Hindawi Publishing Co po a ion
Ma hema ical P oblems in Enginee ing
Volume 2013, A icle ID 418678, 14 pages
h p://dx.doi.o g/10.1155/2013/418678
Resea ch A icle
Op imal Fusion Fil e ing in Mul isenso S ochas ic Sys ems wi h
Missing Measu emen s and Co ela ed Noises
R. Caballe o-Águila,1I. Ga cía-Ga ido,2and J. Lina es-Pé ez2
1Depa amen o de Es ad´
ıs ica, Uni e sidad de Ja´
en, Pa aje Las Lagunillas, 23071 Ja´
en, Spain
2Depa amen o de Es ad´
ıs ica, Uni e sidad de G anada, A enida Fuen enue a, 18071 G anada, Spain
Co espondence should be add essed o R. Caballe o-´
Aguila; [email protected]
Recei ed 30 Janua y 2013; Accep ed 28 Ap il 2013
Academic Edi o : Weihai Zhang
Copy igh © 2013 R. Caballe o-´
Aguila e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion
License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
The op imal leas -squa es linea es ima ion p oblem is add essed o a class o disc e e- ime mul isenso linea s ochas ic sys ems
wi h missing measu emen s and au oco ela ed and c oss-co ela ed noises. The s ochas ic unce ain ies in he measu emen s
coming om each senso (missing measu emen s) a e desc ibed by scala andom a iables wi h a bi a y disc e e p obabili y
dis ibu ion o e he in e al [0,1]; hence, a each single senso he in o ma ion migh be pa ially missed and he di e en senso s
may ha e di e en missing p obabili ies. The noise co ela ion assump ions conside ed a e (i) he p ocess noise and all he senso
noises a e one-s ep au oco ela ed; (ii) di e en senso noises a e one-s ep c oss-co ela ed; and (iii) he p ocess noise and each
senso noise a e wo-s ep c oss-co ela ed. Unde hese assump ions and by an inno a ion app oach, ecu si e algo i hms o
he op imal linea il e a e de i ed by using he wo basic es ima ion usion s uc u es; mo e speci ically, bo h cen alized and
dis ibu ed usion es ima ion algo i hms a e p oposed. The accu acy o hese es ima o s is measu ed by hei e o co a iance
ma ices, which allow us o compa e hei pe o mance in a nume ical simula ion example ha illus a es he easibili y o he
p oposed il e ing algo i hms and shows a compa ison wi h o he exis ing il e s.
1. In oduc ion
Fo a long ime, he leas -squa es (LS) es ima ion p oblem
in linea s ochas ic sys ems om measu emen s pe u bed
by addi i e noises has ecei ed conside able a en ion in he
scien i ic communi y due o i s wide applicabili y in many
p ac ical si ua ions (e.g., ideo and lase acking sys ems,
sa elli e na iga ion, ada and me eo ological applica ions,
e c. [1]). As i is well known, one o he majo con ibu ions
made o sol e his p oblem is he Kalman il e , which p o-
ides a ecu si e algo i hm o he op imal LS es ima o when
headdi i ewhi enoisesand heini ials a ea eGaussian
and mu ually independen (o , equi alen ly, unco ela ed due
o he Gaussiani y assump ion) and, he e o e, he op imal
LS es ima o is he op imal LS linea es ima o . F om he
publica ion o he Kalman il e [2] in 1960, nume ous esul s
and se e al solu ion me hods ha e been epo ed in he
li e a u e o add ess he s a e es ima ion p oblem om noisy
obse a ions, which depend on models ep esen ing possible
ela ionships be ween he unknown s a e and he obse able
a iables and also on he noise p ocesses assump ions.
Speci ically, du ing he pas decades, he e has been an
inc easing in e es in he il e ing p oblem in mul isenso
sys ems, whe e senso ne wo ks a e used o ob ain he whole
a ailable in o ma ion on he sys em s a e and i s es ima ion
mus be ca ied ou om he obse a ions p o ided by all
he senso s. A basic ma e o his class o sys ems is how
o use he measu emen da a om he di e en senso s o
add ess he es ima ion p oblem. Commonly, wo me hods
a eused op ocess hemeasu edda acoming ommul iple
senso s:cen alizedanddis ibu ed usionme hods.In he
cen alized usionme hodall hemeasu edda a omsenso s
a e communica ed o he usion cen e o being p ocessed;
ne e heless, as is widely known, cen alized es ima o s
ha e many compu a ional disad an ages, which mo i a e
he esea ch in o o he usion me hods. In he dis ibu ed
usion me hod, each senso es ima es he s a e based on
i s own single measu emen da a, and hen i sends such
2Ma hema ical P oblems in Enginee ing
es ima e o he usion cen e o usion acco ding o a ce ain
in o ma ion usion c i e ion. Al hough he use o senso
ne wo ks o e s se e al ad an ages, he un eliable ne wo k
cha ac e is ics usually cause p oblems du ing da a ans-
mission om senso s o he usion cen e , such as missing
measu emen s, andom communica ion packe losses and/o
delays. Taking in o accoun hese ne wo k unce ain ies, he
models ep esen ing he ela ionships be ween he s a e and
measu emen s do no allow o apply he Kalman il e , and
modi ica ions o con en ional es ima ion algo i hms ha e
been p oposed (see e.g., [3–9] and e e ences he ein).
As in he Kalman il e , independen whi e noises a e con-
side ed in all he men ioned pape s; howe e , his assump ion
may no be ealis ic and can be a limi a ion in many eal-
wo ld p oblems in which noise co ela ion may be p esen .
This p oblem a ises, o example, when a a ge is aking
an elec onic coun e measu e, o example, noise jamming
[10], o i he p ocess noise and he senso measu emen
noises a e dependen on he sys em s a e, hen he e may be
c oss-co ela ion be ween di e en senso noises and c oss-
co ela ion be ween p ocess noise and senso noises. Also, i
all he senso s a e obse ed in he same noisy en i onmen ,
he measu emen noises o di e en senso s a e usually
co ela ed.
Fo hese easons, he es ima ion p oblem in sys ems wi h
co ela ed noises has ecei ed signi ican esea ch in e es in
ecen yea s. Fo example, he op imal Kalman il e ing usion
p oblem in sys ems wi h c oss-co ela ed senso noises is
add essed in [10], while [11,12]s udy hesamep oblemin
sys ems wi h c oss-co ela ed p ocess noises and measu e-
men noises; in hese pape s co ela ed noises a he same
sampling ime a e conside ed. In gene al, he assump ion o
co ela ion and c oss-co ela ion o he noise p ocess and
measu emen noises in di e en sampling imes makes di -
icul he iden i ica ion o op imal es ima o s; his limi a ion
has encou aged a wide esea ch in o subop imal Kalman-
ype es ima ion p oblems. In [13], a Kalman- ype ecu si e
il e is p esen ed o sys ems wi h ini e-s ep co ela ed
p ocess noises, and he il e ing p oblem wi h mul is ep
co ela ed p ocess and measu emen noises is in es iga ed in
[14]. The op imal obus non agile Kalman- ype ecu si e
il e ing p oblem is s udied in [15] o aclasso unce ain
sys ems wi h ini e-s ep au oco ela ed measu emen noises
and mul iple packe d opou s. The p oblem o dis ibu ed
weigh ed obus Kalman il e usion is s udied in [16] o
a class o unce ain sys ems wi h au oco ela ed and c oss-
co ela ed noises. In [17], a s ochas ic singula sys em wi h
co ela ed noises a he same sampling ime is ans o med
in o an equi alen nonsingula sys em wi h co ela ed noises
a he same and neighbo ing sampling imes. Also, in [18],
an augmen ed pa ame e ized sys em wi h co ela ed noises a
he same and neighbo ing sampling imes is used o desc ibe
he senso delay, packe d opou , and unce ain obse a ion
phenomenons.
On he o he hand, as no ed abo e, he use o communi-
ca ion ne wo ks o ansmi ing measu ed da a mo i a es he
need o conside ing s ochas ic unce ain ies. Missing mea-
su emen s ha e been widely ea ed due o i s applicabili y
o model a la ge class o eal-wo ld p oblems, such as ading
phenomena in p opaga ion channels, a ge acking o , in
gene al, si ua ions whe e he e exis in e mi en ailu es in
he obse a ion mechanism, acciden al loss o some mea-
su emen s, o inaccessibili y o he da a du ing ce ain imes.
The s a e es ima ion p oblem om missing measu emen
ansmi ed by mul iple senso s has been s udied based on
he assump ion ha all he senso s a e iden ical (see, e.g.,
[19–22]);howe e , hisassump ioncanbeun easonable
since some eal sys ems usually in ol e mul iple senso s
wi h di e en cha ac e is ics. Recen ly, he il e ing p oblem
using missing measu emen s whose s a is ical p ope ies a e
assumed no o be he same in all he senso s has been
add essed by se e al au ho s unde di e en app oaches
andhypo heseson hep ocessesin ol ed(see,e.g.,[23–
27]). In all he abo e pape s, Be noulli andom a iables
a e used o model he missing measu emen s phenomenon,
and hence, i is assumed ha he measu emen signal
is ei he comple ely los (i he co esponding Be noulli
a iable akes he alue ze o) o success ully ans e ed
(when he Be noulli a iable is equal o one). Recen ly, his
missing measu emen model has been gene alized consid-
e ing any disc e e dis ibu ion on he in e al [0,1],which
allows o co e some p ac ical applica ions whe e only
pa ial in o ma ion is missing (see [28,29] and e e ences
he ein).
