4 illus a es such a inge p in ing based sys em. BS1 BS2 BS3BS4 M Re e ence Poin Figu e 2.4.: Finge p in ing based posi ioning: he illed ci cles indica es he e e ence (ancho ) poin s, while he iangles show base s a ion loca ions. The use posi ion is he posi i- on o he ancho poin whose aining da a bes ma ch he online measu emen Finge p in ing based me hods can be well o mula ed as a machine lea ning and pa e n ecogni ion app oach. These echniques gene ally consis o wo phases: aining phase and classi ica ion phase (also e e ed as o line phase and online phase). In he aining phase, Fundamen als o Indoo Posi ioning and S a e o Resea ch 13 he aining da a, i.e., RSSI, a e collec ed a he ancho poin s and used o build he da abase which is o en called adio map which desc ibes he RSSI-posi ion ela ionship. The adio map can be de ined as ollows R={(ℓk,Fk)|k= 1,···, K}(2.23) whe e ℓkis he k- h e e ence posi ion, Kis he numbe o e e ence posi ions in he deploy- men a ea, Fkcan be ei he he se o aw measu emen s, i.e., Fk=Xk=xk,1,···,xk,N , whe e xk,n = [xk,n,1,···xk,n,NAP ]Tis he n- h measu emen ec o which con ains he RS- SIs om NAP base s a ions, o i con ains he class condi ional p obabili y densi y unc ions (PDFs), i.e., Fk=p(x|ℓk), es ima ed om he measu emen s a he k- h posi ion. Du ing he classi ica ion phase, he online measu emen s a e compa ed agains he aining da a a e e y ancho poin . The posi ion o he ancho poin whose aining da a bes ma ch he online da a can be conside ed as he es ima ed posi ion o he ecei e . As discussed in chap e 1, he e a e wo mos common app oaches o calcula ing he simila i y be ween aining da a and online da a: de e minis ic app oaches, e.g., he k-nea es neighbo ule, and p obabili ic app oaches, e.g., he Bayesian classi ica ion ule. Though adio inge p in ing based posi ioning echniques can be applied o bo h indoo and ou doo en i onmen s, i seems ha hese echniques a e mo e sui able o indoo since he dis inc ion o adio signal s eng hs obse ed a di e en posi ions indoo is much highe han ou doo due o he densi y o obs uc ions in indoo en i onmen s. In he e y well-known inge p in ing based posi ioning sys em Rada [18, 21], he k- nea es neighbo me hod is employed. The posi ion o he ecei e is compu ed by a e aging he coo dina es o he kancho poin s which ha e highes simila i y be ween he aining da a and he online obse a ion. The simila i y is measu ed by compu ing he Euclidian dis ance be ween he aining da a and online obse a ion in signal s eng h space as ollows D(o,xk,n) = u u NAP X i=1 (oi−xk,n,i)2,∀k, ∀n(2.24) whe e o= [o1,···, oNAP ]Tand Ddeno e he online obse a ion and he dis ance, espec- i ely. The j- h posi ion is conside ed o be he nea es neighbo , i.e., he aining da a ha bes ma ch he online obse a ion, i he e is a aining measu emen xj,m which sa is ies D(o,xj,m)≤ D(o,xk,n),∀k6=j, ∀n(2.25) In [22], an ex ensi e analysis o he inge p in ing based posi ioning sys em ha employs he Euclidian dis ance is p esen ed. The e ec o he numbe o access poin s, he numbe o aining samples pe posi ion, he densi y o he ancho poin s, e c. on he pe o mance o he posi ioning sys em a e analyzed. The analysis esul s p o ide a guideline on choosing pa ame e s o design and deploy an indoo posi ioning sys em. In p obabili ic app oaches, in he aining phase he s a is ical pa ame e s o he PDF o he signal s eng h a e ained. Du ing he classi ica ion phase, he simila i y be ween ai- ning da a and online da a is compu ed by calcula ing he likelihood o obse ing he online da a gi en he PDF o signal s eng h a ancho poin s [24, 25, 8], o some o he c i e ia o measu e he simila i y such as he Bha acha yya dis ance [52], hen again a k-nea es neighbo algo i hm can be applied o compu e he inal es ima ed posi ion o he ecei e . Fundamen als o Indoo Posi ioning and S a e o Resea ch 14 The e a e wo me hods o es ima e he PDF o he aining da a, pa ame ic and non- pa ame ic densi y es ima ion echniques [3]. Pa ame ic es ima ion me hods assume a mo- del, e.g., a Gaussian, o he densi y unc ion and aim o es ima e he pa ame e s o he model [25, 8], i.e., Fk=θk= (µk,Σk). On he o he hand, non-pa ame ic me hods do no assu- me any p io model bu es ima e he class condi ional PDF by using he his og am me hod [24, 52] o Ke nel densi y es ima o s [23, 53]. A e cons uc ing he class condi ional PDF, he inal posi ion es ima e can be ob ained by compu ing he likelihood o he pos e io using he Bayes’ ule. The posi ion which achie es he highes likelihood o pos e io is hen conside ed as he posi ion o he mobile ecei e . Maximum likelihood es ima o aims o ind he posi ion ha maximizes he p obabili y o obse ing he online measu emen o, gi en he class condi ional PDFs a e e ence poin s, as ollows ˆ ℓ= a gmax ℓk p(o|ℓk).(2.26) A maximum a pos e io i es ima o inds he posi ion which has he highes pos e io p oba- bili y as ˆ ℓ= a gmax ℓk P(ℓk|o).(2.27) Applying Bayes’ ule, P(ℓk|o)can be ob ained as ollows P(ℓk|o) = p(o|ℓk)P(ℓk) p(o) =p(o|ℓk)P(ℓk) PK k′=1 p(o|ℓk′)P(ℓk′),(2.28) assuming ha he p io P(ℓk)is gi en. In [25], wo di e en ypes o class condi ional PDF a e conside ed, one is he pa ame ic model (Gaussian) and he o he is he non-pa ame ic model. In he epo ed esul s, he me hod employing he pa ame ic model ou pe o ms he o he . This is because, as s a ed, he pa ame ic echnique smoo hs he dis ibu ion shape o accoun o missing signal s eng h alues in he aining phase, (due o he ini e numbe o aining samples) which a oids ob aining a ze o p obabili y o any signal s eng h alue ha was no obse ed in he aining phase. 2.5. Dead Reckoning Dead eckoning (DR) echniques employ he in o ma ion om a sys em o ine ial senso s, i.e., accele a ion and gy oscope senso s, in o de o es ima e he posi ion o an objec based on he p e ious es ima ed posi ion. DR echniques had been i s de eloped o he a ia ion and ma ine indus y, nowadays hey a e also used in obo ics [54, 55, 56], indoo posi ioning [57, 58, 59, 32], and many mo e sys ems. Fig. 2.5 illus a es he posi ioning p ocedu e o DR echniques whe e he cu en posi i- on is es ima ed based on he p e ious es ima e. is he mo emen ec o compu ed om displacemen in o ma ion and mo emen heading es ima ion. Fundamen als o Indoo Posi ioning and S a e o Resea ch 15 Wi h he de elopmen o Mic oelec omechanical sys ems (MEMS) echnology, o dina- y sma phones now o en ha e a buil -in ine ial measu emen uni (IMU) which allows o employ he dead eckoning echniques o indoo pedes ian acking wi hou addi ional ha dwa e. Howe e , hey ha e lowe accu acy in compa ison wi h he IMU sys em used on ai planes o in space science. As discussed in chap e 1, hese echniques a e o en de eloped in a combina ion wi h o he me hods using o he sou ces o in o ma ion, such as WiFi signal o GPS, which a e used o egula ly ese he e o o DR echniques which accumula es o e ime and dis an- ce. Displacemen es ima ion o he indoo use in DR based indoo posi ioning is done ia s ep de ec ion and s ep leng h es ima ion whe e he obse ed accele a ion da a a e used. Va- ious me hods ha e been p oposed o s ep de ec ion p ocedu e such as peak de ec ion [57], ze o-c ossing de ec ion as men ioned in [59] o au oco ela ion app oach [32]. S ep leng h es ima ion based on accele a ion da a is a icky ask since he accu acy o buil -in senso s on sma phones a e o en e y low. As a esul , mos o he s ep leng h es ima ion me hods a e based on a calib a ion phase o es ima e some heu is ic ac o s, as summa ized in [60]. A Kalman il e is o en used o mo emen heading es ima ion which u ilizes he in o ma ion om gy oscope and magne ic senso s [59]. 1 2 3 Figu e 2.5.: Dead eckoning: he use posi ion is es ima ed based on he es ima ed mo emen ec o s assuming ha he s a ing poin is gi en. 2.6. Compa ison o Techniques All he abo e men ioned adio signal based echniques o posi ioning ha e hei own ad an- ages and disad an ages and hei sui abili y depends on he deploymen en i onmen . •P oximi y based echniques can be applied o any low cos posi ioning sys em o ei - he ou doo o indoo (no equi emen o addi ional ha dwa e) which does no equi e a high posi ioning accu acy. The posi ioning e o depends on he co e age ange o he base s a ions, e.g., he p oximi y posi ioning sys ems using GSM base s a ions ha e e o in he o de o hund eds o me e s. An indoo posi ioning sys em may use hese echniques employing he wi eless local a ea ne wo k (WLAN) sys em. Howe e he posi ioning e o is abou he co e age ange o an AP, i.e., oughly 100 m which is unaccep able in he indoo en i onmen . Mo eo e , in indoo en i onmen s, he con- ou o signal s eng h adia ed om an AP is ob iously no symme ic due o he p esence o obs uc ions. This leads o misclassi ica ion esul s, because he ecei e may ecei e a highe RSSI om a a he AP because o line o sigh o his AP while i obse es a lowe RSSI om a close AP due o absence o line o sigh . Posi ioning using iBeacon ad e ising packe s is ano he app oach which aims o p oduce a oom p ecise localiza ion sys em. Howe e , he highe he accu acy equi emen s, he mo e ha dwa e (iBeacon ansmi e s) needs o be ins alled. Fundamen als o Indoo Posi ioning and S a e o Resea ch 16 •La e a ion and angula ion based echniques a e able p oduce highly p ecise posi io- ning esul s in ee space en i onmen whe e signals can p opaga e by he di ec pa h om he ansmi e o he ecei e and almos no mul ipa h p oblem occu s. Un o u- na ely, he indoo en i onmen is de ini ely no ee space. I has a lo o obs uc ions such as walls, u ni u e o mo emen o people ha makes he accu acy o he posi- ioning using hese echniques e y limi ed. Because o he special equi emen s o he en i onmen , hese echniques may only be sui able o posi ioning in a single oom. Ano he limi a ion o hese echniques is he equi emen o addi ional speciali- zed ha dwa e, i.e., an enna, mic ophone a ays, e c., and deep ha dwa e access which seems o be no possible o a posi ioning sys em employing he exis ing WLAN in- as uc u e and o dina y mobile de ices, i.e., sma phones. Clock synch oniza ion is ano he challenge o la e a ion based echniques which de e mines he accu acy o posi ioning esul s. •Finge p in ing based me hods as discussed abo e consis o wo phases, aining phase and classi ica ion phase. The i s and he o emos disad an age o hese echniques is he necessi y o a aining phase which is e y ime consuming and need o be done e y ca e ully because he classi ica ion esul s a e s ongly in luenced by he quali y o he aining da a. Ano he disad an age o hese echniques is ha he aining da a need o be upda ed egula ly in o de o adap o he changes o he in as uc u e and en i onmen in o de o main ain he accu acy o he posi ioning esul s. Despi e hese disad an ages, inge p in ing based echniques a e s ill he mos p omising me hods o indoo en i onmen s o h ee main easons: Fi s , hey can p oduce a easonable posi ioning esul wi hou any addi ional special ha dwa e. Second, hese echniques a e sui able o be employed in an al eady ins alled WLAN sys em, and WLAN is a ailable in many places. Thi d, sys ems using hese echniques can be applied in a la ge a ea and ha e a good scalabili y. •Dead eckoning echniques a e able o p oduce p ecise posi ioning esul s gi en he ini ial posi ion wi hou any knowledge o he in as uc u e in he co e age a ea, ho- we e , only o e a sho ime and mo emen dis ance. The high accu acy is no kep long because o he accumula ion o e o o e ime and dis ance o mo emen . The e o s a e caused, o example, by he d i o bias o senso sys ems, and also he noise in he en i onmen s, o example he measu emen om a magne ic senso is s ongly a ec ed by me allic ma e ials a ound he senso and no only he magne ic ield o he ea h. F om he analysis o he he ad an ages and disad an ages o he men ioned adio signal based echniques, o de eloping a eal indoo posi ioning sys em based on he egula sma - phones and he al eady ins alled WLAN sys em, he combina ion o inge p in ing based and dead eckoning echniques appea s o be he bes solu ion because o he ollowing easons: •Su icien accu acy: The posi ioning accu acy mee s he equi emen o indoo posi- ioning o he common pu poses such as use acking and secu i y, in la ge indoo a eas, i.e., uni e si ies, museums, hospi als o s o es. •Cos e iciency: no addi ional ha dwa e is needed o build up a posi ioning sys em. Fundamen als o Indoo Posi ioning and S a e o Resea ch 17 •Deploymen simplici y: WLAN is oday a ailable in almos e e y indoo en i on- men . Consequen ly, he posi ioning sys em can be deployed in many indoo a eas easily. •Robus ness: The combina ion o WiFi inge p in ing based and DR echniques ma- kes he sys em obus , i.e., he mobile de ices can do sel localiza ion based on DR echniques when no WiFi signal is a ailable. Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 24 used wi h he same meaning. The goal is o es ima e he pa ame e s θ={µ, σ2}o he unde lying Gaussian. E-S ep Employing he EM algo i hm we iden i y yand x o be he comple e and he obse ed da a, espec i ely. Thus he expec ed log-likelihood o he comple e da a is gi en by Q(θ;θ(κ)) = Eln (pY(y;θ)) |x;θ(κ)(4.4) = N X n=1 Z∞ −∞ ln (pY(yn;θ)) pyn|xn;θ(κ)dyn(4.5) =: N X n=1 nθ;θ(κ),(4.6) whe e nθ;θ(κ)=R∞ −∞ ln (pY(yn;θ)) pyn|xn;θ(κ)dynand whe e κis he i e a ion in- dex. The e m p(yn|xn;θ(κ))can be de e mined as ollows: p(yn|xn;θ(κ)) = (δ(yn−xn), i xn> c p(xn|yn;θ(κ))p(yn;θ(κ)) p(xn;θ(κ)), i xn=c(4.7) Fo he case xn=cwe know ha yn≤c.p(yn|xn;θ(κ))can hus be calcula ed as ollows p(yn|xn;θ(κ)) = p(xn|yn;θ(κ))p(yn;θ(κ)) p(xn;θ(κ)) =p(xn|yn;θ(κ))p(yn;θ(κ)) Rc −∞ p(xn|yn;θ(κ))p(yn;θ(κ))dyn =δ(xn−c)p(yn;θ(κ)) Rc −∞ δ(xn−c)p(yn;θ(κ))dyn =Nyn;θ(κ) I0(θ(κ)) He e we ha e used he no a ion Ij(θ(κ)) = Zc −∞ yjNy;θ(κ)dy(4.8) wi h j= 0.I0(θ(κ))can be calula ed as ollows (see Appendix A.1.1 o he de ailed com- pu a ion): I0(θ(κ)) = 1 2e c −c−µ(κ) √2σ(κ).(4.9) Then we ob ain p(yn|xn;θ(κ)) = (δ(yn−xn), i xn> c N(yn;θ(κ)) I0(θ(κ)), i xn=c.(4.10) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 25 In oducing he bina y andom a iable Znwi h ealiza ion zn, whe e zn= 0 and zn= 1 indica e ha he n- h measu emen is no censo ed o censo ed, espec i ely, he summand in Eq. (4.6) can be w i en as nθ;θ(κ)=zn I0(θ(κ))Zc −∞ ln (N(yn;θ)) Nyn;θ(κ)dyn + (1 −zn) ln (N(xn;θ)) .(4.11) he e, he ac ha yn=xnwhen zn= 0 is exploi ed. M-S ep The pa ame e e-es ima ion o mulas a e ob ained by compu ing he de i a i es o Eq. (4.11) w. . . he elemen s o θand se hem o ze o: ∂ ∂µ n(θ;θκ) = zn I0(θ(κ))Zc −∞ yn−µ σ2Nyn;θ(κ)dyn + (1 −zn)xn−µ σ2 =zn σ2 I1θ(κ)−µI0θ(κ) I0(θ(κ))+ (xn−µ)1−zn σ2(4.12) ∂ ∂σ n(θ;θκ) = zn I0(θ(κ))Zc −∞ −1 σ+(yn−µ)2 σ3Nyn;θ(κ)dyn + (1 −zn)−1 σ+(xn−µ)2 σ3 =zn σ1 σ2I2(θ(κ)) I0(θ(κ))−2µI1(θ(κ)) I0(θ(κ))+µ2−1 +1−zn σ(xn−µ)2 σ2−1,(4.13) He e, we ha e used he no a ion as de ined in Eq. (4.8). Se ing he suma ions o he abo e de i a i es o ze os, e-es ima ion o mulas a e eadily ob ained: N X n=1 ∂ ∂µ n(θ;θκ)µ=µκ+1 = 0 : µ(κ+1) =1 N I1(θ(κ)) I0(θ(κ)) N X i=1 zn+1 N N X i=1 (1 −zn)xn(4.14) N X n=1 ∂ ∂σ n(θ;θκ)σ2=(σ2)κ+1 = 0 : σ2(κ+1) =I2(θ(κ)) I0(θ(κ))−2µ(κ)I1(θ(κ)) I0(θ(κ))+µ2(κ)1 N N X i=1 zn +1 N N X i=1 (1 −zn)xn−µ(κ)2.