INTRODUCTION TO THE SPATIAL INTERTICLES THE DA
ARTICLES
SPATIAL PREDICTION OF
RADIOACTIVITY USING GENERAL
REGRESSION NEURAL NETWORK
_______________________________________________________________________________________
s~Çáã=qáãçåáå=
kìÅäÉ~ê=p~ÑÉíó=fåëíáíìíÉ=Ef_o^bF=çÑ=o^pI=jçëÅçïI=oìëëá~=
`çêêÉëéçåÇÉåÅÉ=íçW=î~Çáã]áÄê~ÉK~ÅKêì=
bäÉå~=p~îÉäáÉî~=
kìÅäÉ~ê=p~ÑÉíó=fåëíáíìíÉ=Ef_o^bF=çÑ=o^pI=jçëÅçïI=oìëëá~=
`çêêÉëéçåÇÉåÅÉ=íçW=Éë~î]áÄê~ÉK~ÅKêì=
_______________________________________________________________________________________
This work describes an application of General Regression Neural Network (GRNN) to
spatial predictions of radioactivity. GRNN belongs to a class of neural networks
widely used for mapping continuous functions. It is based on a non-parametric (kernel) Parzen-Rosenblatt density estimator. The kernel size is the only tuning
parameter, and it allows the user to implement a GRNN in an automatic mode. An
important advantage of the GRNN is its very simple and fast training procedure. The
most important drawbacks are high smoothing and dependence on the spatial density of the monitoring data set. The current case study is performed on the SIC2004
data sets, and the results obtained here can be compared with those obtained by the
other participants using other approaches.
1. INTRODUCTION
qÜÉ=pf`OMMQ=ÉñÉêÅáëÉ=ÇÉ~äë=ïáíÜ=íÉëíáåÖ=ÇáÑÑÉêÉåí=~ééêç~ÅÜÉë=íç=~ìíçã~íáÅ=ëé~íá~ä=éêÉÇáÅJ