Mo i a ed by he abo e conside a ions, ou a en ion is
ocused on in es iga ing he op imal LS linea cen alized
and dis ibu ed usion es ima ion p oblems in mul isenso
sys ems wi h missing measu emen s and au oco ela ed and
c oss-co ela ed noises. In each senso , he missing measu e-
men phenomenon is go e ned by a scala andom a iable
wi h a bi a y disc e e p obabili y dis ibu ion o e he in e -
al [0,1], and he di e en senso s may ha e di e en missing
p obabili ies. Assume ha he p ocess noise and all he senso
noises a e one-s ep au oco ela ed; di e en senso noises a e
one-s ep c oss-co ela ed; and he p ocess noise and each
senso noise a e wo-s ep c oss-co ela ed. This pape makes
a wo old subs an ial no el con ibu ion: (1) unlike mos
p e ious esul s wi h co ela ed noises, in which subop imal
Kalman- ype es ima o s a e p oposed, in his pape op imal
LS linea es ima o s a e ob ained by using an inno a ion
app oach, which p o ides a simple de i a ion o he es ima-
ion algo i hms due o he ac ha he inno a ions cons i u e
a whi e p ocess; and (2) ou missing measu emen model
conside s a each senso he possibili y o obse a ions con-
aining only pa ial in o ma ion abou he s a e, o e en only
noise.
The pape is o ganized as ollows. In Sec ion 2 he sys em
model wi h au oco ela ed and c oss-co ela ed noises and
missing measu emen s coming om mul iple senso s is
desc ibed.Also, hesui ablep ope ieson hes a eand
noise p ocesses a e speci ied and a b ie desc ip ion o he
inno a ion app oach o he op imal LS linea es ima ion
p oblem is included. In Sec ion 3 a ecu si ealgo i hm
o he cen alized op imal linea il e is p esen ed o
he conside ed model ( he de i a ion has been de e ed
o Appendix 6). Nex , in Sec ion 4, he local LS linea
il e s and hei co esponding e o co a iance ma ices
be ween any wo local es ima es a e p o ided, and hen
Ma hema ical P oblems in Enginee ing 3
he dis ibu ed op imal weigh ed usion es ima o s and
hei e o co a iance ma ices a e ob ained by apply-
ing he op imal in o ma ion usion c i e ion weigh ed by
ma ices in he linea minimum a iance sense. Finally, in
Sec ion 5, a nume ical simula ion example is p esen ed o
show he e ec i eness o he es ima ion algo i hms p oposed
in hecu en pape ,andsomeconclusionsa ed awnin
Sec ion 6.
No a ion. The no a ion used h oughou he pape is s an-
da d. Fo any ma ix 𝐴, heno a ionsymbols𝐴𝑇and 𝐴−1
ep esen i s anspose and in e se, espec i ely; R𝑛deno es
he 𝑛-dimensional Euclidean space and R𝑚×𝑛 is he se
o all eal ma ices o dimension 𝑚×𝑛.Thesho hand
Diag(𝑎1,...,𝑎𝑟)deno es a diagonal ma ix whose diagonal
en ies a e 𝑎1,...,𝑎𝑟. I he dimensions o ma ices a e no
explici ly s a ed, hey a e assumed o be compa ible o
algeb aic ope a ions. 𝛿𝑘−𝑠 is he K onecke del a unc ion,
which is equal o one, i 𝑘=𝑠,andze oo he wise.
Mo eo e , o a bi a y andom ec o s 𝛼and 𝛽,wewill
deno e Co [𝛼,𝛽] = 𝐸[(𝛼−𝐸[𝛼])(𝛽−𝐸[𝛽])𝑇]and Co [𝛼] =
Co [𝛼,𝛼],whe e𝐸[⋅] s ands o he ma hema ical expec a-
ion ope a o . Finally, 
𝛼deno es he es ima o o 𝛼and 
𝛼=
𝛼−
𝛼 he es ima ion e o .
2. P oblem Fo mula ion
Ou aim is o ob ain ecu si e algo i hms o he op imal LS
linea il e ing p oblem in a class o disc e e- ime s ochas ic
sys ems wi h missing measu emen s coming om mul iple
senso s, by using cen alized and dis ibu ed usion me hods.
In his sec ion, i s ly he sys em model and he assump ions
abou he s a e and noise p ocesses a e p esen ed and, sec-
ondly, he op imal LS linea es ima ion p oblem is o mula ed
using an inno a ion app oach.
2.1. S ochas ic Sys em Model. Conside a disc e e- ime linea
s ochas ic sys em wi h au oco ela ed and c oss-co ela ed
noises and missing measu emen s coming om 𝑟senso s.
The phenomenon o missing measu emen s occu s andomly
and, o each senso , a di e en sequence o scala andom
a iables wi h disc e e dis ibu ion o e he in e al [0,1]is
used o model his phenomenon. Speci ically, he ollowing
sys em is conside ed:
𝑥𝑘=𝐹
𝑘−1𝑥𝑘−1 +𝑤𝑘−1,𝑘≥1, (1)
whe e 𝑥𝑘∈R𝑛is he s a e, {𝑤𝑘;𝑘≥0}is he p ocess
noise, and 𝐹𝑘, o 𝑘≥0,a eknownma iceswi hcompa ible
dimensions.
Conside 𝑟senso s which, a any ime 𝑘,p o idescala
measu emen s o he sys em s a e, pe u bed by addi i e and
mul iplica i e noises acco ding o he ollowing model:
𝑦𝑖
𝑘=𝜃𝑖
𝑘𝐻𝑖
𝑘𝑥𝑘+V𝑖
𝑘, 𝑘≥1, 𝑖=1,2,...,𝑟, (2)
whe e {𝑦𝑖
𝑘;𝑘≥1}a e he measu ed da a; {V𝑖
𝑘;𝑘≥1}a e
measu emen noises; {𝜃𝑖
𝑘;𝑘≥1}a e scala andom a iables
sequences; 𝐻𝑖
𝑘, o 𝑘≥1, a e known ime- a ying ma ices
wi h compa ible dimensions; supe sc ip 𝑖deno es he 𝑖 h
senso , and 𝑟is he numbe o senso s.
Nex , he s a is ical p ope ies assumed abou he ini ial
s a e and noise p ocesses in ol ed in (1)and(2) a e speci ied.
(i) The ini ial s a e 𝑥0is a andom ec o wi h 𝐸[𝑥0]=𝑥0
and Co [𝑥0]=𝑃
0.
(ii) The p ocess noise, {𝑤𝑘;𝑘≥0}, and he measu emen
noises, {V𝑖
𝑘;𝑘 ≥ 1},𝑖 = 1,2,...,𝑟, a e ze o-mean
sequences wi h co a iances and c oss-co a iances:
Co [𝑤𝑘,𝑤𝑠]=𝑄𝑘,𝑘𝛿𝑘−𝑠 +𝑄𝑘,𝑠𝛿𝑘−𝑠+1 +𝑄𝑘,𝑠𝛿𝑘−𝑠−1,
Co [V𝑖
𝑘,V𝑗
𝑠]=𝑅𝑖𝑗
𝑘,𝑘𝛿𝑘−𝑠 +𝑅𝑖𝑗
𝑘,𝑠𝛿𝑘−𝑠+1 +𝑅𝑖𝑗
𝑘,𝑠𝛿𝑘−𝑠−1,
Co [𝑤𝑘,V𝑖
𝑠]=𝑆𝑖
𝑘,𝑘𝛿𝑘−𝑠 +𝑆𝑖
𝑘,𝑠𝛿𝑘−𝑠+1 +𝑆𝑖
𝑘,𝑠𝛿𝑘−𝑠+2.
(3)
(iii) The mul iplica i e noises {𝜃𝑖
𝑘;𝑘≥1},𝑖=1,2,...,𝑟,
a e whi e sequences o scala a iables wi h disc e e
dis ibu ion o e he in e al [0,1],wi h𝐸[𝜃𝑖
𝑘]=𝜃𝑖
𝑘
and Va [𝜃𝑖
𝑘]=𝑉𝜃𝑖
𝑘.
(i ) The ini ial s a e 𝑥0and he mul iplica i e noises
{𝜃𝑖
𝑘;𝑘≥1}, o 𝑖 = 1,2,...,𝑟,a emu ually
independen , and hey a e independen o he addi i e
noises {𝑤𝑘;𝑘≥0}and {V𝑖
𝑘;𝑘≥1}, o 𝑖=1,2,...,𝑟.