(4.15) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 26 The compu a ion o I1(θ(κ))and I2(θ(κ))a e gi en in Appendix A.1.2 and Appendix A.1.3, espec i ely. As can be seen in Eq. (4.14) and Eq. (4.15), he unobse able da a, i.e., censo ed da a, also con ibu e o he es ima es beside he obse able ones. In ui i ely, in case o absence o censo ed da a, i.e., zn= 0 o all measu emen s, he e-es ima ion o mulas educe o he ypical ML es ima ion o mulas o mean µand a iance σ2. To ha e a be e in ui ion o he e-es ima ion o mulas, he si ua ion a e con e gence o he es ima es is conside ed. Le µ(κ+1) ≈µ(κ)=: ˆµand (σ2)(κ+1) ≈(σ2)(κ)=: ˆσ2, using his in Eq. (4.14) and Eq. (4.15) and sol ing o he es ima es, we a i e a ˆµ=M N 1 M M X i=1 xn+1−M NRc −∞ ypY(y;ˆ θ)dy Rc −∞ pY(y;ˆ θ)dy,(4.16) ˆσ2=M N 1 M M X i=1 (xn−ˆµ)2+1−M NRc −∞(y−ˆµ)2pY(y;ˆ θ)dy Rc −∞ pY(y;ˆ θ)dy,(4.17) whe e we assumed wi hou loss o gene ali y ha he i s M=Pi(1 −zn)obse a ions a e he uncenso ed ones. These exp essions lend hemsel es o he ollowing in e p e a ion: Mean and a iance es ima es a e he weigh ed a e age be ween hei ML es ima es which a e compu ed om he obse ed da a and he mean and a iance o he assumed unca ed Gaus- sian o he unobse able pa s. The weigh s a e he ela i e equencies o he uncenso ed and censo ed measu emen s, espec i ely. P ope ies o he Es ima es In he ollowing, he main p ope ies o he p oposed es ima o a e in es iga ed. As will be shown, he p oposed EM algo i hm deli e s i ually bias ee and e icien es ima es. Unbiasness and Con e gence In o de o s udy he con e gence p ope ies he expec ed alues o he di e ence be ween he es ima es, Eqs. (4.14) and (4.15), and he ue alues o he pa ame e s a e compu ed. Fi s no e ha Znis a Be noulli andom a iable wi h P(Zn= 1) = Zc −∞ pY(y;θ)dy=I0(θ) and P(Zn= 0) = 1 −I0(θ). Thus: E [Zn] = E[Z2 n] = I0(θ). Taking he expec a ion o mean es ima e Eq. (4.14) we ha e Eµ(κ+1)=1 N N X i=1 E[(1 −Zn)Xn] + 1 N N X i=1 EI1(θ(κ)) I0(θ(κ))Zn.(4.18) I has o be no ed ha he es ima e θ(κ)depends on he da a and is hus andom. Thus he expec a ion ope a o also applies o his quan i y. The i s expec a ion can be e alua ed as ollows: E[(1 −Zn)Xn] = 1 X zn=0 Z∞ −∞ (1 −zn)xnP(zn|xn)pY(xn)dxn =Z∞ c xnN(xn;θ)dxn=µ−I1(θ).(4.19) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 27 No e ha he e we ha e used he no a ion pY(xn)because x=yin case o no censo ed. He e we ha e employed he ac ha P(zn= 0|xn) = 1,i xn> c 0,else .(4.20) Using u he E hI1(θ(κ)) I0(θ(κ))zni≈EhI1(θ(κ)) I0(θ(κ))iE[Zn]and sub ac ing µ om ei he side o Eq. (4.18) we ob ain E˜µ(κ+1)=−I1(θ) + I0(θ)EI1(θ(κ)) I0(θ(κ)).(4.21) whe e ˜µ(κ+1) =µ(κ+1) −µ. In o de o compu e he emaining expec a ion, a Taylo se ies expansion a ound he ue pa ame e alues θ= (µ, σ2)is applied and unca ed a e he linea e m: EI1(θ(κ)) I0(θ(κ))≈I1(θ) I0(θ)+E˜µ(κ)∂ ∂µ I1(θ) I0(θ)+Eh˜σ2(κ)i∂ ∂σ2 I1(θ) I0(θ).(4.22) Expec a ion o a iance es ima e Eq. (4.15) can be calula ed in a simila p ocedu e as ollows: Eσ2(κ+1)=E"1 N N X n=1 (1 −zn)(xn−µ)2# +E"1 N N X n=1 zn I2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))# =1 N N X n=1 E(1 −zn)(xn−µ)2 +1 N N X n=1 Ezn I2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))(4.23) The i s expec a ion can be e alua ed as ollows: E(1 −zn)(xn−µ)2= 1 X zn=0 Z∞ −∞ (1 −zn)(xn−µ)2P(zn|xn)pY(xn)dxn =Z∞ c (xn−µ)2N(xn;θ)dxn =σ2−(I2(θ)−2µI1(θ) + µ2I0(θ)).(4.24) Using u he E hzn I2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))i≈EhI2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))iE[Zn]and sub ac ing σ2 om ei he side o Eq. (4.23) we ob ain Eh˜σ2(κ+1)i=−I2(θ)−2µI1(θ) + µ2I0(θ) +I0(θ)EI2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))(4.25) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 28 whe e (˜σ2)(κ+1) = (σ2)(κ+1) −σ2. In o de o compu e he emaining expec a ion, a Taylo se ies expansion a ound he ue pa ame e alues θ= (µ, σ2)is again applied and unca ed a e he linea e m: EI2(θ(κ))−2I1(θ(κ))µ+I0(θ(κ))µ2 I0(θ(κ))≈I2(θ)−2I1(θ)µ+I0(θ)µ2 I0(θ) +E˜µ(κ)∂ ∂µ I2(θ)−2I1(θ)µ+I0(θ)µ2 I0(θ) +Eh˜σ2(κ)i∂ ∂σ2 I2(θ)−2I1(θ)µ+I0(θ)µ2 I0(θ). (4.26) W i ing Eq. (4.21) and Eq. (4.25) in ma ix o m, we a i e a E˜µ(κ+1) Eh(˜σ2)(κ+1)i!≈WµWµσ Wσµ Wσ E˜µ(κ) Eh(˜σ2)(κ)i! =WµWµσ Wσµ Wσκ+1 E˜µ(0) Eh(˜σ2)(0)i!,(4.27) whe e Wµ=I0(θ)∂ ∂µ I1(µ) I0(µ), Wµσ =I0(θ)∂ ∂σ2 I1(θ) I0(θ), Wσ=I0(θ)∂ ∂σ2 I2(θ)−2I1(θ)µ+I0(θ)µ2 I0(θ), Wσµ =I0(θ)∂ ∂µ I2(θ)−2I1(θ)µ+I0(θ)µ2 I0(θ). The Wen ies can be eadily compu ed by using he de i a i es gi en in Appendix A.1.4. In es iga ing all he possible alues o he magni udes o he eigen alues o he ma ix wi h he Wen ies by changing he clipping h eshold c om −∞ o ∞, we obse ed ha hey a e always less han one (app oaching one o c→ ∞, i.e., no obse able da a). We hus conclude ha he es ima es a e bias ee because he expec a ion o he e o dec eases exponen ially o e i e a ions. Since, howe e , in p ac ice µhas o be eplaced by i s es ima e in he es ima ion o he a iance, we exhibi he same bias as in o dina y ML es ima ion o he a iance. In he applica ion conside ed he e, he bias o he ML es ima e o he a iance can be neglec ed due o he la ge numbe o samples. Fu he , an es ima e o he con e gence speed o he EM algo i hms can also be ob ained om Eq. (4.27), see Fig. 4.3. I can be seen ha he numbe o i e a ions quickly ises once mo e han 50% o he da a a e clipped. P ecision In o de o e alua e he p ecision o ou p oposed es ima o , we calcula ed he C ame -Rao Lowe Bound (CRLB) o he es ima o . CRLB o he es ima ion e o a iance o he mean and a iance es ima es can be ob ained by in e ing he in o ma ion ma ix Iwhich con ains he expec ed alues o he second-o de pa ial de i a i es o he log-likelihood unc ion. F om he de i a ion o he EM algo i hm i can be seen ha he measu emen s consis o wo ypes o da a, he numbe Mo noncenso ed obse a ions and he obse a ions hemsel- es. While he o me ype is binomially dis ibu ed, he la e ype is d awn om a unca ed Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 29 c−µ σ No. o EM i e a ions −2−1.5−1−0.500.51 1.52 0 100 200 300 Figu e 4.3.: Theo e ical numbe o EM i e a ions κ equi ed o educe he es ima ion e o o 10−4 o i s ini ial alue as a unc ion o (c−µ)/σ. Ini ial alues µ(0),σ2(0) ha e been se o he ML es ima es o µ, σ2compu ed om he unclipped obse a ions only. Gaussian. I is no ed ha he d aws om he Gaussian a e independen o he binomial an- dom a iable. Following [65] he likelihood is hus gi en by, whe e I0(θ)is he p obabili y o no obse ing a sample and 1 √2πσ2exp −1 2xj−µ σ2!is he p obabili y o obse ing xj>−100 dBm p(x;θ) = N! M!(N−M)!IN−M 0(θ)1 √2πσ2M exp −1 2 M X j=1 xj−µ σ2!,(4.28) and he log-likelihood unc ion is ob ained as ollows: ln p(x;θ) = ln N! M!(N−M)!+ (N−M) ln (I0(θ)) −M 2ln 2πσ2−1 2 M X j=1 xj−µ σ2 .(4.29) The in o ma ion ma ix Iis de ined as: I= Eh−∂2 ∂µ2ln p(x;θ)iEh−∂2 ∂µ∂σ2ln p(x;θ)i Eh−∂2 ∂µ∂σ2ln p(x;θ)iEh−∂2 ∂σ2∂σ2ln p(x;θ)i .(4.30) The en ies o ma ix Ican be calcula ed as ollows (see Appendix A.1.5 o he de ailed compu a ion): E−∂2 ∂µ2ln p(x;θ)=N σ2I2 1−I2I0 I0 + 1(4.31) E−∂2 ∂µ∂σ2ln p(x;θ)=N σ2 − ∂I1 ∂σ2I0−I1∂I0 ∂σ2 I0!(4.32) E−∂2 ∂σ2∂σ2ln p(x;θ)=N σ2 1 2σ2−1 2σ2 ∂I2 ∂σ2I0−I2∂I0 ∂σ2 I0−2µ ∂I1 ∂σ2I0−I1∂I0 ∂σ2 I0!! (4.33) whe e he compu a ion o he de i a i e e ms a e gi en in Appendix A.1.4. Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 30 CRB (σ2) CRB (µ) Simula ion (σ2) Simula ion (µ) log10(MSE) c−µ σ −2−1.5−1−0.500.51 1.52 −2 −1 0 1 2 3 Figu e 4.4.: Compa ison o CRLB o mean and a iance wi h MSE ob ained om simula ion o σ2= 25 and N= 1000. Compu ing he I−1, we ob ain CRLB on es ima ion e o a iance o µand σ2: Va (ˆµ)≥I−1(1,1) (4.34) Va (ˆσ2)≥I−1(2,2) (4.35) Fig. 4.4 compa es he CRLB wi h he mean squa ed e o (MSE) o he p oposed es ima- o s o mean µand a iance σ2ob ained om a simula ion. I can be seen ha he es ima o p ac ically achie es he bound. The di e ences a e so small ha hey a e no isible in he g aph. We he e o e conclude ha he es ima o is e icien . Fu he mo e, o he limi ing ca- se o comple ely uncenso ed da a he well-known esul s o ML pa ame e es ima ion om a no mal popula ion a e ob ained: MSE(µ) = σ2/N; MSE(σ2) = σ4(2N−1)/N2. 4.2.3. EM algo i hm o Censo ed and D opped Gaussian Da a In his subsec ion, he p oposed EM algo i hm o pa ame e es ima ion o censo ed Gaussian da a is ex ended o be able o cope no only wi h he censo ed da a bu also he d opped da a. No a ions The measu emen model o censo ed and d opped da a is shown in Fig. 4.5. He e, he hidden a iables dn,n= 1,...N, indica e whe he an obse a ion is d opped (dn= 1) o no (dn= 0), whe e P(dn= 1) is he d opping a e, in he ollowing, P(dn= 1) is deno ed by π. The a iables a e ga he ed in he bina y ec o d= [d1,...,dN], whe e Nis he numbe o measu emen s. Fu he mo e, de ine y=y1, ..., yN, whe e he yna e i.i.d. wi h Gaussian p obabili y densi y unc ion (PDF) pY(yn) = N(yn;µ, σ2)i dn= 0, and yn=ci dn= 1, whe e we se c o he smalles measu able RSSI alue (e.g., c=−100 dBm). I should be Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 31 no ed ha yis no he comple e da a anymo e since he da a a e possibly d opped. Obse able da a a e he censo ed, possibly d opped da a x=x1, ..., xN, whe e xn= max(yn, c), wi h he censo ing h eshold cas abo e. xn yn max(yn, c) ∼ N(µ, σ2) c dn dn= 0 dn= 1 D opping Censo ing Figu e 4.5.: Measu emen model. The di e ence be ween censo ing and d opping can be exp essed as ollows: E en i he pa ame e s o he Gaussian a e such ha p ac ically no censo ing occu s (because µ≫c), d opping will s ill possibly occu . The goal is o es ima e he pa ame e s θ={µ, σ2, π}. This will be achie ed by he Expec- a ion Maximiza ion (EM) algo i hm, whe e he comple e da a a e {y,d}and he obse able a e {x}. E-S ep I is no ed ha xndoes no con ey mo e in o ma ion abou θ han yn. So, he comple e da a a e {y,d} a he han {x,y,d}. The expec ed log-likelihood o he comple e da a, gi en he obse ed one, is gi en by: Q(θ;θ(κ)) = Eln (p(y,d;θ)) |x;θ(κ) = N X n=1 1 X dn=0 Z∞ −∞ ln (p(yn, dn;θ)) p(yn, dn|xn;θ(κ))dyn,(4.36) whe e κis he i e a ion index. p(yn, dn|xn)can be w i en as p(yn, dn|xn;θ(κ)) = p(yn|dn, xn;θ(κ))P(dn|xn;θ(κ)).(4.37) Fo calcula ing p(yn|dn, xn;θ(κ)), wo cases a e conside ed: •Fo da a ha a e no d opped (dn= 0), see Eq. (4.10), p(yn|dn= 0, xn;θ(κ)) = (δ(yn−xn),i xn> c N(yn;θ(κ)) I0(θ(κ)),i xn=c(4.38) •Fo da a ha a e d opped (dn= 1) p(yn|dn= 1, xn;θ(κ)) = 0,i xn> c δ(yn−c),i xn=c.(4.39) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 32 Fu he mo e, he pos e io o he hidden a iable dncan be compu ed using Bayes’ ule P(dn|xn;θ(κ)) = p(xn|dn;θ(κ))P(dn) P1 dn=0 p(xn|dn;θ(κ))P(dn).(4.40) Since he e a e wo a iables, ou cases can be disce ned. Using he no a ion βn(d, z) := P(dn|zn;θ(κ)), whe e zn= 1 indica es ha xn=cand zn= 0 ha xn> c, we ob ain: βn(1,1) = P(dn= 1|xn=c;θ(κ)) =p(xn=c|dn= 1)P(dn= 1) p(xn=c|dn= 0)P(dn= 0) + p(xn=c|dn= 1)P(dn= 1) =π(κ) I0(θ(κ))(1 −π(κ)) + π(κ),(4.41) whe e he esul om he p e ious sec ion is employed: p(xn=c|dn= 0) = I0(θ(κ))is he p obabili y ha a measu emen is censo ed, i.e., unobse able, gi en ha i is no d opped. In addi ion, p(xn=c|dn= 1) is always equal o 1since i he measu emen is d opped, i is assigned he alue o c.βn(1,1) is he p obabili y ha he measu emen is d opped gi en i is unobse able. The p obabili y ha he measu emen is no d opped gi en i is unobse able, βn(0,1), is hen easily ob ained since βn(0,1) + βn(1,1) = 1. Fu he mo e, i is ob ious ha i a measu emen is obse able, i canno be d opped, so βn(1,0) = 0, and hus βn(0,0) = 1. We u he no e ha p(yn, dn;θ) = P(dn;θ)p(yn|dn;θ) =πδ(yn−c),i dn= 1 (1 −π)N(yn;θ),i dn= 0 (4.42) Using all he abo e in (4.36) we a i e a Q(θ;θ(κ)) = N X n=1 znZc −∞ ln ((1 −π)N(yn;θ)) N(yn;θ(κ)) I0(θ(κ))βn(0,1)dyn + (1 −zn)Z∞ c ln ((1 −π)N(yn;θ)) δ(yn−xn)βn(0,0)dyn +znZc −∞ ln (πδ(yn−c)) δ(yn−c)βn(1,1)dyn = N X n=1 znln(1 −π)βn(0,1) +zn I0(θ(κ))Zc −∞ ln (N(yn;θ)) N(yn;θ(κ))dynβn(0,1) + (1 −zn) ln(1 −π) + (1 −zn) ln (N(xn;θ)) +znln(π)βn(1,1).(4.43) Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a 33 M-S ep Compu ing he de i a i e o he auxilia y unc ion Eq. (4.36) he ollowing i e a i e pa ame- e es ima ion o mulas can be eadily de i ed: µ(κ+1) =PN n=1(1 −zn)xn+I1(θ(κ)) I0(θ(κ))PN n=1 znβn(0,1) N−PN n=1 znβn(1,1) (4.44) σ2(κ+1) =PN n=1 znI2(θ(κ)) I0(θ(κ))−2µI1(θ(κ)) I0(θ(κ))+µ2βn(0,1) N−PN n=1 znβn(1,1) +PN n=1 (1 −zn)(xn−µ)2 N−PN n=1 znβn(1,1) (4.45) π(κ+1) =PN n=1 znβn(1,1) N.(4.46) These o mulas educe o he e-es ima ion o mulas o censo ed da a p esen ed in he p e- ious sec ion i he d opping a e is se o π= 0. Then βn(1,1) = 0 and βn(0,1) = 1. As can be seen in Eqs. (4.44), (4.45) and (4.46), no only obse able and censo ed da a, bu also d opped da a con ibu e o he es ima es. 4.3. Op imal Classi ica ion Rule o Censo ed and D opped Gaussian Da a Abo e ML pa ame e es ima ion is done o all possible use loca ions ℓkand i is done o he measu emen s o each AP sepe a ely, assuming ha da a om di e en APs a e independen . The inal pa ame e es ima es a e deno ed by ˆ µk= [ˆµk,1,···,ˆµk,NAP ]Tand ˆ Σk=diag ˆσ2 k,1,···,ˆσ2 k,NAP , whe e NAP is numbe o obse able APs. Indoo localiza ion can be o mula ed as a classi ica ion p oblem, whe e he classes a e he posi ions om which RSSI measu emen s a e aken du ing he o line aining phase. Fo each posi ion ℓk he pa ame e s o a Gaussian class-condi ional densi y pY(x|ℓk)o RS- SI measu emen s a e es ima ed using he EM algo i hm o he las sec ions. Du ing online classi ica ion, o es ima e he use ’s loca ion, an op imal classi ica ion ule is de eloped. Le x=x1,···, xNAP be he ec o o he online measu emen . Fi s he pos e io is calcula ed as ollows p(ℓk|x) = QNAP i=1 p(xi|ℓk)P(ℓk) PK k′=1 QNAP i=1 p(xi|ℓk′)P(ℓk′)(4.47) whe e Kis he numbe o o line aining loca ions and xiis he RSSI o i- h AP. P(ℓk)is he p io on he posi ion. He e we ha e employed he assump ion o he independence o he RSSIs o di e en APs. I is no ed ha he es da a a e also subjec o censo ing. The likelihood p(xi|ℓk)in Eq. (4.47) can be calcula ed as ollows o censo ed Gaussian da a p(xi|ℓk) = Z∞ −∞ p(xi|yi)pY(yi|ℓk)dyi,(4.48) Sma phone Adap a ion wi hin he MLLR F amewo k 40 w(c(k)) i(κ+1) can be ob ained as ollows w(c(k)) i(κ+1) = R X =1 N X n=1 γk (n) σ2 k ,i zn,i I1(λ(κ) k ,i) I0(λ(κ) k ,i)βk ,i(0,1) + (1 −zn,i)xn,i!ξT k R X =1 N X n=1 γk (n) σ2 k ,i (1 −zn,iβk ,i(1,1)) ξk ξT k !−1 (5.20) 5.2.2. Va iance Adap a ion Va iance adap a ion is pe o med a e mean adap a ion. I is no ed ha we assume he RSSI measu emen s om di e en APs a e independen . Le Σkbe he diagonal co a iance ma ix o he Gaussian andom ec o modelling he RSSI eadings o he APs a posi ion ℓk. The adap ed a iance can be calcula ed as ollows [70]: ˆ Σk=BT kH(c(k))Bk(5.21) whe e H(c(k)) is he ans o m o be es ima ed and Bkis he Choleski ac o o Σk. Since Σk a e diagonal co a iance ma ices, he e-es ima ion o mula Eq. (5.21) simpli ies o ˆσ2 k,i = h(c(k)) iσ2 k,i;i= 1,...,NAP . To es ima e H(c(k)), we again employ he EM algo i hm. The expec ed log-likelihood o he comple e da a, gi en he obse ed, has he same o m as in mean adap a ion. Howe e , he e λk,i ={πk,i, h(c(k)) i, θk,i}wi h h(c(k)) iis he i- h componen o he main diagonal o H(c(k)). Using all in he expec ed log-likelihood unc ion Eq. (5.12), we a i e a : Q(λ;λ(κ)) = K X k=1 N X n=1 γk(n) NAP X i=1 (zn,iβk,i(0,1) ln(1 −πk,i) +zn,iβk,i(0,1) I0(λ(κ) k,i )Zc −∞ ln  1 q2πh(c(k)) iσ2 k,i exp −(yn,i −ˆµk,i)2 2h(c(k)) iσ2 k,i ! N(yn,i;λ(κ) k,i )dyn,i + (1 −zn,i) ln(1 −πk,i) + (1 −zn,i) ln  1 q2πh(c(k)) iσ2 k,i exp −(xn,i −ˆµk,i)2 2h(c(k)) iσ2 k,i !  +zn,iβk,i(1,1) ln(πk,i))(5.22) The e-es ima ion omula o h(c(k)) iis eadily ob ained by compu ing he de i a i e o Sma phone Adap a ion wi hin he MLLR F amewo k 41 Eq. (5.22) w. . . h(c(k)) iand se ing i o ze o: h(c(k)) i(κ+1) =1 σ2 k,i PN n=1 γk(n) (1 −zn,iβk,i(1,1)) N X n=1 γk(n) ((1 −zn,i)(xn,i −ˆµk,i)2βk,i(0,0) +zn,i I2(λ(κ) k,i ) I0(λ(κ) k,i )−2I1(λ(κ) k,i ) I0(λ(κ) k,i )ˆµk,i + ˆµ2 k,i!βk,i(0,1)).(5.23) He e ˆµk,i a e he new upda ed means which we e discussed in he p e ious subsec ion. As can be seen in he de i a ion, beside es ima ing he adap a ion ma ices o means Eq. (5.18) and a iances Eq. (5.23), he d opping a es o he adap a ion da a a e simul a- neously es ima ed. Eq. (5.18) and Eq. (5.23) show ha no only he obse able measu e- men s, bu also he unobse able ones con ibu e o he es ima e o he adap a ion ma ices. I nei he censo ing no d opping occu s, i.e., πk,i = 0 and zn,i = 0, Eq. (5.18) and Eq. (5.23) educe o he o mulas o mean and a iance adap a ion p esen ed in [70]. 5.3. Reg ession Classes In he p e ious sec ion, he me hod o aligning aining model wi h adap a ion da a in he p esence o clipped and d opped da a has been p esen ed o he gene al case o mul iple ans o ma ion ma ices. I he ela ionship be ween wo se s o aining da a and adap a ion da a ollows only one linea ule as men ioned in [61], only one se o adap a ion ma ices, i.e., mean adap a ion ma ix and a iance adap a ion ma ix, o all s a es is needed. Howe e , acco ding o ou da a p elimina y s udy, he ela ionship o RSSI da a measu- ed by any pai o de ices does no ollow only one linea ule. Two possible main ac o s which in luence he ela ionship be ween wo se s o da a a e signal s eng h ange and signal equency. This is ob ious because o he ac ha he sensi i i ies o adio signal senso s de- pend on he s eng h and he equency o he measu ed adio signal. To elax he linea i y assump ion, a clus e ing app oach has been applied o sepe a e he use posi ions in mul iple eg ession classes ha sha e he same linea ela ionship, howe e di e en om he o he clus e s. Howe e , a c i ical issue is how o de ine he eg ession classes, wi hin which he pa a- me e s o all s a es, i.e., inge p in posi ions, change in he same manne . Since no p io knowledge o which models should sha e he same ans o ma ion ule is a ailable, a c i e i- on o inding eg ession classes mus be ound. Va ious c i e ia could be used o clus e ing, o example, he posi ions wi h simila RSSI PDFs could sha e he same linea ule. The si- mila i y o he PDFs can be measu ed by compu ing he likelihood o obse ing he aining da a collec ed a one posi ion gi en he aining models o he o he s. Using hese c i e ia, means and a iances o he aining models a e employed. Howe e , om ou obse a ion, he s a es, i.e., posi ions, which ans o m in he same manne ha e simila RSSI mean a- lues, i espec i e o he ac ual iden i y o he APs. As a esul , hose posi ions should be assigned in o he same clus e . We hus apply a k-means clus e ing on he mean ec o s µk Sma phone Adap a ion wi hin he MLLR F amewo k 42 o he p obabili y densi y unc ion o all loca ion ℓk o ob ain Cclus e s. Fo each clus e , ans o ma ion ma ices a e es ima ed as p esen ed in he p e ious sec ion. 6. Hidden Ma ko Model o Indoo Use T acking 6.1. Mo i a ion In chap e 4, we ha e discussed he me hod o es ima e he pa ame e s o censo ed and d op- ped Gaussian da a and he op imal classi ica ion ule o localizing indoo use s. The p o- posed algo i hms belong o he mos common app oach o indoo posi ioning which is he inge p in ing based app oach. Finge p in ing echniques a e able o p oduce a s able posi- ioning accu acy o e ime and mo emen dis ance o he use . Howe e , he e is ano he app oach which can also be applied success ully in indoo posi ioning, namely he Dead Reckoning echnique. This echnique uses da a om ine ial senso s o loca e he use posi- ion. Howe e , as discussed in chap e 2, he Dead Reckoning echnique is o en applied in a combina ion wi h o he posi ioning echniques since i can only p oduce p ecise posi ion es ima es in a sho pe iod o ime and small dis ance. Fo long e m use and la ge dis ance, he posi ion e o is un easonably la ge because he e o s a e accumula ed o e ime and dis ance. As discussed, he s abili y o inge p in ing based indoo posi ioning echniques make hem become he common candida es o accompany wi h DR echniques in indoo posi io- ning. Va ious combina ions ha e been p oposed o keep he ad an ages and o educe he disad an ages o hose wo app oaches. Senso usion can be achie ed by a Kalman il e [31], pa icle il e [32] o wi h he use o a Hidden Ma ko Model (HMM) [29, 30]. Among hose possible solu ions, we decided o employ he HMM in ou sys em because i gi es us he possibili y o employ he knowledge o possible walking pa h which may imp o e he posi ioning accu acy. 6.2. Hidden Ma ko Model o Indoo Use T acking In his sec ion, he me hod o employ a HMM o use RSSI measu emen s and s ep de ec ion in o ma ion o posi ion es ima ion is p esen ed. Fi s , i should be ecalled ha he s anda d HMM can be depic ed as in Fig. 6.1. He e, s is he hidden s a e a iable and x is he obse a ion a ime . I is no ed ha he hidden s a es canno be obse ed, howe e , he mos likely sequence o s a es can be in e ed om he sequence o obse a ions gene a ed by HMM. The HMM is de e mined by h ee se s o pa ame e s: he emission p obabili ies, he s a e ansi ion p obabili ies and he ini ial s a e p obabili ies. Fig. 6.2 shows he un olding o a s a e diag am o e ime, called ellis diag am. He e i is assumed ha he HMM has K= 4 di e en s a es, i.e., K= 4 di e en alues 43 Hidden Ma ko Model o Indoo Use T acking 44 s −1s s +1 x −1x x +1 Figu e 6.1.: S anda d Hidden Ma ko Model: s is he hidden s a e a iable and x is he obse a ion a ime o s :s ∈ {1,2,3,4}. Each column in he g aph (co esponding o a ime ins an ) con ains nodes ep esen ing he s a es o he HMM. Each node in he g aph has connec ions o a leas one node a an ea lie and one node a a la e ime. The connec ions ep esen he possible ansi ions be ween s a es. The ansi ions om one s a e a one ime ins an o he possible s a es a he la e ime ins an ha e p obabili ies which sum up o 1, i.e., PK j=1 Pi,j = 1, whe e Pi,j =P(s +1=j|s =i), assuming he ansi ion p obabili ies a e independen o ime, gi en any i= 1,...,K. Wi hou any p io knowledge, he ansi ion p obabili ies om one s a e o he nex can be assumed o ollow a uni o m dis ibu ion. 1 1 1 2 2 2 3 3 3 4 4 4 P1,1 P1,2 P1,3 Figu e 6.2.: T ellis diag am wi h ou di e en s a es s ∈ {1,2,3,4}and Pi,j a e he ansi ion p obabili ies om s a e ia one ime ins an o s a e ja he la e ime ins an The emaining pa ame e s o a HMM a e he emission p obabili y dis ibu ions. Fo each HMM s a e, he emission p obabili y dis ibu ion p(x|s )gi es he likelihood o obse ing x a s a e s . Those emission PDFs a e supposed o be di e en amongs s a es. Fo posi ioning pu pose, we iden i y s wi h he posi ion o he use a ime : i s =k hen he use is a he loca ion ℓka ime , whe e ℓkis a wo-dimensional ec o desc ibing he use ’s loca ion. In WiFi inge p in ing based indoo posi ioning sys em, he RSSI aining models and he online RSSI measu emen s can be conside ed as emission p obabili ies and obse a ion sequence, espec i ely. Howe e , wi h s ep de ec ion in o ma ion, he e is ano he se o obse a ions, i.e., mo e- Hidden Ma ko Model o Indoo Use T acking 45 men obse a ions. To inco po a e he s ep de ec ion in o ma ion, he emission p obabili y dis ibu ions o his kind o obse a ion need o be de ined. In ui i ely s ep de ec ion in o - ma ion gi es us he in o ma ion abou mo ing om one s a e o ano he s a e, i.e., he s a e ansi ions, ins ead o he in o ma ion abou a single s a e. So he solu ion he e is o associa e an obse a ion wi h a s a e ansi ion a he han a s a e. This ends up wi h a modi ica ion o he HMM o combine RSSI and s ep de ec ion obse a ions as depic ed in Fig. 6.3. In he ollowing, his model is called modi ied HMM. s −1s s +1 x −1x x +1 +1 Figu e 6.3.: Modi ied HMM o posi ion es ima ion based on he usion o RSSI and mo emen ec- o obse a ions xand , espec i ely. He e, is he wo-dimensional mo emen ec o which deno es he a e sed ou e om s −1 o s . is ob ained om ine ial senso measu emen s. I s compu a ion will be discus- sed in sec ion 6.4. In he modi ied HMM, RSSI measu emen s and mo emen ec o s a e aken as obse a ions a ached o he hidden s a es and he s a e ansi ions, espec i ely. The ansi ion p obabili ies be ween he s a es a e chosen o e lec which posi ions a e accessi- ble om a gi en s a e wi hin one measu emen in e al. Only ansi ions o hose posi ions which a e accessible wi hin one measu emen in e al will be gi en a p obabili y g ea e han ze o. Wi h he Samsung And oid sma phones we ha e a hand, he measu emen in e al is oughly 1,5 s which is he equi ed ime o upda e a new WiFi scan o he de ices. 6.3. Fo wa d Algo i hm Wi h he employmen o a HMM, he es ima ion o he use posi ion can hen be ca ied ou ei he by he Fo wa d algo i hm, he Vi e bi algo i hm o he Fo wa d-Backwa d algo i hm. While Fo wa d algo i hm compu es he p obabili y o being in a ce ain s a e by ga he ing he p obabili ies o e all possible p edecesso s a es, he Vi e bi algo i hm conside s only he mos p obable p edecesso . The Fo wa d-Backwa d algo i hm is only o academic in e es because he induced la ency is no accep able o an online posi ioning sys em. In he ollo- wing, he Fo wa d algo i hm which aims o es ima e he mos p obable s a e a ime ins an gi en he his o y o obse a ions was chosen o decode he use posi ion. Le x1: = [x1,...,x ]be he sequence o RSSI measu emen s up o ime . The s ep de ec ion in o ma ion is ga he ed in he sequence 1: = [ 1,..., ]. Ou goal is o compu e P(s =k| 1: ,x1: ), i.e., he p obabili y o being in he k- h s a e o all possible use posi ions ℓk, k = 1,...,K, gi en all RSSI alues and s ep de ec ion ec o s measu ed so a . Using Bayes’ ule, he p obabili y can be exp essed as ollows: P(s =k| 1: ,x1: ) = p(s =k, 1: ,x1: ) p( 1: ,x1: ) ∝p(s =k, 1: ,x1: ) =: α (k),(6.1) Hidden Ma ko Model o Indoo Use T acking 46 whe e he so-called Fo wa d a iable α (k)is he p obabili y o being a ime in s a e k, while ha ing obse ed he sequence o x1: and 1: . Using ma ginal p obabili y and chain ule, he o wa d a iable can be w i en as ollows α (k) = X i p(s =k, s −1=i, 1: ,x1: ) =X i p( |s =k, s −1=i, 1: −1,x1: −1,x ) ·p(x |s =k, s −1=i, 1: −1,x1: −1) ·P(s =k|s −1=i, 1: −1,x1: −1) ·p(s −1=i, 1: −1,x1: −1).(6.2) Applying he p ope ies o he HMM, which a e depic ed in he g aphical model o Fig. 6.3, i.e., x is independen o all o he a iables i s is gi en, s is independen o x1,...,x −1 and 1,..., −1i s −1is gi en, and is independen o e e y hing i s −1and s a e gi en, and assuming he s ep de ec ion and RSSI in o ma ion o be s a is ically independen o each o he gi en he use loca ion, he abo e o mula simpli ies o α (k) = X i p( |s =k, s −1=i)·p(x |s =k) ·P(s =k|s −1=i)·p(s −1=i, 1: −1,x1: −1) |{z } =α −1(i) (6.3) which is a ecu sion o he o wa d a iable. The inal loca ion es ima e ˆ ℓis ob ained by he weigh ed a e age o e he se Po he mos likely posi ions: ˆ ℓ=1 P k∈P α (k)X k∈P α (k)·ℓk.(6.4) Equa ion (6.3) shows how he di e en knowledge sou ces a e combined. The ansi ion p obabili ies P(s =k|s −1=i)a e nonze o only o hose loca ions ℓk ha can be eached om posi ion ℓiwi hin one ime s ep. The choice o he ansi ion p obabili ies hus enco- des ou knowledge abou he loo plan. The e m p(x |s =k)is he likelihood o he RSSI measu emen x , assuming ha he use ’s posi ion is ℓk. The compu a ion o emission PDF es ima ion and likelihood calcula ion we e discussed in Chap e 4. The mo emen in o ma- ion ga he ed om he s ep de ec ion is cap u ed by he e m p( |s =k, s −1=i), i.e., he likelihood o obse ing he mo emen ec o when mo ing om posi ion ℓi o ℓk. I is no ed ha his de i a ion equi es ha RSSI measu emen s and he mo emen in o - ma ion a e ob ained a he same a e. In ac , he a e o use ’s s eps is o en highe han RSSI measu emen s. To synch onize hese 2sou ces o da a, mo emen ec o is he accumu- la ed esul s o he s ep in o ma ion du ing 1RSSI measu emen in e al. 6.4. Mo emen Vec o Es ima ion In his sec ion, he me hod o in e he s ep de ec ion in o ma ion om da a measu ed by accele a ion senso , gy oscope senso and magne ic senso is summa ized. Mo e de ailed in o ma ion abou mo emen ec o es ima ion can be ound in [71]. Hidden Ma ko Model o Indoo Use T acking 47 The sys em o es ima e he mo emen ec o s is depic ed in Fig. 6.4. a m g gy kak ka′k − G ka′kLP Rdψ b ψ S eps ψ S ep leng h Ls ep 1: k·k Lowpass il e O ien a ion Angula eloci y Kalman il e Mo emen calcula ion S ep de ec ion Figu e 6.4.: S ep de ec ion and posi ion es ima ion sys em o e iew He e, ais he 3-dimen ional accele a ion ec o om he accele ome e o he sma phone, G= 9,81 m/s2is he g a i y cons an , which will be used o pe o m he s ep de ec ion p ocedu e. m,gand gy a e he magne ome e da a, g a i y in o ma ion and gy oscope da a, espec i ely, will be used o es ima e he mo emen heading o he use . The g a i y in o ma ion ec o gis he 3-dimen ional ec o which con ains he o ce o g a i y along h ee axes o he sma phone. Ris he o a ion ma ix, ψis he angle be ween mo emen heading and he magne ic no h di ec ion, and dψ deno es he changes o ψo e ime. The de ail in o ma ion abou hese da a and how o e ie e hem by an And oid applica ion can be ound on he websi e o and oid de elope s [72]. Mo emen ec o is compu ed by accumula ing he es ima ed mo emen o all de ec ed s eps wi hin one RSSI measu emen in e al. 6.4.1. S ep De ec ion The s ep de ec ion p ocedu e is based on he measu ed accele a ion da a om he sma pho- ne. The ob ained samples a e s o ed in 3-dimensional ec o s a= [ax, ay, az]Twhe e x, y, z a e he axes in he coo dina e sys em. This is de ined in he ela ion wi h he sc een o he sma phone as shown in Fig. 6.5(a) (a) De ice coo dina es sys em (b) Wo ld coo dina e sys em Figu e 6.5.: De ice coo dina e sys em (a) and wo ld coo dina e sys em (b) Hidden Ma ko Model o Indoo Use T acking 48 The cha ac e is ics o he use mo emen is cap u ed in he accele a ion da a as depic ed in Fig. 6.6. As can be seen in he example he accele a ion componen along he z-axis clea ly shows he up and down mo ion caused by walking. I is no ed ha he measu ed da a a e always in luenced by he o ce o g a i y. The e o e, o ob ain he eal accele a ion da a, he con ibu ion o he o ce o g a i y mus be elimina ed. One can use his az o s ep de ec ion pu pose i ha is he s able ea u e o he ob ained da a. Howe e , his is jus an example o accele a ion da a when he sma phone was held in he way ha i s sc een is in pa allel wi h he ea h su ace. I he sma phone, howe e , is held in an a bi a y a i ude, he obse ed da a will no ollow he example da a in he igu e any mo e. Because o ha eason, ins ead o using da a om one speci ic componen , he Euclidean no m o he accele a ion da a om all h ee componen s is used o pe o m s ep de ec ion, see Eq. (6.5). Fig. 6.7 shows he calcula ed no m alues a e subs ac ing he o ce o g a i y cons an G=9.81. kak=qa2 x+a2 y+a2 z(6.5) az Samples ay Samples ax Samples Raw Accele a ion Da a [m/s2] 128 192 256 320 128 192 256 320 128 192 256 320 −10 0 10 20 30 −20 0 20 −20 0 20 Figu e 6.6.: Example o aw accele a ion da a i he sma phone is held wi h display in pa allel o ea h su ace when walking To educe he a ia ion caused by senso e o s, a 5-Hz lowpass il