íáçåë= çÑ= ê~Çáç~ÅíáîÉ= Åçåí~ãáå~íáçåK= qÜÉëÉ= ~ééêç~ÅÜÉë= ~êÉ= ÅçåëáÇÉêÉÇ= ~ë= éçëëáÄäÉ=
Å~åÇáÇ~íÉë=Ñçê=áãéäÉãÉåí~íáçå=áåíç=~=ëóëíÉã=Ñçê=ãçåáíçêáåÖ=êÉÖìä~êäó=ê~Çáç~Åíáîáíó=äÉîÉäë=
çÑ= éçíÉåíá~ääó= Ç~åÖÉêçìë= çÄàÉÅíëK= pìÅÜ= ãÉíÜçÇë= êÉèìáêÉ= çåJäáåÉ= ÑìåÅíáçåáåÖI= áå= çíÜÉê=
ïçêÇëW= ÜáÖÜ=êÉ~Åíáçå=ëéÉÉÇ=~åÇ=~ÄëçäìíÉ=~ìíçã~íáçåK=^äëçI=áí=áë=îÉêó=áãéçêí~åí=íÜ~í=íÜÉ=
éêÉÇáÅíáçå= ÄÉ= ìåáèìÉW= ïÉ~â= Eçê= ÉîÉå= åçF= ÇÉéÉåÇÉåÅÉ= çÑ= íÜÉ= êÉëìäíë= çå= íÜÉ= áåáíá~ä= ÅçåÇáJ
íáçåë=çÑ=íÜÉ=íê~áåáåÖ=éêçÅÉÇìêÉK=qÜáë=áë=åçí=íÜÉ=Å~ëÉ=Ñçê=íÜÉ=ÄÉëíJâåçïå=kÉìê~ä=kÉíïçêâ=Ó=
~= jìäíái~óÉê= mÉêÅÉéíêçå= EjimFI= ïÜáÅÜ= áë= ~= îÉêó= éçïÉêÑìä= ~åÇ= ÑäÉñáÄäÉ= áåëíêìãÉåí= Ñçê=
ëé~íá~ä=éêÉÇáÅíáçåë=Eh~åÉîëâó=NVVRFK=pç=áå=íÜÉ=ÅìêêÉåí=ëíìÇó=~åçíÜÉê=âáåÇ=çÑ=åÉìê~ä=åÉíJ
ïçêâë= ï~ë= ìëÉÇ= Ó= dÉåÉê~ä= oÉÖêÉëëáçå= kÉìê~ä= kÉíïçêâ= EdokkFK= ^= Å~ëÉ= ëíìÇó= ïáíÜ=
dokk= Ñçê= ãçåáíçêáåÖ= ÉåîáêçåãÉåí~ä= î~êá~ÄäÉë= ï~ë= ~äêÉ~Çó= éêÉëÉåíÉÇ= Äó= h~åÉîëâó=
ENVVUFK=
dokk= áë= ~= åçåäáåÉ~ê= áåíÉêéçä~íçê= Ä~ëÉÇ= çå= ~= ëçìåÇ= ã~íÜÉã~íáÅ~ä= ÑçìåÇ~íáçåK= fíë=
ã~áå=~Çî~åí~ÖÉ=áë=áíë=ëáãéäáÅáíó=~åÇ=Ñ~ëí=íê~áåáåÖI=ÄÉáåÖ=áåÇÉéÉåÇÉåí=çÑ=íÜÉ=áåáíá~ä=ÅçåÇáJ
íáçåK= qÜÉ= ãçëí= áãéçêí~åí= Çê~ïÄ~Åâ= áë= ëãççíÜáåÖK= fí= ãÉ~åë= íÜ~í= éêÉÇáÅíáçå= ~äï~óë=
çîÉêÉëíáã~íÉë= íÜÉ=ãáåáãìã= ~åÇ=~äï~óë=ìåÇÉêÉëíáã~íÉë=íÜÉ=ã~ñáãìã= î~äìÉëK=qÜìëI=áí=áë=
åçí= ~= îÉêó= ÖççÇ= áåëíêìãÉåí= Ñçê= éêÉÇáÅíáåÖ= ÉñíêÉãÉ= ~åÇ= çìíäáÉê= î~äìÉëI= Äìí= áí= ~ääçïë= íÜÉ=
ìëÉê= íç= ÇÉíÉÅí= íÜÉáê= éêÉëÉåÅÉK= oÉëìäíë= Ñêçã= íÜÉ= ëÉÅçåÇ= E“àçâÉêÒF= Ç~í~= ëÉí= ÅçåÑáêã= íÜáë=