Rema k 1. F om assump ion (ii) he ollowing co ela ion
p ope ies o he addi i e noises a e easily deduced.
(1) The noise ec o s 𝑤𝑘and 𝑤𝑠a eco ela eda consec-
u i e sampling imes, |𝑘 − 𝑠| = 1, and independen
o he wise; he co a iance ma ices o 𝑤𝑘wi h 𝑤𝑘−1,
and 𝑤𝑘+1 a e 𝑄𝑘,𝑘−1,and𝑄𝑘,𝑘+1, espec i ely.
(2) Fo 𝑖,𝑗=1,2,...,𝑟, he measu emen noises V𝑖
𝑘and
V𝑗
𝑠a e c oss-co ela ed a he same sampling ime and
a consecu i e sampling imes, |𝑘−𝑠| = 0,1,and
independen o he wise; he c oss-co a iances o V𝑖
𝑘
wi h V𝑗
𝑘,V𝑗
𝑘−1 and V𝑗
𝑘+1 a e 𝑅𝑖𝑗
𝑘,𝑘,𝑅𝑖𝑗
𝑘,𝑘−1 and 𝑅𝑖𝑗
𝑘,𝑘+1,
espec i ely.
(3) Fo 𝑖 = 1,2,...,𝑟, he measu emen noises V𝑖
𝑘
a e co ela ed wi h he noise ec o s 𝑤𝑠, o 𝑠=
𝑘, 𝑘−1, 𝑘−2, and independen o he wise; he c oss-
co a iance ma ices o V𝑖
𝑘wi h 𝑤𝑘,𝑤𝑘−1 and 𝑤𝑘−2 a e
𝑆𝑖
𝑘,𝑘,𝑆𝑖
𝑘−1,𝑘 and 𝑆𝑖
𝑘−2,𝑘, espec i ely.
The co ela ion condi ions o he p ocess noise and he
measu emen noises conside ed in his pape a e he same as
hose in [16]. Sys ems wi h only ini e-s ep co ela ed p ocess
noises o mul is ep co ela ed p ocess and measu emen
noises a e conside ed in [13–15], among o he s. The cu en
s udy can be ex ended o mo e gene al sys ems in ol ing
ini e-s ep au oco ela ed and c oss-co ela ed noises wi h no
di icul y, excep o a g ea e complexi y in he ma hema ical
de i a ions.
4Ma hema ical P oblems in Enginee ing
Rema k 2. F om he s a e equa ion (1) and assump ions (ii)
and (i ), i is easy o deduce ha 𝐷𝑘=𝐸[𝑥𝑘𝑥𝑇
𝑘]is ecu si ely
calcula ed by
𝐷𝑘=𝐹
𝑘−1𝐷𝑘−1𝐹𝑇
𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹
𝑘−1𝑄𝑘−2,𝑘−1
+𝑄𝑘−1,𝑘−2𝐹𝑇
𝑘−1, 𝑘≥2,
𝐷1=𝐹
0𝐷0𝐹𝑇
0+𝑄0,0,𝐷
0=𝑃
0+𝑥0𝑥𝑇
0.
(4)
Also, i is easy o see ha he s a e 𝑥𝑘is co ela ed wi h
he measu emen noises V𝑖
𝑘, o 𝑖 = 1,2,...,𝑟,and he
expec a ions 𝐸𝑖
𝑘=𝐸[𝑥𝑘V𝑖
𝑘]sa is y
𝐸𝑖
𝑘=𝐹
𝑘−1𝑆𝑖
𝑘−2,𝑘 +𝑆𝑖
𝑘−1,𝑘, 𝑘≥2; 𝐸𝑖
1=𝑆𝑖
0,1.(5)
Rema k 3. Acco ding o assump ion (iii), he scala andom
a iables 𝜃𝑖
𝑘 ake alues o e he in e al [0,1] and hey can
sa is y any a bi a y disc e e p obabili y dis ibu ion o e
such in e al, o ins ance, a Be noulli dis ibu ion. Usually,
Be noulli andom a iables ha e been used o model he
phenomenon o missing measu emen s (see, e.g., [25]and
e e ences he ein), wi h 𝜃𝑖
𝑘=1meaning ha he s a e
𝑥𝑘is p esen in he measu emen 𝑦𝑖
𝑘coming om he 𝑖 h
senso a ime 𝑘, while 𝜃𝑖
𝑘=0means ha he s a e is
missing in he measu ed da a a ime 𝑘o , equi alen ly, ha
such obse a ion only con ains addi i e noise V𝑖
𝑘.Howe e ,
in p ac ice, he in o ma ion ansmi ed a a sampling ime
can usually be nei he comple ely missing no comple ely
success ul, bu only pa o he in o ma ion can go h ough;
in such si ua ions, only pa ial in o ma ion is missing and he
p opo ion o missed da a a one momen is a ac ion o he
han 0 o 1 (see, e.g., [28,29] and e e ences he ein).
2.2. S acked Measu emen Equa ion. As no ed abo e, ou
aimis osol e heop imalLSlinea es ima ionp oblem
o he s a e 𝑥𝑘basedon hemeasu emen s{𝑦𝑖
1,𝑦𝑖
2,...,𝑦𝑖
𝑘},
o 𝑖 = 1,2,...,𝑟, by using cen alized and dis ibu ed
usion me hods o p ocess he measu ed senso da a. The
cen alized usion me hod conside s ha all he measu emen
da a coming om 𝑟senso s a e ansmi ed o a usion cen e
o being p ocessed; o his pu pose and o simpli y he
no a ion, he measu emen equa ion (2)is ew i enina
s acked o m as ollows:
𝑦𝑘=Θ𝑘𝐻𝑘𝑥𝑘+V𝑘, 𝑘≥1, (6)
whe e 𝑦𝑘=(𝑦
1
𝑘,...,𝑦𝑟
𝑘)𝑇,V𝑘=(V1
𝑘,...,V𝑟
𝑘)𝑇,𝐻𝑘=(𝐻
1𝑇
𝑘,...,
𝐻𝑟𝑇
𝑘)𝑇,andΘ𝑘=Diag(𝜃1
𝑘,...,𝜃𝑟
𝑘).
The ollowing p ope ies o he noises in (6)a eeasily
in e ed om he model assump ions (ii)–(i ) p e iously
s a ed.
(i) The addi i e noise {V𝑘;𝑘≥1}is a ze o-mean p ocess
sa is ying:
Co [V𝑘,V𝑠]=𝑅𝑘,𝑘𝛿𝑘−𝑠 +𝑅𝑘,𝑠𝛿𝑘−𝑠+1 +𝑅𝑘,𝑠𝛿𝑘−𝑠−1,
Co [𝑤𝑘,V𝑠]=𝑆𝑘,𝑘𝛿𝑘−𝑠 +𝑆𝑘,𝑠𝛿𝑘−𝑠+1 +𝑆𝑘,𝑠𝛿𝑘−𝑠+2,(7)
whe e 𝑅𝑘,𝑠 =(𝑅𝑖𝑗
𝑘,𝑠)𝑖,𝑗=1,2,...,𝑟 and 𝑆𝑘,𝑠 =(𝑆1
𝑘,𝑠,...,𝑆𝑟
𝑘,𝑠).
(ii) The s a e ec o 𝑥𝑘and he measu emen noise ec o
V𝑘a e co ela ed wi h 𝐸𝑘=𝐸[𝑥𝑘V𝑇
𝑘]sa is ying
𝐸𝑘=𝐹
𝑘−1𝑆𝑘−2,𝑘 +𝑆𝑘−1,𝑘,𝑘≥2,𝐸
1=𝑆0,1.(8)
(iii) The andom ma ices {Θ𝑘;𝑘≥1}sa is y 𝐸[Θ𝑘]=
Θ𝑘=Diag(𝜃1
𝑘,...,𝜃𝑟
𝑘)and 𝐸[(Θ𝑘− Θ𝑘)2]=
Diag(𝑉𝜃1
𝑘,...,𝑉𝜃𝑟
𝑘); also, deno ing 𝜃𝑘=(𝜃
1
𝑘,...,𝜃𝑟
𝑘)𝑇,
i is clea ha Co [𝜃𝑘]=Diag(𝑉𝜃1
𝑘,...,𝑉𝜃𝑟
𝑘).
Mo eo e , o any andom ma ix 𝐺independen o
{Θ𝑘;𝑘≥1}, i is easily deduced ha
𝐸[(Θ𝑘−Θ𝑘)𝐺(Θ𝑘−Θ𝑘)]= Co [𝜃𝑘]∘𝐸[𝐺],(9)
whe e ∘deno es he Hadama d p oduc [23].
(i ) The ini ial s a e 𝑥0and {Θ𝑘;𝑘≥1}a e independen ,
and hey a e independen o {𝑤𝑘;𝑘≥0}and {V𝑘;𝑘≥
1}.