e is applied, as is sugges ed in [57]. Fig. 6.8 shows he ou pu signal o he lowpass il e . He e, a Bu e wo h low-pass o o de 20, wi h a cu o equency o 0.2· n= 5Hz, whe e nco esponds o a hal o he sampling a e (50Hz/2) is used. In he li e a u e, wo common app oaches which ha e been employed in many esea ch o de ec s eps using accele a ion da a a e peak de ec ion and ze o c ossing coun . In [57] a peak de ec ion wi h a minimum h eshold is used o coun he s eps. This has he disad an age ha some imes he maximum is spli in o wo local maxima and hus an addi ional s ep is de ec ed. The o he me hod o de ec ing s eps calcula es he ze o c ossing a e o he Hidden Ma ko Model o Indoo Use T acking 49 ka′k=kak−G Samples 128 192 256 320 −10 −5 0 5 10 Figu e 6.7.: No m o accele a ion da a a e subs ac ing he o ce o g a i y cons an G=9.81 accele a ion da a [59], which in case o noisy da a may also esul in addi ionally de ec ed s eps. The p oposed app oach combines he s eng hs o peak de ec ion and ze o c ossing coun app oaches by coun ing he c ossing a e o accele a ion da a o e a ce ain h eshold. Ou me hod is hen able o cope wi h mul iple local maxima and noisy da a in he measu ed accele a ion da a. Fig. 6.9 illus a es he h eshold-c ossing app oach. The s ep coun e is only inc eased i he h eshold (magen a line) is la ge han he sa e y h eshold ( ed line) o a oid an e oneous coun ing when he use is no mo ing bu he e is s ill some luc ua ion o he obse ed accele a ion da a. The h eshold is no a ixed alue h ough he whole p ocess, ins ead i is calcula ed o e e y single da a block. He e a block size o 64 samples is chosen ega ding he sampling a e o accele a ion senso and p ocessing speed. Fi s , o each da a block, he peak mean is de e mined by calcula ing he mean o all he de ec ed peaks. Second, he h eshold is compu ed by mul iplying he peak mean wi h an expe imen ally de e mined weigh ing ac o . In ou expe imen , he weigh ing ac o o 0.6p oduces he bes esul s. Recen ly, ano he me hod, namely au oco ela ion me hod, o de ec ing s eps has been p oposed in [32]. This me hod employs he ac ha he accele a ion da a exhibi s a e y epe i i e pa e n. The ad an age o he au oco ela ion me hod compa ed o peak de ec ion and ze o c ossing coun me hods is ha i is able o elimina e he hand ges u es when he use is no mo ing, ansi ion om si ing o s anding and ice e sa, and so on. Acco ding o ou expe imen al esul s, he au oco ela ion me hod ou pe o ms he o he wo adi ional me hods. The e o e, a he cu en s a e o ou p ojec , his me hod is employed in ou sys em o de ec ing s eps. Howe e , his sec ion aims o p esen ou p oposed app oach o de ec s eps which has be- Expe imen al Resul s on Indoo Posi ioning 56 Table 7.1.: Mean and s anda d de ia ion o APs a 2 posi ions o a i ical da a expe imen AP index AP1 AP2 AP3 AP4 AP5 µ1,i -102 -103 -97 -89 -95 µ2,i -105 -100 -99 -86 -101 σ1,i 4.8 4.9 5.0 5.2 5.1 σ2,i 5.0 4.8 4.8 5.4 5.0 Table 7.2.: Classi ica ion e o a e on a i icial censo ed da a Me hod E o a e (%) Plain ng + ecog 30.7 EM ng + plain ecog 26.9 EM ng + censo ed ecog 22.5 3-s onges APs 35.1 1-nea es neighbo 36.8 •Plain aining ( ng) + ecogni ion ( ecog): ML pa ame e es ima ion is ca ied ou assuming no mally dis ibu ed, uncenso ed da a. Also ecogni ion is pe o med dis e- ga ding any censo ing. •EM ng + plain ecog: ML pa ame e es ima ion in aining accoun s o he censo ed da a using he p oposed EM algo i hm, while he p esence o censo ed da a is s ill dis ega ded in ecogni ion. •EM ng + censo ed ecog: T aining wi h he p oposed EM algo i hm and ecogni ion employing eq. (4.52). •3-s onges APs: Selec h ee s onges APs o each loca ion in he aining phase, hen apply EM ng + censo ed ecog. •1-nea es neighbo classi ica ion ule. Table 7.2 clea ly shows he supe io i y o he schemes which a e awa e o he censo ing. Conside ing he p esence o censo ed da a in aining imp o ed he e o a e om 30.7% o 26.9%, and a u he imp o emen o 22.5% is ob ained by accoun ing o censo ed da a also in ecogni ion. We can also see he impo an ole o weak APs o he ecogni ion accu acy: using only he h ee s onges APs aises he e o a e o 35.1%.1-nea es neighbo pe o ms wo s wi h he e o a e o 36.8%. Field Da a In o de o e alua e he pe o mance o he p oposed EM algo i hm on eal wo ld da a, eal WiFi RSSI measu emen s we e ga he ed on a loo o an o ice building consis ing o 10 o ice ooms and a long aisle ha ing an o e all size o 12 m by 30 m (see Fig. 7.1). RSSI alues we e aken a 25 di e en posi ions, oughly e enly dis ibu ed, esul ing in an a e age dis ance o 2,7 m be ween wo loca ions. Two measu emen campaigns we e ca ied ou using a sma phone, wi h 100 measu emen s aken pe posi ion pe campaign. Da a o he Expe imen al Resul s on Indoo Posi ioning 57 i s measu emen campaign se ed he pu pose o aining and he second one was o es ing pu poses. Fo he aining da a se , he pe cen age o uncenso ed obse a ions, a e aged o e all APs which we e obse able a each loca ion, was ound o be 36.7%. Figu e 7.1.: Floo plan o he a ea whe e ield da a has been conduc ed. To compa e he p oposed app oach o a s a e-o - he-a sys em, he algo i hm om [52] was implemen ed. The e, o each loca ion ℓk he p obabili y dis ibu ions o he 10 s onges APs a e de e mined du ing he aining phase and compa ed o hose o he online measu- emen s employing he Bha acha yya coe icien . A 3-nea es neighbo ule is hen applied o decide on he use loca ion. Fu he mo e, we compa ed wi h he well-known sys em, RA- DAR [18], whe e classi ica ion is pe o med wi h a 3-nea es neighbo ule, employing he Euclidian dis ance. When applying he p oposed algo i hm, 5online measu emen s we e used o each es ima e, he inal loca ion es ima e was hen compu ed using he 3mos likely posi ions, see Eq. (4.53). I is no ed ha 5online measu emen s we e also employed in he implemen a ion o he algo i hm o [52]. Fig. 7.2 shows he cumula i e dis ibu ion unc ion (CDF) o he e o as a unc ion o he dis ance o each me hod. The CDF is de ined as he p obabili y ha he posi ioning e o ǫ is lowe han a ce ain dis ance d: CDFǫ(d) = P(ǫ≤d)d≥0.(7.1) The esul s in Fig. 7.2 show ha he p oposed me hod ou pe o ms he o he wo, especial- ly o he 40% e o quan ile. No e, also, ha he compu a ional cos o he p oposed me hod du ing he online phase is smalle han hose o compu ing he Bha acha yya dis ances be - ween p obabili y dis ibu ions [52] o he nea es -neighbo based [18] me hods. A e [18] A e [52] P oposed algo i hm Dis ance d[m] CDFǫ(d) 0 2 46 8 10 0 0.2 0.4 0.6 0.8 1 Figu e 7.2.: CDF o he posi ioning e o o di e en sys ems. Expe imen al Resul s on Indoo Posi ioning 58 Table 7.3.: Classi ica ion e o a e on a i icial censo ed and d opped da a Me hod E o a e (%) EM ng + censo ed ecog 29.7 Ad . EM ng + censo ed & d opped ecog 25.7 7.1.2. EM algo i hm o censo ed and d opped da a In he ollowing, expe imen al esul s showing he e ec i eness o EM algo i hm when being awa e o d opped da a in addi ion o censo ed da a a e p esen ed. Fo con enience, le us call he EM algo i hm o pa ame e es ima ion o censo ed and d opped da a as he ad anced EM algo i hm. A i icial Da a The a i icial da a we e gene a ed acco ding o he pa ame e s which we e used in Sec i- on 7.1.1, howe e , he means o all APs a all posi ions a e inc eased by 10 dBm o show mo e impac o d opped da a on he es ima ed pa ame e s and consequen ly on classi ica ion esul s. The gene a ed Gaussian da a we e i s censo ed and hen d opped wi h he d opping a e o 20%. Since he e ec i eness o EM algo i hm o censo ed da a has been p o ed in he p e ious sec ions, his sec ion aims o compa e he classi ica ion esul s be ween wo p oposed EM algo i hms o show he impo ance o he awa eness o d opped da a besides censo ed da a. Table 7.3 shows he classi ica ion esul s on censo ed and d opped da a using he p oposed EM algo i hms. I can be seen ha he pa ame e es ima ion o mulas de i ed in Sec ion 4.2.3 a e able o cope wi h an unknown d op-ou a e, which leads o he imp o emen in he classi ica ion e o a e om 29.7% o 25.7% i indeed d op-ou s occu . The eason is i all unobse able da a a e conside ed as censo ed da a, he pa ame e s es ima ed by he EM algo i hm which is awa e o censo ed da a only, a e inaccu a e, i.e., he es ima ed means a e biased o he le o he ue means and he es ima ed a iances a e highe han he ue a iances. These inaccu a e es ima ed pa ame e s cause he deg ada ion o he classi ica ion pe o mance. Field Da a To demons a e he e ec i eness o he ad anced EM algo i hm, an expe imen on eal ield da a has been done. This expe imen was done by using he same da a se s as used in he ex- pe imen o he EM algo i hm o censo ed Gaussian da a. Fig. 7.3 shows he expe imen al esul s on eal ield da a using he wo p oposed EM algo i hms. As can be seen, being awa e o d opped da a imp o es he posi ioning accu acy, hough he imp o emen is mode a e. The easons migh be ei he he amoun o aining da a, 100 samples pe posi ion, a e no su icien o he ad anced EM algo i hm since i ies o es ima e mo e pa ame e s, o he ue d opping a es a e low. Since he da a se s we e ga he ed in a eal indoo en i onmen , ue d opping a es a e unknown. As ou s udy on he esul s, he es ima ed means and a- iances om ad anced EM a e be e han he EM algo i hm awa ing o censo ed da a only, and he es ima ed d opping a e a e aged o e all APs and all loca ions was app oxima ely Expe imen al Resul s on Indoo Posi ioning 59 0.37, we concluded ha he limi ed imp o emen o posi ioning accu acy is because o he no e y accu a e es ima ed d opping a es. As p esen ed in sec ion 4.3, d opping a e plays an impo an ole in likelihood calcula ion o he unobse able da a, so he inaccu a e es i- ma ed d opping a es limi he imp o emen in classi ica ion esul s. In he bad case o low accu acy o he es ima ed d opping a es, he posi ioning accu acy migh e en dec ease. EM Censo ed Ad . EM CDFǫ(d) Dis ance d[m] 01 2 3 4 56 7 8910 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figu e 7.3.: Compa ing he expe imen al esul s on eal da a using 2p oposed EM algo i hms 7.2. Sma phone Adap a ion This sec ion in es iga es he impac o he p oposed adap a ion app oach on he accu acy o inge p in ing based indoo posi ioning. Some a ia ions o he adap a ion app oach wi hin he MLLR amewo k such as mean and a iance adap a ion, mean adap a ion only wi h ull o diagonal adap a ion ma ix ha e been implemen ed. In addi ion, he impac o he numbe o adap a ion da a and o he numbe o eg ession classes on he pe o mance o adap a ion, and consequen ly on he posi ioning accu acy, a e also in es iga ed. 7.2.1. Classi ica ion on A i icial Da a In his se o expe imen s, a i icial da a we e gene a ed o assess he impac o he d op-ou a e and he amoun o adap a ion da a on he classi ica ion pe o mance. 1000 samples o aining da a we e gene a ed o each o K= 6 posi ions and o NAP = 2 access poin s. A single a ine ans o ma ion o he means o he aining da a is used o each loca ion acco ding o ˆ µk=Aµk+b, whe e A= [0.9,0; 0,0.9] and b= [−3,2]T. Adap a ion and es da a we e hen gene a ed by sampling om he ans o med Gaussians wi h means ˆ µkand a iances as hose o he aining da a. While he amoun o adap a ion da a was g adually inc eased om 1% o 5, 10, 50 and 100% o he numbe o aining samples, he numbe o es samples was ixed a 100 obse a ions pe posi ion. The alue o he d op-ou a e is se o 0% (no d op-ou s, π= 0) and 20% (π= 0.2). Al hough he e ec i eness o being awa e o d opped da a on classi ica ion esul s was demons a ed in sec ion 7.1.2, o he con enience o e alua ing he e ec i eness o he adap a ion app oach, classi ica ion esul s employing ad anced EM wi h he cu en se o pa ame e s a e s ill p esen ed. Table 7.4 discusses he impac o he d op-ou a e wi hou Expe imen al Resul s on Indoo Posi ioning 60 Table 7.4.: Classi ica ion esul s (% posi ions co ec ly classi ied) when aining and es da a a e gene a ed om he same pa ame e se Awa e o d opping yes no π= 0 86 86 π= 0.275 64 Table 7.5.: Adap a ion pe o mance (% posi ions co ec ly classi ied) on a i ical da a as a unc ion o amoun o adap a ion da a No. Pos wi h adap . da a π0% 1% 5% 10% 50% 100% 3 0 68 81 85 85 86 86 0.2 56 70 73 74 75 75 6 0 68 85 86 86 86 86 0.2 56 72 75 75 75 75 adap a ion. He e, he ’adap a ion’ da a we e used o a comple ely new aining om sc a ch o each o he 6posi ions. 1000 samples o adap a ion da a we e used in his expe imen . As can be seen, i no d opping occu s, he wo EM algo i hms p oduce he same classi ica ion esul . Once he d opping occu s, he ad anced EM algo i hm ou pe o ms he o he . Table 7.5 shows he e ec i eness o applying he p oposed adap a ion algo i hm. As can be seen, he esul s when only 3loca ions ha e adap a ion da a is compa able o he case when adap a ion da a a e a ailable o all 6loca ions. The eason is e en i he e a e only 3loca ions wi h adap a ion da a, s ill he PDFs o all loca ions a e well adap ed, since hey a e om a single eg ession class. Howe e , i is also no iceable ha when he numbe o adap a ion da a pe posi ion is small, i.e., less han 10% o he aining da a, classi ica ion esul s using 6posi ions wi h adap a ion da a a e abou 1 o 4% be e han hose when adap a ion da a a e only a ailable o 3posi ions. The column wi h 0% o adap a ion da a shows he esul s when no adap a ion is pe o med. Compa ing wi h he esul s o Table 7.4 i is clea ha he classi ica ion esul s a e adap a ion app oach hose o e aining, excep when he e is only 1% o adap a ion da a, e en i adap a ion da a a e only a ailable o 3ou o 6posi ions. Fo he case o only 1% o adap a ion da a a ailable, he e is s ill a d ama ic imp o emen in classi ica ion esul s when adap a ion is employed compa ed o he case i no adap a ion is pe o med. 7.2.2. Classi ica ion on Field Da a To examine he e ec i eness o he adap a ion app oach on eal wo ld da a, measu emen s we e collec ed on 3 loo s o an o ice building wi h oughly 30 ooms (lec u e halls, o ice and labo a o y ooms), whe e each loo has an o e all size o 35 m by 35 m, RSSI alues we e aken a 60 di e en posi ions wi h an a e age dis ance o 5.0m be ween 2posi ions. 200 measu emen s we e aken pe posi ion wi h 2di e en sma phones a each posi ion. Da a o he i s sma phone we e used o es ima e he aining models while da a om he second sma phone we e di ided in o 2se s a each posi ion. The i s was used o adap a ion (0, 5, 25, o 75 samples) o e aining (150 samples) om sc a ch and he second was o es ing (50 samples). Fo he measu ed da a, he es ima ed d opping a e a e aged o e all Expe imen al Resul s on Indoo Posi ioning 61 Table 7.6.: RMS posi ioning e o (in [m]) as a unc ion o he amoun o posi ions ha ing adap a ion da a and he amoun o ada a ion da a Condi ion Adap a ion Me hod No. Pos wi h Amoun o µ&σ2,µonly, µonly, adap . da a adap . da a A ull A ull Adiag 0 6.15 15 5 5.08 4.76 3.93 25 4.60 4.41 3.93 75 4.62 4.52 3.93 30 5 3.86 3.96 3.84 25 3.82 3.96 3.85 75 3.76 3.91 3.85 60 5 3.74 3.93 3.84 25 3.67 3.87 3.83 75 3.65 3.87 3.84 e ain 2.43 APs and all posi ions was app oxima ely 0.3. Fo he adap a ion p ocedu e, he es ima ed aining means we e so ed a each posi ion in descending o de , and only he 8s onges APs we e used o es ima e he adap a ion ma ices since he con ibu ion o he emaining APs o he likelihood was negligible. The adap a ion ma ices a e hen used o calcula e he adap ed pa ame e s o he 8s onges APs a all posi