éçáåíK=^åÇ=áí=ëÉÉãë=íç=ÄÉ=ÉåçìÖÜ=Ñçê=~å=~ìíçã~íáÅ=ãçåáíçêáåÖ=ëóëíÉã=áÑ=áí=áë=~ÄäÉ=íç=ÇÉíÉÅí=
APPLIED GIS, VOLUME 1, NUMBER 2, 2005 MONASH UNIVERSITY EPRESS
19-1
íÜ~í=“ëçãÉíÜáåÖ=áë=ïêçåÖÒ=áå=íÜÉ=ëíìÇó=~êÉ~I=éçëáíáçå=íÜÉ=Ñäìñ=~åÇ=éêçîáÇÉ=~=éêÉäáãáå~êó=
äÉîÉä=çÑ=î~äìÉëK=jçêÉ=ÇÉí~áäÉÇ=éêÉÇáÅíáçå=çÑ=ÉñíêÉãÉë=Å~å=ÄÉ=ã~ÇÉ=ä~íÉê=EáÑ=åÉÉÇÉÇF=ìëáåÖ=
çíÜÉê=~ééêç~ÅÜÉë=~åÇLçê=Äó=áåÅäìÇáåÖ=~ÇÇáíáçå~ä=ãÉ~ëìêÉãÉåíëK=
gìëí=~ë=Ñçê=~ää=áåíÉêéçä~íáçå=íÉÅÜåáèìÉëI=dokk=êÉëìäíë=ÇÉéÉåÇ=çå=íÜÉ=ëé~íá~ä=ÇÉåëáíó=çÑ=
íÜÉ=ãçåáíçêáåÖ=Ç~í~=ëÉíK=qÜÉ=ãçêÉ=ÜçãçÖÉåÉçìëäó=íÜÉ=áåéìí=ë~ãéäÉë=~êÉ=ÇáëíêáÄìíÉÇI=íÜÉ=
ÄÉííÉê=éÉêÑçêãë=íÜÉ=dokkK=aÉåëÉ=ë~ãéäÉë=ÅäìëíÉêë=Eçê=Éãéíó=ÜçäÉëF=ã~ó=Çáëíçêí=íÜÉ=ÉÑÑáJ
ÅáÉåÅó= çÑ= dokkK= qÜÉ= pf`OMMQ= ÉñÉêÅáëÉ= ë~ãéäÉë= EOMM= íê~áåáåÖ= éçáåíëF= ~êÉ= ÇáëíêáÄìíÉÇ=
ê~íÜÉê=ÜçãçÖÉåÉçìëäóI=ëç=íÜáë=ÑÉ~íìêÉ=çÑ=dokk=ÇçÉë=åçí=áåÑäìÉåÅÉ=íÜÉ=ÖççÇ=êÉëìäíK=
qÜÉ= pf`OMMQ= ÉñÉêÅáëÉ= áåÅäìÇÉÇ= ëçãÉ= éêáçê= áåÑçêã~íáçåK= ^= èìáÅâ= äççâ= ~í= íÜÉëÉ= Ç~í~=
ëÉíë= äÉÇ= íç= íÜÉ= ÅçåÅäìëáçå= íÜ~í= íÜÉ= dokk= ÅçìäÇ= ÄÉ= ~ééäáÉÇ= áå= íÜáë= ÉñÉêÅáëÉK= ^äëç= éêáçê=
áåÑçêã~íáçå= ~ääçïÉÇ= ìë= íç= ÇÉÑáåÉ= áåéìí= é~ê~ãÉíÉêë= Ñçê= íÜÉ= dokk= çéíáãáë~íáçå= éêçÅÉJ
ÇìêÉK=qÜÉ=ÇÉí~áäÉÇ=é~ê~ãÉíÉê=íìåáåÖ=ï~ë=éÉêÑçêãÉÇ=çå=êÉ~ä=ÉñÉêÅáëÉ=Ç~í~=áå=~å=~ìíçã~íáÅ=
ãçÇÉK=
2. METHODOLOGY
2.1. KERNEL REGRESSION WITH NEURAL NETWORK: GENERAL REGRESSION
NEURAL NETWORK
`çåëáÇÉê=~=åçåJäáåÉ~ê=êÉÖêÉëëáçå=éêçÄäÉãI=ÇÉëÅêáÄÉÇ=Äó=~=ãçÇÉä=ïÜçëÉ=çÄëÉêî~ÄäÉ=çìíéìí=