2.3. Inno a ion App oach o he Op imal LS Linea Es i-
ma ion P oblem. To add ess he op imal LS linea es ima-
ion p oblem o he s a e 𝑥𝑘based on he measu emen s
{𝑦𝑖
1,𝑦𝑖
2,...,𝑦𝑖
𝑘},𝑖=1,2,...,𝑟, he cen alized and dis ibu ed
usion me hods will be used. In bo h cases, ecu si e algo-
i hms o he LS linea es ima o s will be es ablished using an
inno a ion app oach and he o hogonal p ojec ion Lemma
(OPL); mo e speci ically we ha e he ollowing.
Cen alized Fusion Es ima ion P oblem. Ou aim is o ob ain
he op imal LS linea il e , 
𝑥𝑘/𝑘,o hes a e𝑥𝑘based on
he measu emen s {𝑦1,𝑦2,...,𝑦𝑘},gi enin(6), by ecu si e
algo i hms.
Asknown, heLSlinea il e 
𝑥𝑘/𝑘 is he o hogonal
p ojec ion o he s a e 𝑥𝑘o e he linea space spanned by
{𝑦1,𝑦2,...,𝑦𝑘}. These obse a ions a e gene ally nono hog-
onal ec o s, bu he G am-Schmid o hogonaliza ion p o-
cedu e allows us o subs i u e hem by a se o o hogonal
ec o s, called inno a ions, de ined as he di e ence be ween
each obse a ion and i s one-s age p edic o . Due o he
o hogonali y p ope y o he inno a ions and since he
inno a ion p ocess is uniquely de e mined by he obse a-
ions, he LS linea il e , 
𝑥𝑘/𝑘, can be calcula ed as linea
combina ion o he inno a ions; namely,

𝑥𝑘/𝑘 =𝑘
∑
𝑠=1
X𝑘,𝑠Π−1
𝑠,𝑠𝜇𝑠, 𝑘≥1, (10)
whe e 𝜇𝑠=𝑦
𝑠−
𝑦𝑠/𝑠−1 a e he inno a ion ec o s, wi h 
𝑦𝑠/𝑠−1
he one-s age obse a ion p edic o , Π𝑠,𝑠 = 𝐸[𝜇𝑠𝜇𝑇
𝑠],and
X𝑘,𝑠 =𝐸[𝑥𝑘𝜇𝑇
𝑠].
Dis ibu ed Fusion Es ima ion P oblem. To add ess he dis-
ibu ed usion es ima ion p oblem, i s ly, ecu si e algo-
i hms o ob ain local LS linea il e s, 
𝑥𝑖
𝑘/𝑘, o 𝑖=1,2,...,𝑟,
and he e o c oss-co a iance ma ices be ween any wo
local es ima es, a e de i ed. Secondly, he dis ibu ed usion
il e , 
𝑥𝐷
𝑘/𝑘, is es ablished by applying he op imal in o ma ion
Ma hema ical P oblems in Enginee ing 5
usion c i e ion weigh ed by ma ices in he linea minimum
a iance sense [30].
Analogously o (10), deno ing 𝜇𝑖
𝑠=𝑦
𝑖
𝑠−
𝑦𝑖
𝑠/𝑠−1,Π𝑖𝑖
𝑠,𝑠 =
𝐸[𝜇𝑖
𝑠𝜇𝑖
𝑠],andX𝑖
𝑘,𝑠 = 𝐸[𝑥𝑘𝜇𝑖
𝑠], helocal il e 
𝑥𝑖
𝑘/𝑘 is exp essed
as

𝑥𝑖
𝑘/𝑘 =𝑘
∑
𝑠=1
X𝑖
𝑘,𝑠(Π𝑖𝑖
𝑠,𝑠)−1𝜇𝑖
𝑠, 𝑘≥1. (11)
3. Op imal LS Linea Cen alized
Fusion Es ima ion
In his sec ion a ecu si e algo i hm o he cen alized
op imal (unde he LS c i e ion) linea il e , 
𝑥𝑘/𝑘 is de i ed.
Such algo i hm is deduced using (10) and he OPL, and i is
p esen ed in Theo em 5. Fi s ly, in o de o simpli y he p oo
o Theo em 5, he ollowing lemma is es ablished.
Lemma 4. Unde assump ions (i)–(i ), he ollowing esul s
hold:
W𝑘,𝑘 =𝐸[𝑤
𝑘𝜇𝑇
𝑘]=𝑄𝑘,𝑘−1𝐻𝑇
𝑘Θ𝑘+𝑆𝑘,𝑘,𝑘≥1, (12)
V𝑘,𝑘−1 =𝐸[V𝑘𝜇𝑇
𝑘−1]=𝑆𝑇
𝑘−2,𝑘𝐻𝑇
𝑘−1Θ𝑘−1 +𝑅𝑘,𝑘−1, 𝑘≥2.
(13)
P oo . Since 𝑤𝑘is independen o 𝑦1,...,𝑦𝑘−1,𝐸[𝑤𝑘
𝑦𝑇
𝑘/𝑘−1]=
0and hence W𝑘,𝑘 = 𝐸[𝑤𝑘𝑦𝑇
𝑘].Now,using(1)and(6), W𝑘,𝑘
canbecalcula edas ollows:
W𝑘,𝑘 =𝐸[𝑤
𝑘(Θ𝑘𝐻𝑘𝑥𝑘+V𝑘)𝑇]
=𝐸[𝑤
𝑘𝑥𝑇
𝑘]𝐻𝑇
𝑘Θ𝑘+𝑆𝑘,𝑘
=𝐸[𝑤
𝑘(𝐹𝑘−1𝑥𝑘−1 +𝑤𝑘−1)𝑇]𝐻𝑇
𝑘Θ𝑘+𝑆𝑘,𝑘
=𝑄
𝑘,𝑘−1𝐻𝑇
𝑘Θ𝑘+𝑆𝑘,𝑘.
(14)
Taking in o accoun ha V𝑘is independen o 𝑦1,...,𝑦𝑘−2, he
calcula ion o V𝑘,𝑘−1 is simila o ha o W𝑘,𝑘, and hence he
p oo is omi ed.
Theo em 5. Fo he sys em model (1)and measu emen model
(6), unde assump ions (i)–(i ), he op imal LS linea il e 
𝑥𝑘/𝑘
is ob ained as

𝑥𝑘/𝑘 =
𝑥𝑘/𝑘−1 +X𝑘,𝑘Π−1
𝑘,𝑘𝜇𝑘, 𝑘≥1, 
𝑥0/0 =𝑥0,(15)
whe e hes a ep edic o ,
𝑥𝑘/𝑘−1,sa is ies

𝑥𝑘/𝑘−1 =𝐹
𝑘−1 
𝑥𝑘−1/𝑘−1 +W𝑘−1,𝑘−1Π−1
𝑘−1,𝑘−1𝜇𝑘−1, 𝑘≥2,

𝑥1/0 =𝐹
0
𝑥0/0.(16)
The inno a ion, 𝜇𝑘,isgi enby
𝜇𝑘=𝑦
𝑘−Θ𝑘𝐻𝑘
𝑥𝑘/𝑘−1 −V𝑘,𝑘−1Π−1
𝑘−1,𝑘−1𝜇𝑘−1, 𝑘≥2,
𝜇1=𝑦
1−Θ1𝐻1
𝑥1/0.(17)
The ma ix X𝑘,𝑘 =𝐸[𝑥𝑘𝜇𝑇
𝑘]is calcula ed by
X𝑘,𝑘 =𝑃
𝑘/𝑘−1𝐻𝑇
𝑘Θ𝑘+𝐸𝑘−X𝑘,𝑘−1Π−1
𝑘−1,𝑘−1V𝑇
𝑘,𝑘−1, 𝑘≥2,
X1,1 =𝑃
1/0𝐻𝑇
1Θ1+𝐸1,(18)
whe e X𝑘,𝑘−1 =𝐸[𝑥𝑘𝜇𝑇
𝑘−1]sa is ies
X𝑘,𝑘−1 =𝐹
𝑘−1X𝑘−1,𝑘−1 +W𝑘−1,𝑘−1, 𝑘≥2. (19)
The p edic ion e o co a iance ma ix, 𝑃𝑘/𝑘−1,isob ainedby
𝑃𝑘/𝑘−1 =𝐹
𝑘−1𝑃𝑘−1/𝑘−1𝐹𝑇
𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹
𝑘−1J𝑘−1 +J𝑇
𝑘−1𝐹𝑇
𝑘−1
−W𝑘−1,𝑘−1Π−1
𝑘−1,𝑘−1W𝑇
𝑘−1,𝑘−1, 𝑘≥2,
𝑃1/0 =𝐹
0𝑃0/0𝐹𝑇
0+𝑄0,0,(20)
whe e J𝑘=𝐸[
𝑥𝑘/𝑘𝑤𝑇
𝑘]is calcula ed by
J𝑘=𝑄𝑘−1,𝑘 −X𝑘,𝑘Π−1
𝑘,𝑘W𝑇
𝑘,𝑘,𝑘≥1. (21)
The il e ing e o co a iance ma ix, 𝑃𝑘/𝑘,isgi enby
𝑃𝑘/𝑘 =𝑃
𝑘/𝑘−1 −X𝑘,𝑘Π−1
𝑘,𝑘X𝑇
𝑘,𝑘, 𝑘≥1, 𝑃
0/0 =𝑃
0.(22)
The inno a ion co a iance ma ix, Π𝑘,𝑘,sa is ies
Π𝑘,𝑘 =Co (𝜃𝑘)∘(𝐻𝑘𝐷𝑘𝐻𝑇
𝑘)+𝑅𝑘,𝑘 +Θ𝑘𝐻𝑘X𝑘,𝑘
+X𝑇
𝑘,𝑘𝐻𝑇
𝑘Θ𝑘−Θ𝑘𝐻𝑘𝑃𝑘/𝑘−1𝐻𝑇
𝑘Θ𝑘
−V𝑘,𝑘−1Π−1
𝑘−1,𝑘−1V𝑇
𝑘,𝑘−1, 𝑘≥2,
Π1,1 =Co (𝜃1)∘(𝐻1𝐷1𝐻𝑇
1)+𝑅1,1 +Θ1𝐻1X1,1
+X𝑇
1,1𝐻𝑇
1Θ1−Θ1𝐻1𝑃1/0𝐻𝑇
1Θ1.