ions being in he same eg ession class. Table 7.6 shows he dependency o he posi ioning accu acy on he numbe o loca ions o which adap a ion da a a e a ailable and he amoun o a ailable adap a ion da a a each posi ion. The esul s in Table 7.6 a e he a e age o he oo mean squa e (RMS) posi ioning e o o 50 expe imen s. In each expe imen , es da a, adap a ion da a and he posi ions wi h adap a ion da a a e andomly selec ed. As expec ed, he mo e posi ions he e a e wi h adap- a ion da a and he mo e adap a ion samples pe posi ion, he be e he posi ioning esul s. Fo all conside ed adap a ion me hods, imp o emen s in posi ioning accu acy a e ob ai- ned, e en when e y ew adap a ion da a a e a ailable. Howe e , mean and a iance adap a- ion wi h a comple ely illed ma ix Adeli e s only he bes esul s i he e a e su icien ly many adap a ion da a, while he use o only mean adap a ion wi h a diagonal Ais supe io i only ew adap a ion da a a e a ailable, since ewe pa ame e s need o be es ima ed. F om he p esen ed esul s, indoo posi ioning sys em de elope s could ha e an idea o which and how pa ame e s can be e icien ly adap ed gi en he a ailable adap a ion da a. Howe e , he posi ioning accu acy when all 60 posi ions ha e adap a ion da a is s ill well below he accu acy achie able, when aining and es da a a e collec ed by he same sma - phone, which is 2.43 m. The eason could be one ans o ma ion ule could no desc ibe well he ela ionship be ween aining and adap a ion da a o all posi ions which ac ually ollows a nonlinea ule as discussed in chap e 5. To elax he linea i y assump ion be ween aining and adap a ion da a, mul iple eg essi- on classes a e employed. Fo es ima ing he eg ession classes, he means o he 8s onges APs a each posi ion a e so ed in descending o de , he esul ing 8-dimensional ec o s a e Expe imen al Resul s on Indoo Posi ioning 62 Table 7.7.: E ec o numbe o clus e s on RMS posi ioning e o No. o clus e s 1 2 3 RMS pos. e o [m] 3.76 3.48 3.25 clus e ed using k-means, as discussed in sec ion 5.3. Table 7.7 shows he posi ioning esul s when doing clus e ing and es ima ing di e en adap a ion ma ices o each clus e , assu- ming ha in each clus e 50% o posi ions ha e adap a ion da a wi h 75 adap a ion samples pe posi ion. I has o be no ed ha he op imal numbe o eg ession classes depends on he amoun o a ailable adap a ion da a. The mo e adap a ion da a he mo e ans o ma ion ma- ices can be eliably es ima ed. In ou se up he bes esul s we e achie ed wi h 3clus e s, which led o a educ ion o he RMS posi ioning e o om 3.76 m o 3.25 m. 7.3. HMM o Indoo Use T acking This sec ion e alua es he e ec i eness o he p oposed algo i hms discussed in Chap e 6 o an indoo posi ioning p oblem bo h on a i icially gene a ed da a and eal ield da a, especially he impac o he in oduc ion o pseudo s a es on he classi ica ion/posi ioning esul s. 7.3.1. A i icial Da a Fig. 7.4 shows he HMM s a es wi h he loca ions o he egula and pseudo s a es ma ked wi h ed and g een ci cles, espec i ely. 530 535 540 545 550 555 560 565 985 990 995 1000 1005 1010 1015 x[m] y[m] Figu e 7.4.: The modi ied HMM s a es, i.e., he allowable use posi ions, and he ansi ions be ween hem. Red and g een ci cles indica e egula and pseudo s a es, espec i ely. The RSSI measu emen s o 15 andomly placed APs o aining he Gaussian emission PDF o he egula HMM s a es a e gene a ed a i icially as ollows: The signal s eng h ol- Expe imen al Resul s on Indoo Posi ioning 63 Table 7.8.: Mean posi ioning e o on a i icial da a Me hod Mean e o [m] RSSI only [74] 1.74 RSSI + s ep de . [73] 1.37 RSSI + s ep de . + pseudo s a es [75] 1.02 lows a la ge-scale log-no mal ading model wi h an addi ional ze o mean Gaussian andom a iable wi h s anda d de ia ion σL= 5 o model small-scale ading. A each posi ion, we gene a ed a se o 300 RSSI measu emen s as he aining da a, hen es ima ed he pa ame e s o RSSI dis ibu ion o each AP a each posi ion as desc ibed in Chap e 4. The s ep de ec ion in o ma ion is modeled as a bi a ia e Gaussian wi h he mean µi,j = ℓj−ℓiand a diagonal co a iance ma ix Σ wi h en ies Σ [1,1] = Σ [2,2] = 0,25 m2. This alue has been de e mined in o line expe imen s. In he expe imen , pseudo s a es we e in oduced in he way ha he Euclidean dis ance be ween neighbo ing s a es is no mo e han a mos 0,75 m (which is close o he measu ed a e age s ep leng h wi hin he expe imen al da a), while he dis ance be ween wo neighbo- ing egula s a es is abou 3-5m excep o some special a eas such as he s ai s. The o al numbe o pseudo s a es is 125, which has o be compa ed o he numbe o 81 egula s a es. In he expe imen s we assume ha a use canno mo e as e han 3 m/s. Use mo emen is simula ed by a andom walk on he HMM g aph. Since mo emen ec o s and RSSI mea- su emen s a e gene a ed e e y 1,5 s, only a limi ed numbe Uo HMM s a es can be eached om any gi en s a e s −1=i(on he a e age U=15 s a es) and he co esponding ansi ion p obabili y P(s =j|s −1=i)is se o 1 U. Fo all s a es ou side his neighbo hood he co e- sponding ansi ion p obabili ies a e se o ze o. Table 7.8 p esen s he mean posi ioning e o in me e s, a e aged o e 100 expe imen s, whe e each expe imen co esponds o a di e en andom walk on he HMM g id o leng h o abou 200 m. We compa e he pe o mance o he p oposed algo i hm wi h ou ea lie wo k o [73], which also used RSSI and s ep de ec ion in o ma ion, howe e wi hou he in oduc ion o pseudo s a es. Using pseudo s a es imp o es he mean posi ioning e o om 1,37 m o 1,02 m. I s ep de ec ion in o ma ion is neglec ed and use posi ioning elies only on RSSI in o ma ion, a mean posi ioning e o o 1,74 m is ob ained. 7.3.2. Field Da a The p oposed app oach is also e alua ed wi h he ield da a eco ded in he o ice building depic ed in Fig. 7.1. In he aining phase, 100 RSSI measu emen s pe posi ion we e collec- ed and he RSSI dis ibu ion we e es ima ed as desc ibed in Chap e 4. In he online es ing phase, he sma phone use andomly wen h ough he whole loo a ea o collec he es da a, i.e., WiFi da a and ine ial senso da a. Two di e en ajec o ies we e eco ded, each ajec o y consis s o app oxima ely 140 es po i ions. The posi ion es ima e was pe o med e e y 1,5 s using he p oposed app oach. The s ep de ec ion in o ma ion is modeled in he same ashion as in he expe imen s using a i icial da a. Acco ding o he ob ained da a and expe imen al esul s, i seemed ha he s ep de ec- ion in o ma ion is mo e eliable han he RSSI in o ma ion, as a consequence, a heu is ic Expe imen al Resul s on Indoo Posi ioning 64 weigh ing ac o λwas in oduced in he calcula ion o he o wa d a iable as ollows α (j) = [p(o |s =j)]λX i [p( |s =j, s −1=i)](1−λ)P(s =j|s −1=i)·α −1(i).(7.2) Fo he de e mina ion o λa jackkni e p ocedu e was employed: The da a o 1ou o he 2 ajec o ies was used o he es ima ion o λ, whe eas es s we e conduc ed on he held-ou da a. This was epea ed 2 imes, e e y ime, one ajec o y was used o es ima e λ. Wi h ou collec ed da a, he es ima ed alue was always λ≈0.003. The e y small alue o λ can be explained as ollows: since he likelihood o RSSI da a is much smalle han he likelihood o s ep de ec ion in o ma ion, his alue o λwould help o a oid he p oblem ha he con ibu ion o he likelihood o s ep de ec ion in o ma ion o he calcula ed o wa d a iable is domina ed by he likelihood o RSSI da a. The p oposed me hod was compa ed o ou ealie wo k: i s , using RSSI only o use posi ioning [74], and second, using HMM model as desc ibed in Chap e 6, howe e wi hou he in oduc ion o pseudo s a es [73]. Fo bo h ajec o ies he expe imen al esul s showed ha he new app oach ou pe o ms he o he s, especially o he 90% e o quan ile, in e ms o he CDF o he posi ioning e o Eq. (7.1). Al hough he es a ea is limi ed, he expe imen al esul s in Fig. 7.5 indica e ha he p oposed app oach is signi ican ly be e han he o he app oaches. RSSI only RSSI + S ep De . RSSI + S ep De . + Pseudo S a es CDFǫ(d) Dis ance d[m] 0 1 2345 6 78 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figu e 7.5.: CDF o he posi ioning e o o di e en sys ems. A e age o e 2 es ajec o ies. 8. Se e Based Indoo Na iga ion Sys em Na iga ion Se e Clien s Figu e 8.1.: Se e based Indoo Na iga ion Sys em. Toge he wi h de eloping heo e ical algo i hms o he enhancemen o posi ioning accu- acy, a eal indoo na iga ion sys em has been de eloped o o e a pe iod o h ee yea s wi h he con ibu ions om many employees and s uden s a he Depa men o Communica ions Enginee ing, Uni e si y o Pade bo n. The sys em is he esul o ou a emp o b ing he heo e ical esea ch esul s in o eali y. Be o e building he indoo na iga ion sys em, one needs o answe he ques ion: Which c i e ia mus he sys em mee ? To ou unde s anding, he mos impo an c i e ia o e alua e a eal sys em a e cos s, ease o deploymen , s abili y, secu i y, scalabili y and main ainabili y. The be e he c i e ia a e ull illed, he be e he sys em is. Sys em a chi ec u e has he mos impo an impac on hose c i e ia. As a consequence, o achie e he equi ed c i e ia he bes , di e en sys em a chi ec u es ha e been analysed. The e exis wo common a chi ec u- es o posi ioning sys ems: Fi s , sma phone based sys em in which all da a a e s o ed and p ocesses a e pe o med on he sma phone i sel , e.g., he solu ions om WIFARER [77], 65 Se e Based Indoo Na iga ion Sys em 72 ma ke id lon la oom laye e ma ke loca ions connec ionid ma ke id linkedma ke dis ance highway access ma ke connec ions Table 8.1.: Map in o ma ion ables in he da abase: s o age o he geog aphical in o ma ion o in- ge p in posi ions, and he in o ma ion abou he di ec connec ion be ween any pai o posi ions. we decided o combine all he sepe a e OSM map iles in o one global OSM map ile. The na iga ion da abase is hen buil up by impo ing he in o ma ion om he global map ile o he Pos GIS da abase. The ende ed map iles a e s o ed on he se e and deli e ed o he clien s o map dis- playing pu poses once he se e ecei es map eques s om clien s. Map in o ma ion s o ed in he na iga ion da abase is he in o ma ion o inge p in ing posi ions and he connec ions among hose. This in o ma ion is ga he ed om he s anda d ables by a C++ p og am and s o ed in wo new ables (see Table 8.1 o an illus a ion). The bold ields a e he so-called unique keys o mas e keys o he ables. Table ma ke loca ions con ains all he in- o ma ion abou any inge p in ing posi ion such as longi ude, la i ude, oom numbe , laye ( loo le el) and e (co ido , oile , ec .). These inge p in ing posi ions a e he posi ions whe e aining da a will be collec ed. I is e y con enien o dis ibu e he aining posi ions du ing he map edi ing p ocedu e since one can choose he c i ical and in e es ing poin s on he maps o ma k hem as inge p in ing poin s. I is also easy o manage he densi y o he inge p in g id. Table ma ke connec ions con ains he in o ma ion abou he di ec connec ion be ween any pai o posi ions, e.g., ID o cu en conside ed posi ion (ma ke id), ID o he connec ed posi ion (linkedma ke ), spa ial dis ance be ween hese posi ions (di- s ance), ype o connec ion (highway), i.e., oo way o bicycle, and accessibili y in o ma ion (access) o indica e whe he he connec ion is blocked o i he e is ee access. The in o ma- ion in Table ma ke connec ions is needed o he ou e es ima ion p ocedu e and he employmen o he HMM which was discussed in Chap e 6. 8.2.2. RSSI da a RSSI da a a e di ided in o wo ypes: aw da a (RSSI measu emen s) and p ocessed da a ( he es ima ed aining models). Table 8.2 shows he ables in he da abase which s o e he measu emen da a. The e a e se e al easons o s o ing he aw da a. Fo example, aw da a can be used o u u e esea ch, i.e., i new algo i hms a e used o es ima e he aining model om measu ed da a, o o compa ison pu poses, i.e., e alua ing he posi ioning accu acy o o he algo i hms on he same da a se . By s o ing he aw da a in he way as desc ibed in Table 8.2, i is possible o e-gene a e he o iginal measu emen s. Tables scan esul s, measu emen s and ha dwa ein o ma ion a e used oge he o s o e he RSSI mea- su emen s, he posi ion, ID and o ien a ion o he de ice, and he ime s amp o he measu- Se e Based Indoo Na iga ion Sys em 73 scanid measu emen id mac ssi scan esul s measu emen id lon la laye hwid o ien a ion measu emen ime measu emen s hwid modelid uniqueid ha dwa ein o ma ion Table 8.2.: Measu emen ables in he da abase: con ain he RSSI measu emen s, he posi ion, ID and o ien a ion o he de ice which has been used o collec da a, and he ime s amp o he measu emen s pa ame e id ma ke id macid numbe o measu emen numbe o obse a ion mean a iance a e sumo obse a ions sumo squa e upda ed ime pa ame e s macid bssid macadd esses Table 8.3.: T aining model in o ma ion ables in he da abase: con ain he es ima ed pa ame e s and he su icien s a is ics in o ma ion. emen s. The in o ma ion abou he sma phone ha dwa e is s o ed o suppo he adap a ion p ocedu e as desc ibed in chap e 5. In addi ion o s o ing he aw da a, he RSSI da abase also con ains he ables o s o e he in o ma ion o he es ima ed aining models as well as he su icien s a is ics in o ma ion as depic ed in Table 8.3. Table pa ame e s s o es he es ima ed pa ame e s as well as he su icien s a is ics in o ma ion. The es ima ed pa ame e s a e he means, a iances and a e (d opping a e) o he obse ed RSSI da a o he AP, speci ied by macid, a a loca i- on, speci ied by ma ke id. Su icien s a is ics pa ame e s such as numbe o measu emen s, numbe o obse a ions, sum o obse a ions and sum o obse a ions squa ed, a e used o inc emen ally upda ing he aining model o educe he es ima ion ime using he EM algo- i hm. Ob iously, es ima ion ime is no a se ious p oblem when he amoun o aining da a is small. Howe e , aining da a a e collec ed o e he cou se o ime and model es ima i- on om sc a ch would ake much mo e ime han applying he inc emen al upda e me hod. Fo con enience o AP sea ching, he able macadd esses is c ea ed o s o e he MAC add esses o all he obse ed access poin s. Se e Based Indoo Na iga ion Sys em 74 8.3. Sha ed Memo y As men ioned abo e, sha ed memo y is used o ep esen he da abase in o de o speed up he da a eading p ocedu e. The s uc u e o he da a s o ed in sha ed memo y is simila o he s uc u e used in he Pos GIS da abase as p esen ed in he p e ious sec ion. Sepe a e sha ed memo y a eas a e c ea ed o s o e he in o ma ion o inge p in ing posi ions, connec ions, MAC add esses and ained models. In addi ion, o speed up he localiza ion p ocedu e, o he a eas a e c ea ed o s o e he in o ma ion o he p e-compu ed posi ions a which a gi en MAC add ess was obse ed in he aining da a. The eason is ha in inge p in ing based posi ioning, online measu ed da a a e compa ed wi h he aining da a in he da abase o come up wi h he decision o he use posi ion es ima e. Ob iously he bigge he da abase becomes, he mo e compu a ion ime is needed o comple e he compa ison p ocedu e. To ge id o his p oblem, ins ead o compu ing he simila i y be ween he online measu emen wi h he whole da abase, he localiza ion module has o compu e i a he possible posi ions only, whe e a leas one o he APs, which is p esen in he online measu emen , was obse ed in he aining da a. 8.4. O e iew o Sma phone Applica ion This sec ion p esen s an o e iew o he sma phone applica ion. Al hough he de eloped applica ion is able o suppo many asks, e.g., da a ga he ing, localiza ion, na iga ion, social