òá=áå=êÉëéçåëÉ=íç=~å=áåéìí=îÉÅíçê=ñá=áë=ÇÉÑáåÉÇ=Äó=
z i = f ( xi ) + ε i
i = 1,2K, N
(1)
ïÜÉêÉ=ÑEñiF=áë=~=“ëãççíÜ=ÅìêîÉÒI=~åÇ= ε i áë=~=ê~åÇçã=êÉëáÇì~ä=Çê~ïå=Ñêçã=~=ïÜáíÉ=åçáëÉ=
éêçÅÉëë=çÑ=òÉêç=ãÉ~å=~åÇ=î~êá~åÅÉ=σ K=qÜ~í=áë=
2
E[ε i ] = 0 ∀i
and
⎧σ 2 i = j
E[ε i ε j ] = ⎨
⎩0 otherwise
(2)
qÜÉ=éêçÄäÉã=áë=íç=êÉÅçåëíêìÅí=íÜÉ=ìåÇÉêäóáåÖ=ÑìåÅíáçåI=ÑEñáFI=ÖáîÉå=íÜÉ=íê~áåáåÖ=ë~ãéäÉ=
{( x i , z i )}iN=1 K=^ë=~=êÉ~ëçå~ÄäÉ=Éëíáã~íÉ=çÑ=~å=ìåâåçïå=êÉÖêÉëëáçå=ÑìåÅíáçå=ÑEñFI=çåÉ=ã~ó=
í~âÉ=~=ãÉ~å=çÑ=çÄëÉêî~íáçåë=EáKÉK=î~äìÉë=çÑ=íÜÉ=ãçÇÉä=çìíéìí=òF=Ñêçã=~=åÉáÖÜÄçêÜççÇ=çÑ=
íÜÉ=éçáåí=ñK=qÜáë=~ééêç~ÅÜ=áë=ëìÅÅÉëëÑìä=áÑ=íÜÉ=äçÅ~ä=~îÉê~ÖÉ=áë=ÅçåÑáåÉÇ=íç=çÄëÉêî~íáçåë=áå=
~=ëã~ää=åÉáÖÜÄçêÜççÇ=EáKÉK=êÉÅÉéíáîÉ=ÑáÉäÇF=çÑ=íÜÉ=éçáåí=ñ=~ë=çÄëÉêî~íáçåë=ÅçêêÉëéçåÇáåÖ=íç=
éçáåíë= ~ï~ó= Ñêçã= ñ= ïáää= ÖÉåÉê~ääó= Ü~îÉ= ÇáÑÑÉêÉåí= ãÉ~å= î~äìÉëK= jçêÉ= éêÉÅáëÉäóI= ÑEñF= áë=
Éèì~ä=íç=íÜÉ=ÅçåÇáíáçå~ä=ãÉ~å=çÑ=ò=ÖáîÉå=ñ=EáKÉKI=íÜÉ=êÉÖêÉëëáçå=çÑ=ò=çå=ñFK=rëáåÖ=íÜÉ=ÑçêJ
ãìä~=Ñçê=íÜÉ=ÉñéÉÅí~íáçå=çÑ=~=ê~åÇçã=î~êá~ÄäÉI=áí=Å~å=ÄÉ=ïêáííÉå=
∞
f ( x ) = E[ z | x ] =
∫ zf
Z
( z | x)dz
(3)
−∞
ïÜÉêÉ= ÑwEòöñF= áë= íÜÉ= ÅçåÇáíáçå~ä= éêçÄ~Äáäáíó= ÇÉåëáíó= ÑìåÅíáçå= EéKÇKÑKF= çÑ= òI= ÖáîÉå= ñK= cêçã=
éêçÄ~Äáäáíó=íÜÉçêóI=ïÉ=Ü~îÉ=
f Z ( z | x) =
19-2
f X ,Z ( x , z )
f X ( x)
(4)
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
ïÜÉêÉ=ÑuIIwEñIòF=áë=íÜÉ=àçáåí=éKÇKÑK=çÑ=ñ=~åÇ=ò=~åÇ=ÑuEñF=áë=íÜÉ=ã~êÖáå~ä=éKÇKÑK=çÑ=ñK=eÉåÅÉI=