(23)
The ma ices 𝐷𝑘,𝐸𝑘,W𝑘,𝑘,andV𝑘,𝑘−1 a e gi en in (4),(8),
(12),and(13), espec i ely.
P oo . See Appendix 6.
Rema k 6. In con en ional es ima ion p oblems in sys ems
wi h missing measu emen s and unco ela ed addi i e whi e
noises, he one-s age s a e and obse a ion p edic o s a e
calcula ed as 
𝑥𝑘/𝑘−1 =𝐹
𝑘−1 
𝑥𝑘−1/𝑘−1 and 
𝑦𝑘/𝑘−1 =Θ𝑘𝐻𝑘
𝑥𝑘/𝑘−1,
espec i ely.Howe e , hisisno ue o hep oblema hand
since, due o he co ela ion assump ion (ii), he noise es i-
ma o s 
𝑤𝑘−1/𝑘−1 and 
V𝑘/𝑘−1 mus be aken in o accoun o he
de i a ion o he p edic o s. Besides he ac o conside ing
missing measu emen s, his is he main di e ence be ween
he op imal es ima o s p oposed in he cu en pape and he
subop imal Kalman- ype ones p oposed in [16], whe e he
noise es ima o s a e conside ed o be equal o ze o.

6Ma hema ical P oblems in Enginee ing
4. Dis ibu ed Fusion Es ima ion
One o he main disad an ages o he cen alized usion
es ima o s de i ed in Sec ion 3 is ha hey may ha e a
high compu a ional cos due o augmen a ion. Mo eo e ,
as is widely known, he cen alized app oach has se e al
o he d awbacks, such as aul de ec ion, isola ion, poo
eliabili y,andso o h.Too e come hesedisad an ages,
ou aim in his sec ion is o add ess he op imal dis ibu ed
usion es ima ion p oblem, in which each single senso
p o ides i s local LS linea es ima o and hei es ima ion
e o co a iance ma ices, and hen hese local es ima o s
along wi h he co a iances and c oss-co a iance ma ices o
he es ima ion e o s be ween any wo senso s a e sen o
he usion cen e o usion based on he ma ices-weigh ed
usion es ima ion c i e ion in he linea minimum a iance
sense [30].
4.1. Local LS Linea Fil e ing Algo i hms. Fo each single sen-
so subsys em o sys ems (1)and(2), he ollowing heo em
p o ides ecu si e o mulas o he local LS linea il e s, 
𝑥𝑖
𝑘/𝑘,
and hei co esponding e o co a iance ma ices, 𝑃𝑖𝑖
𝑘/𝑘.
Theo em 7. Fo he 𝑖 h senso subsys em o sys ems (1)and
(2)unde assump ions (i)–(i ), he local LS linea il e , 
𝑥𝑖
𝑘/𝑘,is
calcula ed by

𝑥𝑖
𝑘/𝑘 =
𝑥𝑖
𝑘/𝑘−1 +X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1𝜇𝑖
𝑘, 𝑘≥1, 
𝑥𝑖
0/0 =𝑥0,
(24)
whe e he local LS linea p edic o , 
𝑥𝑖
𝑘/𝑘−1,sa is ies

𝑥𝑖
𝑘/𝑘−1 =𝐹
𝑘−1 
𝑥𝑖
𝑘−1/𝑘−1 +W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝜇𝑖
𝑘−1, 𝑘≥2,

𝑥𝑖
1/0 =𝐹
0
𝑥𝑖
0/0,(25)
wi h W𝑖
𝑘,𝑘 =𝜃𝑖
𝑘𝑄𝑘,𝑘−1𝐻𝑖𝑇
𝑘+𝑆𝑖
𝑘,𝑘,𝑘≥1.
The inno a ion, 𝜇𝑖
𝑘,isgi enby
𝜇𝑖
𝑘=𝑦𝑖
𝑘−𝜃𝑖
𝑘𝐻𝑖
𝑘
𝑥𝑖
𝑘/𝑘−1 −V𝑖𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝜇𝑖
𝑘−1,𝑘≥2,
𝜇𝑖
1=𝑦𝑖
1−𝜃𝑖
1𝐻𝑖
1
𝑥𝑖
1/0,(26)
wi h V𝑖𝑖
𝑘,𝑘−1 =𝜃𝑖
𝑘−1𝑆𝑖𝑇
𝑘−2,𝑘𝐻𝑖𝑇
𝑘−1 +𝑅𝑖𝑖
𝑘,𝑘−1,𝑘≥2.
The ec o X𝑖
𝑘,𝑘 =𝐸[𝑥𝑘𝜇𝑖
𝑘]is calcula ed om he ollowing
exp ession
X𝑖
𝑘,𝑘 =𝜃𝑖
𝑘𝑃𝑖𝑖
𝑘/𝑘−1𝐻𝑖𝑇
𝑘+𝐸𝑖
𝑘−X𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1V𝑖𝑖
𝑘,𝑘−1,
𝑘≥2,
X𝑖
1,1 =𝜃𝑖
1𝑃𝑖𝑖
1/0𝐻𝑖𝑇
1+𝐸𝑖
1,(27)
whe e X𝑖
𝑘,𝑘−1 =𝐹
𝑘−1X𝑖
𝑘−1,𝑘−1 +W𝑖
𝑘−1,𝑘−1,𝑘≥2.
The local p edic ion e o co a iance ma ix, 𝑃𝑖𝑖
𝑘/𝑘−1,is
ob ained by
𝑃𝑖𝑖
𝑘/𝑘−1 =𝐹
𝑘−1𝑃𝑖𝑖
𝑘−1/𝑘−1𝐹𝑇
𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹
𝑘−1J𝑖
𝑘−1 +J𝑖𝑇
𝑘−1𝐹𝑇
𝑘−1
−W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1W𝑖𝑇
𝑘−1,𝑘−1, 𝑘≥2,
𝑃𝑖𝑖
1/0 =𝐹
0𝑃𝑖𝑖
0/0𝐹𝑇
0+𝑄0,0,(28)
whe e J𝑖
𝑘=𝑄
𝑘−1,𝑘 −X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1W𝑖𝑇
𝑘,𝑘,𝑘≥1,and𝑃𝑖𝑖
𝑘/𝑘, he
il e inge o co a iancema ix,isgi enby
𝑃𝑖𝑖
𝑘/𝑘 =𝑃𝑖𝑖
𝑘/𝑘−1 −X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1X𝑖𝑇
𝑘,𝑘, 𝑘≥1, 𝑃𝑖𝑖
0/0 =𝑃
0.
(29)
The inno a ion a iance, Π𝑖𝑖
𝑘,𝑘,sa is ies
Π𝑖𝑖
𝑘,𝑘 =𝑉
𝜃𝑖
𝑘𝐻𝑖
𝑘𝐷𝑘𝐻𝑖𝑇
𝑘+𝑅𝑖𝑖
𝑘,𝑘 +𝜃𝑖
𝑘𝐻𝑖
𝑘X𝑖
𝑘,𝑘 +𝜃𝑖
𝑘X𝑖𝑇
𝑘,𝑘𝐻𝑖𝑇
𝑘
−(𝜃𝑖
𝑘)2𝐻𝑖
𝑘𝑃𝑖𝑖
𝑘/𝑘−1𝐻𝑖𝑇
𝑘−(V𝑖𝑖
𝑘,𝑘−1)2(Π𝑖𝑖
𝑘−1,𝑘−1)−1,
𝑘≥2,
Π𝑖𝑖
1,1 =𝑉
𝜃𝑖
1𝐻𝑖
1𝐷1𝐻𝑖𝑇
1+𝑅𝑖𝑖
1,1 +𝜃𝑖
1𝐻𝑖
1X𝑖
1,1
+𝜃𝑖
1X𝑖𝑇
1,1𝐻𝑖𝑇
1−(𝜃𝑖
1)2𝐻𝑖
1𝑃𝑖𝑖
1/0𝐻𝑖𝑇
1.