ea u es (pee g oup inding) and 3-D ep esen a ion o he ou ing pa h, in he ollowing, only he s uc u e which is ela ed o he main ea u es, which a e localiza ion and na iga ion, is p esen ed. Fig. 8.5 gi es an o e iew o he educed e sion o he sma phone applica ion a chi ec- u e as a class diag am. I should be no ed ha in Fig. 8.5, he de ails o each class, i.e., a iables and me hods, a e no p esen ed since his would ex emely ex end he size o he g aph. In he ollowing, he ole o each class in he applica ion and he ela ion o classes will be summa ized. As can be seen in Fig. 8.5, besides he egula classes, he e a e se e al special ypes o classes in an And oid applica ion such as ac i i y, se ice, applica ion which ope a e di - e en ly [72] as summa ized below: •Ac i i y: An ac i i y is an applica ion componen ha p o ides a single window wi h a use in e ace ha in e ac s wi h he use s in o de o do an ac ion. An applica ion may consis o se e al ac i i ies o handle di e en asks. An ac i i iy can s a ano- he ac i i y. Once a new ac i i y s a s, he p e ious ac i i y is s opped. The sys em p ese es he p e ious ac i i y in a “las in, i s ou ” s ack. The e o e, when he use is done wi h he cu en ac i i y and p esses he “Back” bu on o he sma phone, he p e ious ac i i y will esume. •Se ice: A se ice is an applica ion componen ha can ope a e in he backg ound and does no p o ide a use in e ace. A se ice can be s a ed by ano he applica ion componen , i.e., ac i i y, and keeps unning in he backg ound e en i he use swi ches o ano he applica ion. Se e al ypes o se ice implemen a ion can be used o manage Se e Based Indoo Na iga ion Sys em 75 Figu e 8.5.: Class diag am o Ja a applica ion a chi ec u e he li ecycle o a se ice ha he se ice may s op i sel when i comple es he ask o i s binding componen s ops o an ac i i y s ops i . •Applica ion: An applica ion is a base class o main ain global applica ion s a e. In ou applica ion, he main ac i i y is “Indoo Na Map” which is launched when he use s a s he applica ion. All o he ac i i ies, i.e., WiFi scanning, se ing, e c., can be ac i a ed om his ac i i y, see he mani es ile o he applica ion in Appendix A.3.1 o in o ma ion o all ac i i ies. Fig 8.6 shows a snapsho o he applica ion when i is launched by he use s. In he ollowing discussion, he ela ion be ween he classes in Fig. 8.5 and he ole o each class a e summa ized: •“Indoo Na Map” uses he se ing in o ma ion and ini ialized in o ma ion s o ed in “MapSc ollApplica ion” and he app op ia e me hods implemen ed in “CellMapSu a- Se e Based Indoo Na iga ion Sys em 76 Figu e 8.6.: Ja a applica ion showing loo plan in o ma ion and use posi ion ( ed do ). ceView” o display map and o he in o ma ion such as cu en use posi ion, na iga ion pa h, e c. . •“MapSc ollApplica ion”, he applica ion class, is implemen ed o main ain all he glo- bal s a es and he da a which a e needed o map displaying, o line localiza ion, na i- ga ion, and so on. Fo map d awing, posi ioning and na iga ion, “MapSc ollApplica i- on” s o es he me ada a o map da a, i.e., objec s o “Map” and “MapTile” classes, and he in o ma ion o localiza ion and na iga ion pu poses, i.e., “Bina yModel“, ”Ma- cObjec “, ”NodeObjec “ and ”Rou ingPa hElemen “, as descibed in sec ion 8.2. •“CellMapSu aceView” ex ends he “Su aceView” class p o ided by And oid APIs, and con ains all me hods o manage map d awing and use in e ac ion. •“ISe e Connec ion” is he in e ace decla ing all he me hods which a e implemen- ed in “H pSe e Connec ion” o suppo he communica ion p ocedu e be ween he applica ion and he se e . These me hods can be called by “Indoo Na Map” o da a downloading o by “Localiza ionSe ice” in case o doing localiza ion in online mode. •As all he messages a e in he p e-de ined XML o ma , me hods o w i ing/pa sing da a o/ om XML o ma messages a e implemen ed in “XmlW i e ” and “XmlRea- de ” o communica ing wi h he se e . •“Localiza ionSe ice” is he se ice o handling he asks ela ed o he localiza i- on p ocedu e. This se ice is s a ed by “Indoo Na Map” and calls he me hods im- plemen ed in “Localiza ionManage ” o pe o m localiza ion. This se ice is s opped au oma ically by he sys em once he use e mina es he applica ion. •“Indoo Na MapB oadcas Re ei e ” and “Localiza ionSe iceB oadcas Recei e ” a e implemen ed o suppo he da a exchange be ween “Localiza ionSe ice” and “In- doo Na Map”. These wo classes ex end he “B oadcas Recei e ” p o ided by he And oid APIs. •“WiFiScanne ” is esponsible o WiFi da a acquisi ion o localiza ion p ocedu e. Se e Based Indoo Na iga ion Sys em 77 •“Rou eCalcula e” con ains all he me hods ela ed o na iga ion pa h calcula ion. The- se me hods a e ac i a ed by “Indoo Na Map” ia “MapSc ollApplica ion” whe e he in o ma ion o he calcula ed ou e is s o ed. By doing his, i he use swi ches o any o he ac i i ies o applica ions, once use ge s back o he “Indoo Na Map” ac i i y, he na iga ion pa h will be displayed wi hou e-calcula ion. Mo e de ails abou he na iga ion p ocedu e will be p esen ed in sec ion 8.5.3. 8.5. Sys em Ope a ion This sec ion p esen s he ope a ion o he main ea u es o ou indoo posi ioning sys em om a use ’s iew, as well as he low sequence o he sma phone applica ion o each ea u e. The de eloped indoo na iga ion sys em is able o suppo se e al se ices such as ga he ing o aining da a, localiza ion, na iga ion, some ini ial e sions o social ea u es such as pee g oup inding, and 3-D ep esen a ion o he ou ing pa h. The main ea u es, i.e., ga he ing o aining da a, localiza ion, and na iga ion, a e discussed in de ail, a sho in oduc ion abou he o he ea u es is gi en a he end o he sec ion. 8.5.1. Da a Ga he ing This ea u e is esponsible o ga he ing aining da a. To do his, he sma phone applica ion pe o ms WiFi scans and w i es he measu ed da a o a “Se Finge p in Reques ”. I should be no ed ha on he sma phones, a he momen , his ea u e is only a ailable when he applica ion uns in he de elopmen mode in o de o a oid p oblems in case someone ies o ha m he da abase. In u u e, o ex ending he sys em owa ds online lea ning, his ea u e could be pe o med in he backg ound o he sma phone applica ion du ing he usage by use s. The de elopmen mode is ac i a ed by he de elope s by choosing he op ion “Ac i a e de elope iew”, as shown in Fig. 8.7. Figu e 8.7.: Ja a applica ion showing se ing op ions. The de elopmen iew o he sma phone applica ion is illus a ed in Fig 8.8. In he igu e, he ed ci cles indica e he posi ions a which RSSI aining da a ha e al eady been collec ed. Se e Based Indoo Na iga ion Sys em 78 By doing so, he de elope s can hen choose he inge p in posi ions which ha e no been ained o collec RSSI da a. The ed “+” sign on he map shows he posi ion whe e WiFi da a a e going o be collec ed. The de elope s can modi y he posi ion o he ed “+” sign by mo ing he map. To collec he RSSI da a, since each posi ion needs a su icien amoun o measu emen s o accu a ely es ima e aining models, an ac i i y named “WLANDa aAs- semble Ac i i y” was de eloped o suppo his equi emen . Figu e 8.8.: Ja a applica ion showing he de elopmen mode: he ed ci cles a e he posi ions whe e aining da a we e al eady collec ed, he ed “+” sign on he map shows he posi ion whe e WiFi da a a e going o be collec ed. Once he de elope s a s he da a ga he ing p ocedu e, a window appea s which allows he de elope o e i y he in o ma ion o he cu en posi ion and en e addi ional in o ma- ion, i.e., name o he posi ion, amoun o scans and scan in e al, see Fig. 8.9(a). Depending on he sampling a e o he WiFi senso o he es sma phone, he scan in e al need o be la ge enough o a oid duplica ion o he measu emen da a, We obse ed he su icien scan in e al o abou 1,5 s o Samsung sma phones, while o a Sony de ice i is oughly 5 s. Once he in o ma ion is alida ed and en e ed, he measu emen p ocess will be s a ed by simply clicking he “S a scan” bu on. Fig. 8.9(b) shows he sc eensho when he sma pho- ne pe o ms a WiFi scan, whe e he numbe o measu emen s and scanned da a a e shown. The measu ed da a a e hen w i en in o an XML ile which is s o ed on he local s o age o sma phone once he scanning p ocedu e is comple ed. The XML ile con ains he measu ed RSSIs, MAC add esses o he obse ed APs, he sma phone in o ma ion, as well as he in o ma ion o he loca ion whe e he measu emen s a e aken. I should be no ed ha each XML ile con ains he aining da a a one posi ion only. A e he measu emen campaign, all XML iles a e impo ed in o he da abase on he se - e using a C++ module named se inge p in . This module pa ses he eques and accesses he Pos GIS da abase o inse he measu ed da a. Fo each ile, se inge p in p oduces a esponse message o in o m whe he he da a a e success ully inse ed o he da abase o no . Appendix A.4.1 shows an example o he “Se Finge p in Reques ” and esponse messages. Ou sys em is also able o suppo he online mode o he measu emen campaign, i.e., he measu ed da a a e sen di ec ly o he se e a e each measu emen ia a wi eless ne wo k, Se e Based Indoo Na iga ion Sys em 79 (a) Se ing o WiFi scan (b) WiFi scanning Figu e 8.9.: WiFi aining da a ga he ing i we use he se inge p in module as a Fas CGI module. This op ion p o ides he possibi- li y o ga he ing da a du ing he usage o use s o an online lea ning app oach. Howe e , a he momen , no algo i hm has been de eloped o measu e he eliabili y o a measu emen . The e o e, we empo a ily disable his op ion o a oid acciden al measu emen s con aining w ong in o ma ion which may ha m he da abase. The indoo en i onmen and WLAN ne - wo k a e subjec o change o e ime, his equi es egula da abase main enance (upda e) o keep he posi ioning accu acy. I is in easible o e ain he da abase a e e e y change o he en i onmen o he ne wo k in as uc u e. The eason is collec ing aining da a is e y ime consuming, especially o he la ge deploymen a ea. The e o e, a me hod o au oma ically upda ing da abase, i.e., he online lea ning app oach, could be a solu ion. The sequence diag am o Fig. 8.10 shows how he sma phone applica ion pe o ms WiFi scanning in o line mode, i.e., he scanned da a a e w i en in an XML ile and s o ed in he local s o age o he sma phone. 8.5.2. Localiza ion Fo localiza ion, use s can choose ei he he o line localiza ion mode o he online localiza- ion mode in he se ings o he sma phone applica ion . In o line mode, posi ion es ima ion is pe o med locally on he sma phone using he da abase which is s o ed on i s local s o- age wi hou any need o in e ne connec ion. In online localiza ion mode, which equi es ne wo k connec ion o da a exchange, posi ion es ima ion is pe o med on he se e . Figu- e 8.11 shows he op ions o localiza ion, i.e., ope a ion mode and he da a sou ce. The sequence diag ams shown in Fig. 8.12 and Fig. 8.13 p esen he communica ion among he componen s o he sma phone applica ion o pe o m localiza ion in o line mode and online mode using WiFi in o ma ion, espec i ely. To pe o m he localiza ion using WiFi da a, sma phones pe iodically log he WiFi da a which a e he MAC add esses o he obse ed APs and hei measu ed RSSIs. An es ima o is hen used o p ocess he scanned da a o de e mine he use posi ions In o line localiza- Se e Based Indoo Na iga ion Sys em 80 Figu e 8.10.: Sequence diag am showing he da a ga he ing p ocedu e. ion, he es ima o is implemen ed as a me hod in he sma phone Ja a applica ion which uses he local da abase o compa e wi h obse ed da a. Se e al issues may a ise in o line localiza ion, o example, he local da abase o he posi ioning algo i hms migh be ou o da e, as discussed in he beginning o his chap e . To sol e his p oblem, he local RSSI da abase can be eloaded om se e egula ly, i.e., once a week, au oma ically o manually i ne wo k connec ion is a ailable. The p oblem wi h posi ioning algo i hms is unsol able unless he use upda es he applica ion. Wi hin his wo k, an au oma ic upda e p ocedu e o sma phone applica ion has no been de eloped ye . Fo online localiza ion, scanned da a a e sen o he se e ia he WLAN connec ion in an XML o ma . On he se e , a Fas CGI module named “posi iones ima e” was de eloped o pa se he eques and pe o m he localiza ion p ocedu e. The posi ion es ima ion algo i hm was implemen ed using he classi ica ion ule which was discussed in Chap e 4, he same algo i hm is used o o line localiza ion. This module uses he aining models s o ed in he sha ed memo y o ca y ou he localiza ion p ocedu e. Senso usion o he imp o emen in posi ioning accu acy as discussed in Chap e 6, is no implemen ed on he se e side, bu on he clien side. The eason is ha he se e mus be a s a eless se e since he e would be an explosion o he amoun o HMM s a es which need o be s o ed i many clien s eques he localiza ion se ice a he same ime using RSSI Se e Based Indoo Na iga ion Sys em 81 (a) Se ing op ions (b) Localiza ion modes Figu e 8.11.: Se ing sc een showing op ions o localiza ion. and s ep de ec ion in o ma ion. This would d ama ically deg ade he esponse ime o he se e . Once he posi ion es ima e is ob ained, a localiza ion esponse is sen om he se e o he clien which con ains he in o ma ion o he es ima ed use posi ion. The clien pa ses he esponse and shows he use loca ion on he sma phone sc een, see Fig. 8.6 whe e he ed do ep esen s he cu en use posi ion. 8.5.3. Na iga ion Fo na iga ion pu pose, he use can inpu he in o ma ion o he sou ce and des ina ion posi ions, i.e., oom numbe s, and he op ions o he expec ed ou e, i.e., using ele a o s o s ai s. Fig. 8.14 shows an example o a sma phone sc een when he use s a s he ou ing unc ion o he sma phone applica ion. Na iga ion, simila o localiza ion, can be pe o med in ei he o line mode o online mo- de. In online mode, a na iga ion eques wi h he in o ma ion inse ed by he use is gene a ed by he sma phone applica ion and sen o he emo e se e . On he se e , a Fas CGI modu- le named “ ou ees ima e” was de eloped which is able o pa se he eques and pe o m he ou e calcula ion. Da a o he ou e calcula ion p ocess a e ob ained by eading he sha ed memo y. To pe o m he pa h sea ch, se e al pa h inding algo i hms ha e been conside ed such as he well known Dijks a’s algo i hm, he A* algo i hm (a a ian o Dijks a’s algo- i hm) and he jump poin algo i hm. In na iga ion, he pa h cos is simply he geog aphical dis ance o any pai o connec ed posi ions. While Dijks a’s algo i hm examines all nodes o ind he sho es pa h be ween he sou ce and he des ina ion, he A* algo i hm is ying o examine he nodes which a e po en ially on he sho es pa h i s o op imize he compu a io- nal demand. This is he eason why he A* is also called goal-o ien ed Dijks a’s algo i hm. Howe e , he A* algo i hm needs heu is ic weigh s o gua an ee he solu ion is he sho es pa h. The jump poin algo i hm ies o op imize A* in case o uni o m-cos g ids, which is no mally no sui able o an indoo en i onmen since dis ibu ing he indoo inge p in ing 9. Conclusions Wi hin his hesis, he echniques o imp o e he accu acy o WiFi inge p in ing based in- doo posi ioning a e p esen ed. The accu acy o indoo posi ioning employing WLAN in o ma ion can be enhanced by a s a is ical app oach which is able o accoun o he a ia ion o he measu ed RSSIs in he indoo en i onmen . The posi ioning accu acy depends on wo ac o s: aining models and classi ica ion ule. As ou ca e ul s udy on WiFi da a in indoo en i onmen