ìëáåÖ=EQF=áå=EPFI=ïÉ=çÄí~áå=íÜÉ=ÑçääçïáåÖ=Ñçêãìä~=Ñçê=íÜÉ=ÖÉåÉê~ä=êÉÖêÉëëáçå=ÑìåÅíáçå=
∞
f ( x) =
∫ zf
X ,Z
( x , z )dz
−∞
(5)
f X ( x)
kçíÉ=íÜ~í=ïÉ=Å~å=ÇÉêáîÉ=íÜÉ=ã~êÖáå~ä=éKÇKÑK=ÑuEñF=Äó=áåíÉÖê~íáåÖ=íÜÉ=àçáåí=éKÇKÑK=ÑuIwEñIòF=
çîÉê=ÇòW=
∞
f X ( x) =
∫f
X ,Z
( x , z )dz
(6)
−∞
lìê=é~êíáÅìä~ê=áåíÉêÉëí=áë=áå=~=ëáíì~íáçå=ïÜÉêÉ=íÜÉ=àçáåí=éKÇKÑK=ÑuIwEñIòF=áë=ìåâåçïåK=^ää=
N
íÜ~í= ïÉ= Ü~îÉ= ~î~áä~ÄäÉ= áë= ~= ëÉí= çÑ= íê~áåáåÖ= ë~ãéäÉëÄ= {( x i , z i )}i =1 K= qç= Éëíáã~íÉ= ÑuIwEñIòF=
~åÇ= ÑuEñF= íÜÉêÉÑçêÉI= ïÉ= ã~ó= ìëÉ= ~= åçåJé~ê~ãÉíêáÅ= Éëíáã~íçê= âåçïå= ~ë= íÜÉ= m~êòÉåJ
oçëÉåÄä~íí= ÇÉåëáíó= Éëíáã~íçê= EoçëÉåÄä~íí= NVRSI= NVTMX= m~êòÉå= NVSOFK= qÜÉ= Ä~ëáë= Ñçê= íÜÉ=
Ñçêãìä~íáçå=áë=~=âÉêåÉäI=ÇÉåçíÉÇ=Äó=hEñFI=ïÜáÅÜ=Ü~ë=éêçéÉêíáÉë=ëáãáä~ê=íç=íÜçëÉ=~ëëçÅá~íÉÇ=
ïáíÜ=~=éKÇKÑK=E~=ÅçåíáåìçìëI=ÄçìåÇÉÇI=~åÇ=êÉ~ä=ÑìåÅíáçå=çÑ=ñ=ïáíÜ=íÜÉ=íçí~ä=îçäìãÉ=ìåÇÉê=
íÜÉ=ëìêÑ~ÅÉ=Éèì~ä=íç=~=ìåáíFK=tÜÉêÉ~ë=íÜÉ=âÉêåÉä=áë=~=ëóããÉíêáÅ=ÑìåÅíáçå=~êçìåÇ=íÜÉ=çêáÖáå=
ïÜÉêÉ=áí=~íí~áåë=áíë=ã~ñáãìã=î~äìÉI=íÜáë=áë=åçí=~=êÉèìáêÉãÉåí=Ñçê=~=éKÇKÑK=
^ëëìãáåÖ=íÜ~í=EñiIòiF=~êÉ=áåÇÉéÉåÇÉåíI=áÇÉåíáÅ~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=îÉÅíçêëI=ïÉ=ã~ó=
Ñçêã~ääó=ÇÉÑáåÉ=íÜÉ=m~êòÉåJoçëÉåÄä~íí=ÇÉåëáíó=Éëíáã~íçê=çÑ=àçáåí=éKÇKÑK=ÑuIwEñIòF=áå=íÜÉ=àçáåí=
áåéìíJçìíéìí=ëé~ÅÉ=~ë=Em~êòÉå=NVSOX=oçëÉåÄä~íí=NVRSFW=
ˆf X ,Z ( x , z ) =
1
σ
p +1
N
⎛ x − xi ⎞ ⎛ z − zi ⎞
⎟K ⎜
⎟
σ ⎠ ⎝ σ ⎠
∑ K ⎜⎝
N
i =1
for
x ∈ ℜ p and z ∈ ℜ1
(7)
ïÜÉêÉ=íÜÉ=ëãççíÜáåÖ=é~ê~ãÉíÉê=σ=áë=~=éçëáíáîÉ=åìãÄÉê=Å~ääÉÇ=Ä~åÇïáÇíÜ=çê=ëáãéäó=ïáÇíÜK=