(30)
The ma ix 𝐷𝑘and he ec o 𝐸𝑖
𝑘a e gi en in (4)and (5),
espec i ely.
P oo . Thep oo ,basedon heinno a ionapp oachand he
OPL, is omi ed o being analogous o ha o Theo em 5.
Ne e heless, i should be indica ed ha , in his p oo , he
Hadama d p oduc is no used since, ins ead o he diagonal
s ochas ic ma ix Θ𝑘, hescala a iable𝜃𝑖
𝑘is now in ol ed in
he de i a ion o he es ima o s.
Rema k 8. As indica ed in Rema k 6 o he cen alized
es ima o s, i mus be no ed ha , due o he co ela ion
assump ion (ii) o he addi i e noises {𝑤𝑘}and {V𝑖
𝑘}, he
es ima o s 
𝑤𝑖
𝑘−1/𝑘−1 =W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝜇𝑖
𝑘−1 and 
V𝑖
𝑘/𝑘−1 =
V𝑖𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝜇𝑖
𝑘−1 a e no equal o ze o, and hence he
op imal local s a e p edic o , 
𝑥𝑖
𝑘/𝑘−1 =𝐹
𝑘−1 
𝑥𝑖
𝑘−1/𝑘−1+
𝑤𝑖
𝑘−1/𝑘−1,
and he obse a ion p edic o , 
𝑦𝑖
𝑘/𝑘−1 = 𝜃𝑖
𝑘𝐻𝑖
𝑘
𝑥𝑖
𝑘/𝑘−1 +
V𝑖
𝑘/𝑘−1,
a e qui e di e en om con en ional il e ing algo i hms
wi h unco ela ed whi e noises. This issue, along wi h he
conside a ion o missing measu emen s a each single senso ,
cons i u es he main di e ence be ween he cu en op imal
local es ima o s and he subop imal local es ima o s p o-
posed in [16].
4.2. C oss-Co a iance Ma ices o Local Es ima ion E o s. To
apply he op imal usion c i e ion weigh ed by ma ices in
Ma hema ical P oblems in Enginee ing 7
he linea minimum a iance sense, he il e ing, 𝑃𝑖𝑗
𝑘/𝑘,and
p edic ion, 𝑃𝑖𝑗
𝑘/𝑘−1, e o c oss-co a iance ma ices be ween
local es ima o s o any wo subsys ems mus be calcula ed.
Fo simplici y, besides he no a ion o Theo em 7, o 𝑖 =𝑗,
𝑖,𝑗=1,2...,𝑟, we in oduce he ollowing no a ion:
𝐿𝑖𝑗
𝑘=𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘], Π𝑖𝑗
𝑘,𝑠 =𝐸[𝜇𝑖
𝑘𝜇𝑗
𝑠],
V𝑖𝑗
𝑘,𝑘−1 =𝐸[V𝑖
𝑘𝜇𝑗
𝑘−1]. (31)
Also,ino de osimpli y hecalcula iono hee o c oss-
co a iance ma ices, he ollowing lemmas a e gi en.
Lemma 9. Unde assump ions (i)–(i ), he ollowing esul s
hold.
(a) The expec a ion 𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘−1]sa is ies
𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘−1]=𝐹
𝑘−1𝐿𝑖𝑗
𝑘−1
+X𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1, 𝑘≥2.
(32)
(b) The expec a ion 𝐸[
𝑥𝑖
𝑘/𝑘−1V𝑗
𝑘]sa is ies
𝐸[
𝑥𝑖
𝑘/𝑘−1V𝑗
𝑘]=X𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1V𝑗𝑖
𝑘,𝑘−1, 𝑘≥2, (33)
whe e V𝑗𝑖
𝑘,𝑘−1 =𝜃𝑖
𝑘−1𝐻𝑖
𝑘−1𝑆𝑗
𝑘−2,𝑘 +𝑅𝑖𝑗
𝑘−1,𝑘.
(c) The expec a ion 𝐸[V𝑖
𝑘𝜇𝑗
𝑘]sa is ies
𝐸[V𝑖
𝑘𝜇𝑗
𝑘]=𝜃𝑗
𝑘𝐸𝑖𝑇
𝑘𝐻𝑗𝑇
𝑘+𝑅𝑖𝑗
𝑘,𝑘
−V𝑖𝑗
𝑘,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1(𝜃𝑗
𝑘𝐻𝑗
𝑘X𝑗
𝑘,𝑘−1 +V𝑗𝑗
𝑘,𝑘−1)𝑇,
𝑘≥2.
(34)
P oo . (a) F om (25) o 
𝑥𝑖
𝑘/𝑘−1 and (24) o 
𝑥𝑖
𝑘−1/𝑘−1,weha e
𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘−1]
=𝐹
𝑘−1𝐸[
𝑥𝑖
𝑘−1/𝑘−1𝜇𝑗
𝑘−1]+W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
=𝐹
𝑘−1𝐸[
𝑥𝑖
𝑘−1/𝑘−2𝜇𝑗
𝑘−1]+𝐹
𝑘−1X𝑖
𝑘−1,𝑘−1
×(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
+W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
=𝐹
𝑘−1𝐿𝑖𝑗
𝑘−1 +(𝐹
𝑘−1X𝑖
𝑘−1,𝑘−1 +W𝑖
𝑘−1,𝑘−1)
×(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1,(35)
and since X𝑖
𝑘,𝑘−1 =𝐹
𝑘−1X𝑖
𝑘−1,𝑘−1 +W𝑖
𝑘−1,𝑘−1, exp ession (32)
is p o ed.
(b) Analogously, aking in o accoun ha 𝐸[
𝑥𝑖
𝑘−1/𝑘−2V𝑗
𝑘]=
0,weha e
𝐸[
𝑥𝑖
𝑘/𝑘−1V𝑗
𝑘]=𝐹
𝑘−1𝐸[
𝑥𝑖
𝑘−1/𝑘−1V𝑗
𝑘]
+W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝐸[𝜇𝑖
𝑘−1V𝑗
𝑘]
=(𝐹𝑘−1X𝑖
𝑘−1,𝑘−1+W𝑖
𝑘−1,𝑘−1)(Π𝑖𝑖
𝑘−1,𝑘−1)−1V𝑗𝑖
𝑘,𝑘−1,
(36)
and exp ession (33) is immedia ely ob ained. Finally, he
de i a ion o exp ession V𝑗𝑖
𝑘,𝑘−1 = 𝜃𝑖
𝑘−1𝐻𝑖
𝑘−1𝑆𝑗
𝑘−2,𝑘 +𝑅
𝑖𝑗
𝑘−1,𝑘
is simila o ha o (13) and hence i is omi ed.
(c) Taking in o accoun exp ession (26) o 𝜇𝑗
𝑘,wi h(2)
o 𝑦𝑗
𝑘,weha e
𝐸[V𝑖
𝑘𝜇𝑗
𝑘]=𝜃𝑗
𝑘𝐸𝑖𝑇
𝑘𝐻𝑗𝑇
𝑘+𝑅𝑖𝑗
𝑘,𝑘 −𝜃𝑗
𝑘𝐸[V𝑖
𝑘
𝑥𝑗𝑇
𝑘/𝑘−1]𝐻𝑗𝑇
𝑘
−V𝑖𝑗
𝑘,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1, 𝑘≥2, (37)
and using (33) o 𝐸[V𝑖
𝑘
𝑥𝑗𝑇
𝑘/𝑘−1], exp ession (34)isob ained.
Lemma 10. Unde assump ions (i)–(i ), o 𝑖 =𝑗,𝑖,𝑗 =
1,2...,𝑟, heexpec a ions𝐿𝑖𝑗
𝑘=𝐸[

𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘]a e ecu si ely
ob ained by
𝐿𝑖𝑗
𝑘= 𝜃𝑗
𝑘(𝑃𝑗𝑗
𝑘/𝑘−1 −𝑃𝑖𝑗
𝑘/𝑘−1)𝐻𝑗𝑇
𝑘−𝐹
𝑘−1𝐿𝑖𝑗
𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1
+X𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1
×(V𝑗𝑖
𝑘,𝑘−1 −V𝑗𝑗
𝑘,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1), 𝑘≥2,
(38)
wi h ini ial condi ion 𝐿𝑖𝑗
1=0.
P oo . Taking in o accoun exp ession (26) o 𝜇𝑗
𝑘,wi h(2) o
𝑦𝑗
𝑘,weha e
𝐿𝑖𝑗
𝑘= 𝜃𝑗
𝑘𝐸[
𝑥𝑖
𝑘/𝑘−1𝑥𝑇
𝑘]𝐻𝑗𝑇
𝑘+𝐸[
𝑥𝑖
𝑘/𝑘−1V𝑗
𝑘]
−𝜃𝑗
𝑘𝐸[
𝑥𝑖
𝑘/𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1]𝐻𝑗𝑇
𝑘
−𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘−1](Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1, 𝑘≥2.