showed, RSSI measu emen s su e om wo p oblems namely censo ing and d opping. As discussed in chap e 2 and chap e 4, hese wo p oblems o he indoo WiFi da a ha e no been add essed in any p e ious esea ch. The e o e, wi hin his wo k, me hods o es ima ing he pa ame e s o he aining model and classi ica ion in he p esence o censo ed and d opped da a we e p oposed. Fo pa ame e es ima ion, an EM algo i hm was p oposed in chap e 4 which e - icien ly copes wi h he censo ing and d opping p oblem. The p oposed EM algo i hm o censo ed Gaussian da a was p o ed o be a i ually bias ee and e icien es ima o . Imp o- emen s in posi ioning accu acy a e demons a ed bo h on a i icially gene a ed da a and in eal ield da a expe imen s compa ed o some o he app oaches. The expe imen s p esen ed in chap e 7 showed he supe io i y o he s a is ical app oach compa ed o a de e minis ic app oach. I is no ed ha he compu a ional demand o posi ioning using a pa ame ic s a i- s ical app oach is less han he o he men ioned app oaches. Ano he p oblem ha has been conside ed in his wo k is he misma ch be ween he mea- su ed da a o he aining de ice and he es de ices as discussed in chap e 5 which leads o a se ious educ ion o he posi ioning accu acy. In he li e a u e, we ound only one solu ion which ied o add ess his p oblem using Leas Squa es app oach whe e he au ho s ass- umed a linea ela ionship be ween he RSSI eadings o di e en de ices. Howe e , wha we obse ed is ha his ela ion is no linea which ende s he LS app oach inapp op ia- e. An e ec i e me hod o cope wi h his p oblem mus be de eloped o make WiFi signal based indoo posi ioning ealis ic. The e o e, we p oposed a me hod o aligning he ai- ning model wi h he p ope ies o he es de ices while elaxing he linea i y assump ion in chap e 5. The p oposed aligning me hod called “sma phone adap a ion” was de eloped wi- hin he MLLR amewo k which is a e y well known and success ul echnique o speake adap a ion in au oma ic speech ecogni ion. I has o be no ed ha he p oposed adap a ion me hod is able o cope wi h censo ed and d opped da a in he adap a ion da a, esul ing in eliable adap ed models. Du ing he adap a ion p ocedu e, he d opping a e o he adap a- ion da a is also es ima ed which will be used in he classi ica ion p ocedu e. Expe imen al esul s p esen ed in chap e 7 demons a ed he e ec i eness o doing adap a ion in indoo posi ioning. Assuming he RSSI eadings om di e en de ices ollow one linea ela ion- ship, applying he p oposed app oach showed a big imp o emen in posi ioning accu acy, howe e , s ill well a below he achie able accu acy. Employing clus e ing app oach be o e 88 Conclusions 89 doing adap a ion elaxed he linea i y assump ion esul ing in assuming piecewise linea i y. As a esul , be e posi ion accu acy was ob ained. Imp o emen s in posi ioning accu acy can be ob ained by using he in o ma ion om di - e en sou ces such as WiFi in o ma ion and ine ial senso in o ma ion. These wo kinds o in o ma ion a e ob ainable on mos mode n sma phones. Ine ial na iga ion which u ilizes he da a om he buil -in senso s o he sma phones is able o p oduce p ecise posi ion es i- ma ion in a sho e m ( ime and dis ance) only. Un o una ely he p ecise posi ioning esul s canno be main ained o a longe pe iod o ime due o e o accumula ion o e ime and dis ance, esul ing in un eliable posi ion es ima es. WiFi inge p in ing based posi ioning, on he o he hand, can p o ide a s able posi ioning accu acy wi hou he e o accumula ion p oblem. As discussed in chap e 2, hese wo posi ioning echniques can be combined in o de o p oduce be e posi ioning esul s compa ed o using any indi idual app oach alone. The e o e, chap e 6 p esen ed a modi ied HMM o da a usion o WiFi da a and ine ial senso da a and u ilizing he possible walking pa h o he use o come up wi h posi ion es i- ma es. Mo e accu a e posi ioning esul s we e ob ained by employing HMM in compa ison wi h using WiFi in o ma ion o ine ial senso in o ma ion alone. Fu he mo e, a me hod o educe he quan iza ion e o caused by he coa se g id o ained posi ions was p oposed in chap e 6. By in oducing pseudo s a es o he HMM inbe ween he egula s a es and syn- hesizing he emission PDFs o he pseudo s a es om hose o neighbo ing egula s a es, we ob ained a dense g id o s a es wi hou addi ional aining e o . Expe imen al esul s showed ha employing he ex ended HMM imp o ed he posi ioning accu acy. In addi ion o he heo e ical esea ch, a eal se e based indoo posi ioning and na iga- ion sys em was de eloped as p esen ed in chap e 8. A he momen he sys em is able o p o ide he localiza ion and na iga ion se ices o he s uden s o he Uni e si y o Pade - bo n. The sys em employs he ligh pd web se e and he Fas CGI p o ocol which allows o handle housands o connec ions in pa allel on he se e . In addi ion, he Fas CGI mo- dules, which a e w i en in C++, un in sepe a e p ocesses which make he sys em s able, and u he , easy o scale and easy o main ain. A Ja a applica ion o And oid sma pho- nes was de eloped which is able o un as a s andalone posi ioning sys em o as a clien in he sys em. As discussed in chap e 8, de eloping wo possible ope a ion modes o he sma phone applica ion sol es he p oblem o loss o in e ne connec ion, unsynch oniza ion o map da a and inge p in ing aining model, and so on. As a esul , he sys em seems o sa is y he equi emen s o low cos , high obus ness and simplici y in deploymen , scaling and main enance. Ou look An in e es ing di ec ion o u u e esea ch is how o imp o e/upda e he adio map wi h he measu ed da a epo ed by he use s du ing he online phase. To do ha , a possible solu i- on is o de elop a semi-supe ised online lea ning sys em whe e he al eady buil da abase is con inuously upda ed du ing he sys em ope a ion using he measu emen s epo ed by use s. This kind o sys em can au oma ically handle he changes o he indoo en i onmen o WLAN in as uc u e wi hou e- aining he da abase om sc a ch a e a ce ain amoun o ime. This also helps o imp o e he accu acy o he aining model since mo e aining da a a e a ailable. The challenge is how o de e mine he posi ion whe e he measu emen is Conclusions 90 aken. The REDPIN sys em belie es in he in o ma ion ha is epo ed om any use s which does no seem o be aul ole an since he da abase can be co up ed on pu pose o unin- en ionally h ough w ong use posi ions. Simul aneous Localiza ion And Mapping (SLAM) is a common app oach in he obo ics communi y which mainly elies on ine ial senso in- o ma ion o ack he posi ion o a mobile obo and build he map simul aneously. This app oach can be used o de e mine he posi ion a which a speci ic RSSI measu emen is col- lec ed. Howe e , as discussed be o e, his echnique su e s om he e o accumula ion o e ime and dis ance and, as a consequence, es ima ed posi ions a e no eliable. Map ma ching can be o mula ed as a inge p in ing based localiza ion p oblem whe e he use posi ion is assigned o a ou e segmen based on he online obse a ion and aining da a. The e o e, he combina ion o SLAM and map ma ching app oach would be a ele en solu ion o es ima e he posi ion o he use whe e RSSI measu emen is collec ed since he accumula ed e o in ine ial na iga ion a e co ec ed globally by RSSI da a and loo plan in o ma ion. Ou p oposed EM algo i hms p esen ed in chap e 4 a e e icien o es ima ing he pa a- me e s o censo ed and d opped Gaussian da a. Howe e , we employed he empi ical ixed clipping h eshold o da a measu ed by all de ices. This migh deg ade he pa ame e es i- ma ion pe o mance i he clipping h eshold o di e en de ices a e no iden ical. The e o e, es ima ing clipping h eshold om aining da a should be done be o e pa ame e es ima i- on p ocedu e o ensu e he p ecision o he es ima ed pa ame e s. As a esul , a me hod o clipping h eshold es ima ion is needed. Employing a Gaussian mix u e model es ima ion o es ima e he RSSI dis ibu ion ins ead o he assump ion o a single Gaussian would be an in e es ing y. This me hod migh help o imp o e he p ecision o signal s eng h dis ibu ion es ima ion, since i is able o cap u e all he modes in he aining da a dis ibu ion. Howe e , since censo ing and d opping a e se e e in WiFi da a, me hods o dealing wi h censo ed and d opped da a du ing he pa ame e es ima ion p ocedu e would need o be aken in o conside a ion. A. Appendix A.1. De i a ion o EM Algo i hm A.1.1. Compu a ion o I0 I0(θ(κ)) = Zc −∞ Ny;θ(κ)dy =Zc −∞ 1 √2πσ(κ)exp −(y−µ(κ))2 2(σ2)(κ)dy(A.1) Le =y−µ(κ) √2σ(κ)and change he a iable o he in eg al, we a i e a : I0(θ(κ)) = 1 √πZc−µ(κ) √2σ(κ) −∞ exp(− 2)d (A.2) Since exp(− 2)is an e en unc ion, he limi s o he in eg al in Eq. (A.2) can be modi ied by changing hei signs and swapping lowe and uppe limi s, hen I0(θ(κ))can be easily ob ained by using complemen a y e o unc ion I0(θ(κ)) = 1 2e c −c−µ(κ) √2σ(κ)(A.3) A.1.2. Compu a ion o I1 I1(θ(κ)) = Zc −∞ yNy;θ(κ)dy =Zc −∞ y1 √2πσ(κ)exp −(y−µ(κ))2 2(σ2)(κ)dy(A.4) Employing he in eg a ion by pa s ule, le (u=y d =1 √2πσ(κ)exp −(y−µ(κ))2 2(σ2)(κ)dy⇒(du=dy =1 2e y−µ(κ) √2σ(κ),(A.5) hen we a i e a I1(θ(κ)) = y1 2e y−µ(κ) √2σ(κ) c −∞ −1 2Zc −∞ e y−µ(κ) √2σ(κ)dy |{z } =A1 (A.6) 91 Appendix 92 Again in eg a ion by pa s ule is used o compu ing he emaining in egal A1as ollows, o simpli y he in eg al, le =y−µ(κ) √2σ(κ),A1can be w i en as A1=√2σ(κ)Zc−µ(κ) √2σ(κ) −∞ e ( )d (A.7) Le u=e ( ) d =d ⇒du=2 √πexp (− 2)d = ,(A.8) hen we a i e a A1=√2σ(κ)  e ( ) c−µ(κ) √2σ(κ) −∞ −1 √πZc−µ(κ) √2σ(κ) −∞ 2 exp − 2d   =√2σ(κ) e ( ) + 1 √πexp − 2 c−µ(κ) √2σ(κ) −∞ (A.9) Using A1in Eq. (A.6), I1(θ(κ))is eadily ob ained: I1(θ(κ)) = µ(κ)I0(θ(κ))−1 √2πσ(κ)exp −c−µ(κ) √2σ(κ)2!(A.10) A.1.3. Compu a ion o I2 I2(θ(κ)) = Zc −∞ y2Ny;θ(κ)dy =Zc −∞ y21 √2πσ(κ)exp −(y−µ(κ))2 2(σ2)(κ)dy(A.11) Employing he in eg a ion by pa s ule, le (u=y2 d =1 √2πσ(κ)exp −(y−µ(κ))2 2(σ2)(κ)dy⇒(du= 2ydy =1 2e y−µ(κ) √2σ(κ),(A.12) hen we a i e a I2(θ(κ)) = y21 2e y−µ(κ) √2σ c −∞ −Zc −∞ ye y−µ(κ) √2σ(κ)dy |{z } A2 (A.13) To compu e A2, in eg a ion by pa s is again applied, le (u1=y d 1=e y−µ(κ) √2σ(κ)dy⇒(du1=dy 1=Re y−µ(κ) √2σ(κ)dy,(A.14) Appendix 93 Fo calcula ing 1, le =y−µ(κ) √2σ(κ) 1=√2σ(κ)Ze ( )d (A.15) and applying in eg a ion by pa s u2=e ( ) d 2=d ⇒du2=2 √πexp (− 2)d 2= ,(A.16) we a i e a 1=√2σ(κ) .e ( )−1 √πZ2 exp − 2d  =√2σ(κ) .e ( ) + 1 √πexp − 2.(A.17) Using =y−µ(κ) √2σ(κ)in 1 hen we ob ain 1=√2σ(κ)y−µ(κ) √2σ(κ)e y−µ(κ) √2σ(κ)+1 √πexp −(y−µ(κ) √2σ(κ))2 = (y−µ(κ))e y−µ(κ) √2σ(κ)+σ(κ) 2 πexp −y−µ(κ) √2σ(κ)2!.(A.18) Now, A2can be calcula ed as ollows A2=u1 1 c −∞ −Zc −∞ 1du1 =y((y−µ(κ))e y−µ(κ) √2σ(κ)+σ(κ) 2 πexp −y−µ(κ) √2σ(κ)2!) c −∞ −Zc −∞ ((y−µ(κ))e y−µ(κ) √2σ(κ)+σ(κ) 2 πexp −y−µ(κ) √2σ(κ)2!)dy =y((y−µ(κ))e y−µ(κ) √2σ(κ)+σ(κ) 2 πexp −y−µ(κ) √2σ(κ)2!) c −∞ −Zc −∞ y.e y−µ(κ) √2σ(κ)dy |{z } A2 +µ(κ)Zc −∞ e y−µ(κ) √2σ(κ)dy |{z } 1c −∞ −2σ2(κ)Zc −∞ 1 √2πσ(κ)exp −y−µ(κ) √2σ(κ)2!dy |{z } I0(θ(κ)) (A.19) Appendix 94 Simply using he limi s, A2is eadily ob ained: A2=1 2y2e y−µ(κ) √2σ(κ) c −∞ −1 2(µ(κ)2 +σ2(κ)).e c−µ(κ) √2σ(κ)+ 1 −c+µ(κ)σ(κ) 2 πexp −c−µ(κ) √2σ(κ)2!) =1 2y2e y−µ(κ) √2σ(κ) c −∞ −(µ(κ)2 +σ2(κ))I0(θ(κ)) +c+µ(κ).σ(κ)1 √2πexp −c−µ(κ) √2σ(κ)2!(A.20) Plugging A2in o Eq. (A.13), he calcula ion o I2ends up wi h I2(θ(κ)) = σ2(κ)+µ2(κ)I0(θ(κ))−1 √2πσ(κ)(µ(κ)+c) exp −c−µ(κ) √2σ(κ)2!(A.21) A.1.4. Compu a ion o W En ies ∂ ∂µI0(θ) = ∂ ∂µ Zc −∞ p(y;θ)dy =1 σ2Zc −∞ (y−µ)p(y;θ)dy =1 σ2(I1(θ)−µI0(θ)) (A.22) ∂ ∂µI1(θ) = ∂ ∂µ Zc −∞ y.p(y;θ)dy =1 σ2Zc −∞ y(y−µ)p(y;θ)dy =1 σ2(I2(θ)−µI1(θ)) (A.23) ∂ ∂µI2(θ) = ∂ ∂µ Zc −∞ y2.p(y;θ)dy =1 σ2Zc −∞ y2(y−µ)p(y;θ)dy =1 σ2(I3(θ)−µI2(θ)) (A.24) ∂ ∂σ2I0(θ) = ∂ ∂σ2Zc −∞ p(y;θ)dy =−1 2σ2I0(θ) + 1 2σ4I2(θ)−2I1(θ)µ+I0(θ)µ2(A.25) Appendix 95 ∂ ∂σ2I1(θ) = ∂ ∂σ2Zc −∞ yp(y;θ)dy =−1 2σ2I1(θ) + 1 2σ4I3(θ)−2I2(θ)µ+I1(θ)µ2(A.26) ∂ ∂σ2I2(θ) = ∂ ∂σ2Zc −∞ y2p(y;θ)dy =−1 2σ2I2(θ) + 1 2σ4I4(θ)−2I3(θ)µ+I2(θ)µ2(A.27) whe e I3(θ)and I4(θ)can be compu ed by using in eg a ion by pa s as he compu a ion o I2(θ), howe e leng hie compu a ion I3(θ) = Zc −∞ y3p(y;θ)dy =µ3σ2+µ2I0(θ)−1 √2πσ[2σ2+µ2+µc +c2] exp −c−µ(κ) √2σ(κ)2!(A.28) I4(θ) = Zc −∞ y4p(y;θ)dy = (3σ4+ 6µ2σ2+µ4)I0(θ) −1 √2πσ[σ2(5µ+ 3c) + µ3+µ2c+µc2+c3] exp −c−µ(κ) √2σ(κ)2!(A.29) A.1.5. Compu a ion o In o ma ion Ma ix I Fo con enience, he log-likelihood unc ion and he in o ma ion ma ix a e ecalled as ol- lows ln p(x;θ) = ln N! M!(N−M)!+ (N−M) ln (I0(θ)) −M 2ln 2πσ2−1 2 M X j=1 xj−µ σ2 .(A.30) I= Eh−∂2 ∂µ2ln p(x;θ)iEh−∂2 ∂µ∂σ2ln p(x;θ)i Eh−∂2 ∂µ∂σ2ln p(x;θ)iEh−∂2 ∂σ2∂σ2ln p(x;θ)i .(A.31) In he ollowing, o sho en he no a ion, he pa ame e s o he Ij unc ions a e emo ed. He e he de i a i e compu a ions o he Ijwhich we e gi en in Eq. (A.22) o (A.27) a e em- ployed. Since I0is he p obabili y ha a measu emen is censo ed, in he ollowing de i a i- on, he expec ed numbe o uncenso ed measu emen s is de e mined by E [M] = N(1 −I0), whe e Nis he o al numbe o measu emen s. Appendix 96 •Eh−∂2 ∂µ2ln p(x;θ)i: Fi s de i a i e o log-likelihood unc ion w. . . µ ∂ ∂µ ln p(x;θ) = (N−M)1 I0 ∂ ∂µI0−1 2 M X j=1 −2xj−µ σ2 = (N−M)1 I0 1 σ2I1−µI0+1 σ2 M X j=1 (xj−µ)(A.32) Second de i a i e o log-likelihood unc ion w. . . µ: ∂2 ∂µ2ln p(x;θ) = ∂ ∂µ (N−M)1 I0 1 σ2I1−µI0+1 σ2 M X j=1 (xj−µ)! =N−M σ2 ∂ ∂µ I1 I0−µ−M σ2 =N−M σ2 ∂I1 ∂µ I0−I1∂I0 ∂µ I2 0−1!−M σ2 =N−M σ21 σ2(I2−µI1)I0−I11 σ2(I1−µI0) I2 0−1−M σ2 =N−M σ4I2I0−I2 1 I2 0−σ2−M σ2 =N−M σ4I2I0−I2 1 I2 0−N σ2(A.33) Expec a ion o he second de i a i e o log-likelihood unc ion w. . . µ: E−∂2 ∂µ2ln p(x;θ)=E−N−M σ4I2I0−I2 1 I2 0+N σ2 =−E[N−M]1 σ4I2I0−I2 1 I2 0+N σ2 =−NI0 1 σ4I2I0−I2 1 I2 0+N σ2 =N σ2I2 1−I2I0 I0 + 1(A.34) Appendix 97 •Eh−∂2 ∂µ∂σ2ln p(x;θ)i: ∂2 ∂µ∂σ2ln p(x;θ) = ∂ ∂σ2 (N−M)1 I0 1 σ2I1−µI0+1 σ2 M X j=1 (xj−µ)! = (N−M)∂ ∂σ21 σ2I1 I0−µ−1 σ4 M X j=1 (xj−µ) = (N−M)(−1 σ4I1 I0−µ+1 σ2 ∂I1 ∂σ2I0−I1∂I0 ∂σ2 I2 0!) −1 σ4 M X j=1 (xj−µ)·(A.35) Expec a ion o he second de i a i e o log-likelihood unc ion w. . . µand σ2: E−∂2 ∂µ∂σ ln p(x;θ)=NI0(1 σ4I1 I0−µ−1 σ2 ∂I1 ∂σ2I0−I1∂I0 ∂σ2 I2 0!) +1 σ4E"M X j=1 (xj−µ)#,(A.36) whe e he emaining expec a ion can be compu ed as ollows E"M X j=1 (xj−µ)#=E[M] (E[x]−µ) =N(1 −I0)1 1−I0Z∞ c yp(y;θ)dy −µ =N(1 −I0)1 1−I0 (µ−I1)−µ =N(µI0−I1).(A.37) Using his in Eq. (A.36) we a i e a E−∂2 ∂µ∂σ2ln p(x;θ)=N σ2− ∂I1 ∂σ2I0−I1∂I0 ∂σ2 I0(A.38) •E−∂2 ∂σ2∂σ2ln p(x;θ): Fi s de i a i e o log-likelihood unc ion w. . . σ2 ∂ ∂σ2ln p(x;θ) = (N−M)1 I0 ∂ ∂σ2I0−M 2 1 σ2−1 2−1 σ4M X j=1 (xj−µ)2 = (N−M)−1 2σ2+1 2σ4I2 I0−2µI1 I0 +µ2 −M 2σ2+1 2σ4 M X j=1 (xj−µ)2(A.39) No a ions and Symbols Gene al No a ions and Func ions E[·]. . . . . . . . . . . . . . . . Expec a ion (·)T. . . . . . . . . . . . . . . . T anspose Q(·). . . . . . . . . . . . . . . Auxilia y unc ion (·)−1. . . . . . . . . . . . . . . In e sion ln(·). . . . . . . . . . . . . . . Na u al loga i hm unc ion p(x|ℓk). . . . . . . . . . . . . Class condi ional p obabili y densi y unc ion δ(·). . . . . . . . . . . . . . . . Di ac del a unc ion e (·). . . . . . . . . . . . . . E o unc ion e c (·). . . . . . . . . . . . . Complemen a y e o unc ion N(θ). . . . . . . . . . . . . . Gaussian dis ibu ion pa ame e ized by θ p(x;θ). . . . . . . . . . . . . P obabili y densi y unc ion pa ame e ized by θ P. . . . . . . . . . . . . . . . . Summa ion o a sequence Q. . . . . . . . . . . . . . . . . P oduc s o a sequence (.)