fí= Åçåíêçäë= íÜÉ= ëáòÉ= çÑ= íÜÉ= âÉêåÉä= hEñFK= ^ëëìãáåÖ= íÜ~í= íÜÉ= ëÅ~äáåÖ= Ñçê= áåéìí= ~åÇ= çìíéìí=
î~êá~ÄäÉë=áë=íÜÉ=ë~ãÉ=ÇçÉë=åçí=~ÑÑÉÅí=íÜÉ=êÉëìäíI=~ë=ïáää=ÄÉ=ëÉÉå=ä~íÉêK=rëáåÖ=ESF=çê=áåíÉÖê~íJ
áåÖ=
ˆf X ,Z ( x , z ) = áå= ETF= ïáíÜ= êÉëéÉÅí= íç= òI= ïÉ= Å~å= ÇÉÑáåÉ= íÜÉ= m~êòÉåJoçëÉåÄä~íí= ÇÉåëáíó=
Éëíáã~íçê=çÑ=éKÇKÑK=ÑuEñF=~ë=
fˆX ( x) =
1
σ pN
N
⎛ x − xi ⎞
⎟
σ ⎠
∑ K ⎜⎝
i =1
for
x ∈ℜp
(8)
kçíÉ= íÜ~í= ~å= áãéçêí~åí= éêçéÉêíó= çÑ= íÜáë= Éëíáã~íçê= áë= íÜ~í= áí= áë= ÅçåëáëíÉåí= EáKÉK= ~ëóãéíçJ
íáÅ~ääó=ìåÄá~ëÉÇF=áå=íÜÉ=ëÉåëÉ=íÜ~í=áÑ=σZσEkF=áë=ÅÜçëÉå=~ë=~=ÑìåÅíáçå=çÑ=k=ëìÅÜ=íÜ~í=
lim σ ( N ) = 0
N →∞
íÜÉå=
lim E[ fˆX ( x)] = f X ( x)
N →∞
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-3
cçê=íÜáë=ä~ííÉê=Éèì~íáçå=íç=ÜçäÇI= x =ëÜçìäÇ=ÄÉ=~=éçáåí=çÑ=Åçåíáåìáíó=Ñçê=
fˆX ( x ) K=
qÜìëI=ìëáåÖ=ETF=ïÉ=Å~å=êÉïêáíÉ=íÜÉ=åìãÉê~íçê=çÑ=ERF=~ë=
∞
∫ zˆf X ,Z ( x , z )dz =
−∞
∞
⎛ x − xi ⎞
⎛ z − zi ⎞
K⎜
⎟ ∫ zK ⎜
⎟dz
∑
p +1
σ N i =1 ⎝ σ ⎠ −∞ ⎝ σ ⎠
N
1
(9)
`Ü~åÖáåÖ=íÜÉ=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçå=Äó=ëÉííáåÖ=~=Z=Eò=Ó=òi=F=L=σ=I=~åÇ=ìëáåÖ=íÜÉ=ëóããÉíJ
êáÅ=éêçéÉêíó=çÑ=íÜÉ=âÉêåÉä=hE⋅FI=ïÉ=çÄí~áå=íÜÉ=êÉëìäí=
∞
∫ zˆf X ,Z ( x , z )dz =
−∞
1
σ pN
N
⎛ x − xi ⎞
⎟
σ ⎠
∑ z K ⎜⎝
i =1
i
(10)
qÜÉ=ÇÉåçãáå~íçê=çÑ=ERF=áë=EUFK=`~åÅÉäáåÖ=Åçããçå=íÉêãëI=ïÉ=çÄí~áå=íÜÉ=ÑçääçïáåÖ=ÉëJ
íáã~íÉ=çÑ=íÜÉ=êÉÖêÉëëáçå=ÑìåÅíáçå=ÑEñFW=
⎛ x − xi ⎞
⎟
σ ⎠
ˆf ( x ) = i =1
N
⎛ x − xi ⎞
K⎜
⎟
∑
⎝ σ ⎠
i =1
N
∑ z K ⎜⎝