(39)
F om he OPL, 𝐸[
𝑥𝑖
𝑘/𝑘−1𝑥𝑇
𝑘]=𝐸[

𝑥𝑖
𝑘/𝑘−1 
𝑥𝑖𝑇
𝑘/𝑘−1]; hen, aking
in o accoun (32) o 𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘−1],and(33) o 𝐸[
𝑥𝑖
𝑘/𝑘−1V𝑗
𝑘],
i is enough o p o e ha
𝐸[
𝑥𝑖
𝑘/𝑘−1 
𝑥𝑖𝑇
𝑘/𝑘−1]−𝐸[
𝑥𝑖
𝑘/𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1]=𝑃𝑗𝑗
𝑘/𝑘−1 −𝑃𝑖𝑗
𝑘/𝑘−1,(40)
8Ma hema ical P oblems in Enginee ing
which is easily deduced since
𝐸[
𝑥𝑖
𝑘/𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1]
=𝑃𝑖𝑗
𝑘/𝑘−1 −𝐷𝑘+𝐸[
𝑥𝑖
𝑘/𝑘−1 
𝑥𝑖𝑇
𝑘/𝑘−1]+𝐸[
𝑥𝑗
𝑘/𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1],
𝐸[
𝑥𝑗
𝑘/𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1]=𝐷𝑘−𝑃𝑗𝑗
𝑘/𝑘−1.(41)
Lemma 11. Unde assump ions (i)–(i ), o 𝑖 =𝑗,𝑖,𝑗 =
1,2...,𝑟, he inno a ion c oss-co a iance Π𝑖𝑗
𝑘,𝑘 = 𝐸[𝜇𝑖
𝑘𝜇𝑗
𝑘]
sa is ies
Π𝑖𝑗
𝑘,𝑘 = 𝜃𝑖
𝑘𝐻𝑖
𝑘(X𝑗
𝑘,𝑘 −𝐿𝑖𝑗
𝑘)+𝜃𝑗
𝑘𝐸𝑖𝑇
𝑘𝐻𝑗𝑇
𝑘+𝑅𝑖𝑗
𝑘,𝑘
−V𝑖𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘
−V𝑖𝑗
𝑘,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1(𝜃𝑗
𝑘𝐻𝑗
𝑘X𝑗
𝑘,𝑘−1 +V𝑗𝑗
𝑘,𝑘−1)𝑇,
𝑘≥2,
Π𝑖𝑗
1,1 =𝜃𝑖
1𝐻𝑖
1X𝑗
1,1 +𝜃𝑗
1𝐸𝑖
1𝐻𝑗𝑇
1+𝑅𝑖𝑗
1,1,(42)
whe e Π𝑖𝑗
𝑘−1,𝑘 =𝐸[𝜇𝑖
𝑘−1𝜇𝑗
𝑘]is gi en by
Π𝑖𝑗
𝑘−1,𝑘 = 𝜃𝑗
𝑘(X𝑖
𝑘,𝑘−1 −𝐹
𝑘−1𝐿𝑗𝑖
𝑘−1
−X𝑗
𝑘,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1Π𝑗𝑖
𝑘−1,𝑘−1)𝑇𝐻𝑗𝑇
𝑘
+V𝑗𝑖
𝑘,𝑘−1 −Π𝑖𝑗
𝑘−1,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1,𝑘≥2.
(43)
P oo . Taking in o accoun exp ession (26) o 𝜇𝑖
𝑘,wi h(2) o
𝑦𝑖
𝑘,weha e
Π𝑖𝑗
𝑘,𝑘 = 𝜃𝑖
𝑘𝐻𝑖
𝑘𝐸[𝑥𝑘𝜇𝑗
𝑘]+𝐸[V𝑖
𝑘𝜇𝑗
𝑘]−𝜃𝑖
𝑘𝐻𝑖
𝑘𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘]
−V𝑖𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1𝐸[𝜇𝑖
𝑘−1𝜇𝑗
𝑘]
=𝜃𝑖
𝑘𝐻𝑖
𝑘(X𝑗
𝑘,𝑘 −𝐿𝑖𝑗
𝑘)+𝐸[V𝑖
𝑘𝜇𝑗
𝑘]
−V𝑖𝑖
𝑘,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘, 𝑘≥2,
(44)
and, om (34) o 𝐸[V𝑖
𝑘𝜇𝑗
𝑘], exp ession o Π𝑖𝑗
𝑘,𝑘 is clea .
Analogously, aking in o accoun exp ession (26) o 𝜇𝑗
𝑘,
wi h (2) o 𝑦𝑗
𝑘,weha e
Π𝑖𝑗
𝑘−1,𝑘 = 𝜃𝑗
𝑘𝐸[𝜇𝑖
𝑘−1𝑥𝑇
𝑘]𝐻𝑗𝑇
𝑘+𝐸[𝜇𝑖
𝑘−1V𝑗
𝑘]
−𝜃𝑗
𝑘𝐸[𝜇𝑖
𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1]𝐻𝑗𝑇
𝑘
−𝐸[𝜇𝑖
𝑘−1𝜇𝑗
𝑘−1](Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1,
=𝜃𝑗
𝑘(X𝑖
𝑘,𝑘−1 −𝐸[𝜇𝑖
𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1])𝐻𝑗𝑇
𝑘+V𝑗𝑖
𝑘,𝑘−1
−Π𝑖𝑗
𝑘−1,𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1V𝑗𝑗
𝑘,𝑘−1,𝑘≥2,
(45)
and, om (32) o 𝐸[𝜇𝑖
𝑘−1 
𝑥𝑗𝑇
𝑘/𝑘−1], exp ession o Π𝑖𝑗
𝑘−1,𝑘 is
immedia ely de i ed.
In he ollowing heo em, ecu si e o mulas o calcula e
he il e ing and p edic ion e o c oss-co a iance ma ices,
𝑃𝑖𝑗
𝑘/𝑘 and 𝑃𝑖𝑗
𝑘/𝑘−1, espec i ely, a e de i ed.
Theo em 12. Unde assump ions (i)–(i ), he c oss-co a iance
ma ices, 𝑃𝑖𝑗
𝑘/𝑘, o he il e ing e o s be ween he 𝑖 h and he 𝑗 h
senso subsys ems a e ecu si ely compu ed by
𝑃𝑖𝑗
𝑘/𝑘 =𝑃
𝑖𝑗
𝑘/𝑘−1 +X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1Π𝑖𝑗
𝑘,𝑘(Π𝑗𝑗
𝑘,𝑘)−1X𝑗𝑇
𝑘,𝑘
−(X𝑗
𝑘,𝑘 −𝐿𝑖𝑗
𝑘)(Π𝑗𝑗
𝑘,𝑘)−1X𝑗𝑇
𝑘,𝑘
−X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1(X𝑖
𝑘,𝑘 −𝐿𝑗𝑖
𝑘)𝑇, 𝑘≥1,
𝑃𝑖𝑗
0/0 =𝑃
0,
(46)
whe e 𝑃𝑖𝑗
𝑘/𝑘−1, he c oss-co a iance ma ix o he p edic ion
e o be ween he 𝑖 h and he 𝑗 h senso subsys ems, sa is ies
𝑃𝑖𝑗
𝑘/𝑘−1 =𝐹
𝑘−1𝑃𝑖𝑗
𝑘−1/𝑘−1𝐹𝑇
𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹
𝑘−1J𝑖
𝑘−1
+J𝑗𝑇
𝑘−1𝐹𝑇
𝑘−1 +W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
×(Π𝑗𝑗
𝑘−1,𝑘−1)−1W𝑗𝑇
𝑘−1,𝑘−1
−G𝑖𝑗
𝑘−1(Π𝑗𝑗
𝑘−1,𝑘−1)−1W𝑗𝑇
𝑘−1,𝑘−1
−W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1G𝑗𝑖𝑇
𝑘−1, 𝑘≥2,
𝑃𝑖𝑗
1/0 =𝐹
0𝑃𝑖𝑗
0/0𝐹𝑇
0+𝑄0,0,(47)
whe e G𝑖𝑗
𝑘=W𝑗
𝑘,𝑘 +𝐹𝑘(X𝑗
𝑘,𝑘 −𝐿𝑖𝑗
𝑘−X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1Π𝑖𝑗
𝑘,𝑘),𝑘≥1.
The ec o s 𝐿𝑖𝑗
𝑘and he inno a ion c oss-co a iances Π𝑖𝑗
𝑘,𝑘 a e
gi eninLemmas10 and 11, espec i ely.