! . . . . . . . . . . . . . . . . . Fac o ial o a non nega i e in ege Fundamen als o Indoo Posi ioning and S a e o Resea ch R. . . . . . . . . . . . . . . . . . Ma hema ical exp ession o adio map in inge p in ing based ech- niques ℓk. . . . . . . . . . . . . . . . . . Vec o consis s o coo dina es o he k- h posi ion Fk. . . . . . . . . . . . . . . . . Finge p in a he k- h posi ion Xk. . . . . . . . . . . . . . . . . Se o measu emen ec o s a he k- h posi ion xk,n ................ Then- h measu emen ec o a he k- h posi ion D(o,xk,n). . . . . . . . . . Euclidian dis ance be ween online sample oand aining sample xk,n θk. . . . . . . . . . . . . . . . . . Se o he pa ame e s o he Gaussian desc ibing he signal s eng h dis ibu ion a he k- h posi ion µk. . . . . . . . . . . . . . . . . Mean ec o o he Gaussian desc ibing he signal s eng h dis i- bu ion a he k- h posi ion Σk. . . . . . . . . . . . . . . . . Co a iance ma ix o he Gaussian desc ibing he signal s eng h dis ibu ion a he k- h posi ion 104 No a ions and Symbols 105 Pa ame e Es ima ion o Censo ed and D opped Gaussian Da a c. . . . . . . . . . . . . . . . . . . Clipping h eshold Θ. . . . . . . . . . . . . . . . . . Se o pa ame e s o a GMM πk. . . . . . . . . . . . . . . . . Mixing weigh o he k- h componen o a GMM θk. . . . . . . . . . . . . . . . . . Se o he pa ame e s o he k- h componen o a GMM X. . . . . . . . . . . . . . . . . . Se o obse able da a Z. . . . . . . . . . . . . . . . . . Se o hidden a iables y. . . . . . . . . . . . . . . . . . Se o unobse able, non-censo ed, possibly d opped da a yn................. Then- h measu emen , scala alue x. . . . . . . . . . . . . . . . . . Se o obse able da a θ. . . . . . . . . . . . . . . . . . . Se o he pa ame e s o a uni a ia e single Gaussian µ. . . . . . . . . . . . . . . . . . Mean o a uni a ia e single Gaussian σ. . . . . . . . . . . . . . . . . . Va iance o a uni a ia e single Gaussian θ(κ). . . . . . . . . . . . . . . . Se o he es ima ed pa ame e s o a uni a ia e single Gaussian a e he κ- h i e a ion o an EM algo i hm Ijθ(κ)............ Thej- h momen o he unca ed pa o a censo ed Gaussian, de- ined in Eq. (4.8) zn. . . . . . . . . . . . . . . . . . Realiza ion o he bina y andom a iable Znindica e whe he he n- h measu emen is censo ed (zn= 1) o no (zn= 0) N. . . . . . . . . . . . . . . . . . Numbe o measu emen s M. . . . . . . . . . . . . . . . . Numbe o obse able measu emen s ˜µ(κ). . . . . . . . . . . . . . . . Di e ence be ween he es ima ed mean a e he κ- h EM i e a ion and he ue mean (˜σ2)(κ). . . . . . . . . . . . . Di e ence be ween he es ima ed a iance a e he κ- h EM i e a- ion and he ue a iance I. . . . . . . . . . . . . . . . . . . In o ma ion ma ix dn. . . . . . . . . . . . . . . . . Hidden a iables indica e whe he he n- h measu emen is d opped (dn= 1) o no (dn= 0) π. . . . . . . . . . . . . . . . . . D opping a e, de ined as π=P(dn= 1) βn(d, z). . . . . . . . . . . . P obabili y ha dn= 0 o dn= 1 gi en zn= 0 o zn= 1 Sma phone Adap a ion wi hin MLLR F amewo k µk. . . . . . . . . . . . . . . . . Mean ec o consis s o means o all APs a he k- h posi ion ξk. . . . . . . . . . . . . . . . . Ex ended mean ec o consis s o means o all APs a he k- h po- si ion and a o se e m ac o ˆ µk. . . . . . . . . . . . . . . . . Adap ed mean ec o a he k- h posi ion NAP . . . . . . . . . . . . . . . To al numbe o obse able APs c(k). . . . . . . . . . . . . . . . Reg ession class c(k)whe e he Gaussian desc ibing k- h posi ion belongs o C. . . . . . . . . . . . . . . . . . To al numbe o eg ession classes W(c(k)) . . . . . . . . . . . . . Mean ans o ma ion ma ix o be applied o he eg ession class c(k) γk(n). . . . . . . . . . . . . . The pos e io p obabili y ha RSSI measu emen ec o xnis om posi ion ℓk No a ions and Symbols 106 Y. . . . . . . . . . . . . . . . . . Se o measu emen ec o s yn................. Then- h measu emen ec o consis s o RSSI om NAP APs yn,i . . . . . . . . . . . . . . . . The non-censo ed, possibly d opped measu ed da a o he n- h mea- su emen om he i- h AP D. . . . . . . . . . . . . . . . . . Se o hidden a iable ec o s dn................. Then- h hidden a iable ec o consis s o andom a iables, each indica es whe he he measu ed da a om co esponding AP is d opped o no dn,i . . . . . . . . . . . . . . . . Random a iable indica es whe he he n- h measu emen om he i- h AP is d opped o no X. . . . . . . . . . . . . . . . . . Se o obse able, censo ed, possibly d opped measu emen ec o s xn................. Then- h measu emen ec o consis s o obse able, censo ed, pos- sibly d opped measu ed da a om NAP APs xn,i . . . . . . . . . . . . . . . . The censo ed, possibly d opped measu ed da a o he n- h measu- emen om he i-AP w(c(k)) i. . . . . . . . . . . . . . The ow ec o which is he i- h ow o W(c(k)) λ. . . . . . . . . . . . . . . . . . Se o pa ame e s, i.e., d opping a e, mean and a iance adap a ion ma ices, o be es ima ed o adap a ion pu pose λk,i . . . . . . . . . . . . . . . . Se o pa ame e s, i.e., d opping a e, mean and a iance adap a ion ma ices, o be es ima ed o he i- h AP a he k- h posi ion πk,i . . . . . . . . . . . . . . . . D opping a e o he adap a ion da a o he i- h AP a he k- h posi- ion µk,i . . . . . . . . . . . . . . . . Mean o he Gaussian desc ibing he RSSI dis ibu ion o he i- h AP a he k- h posi ion σk,i . . . . . . . . . . . . . . . . Va iance o he Gaussian desc ibing he RSSI dis ibu ion o he i- h AP a he k- h posi ion zn,i . . . . . . . . . . . . . . . . . Random a iable indica es whe he he n- h measu emen om he i- h AP is obse ed o no βk,n,i(d, z). . . . . . . . . . P obabili y ha dn,i = 0 o dn,i = 1 gi en zn,i = 0 o zn,i = 1 a he k- h posi ion ˆ Σk. . . . . . . . . . . . . . . . . Adap ed co a iance ma ix o he Gaussian desc ibing he RSSI ea- dings o he APs a he k- h posi ion Bk. . . . . . . . . . . . . . . . . The Choleski ac o o Σk H(c(k)) . . . . . . . . . . . . . Va iance ans o ma ion ma ix o be applied o he eg ession class c(k) h(c(k)) i.............. Thei- h componen o he main diagonal o H(c(k)) Hidden Ma ko Model o Indoo Use T acking s . . . . . . . . . . . . . . . . . . Hidden s a e a iable a ime x . . . . . . . . . . . . . . . . . . RSSI obse a ion a ime Pi,j . . . . . . . . . . . . . . . . T ansi ion p obabili y om he i- h o he j- h s a e . . . . . . . . . . . . . . . . . . Two-dimensional mo emen ec o compu ed om ine ial senso da a No a ions and Symbols 107 α (k). . . . . . . . . . . . . . . Fo wa d a iable: The p obabili y ha he use is a ime in s a e kgi en he obse ed sequence o o1: and 1: a.................. 3-dimen ional accele a ion da a ec o ax. . . . . . . . . . . . . . . . . Accele a ion da a along x-axis ay. . . . . . . . . . . . . . . . . Accele a ion da a along y-axis az. . . . . . . . . . . . . . . . . . Accele a ion da a along z-axis G. . . . . . . . . . . . . . . . . . G a i y cons an m. . . . . . . . . . . . . . . . . Magne o da a ec o g. . . . . . . . . . . . . . . . . . G a i y da a ec o gy . . . . . . . . . . . . . . . . Gy oscope da a ec o R. . . . . . . . . . . . . . . . . . Ro a ion ma ix ψ. . . . . . . . . . . . . . . . . . Azimu h angle α(n). . . . . . . . . . . . . . . S a e ec o ha con ains he absolu e Azimu h angle and i s de i- a i e F(n). . . . . . . . . . . . . . . T ansi ion ma ix which indica es a ansi ion o he sys em om ime n o n+ 1 in Kalman il e ν1(n). . . . . . . . . . . . . . Whi e Gaussian sys em noise in Kalman il e ν2(n). . . . . . . . . . . . . . Whi e Gaussian measu emen noise in Kalman il e G(n). . . . . . . . . . . . . . . Ma ix ha desc ibes he in luences o he sys em noise on he s a e ec o in Kalman il e Q1. . . . . . . . . . . . . . . . . Co a iance ma ix o he whi e Gaussian sys em noise in Kalman il e Q2. . . . . . . . . . . . . . . . . Co a iance ma ix o he whi e Gaussian measu emen noise in Kal- man il e H(n). . . . . . . . . . . . . . . Measu emen ma ix ha desc ibes he in luences o he measu e- men on he s a e ec o in Kalman il e Ls ep . . . . . . . . . . . . . . . S ep leng h Expe imen al Resul s on Indoo Posi ioning λ. . . . . . . . . . . . . . . . . . Weigh ing ac o be ween ine ial senso in o ma ion and WiFi in- o ma ion in he calcula ion o o wa d a iable Lis o igu es 2.1. P oximi y-based posi ioning: he es ima ed posi ion o he mobile use Miis he posi ion o he base s a ion BSji Mide ec s only signal om BSj, o he obse ed signal s eng h o BSjis he s onges among he de ec ed singals 6 2.2. Cicula la e a ion based posi ioning: he es ima ed posi ion o he mobile use Mis de e mined based on he es ima ed dis ances be ween he mobile use and he e e ence poin s. A leas h ee e e ence poin s a e needed o calcula e he use posi ion. . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3. Angula ion based posi ioning: he es ima ed posi ion o he mobile use M is de e mined based on he es ima ed angle αibe ween he mobile use and he e e ence poin s. A leas wo e e ence poin s a e needed o calcula e he use posi ion wi h he condi ion ha he use posi ion does no lie on he line connec ing he e e ence poin s. . . . . . . . . . . . . . . . . . . . . . . . 11 2.4. Finge p in ing based posi ioning: he illed ci cles indica es he e e ence (ancho ) poin s, while he iangles show base s a ion loca ions. The use posi ion is he posi ion o he ancho poin whose aining da a bes ma ch he online measu emen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5. Dead eckoning: he use posi ion is es ima ed based on he es ima ed mo e- men ec o s assuming ha he s a ing poin is gi en. . . . . . . . . . . . . 15 4.1. His og am o eal ield da a illus a es censo ing and d opping p oblem o WiFi da a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2. Censo ing p oblem: le igu e: PDF om whe e yis d awn; igh igu e: PDF om whe e he obse a ion xis d awn. . . . . . . . . . . . . . . . . . 23 4.3. Theo e ical numbe o EM i e a ions κ equi ed o educe he es ima ion e o o 10−4o i s ini ial alue as a unc ion o (c−µ)/σ. Ini ial alues µ(0),(σ2)(0) ha e been se o he ML es ima es o µ, σ2compu ed om he unclipped obse a ions only. . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4. Compa ison o CRLB o mean and a iance wi h MSE ob ained om simu- la ion o σ2= 25 and N= 1000. . . . . . . . . . . . . . . . . . . . . . . 30 4.5. Measu emen model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.1. S anda d Hidden Ma ko Model: s is he hidden s a e a iable and x is he obse a ion a ime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2. T ellis diag am wi h ou di e en s a es s ∈ {1,2,3,4}and Pi,j a e he ansi ion p obabili ies om s a e ia one ime ins an o s a e ja he la e ime ins an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 108 Lis o igu es 109 6.3. Modi ied HMM o posi ion es ima ion based on he usion o RSSI and mo emen ec o obse a ions xand , espec i ely. . . . . . . . . . . . . 45 6.4. S ep de ec ion and posi ion es ima ion sys em o e iew . . . . . . . . . . . 47 6.5. De ice coo dina e sys em (a) and wo ld coo dina e sys em (b) . . . . . . . 47 6.6. Example o aw accele a ion da a i he sma phone is held wi h display in pa allel o ea h su ace when walking . . . . . . . . . . . . . . . . . . . . 48 6.7. No m o accele a ion da a a e subs ac ing he o ce o g a i y cons an G=9.81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.8. ka′ka e lowpass il e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.9. S ep de ec ion p ocedu e: In each da a block, he numbe o s eps is de e - mined based on he c ossing a e o he accele a ion da a o e a h eshold (magen a lines). The sa e y h eshold is he minimum h eshold alue ha he ampli ude o noisy accele a ion da a canno exceed. . . . . . . . . . . . 51 6.10. In oduc ion o pseudo s a es: The ed ci cles a e he egula s a es wi h ai- ning da a, he g een ci cle is he pseudo s a e in oduced inbe ween he egu- la s a es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1. Floo plan o he a ea whe e ield da a has been conduc ed. . . . . . . . . . 57 7.2. CDF o he posi ioning e o o di e en sys ems. . . . . . . . . . . . . . . 57 7.3. Compa ing he expe imen al esul s on eal da a using 2p oposed EM algo- i hms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.4. The modi ied HMM s a es, i.e., he allowable use posi ions, and he ansi i- ons be ween hem. Red and g een ci cles indica e egula and pseudo s a es, espec i ely. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.5. CDF o he posi ioning e o o di e en sys ems. A e age o e 2 es a- jec o ies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.1. Se e based Indoo Na iga ion Sys em. . . . . . . . . . . . . . . . . . . . 65 8.2. Se e a chi ec u e o he indoo na iga ion sys em. . . . . . . . . . . . . . 69 8.3. Examples o o iginal map and edi ed map . . . . . . . . . . . . . . . . . . 71 8.4. Map edi ing wi h JOSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.5. Class diag am o Ja a applica ion a chi ec u e . . . . . . . . . . . . . . . . 75 8.6. Ja a applica ion showing loo plan in o ma ion and use posi ion ( ed do ). 76 8.7. Ja a applica ion showing se ing op ions. . . . . . . . . . . . . . . . . . . . 77 8.8. Ja a applica ion showing he de elopmen mode: he ed ci cles a e he posi- ions whe e aining da a we e al eady collec ed, he ed “+” sign on he map shows he posi ion whe e WiFi da a a e going o be collec ed. . . . . . . . . 78 8.9. WiFi aining da a ga he ing . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.10. Sequence diag am showing he da a ga he ing p ocedu e. . . . . . . . . . . 80 8.11. Se ing sc een showing op ions o localiza ion. . . . . . . . . . . . . . . . 81 8.12. Sequence diag am p esen s he o line localiza ion p ocedu e. . . . . . . . . 82 8.13. Sequence diag am p esen s he online localiza ion p ocedu e. . . . . . . . . 83 8.14. Ja a applica ion showing he ou ing op ions. . . . . . . . . . . . . . . . . 84 8.15. Sequence low p esen s he o line na iga ion p ocedu e. . . . . . . . . . . 85 8.16. Ja a applica ion showing he indoo ou ing pa h. . . . . . . . . . . . . . . 86 8.17. Ja a applica ion showing he indoo and ou doo ou ing pa h. . . . . . . . 86 Lis o igu es 110 8.18. Pee g oup inding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.19. 3-D ep esen a ion o he ou ing pa h . . . . . . . . . . . . . . . . . . . . 87 Lis o ables 7.1. Mean and s anda d de ia ion o APs a 2 posi ions o a i ical da a expe imen 56 7.2. Classi ica ion e o a e on a i icial censo ed da a . . . . . . . . . . . . . . 56 7.3. Classi ica ion e o a e on a i icial censo ed and d opped da a . . . . . . . 58 7.4. Classi ica ion esul s (% posi ions co ec ly classi ied) when aining and es da a a e gene a ed om he same pa ame e se . . . . . . . . . . . . . . . 60 7.5. Adap a ion pe o mance (% posi ions co ec ly classi ied) on a i ical da a as a unc ion o amoun o adap a ion da a . . . . . . . . . . . . . . . . . . . 60 7.6. RMS posi ioning e o (in [m]) as a unc ion o he amoun o posi ions ha- ing adap a ion da a and he amoun o ada a ion da a . . . . . . . . . . . . 61 7.7. E ec o numbe o clus e s on RMS posi ioning e o . . . . . . . . . . . 62 7.8. Mean posi ioning e o on a i icial da a . . . . . . . . . . . . . . . . . . . 63 8.1. Map in o ma ion ables in he da abase: s o age o he geog aphical in o ma- ion o inge p in posi ions, and he in o ma ion abou he di ec connec ion be ween any pai o posi ions. . . . . . . . . . . . . . . . . . . . . . . . . 72 8.2. 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