P oo . By using (24) o 
𝑥𝑖
𝑘/𝑘 and 
𝑥𝑗
𝑘/𝑘,weha e
𝑃𝑖𝑗
𝑘/𝑘 =𝑃
𝑖𝑗
𝑘/𝑘−1 +X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1Π𝑖𝑗
𝑘,𝑘(Π𝑗𝑗
𝑘,𝑘)−1X𝑗𝑇
𝑘,𝑘
−𝐸[(𝑥𝑘−
𝑥𝑖
𝑘/𝑘−1)𝜇𝑗𝑇
𝑘](Π𝑗𝑗
𝑘,𝑘)−1X𝑗𝑇
𝑘,𝑘
−X𝑖
𝑘,𝑘(Π𝑖𝑖
𝑘,𝑘)−1𝐸[𝜇𝑖
𝑘(𝑥𝑘−
𝑥𝑗
𝑘/𝑘−1)𝑇].
(48)
Taking in o accoun ha 𝐸[𝑥𝑘𝜇𝑗
𝑘]=X𝑗
𝑘,𝑘 and 𝐸[
𝑥𝑖
𝑘/𝑘−1𝜇𝑗
𝑘]=
𝐿𝑖𝑗
𝑘, he ecu si e exp ession o he c oss-co a iance ma ices
o he local il e ing e o s is immedia ely deduced.
Ma hema ical P oblems in Enginee ing 9
Following an analogous easoning, using now (25)and
aking in o accoun ha 𝐸[(𝑥𝑘−
𝑥𝑖
𝑘/𝑘)𝑤𝑇
𝑘]=J𝑖
𝑘and
𝐸[𝜇𝑖
𝑘𝑤𝑇
𝑘]=W𝑖𝑇
𝑘,𝑘,i iseasy osee ha
𝑃𝑖𝑗
𝑘/𝑘−1 =𝐹
𝑘−1𝑃𝑖𝑗
𝑘−1/𝑘−1𝐹𝑇
𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹
𝑘−1J𝑖
𝑘−1
+J𝑗𝑇
𝑘−1𝐹𝑇
𝑘−1 +W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
×(Π𝑗𝑗
𝑘−1,𝑘−1)−1W𝑗𝑇
𝑘−1,𝑘−1
−(W𝑗
𝑘−1,𝑘−1 +𝐹
𝑘−1𝐸[
𝑥𝑖
𝑘−1/𝑘−1𝜇𝑗
𝑘−1])
×(Π𝑗𝑗
𝑘−1,𝑘−1)−1W𝑗𝑇
𝑘−1,𝑘−1
−W𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1
×(W𝑖
𝑘−1,𝑘−1 +𝐹
𝑘−1𝐸[
𝑥𝑗
𝑘/𝑘𝜇𝑖
𝑘−1])𝑇.
(49)
Finally, using again (24) o 
𝑥𝑖
𝑘−1/𝑘−1, and since
𝐸[
𝑥𝑖
𝑘−1/𝑘−2𝜇𝑗
𝑘−1]=𝐿𝑖𝑗
𝑘−1,weha e
𝐸[
𝑥𝑖
𝑘−1/𝑘−1𝜇𝑗
𝑘−1]
=X𝑗
𝑘−1,𝑘−1 −𝐿𝑖𝑗
𝑘−1 −X𝑖
𝑘−1,𝑘−1(Π𝑖𝑖
𝑘−1,𝑘−1)−1Π𝑖𝑗
𝑘−1,𝑘−1
(50)
and heexp ession o hec oss-co a iancema iceso he
localp edic ione o siseasilyob ained.
4.3. Dis ibu ed Fusion Fil e ing Es ima o s. Once he local
LS linea il e ing es ima o s 
𝑥𝑖
𝑘/𝑘 and hei e o co a iance
ma ices 𝑃𝑖𝑖
𝑘/𝑘,gi eninTheo em 7, along wi h he e o c oss-
co a iance ma ices, 𝑃𝑖𝑗
𝑘/𝑘,gi eninTheo em 12,a ea ailable,
he dis ibu ed op imal weigh ed usion es ima o s and hei
e o co a iance ma ices a e ob ained by applying he
op imal in o ma ion usion c i e ion weigh ed by ma ices in
he linea minimum a iance sense [30].
Theo em 13. Fo he sys em model (1)and measu emen
model (2), unde assump ions (i)–(i ), he dis ibu ed op imal
usion il e , 
𝑥𝐷
𝑘/𝑘,isgi enby

𝑥𝐷
𝑘/𝑘 =𝐴1
𝑘
𝑥1
𝑘/𝑘 +⋅⋅⋅+𝐴𝑟
𝑘
𝑥𝑟
𝑘/𝑘, 𝑘≥0, (51)
whe e he local es ima o s 
𝑥𝑖
𝑘/𝑘,𝑘 ≥ 0(𝑖 = 1,2...,𝑟)a e
calcula ed by he ecu si e algo i hm es ablished in Theo em 7.
The op imal ma ix weigh s 𝐴𝑖𝑘(𝑖 = 1,2,...,𝑟) a e
compu ed by
𝐴𝑘=Σ−1
𝑘/𝑘𝑒(𝑒𝑇Σ−1
𝑘/𝑘𝑒)−1,(52)
whe e he ma ices 𝐴𝑘=[𝐴1
𝑘,...,𝐴𝑟
𝑘]𝑇and 𝑒=[𝐼,...,𝐼]𝑇a e
bo h 𝑛𝑟×𝑛ma ices, and
Σ𝑘/𝑘 =𝐸[(
𝑥1
𝑘/𝑘,...,
𝑥𝑟
𝑘/𝑘)(
𝑥1
𝑘/𝑘,...,
𝑥𝑟
𝑘/𝑘)𝑇]
=(𝑃𝑖𝑗
𝑘/𝑘)𝑖,𝑗=1,2,...,𝑟
(53)
is an 𝑛𝑟×𝑛𝑟 posi i e de ini e symme ic block ma ix, whose
𝑛×𝑛ma ix en ies 𝑃𝑖𝑗
𝑘/𝑘 a e gi en in Theo ems 7and 12.
The e o co a iance ma ices o he dis ibu ed weigh ed
usion il e ing es ima o s a e compu ed by
𝑃𝐷
𝑘/𝑘 =(𝑒𝑇Σ−1
𝑘/𝑘𝑒)−1,𝑘≥0, (54)
and he ollowing inequali y holds: 𝑃𝐷
𝑘/𝑘 ≤𝑃𝑖𝑖
𝑘/𝑘,𝑖=1,2,...,𝑟.
P oo . The p oo is omi ed because i ollows di ec ly om
he op imal in o ma ion c i e ion weigh ed by ma ices in he
linea minimum a iance sense [30].
Rema k 14. The p oposed dis ibu ed op imal LS linea
usion il e equi es he compu a ion o an 𝑛𝑟 × 𝑛𝑟 in e se
ma ix, wi h 𝑛 he dimension o he sys em s a e and 𝑟
he numbe o senso s. Consequen ly, he p oposed dis-
ibu ed usion me hod has a compu a ional complexi y
o 𝑂[(𝑛𝑟)3], equal o ha o he dis ibu ed Kalman- ype
il e in [16] and less han ha o he dis ibu ed usion
il e sbasedon hes a eaugmen a ionapp oach.Hence,ou
dis ibu ed usionme hodissupe io o he il e p oposed
in [16](sincei has hesamecompu a ionbu denbu
be e accu acy) and also o he dis ibu ed usion il e s
based on s a e augmen a ion (since i has less compu a ional
complexi y).
5. Nume ical Simula ion Example
In his sec ion, a nume ical simula ion example is p esen ed
o illus a e he e ec i eness o he cen alized and dis ibu ed
il e ing algo i hms p oposed in his pape . Conside a
scala i s -o de au o eg essi e model wi h missing mea-
su emen s coming om wo senso s wi h au oco ela ed
and c oss-co ela ed noises. Acco ding o he p oposed
obse a ion model, wo di e en independen sequences
o andom a iables wi h a ce ain p obabili y dis ibu ion
o e he in e al [0,1] a eused omodel hemissingphe-
nomenon. Speci ically, he ollowing model is conside ed as
ollows:
𝑥𝑘=0.95𝑥𝑘−1 +𝑤𝑘−1, 𝑘≥1
𝑦𝑖
𝑘=𝜃𝑖
𝑘𝑥𝑘+V𝑖
𝑘,𝑘≥1,𝑖=1,2, (55)
whe e he ini ial s a e 𝑥0is a ze o-mean Gaussian a iable
wi h a iance 𝑃0=1. The noise p ocesses {𝑤𝑘;𝑘≥0}and
{V𝑖
𝑘;𝑘≥1},𝑖=1,2, a e de ined by
𝑤𝑘=0.6(𝜂𝑘+1 +𝜂𝑘+2),
V𝑖
𝑘=𝑐
𝑖(𝜂𝑘+𝜂𝑘+1), 𝑖=1,2, (56)
whe e he sequence o a iables {𝜂𝑘;𝑘≥1}is a ze o-mean
Gaussian whi e p ocess wi h a iance 0.5. Clea ly, acco ding