i
(11)
tÉ=Å~å=ÇÉÑáåÉ=íÜÉ=åçêã~äáòÉÇ=ïÉáÖÜíáåÖ=ÑìåÅíáçå=~ë=~=ÑìåÅíáçå=çÑ=ñ=
⎛ x − xj ⎞
⎟
K ⎜⎜
σ ⎟⎠
⎝
W j ( x) = N
⎛ x − xi ⎞
K⎜
⎟
∑
⎝ σ ⎠
i =1
j = 1,2K, N
(12)
qÜÉ=ÇÉåçãáå~íçê=çÑ=ENOF=ÖáîÉë=ìë=íÜÉ=åçêã~äáò~íáçå=éêçéÉêíó=
N
∑W ( x ) = 1
j =1
j
∀x
(13)
kçï=ïÉ=Å~å=êÉïêáíÉ=ENPF=áå=íÜÉ=ëáãéäáÑáÉÇ=Ñçêã=
N
ˆf ( x ) = ∑ W j ( x )z j
(14)
j =1
få=íÜáë=ÑçêãI=Éèì~íáçå=ENNF=~åÇ=éêçéÉêíó=ENPF=ÇÉëÅêáÄÉ=
fˆ ( x) =~ë=~=ïÉáÖÜíÉÇ=~îÉê~ÖÉ=çÑ=
íÜÉ= ò= çÄëÉêî~íáçåëK= qÜÉ= é~êíáÅìä~ê= Ñçêã= çÑ= ïÉáÖÜíáåÖ= ÑìåÅíáçå= tjEñF= ÖáîÉå= áå= ENOF= ï~ë=
çêáÖáå~ääó= éêçéçëÉÇ= Äó= k~Ç~ê~ó~= ENVSQF= ~åÇ= t~íëçå= ENVSQFK= ^ÅÅçêÇáåÖäóI= íÜÉ= ~ééêçñáJ
ã~íáçå=ÑìåÅíáçå=áå=ENQF=áë=çÑíÉå=Å~ääÉÇ=íÜÉ=k~Ç~ê~ó~Jt~íëçå=hÉêåÉä=oÉÖêÉëëáçå=bëíáã~íçê=
EkthobFI=çê=êÉÑÉêêáåÖ=íç=íÜÉ=ëí~êíáåÖ=éçáåíë=EPF=~åÇ=ERFI=íÜÉ=ÖÉåÉê~ä=êÉÖêÉëëáçå=Éëíáã~íçêK=
qÜÉêÉ=áë=~=î~êáÉíó=çÑ=éçëëáÄäÉ=âÉêåÉä=ÑìåÅíáçåë=hE⋅FK=qÜÉ=ãçëí=ïáÇÉäó=ìëÉÇ=~ãçåÖ=íÜÉã=
áë=~=ãìäíáî~êá~íÉ=d~ìëëá~å=âÉêåÉäW=
19-4
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
⎛ x 2⎞
1
⎟ x = (x , K , x )
K ( x) = ∏ K i ( xi ) =
exp⎜ −
1
p
p/2
⎜
⎟
(
2
)
2
π
i =1
⎝
⎠
p
(15)
ïáíÜ= ööñöö= ÜÉêÉ= ~åÇ= ÑìêíÜÉê= ãÉ~åáåÖ= íÜÉ= åçêã= ÇÉÑáåÉÇ= áå= íÜÉ= éJëáòÉ= ëé~ÅÉK= `ÉåíÉêáåÖ= íÜÉ=
âÉêåÉä= çå= ~= Ç~í~= éçáåí= ñi= ~åÇ= ëÅ~äáåÖ= íÜÉ= ïáÇíÜ= ïáíÜ= íÜÉ= ëãççíÜáåÖ= é~ê~ãÉíÉê= σI= ïÉ==
çÄí~áå=íÜÉ=ÑçääçïáåÖ=Ñçêã=
⎛ x − xi
1
⎛ x − xi ⎞
⎜−
K⎜
exp
⎟=
2 p/2
⎜ 2σ 2
⎝ σ ⎠ ( 2πσ )
⎝
2
⎞
⎟ i = 1,2K, N
⎟
⎠
(16)
táíÜ=íÜÉ=âÉêåÉä=ENSFI=íÜÉ=k~Ç~ê~ó~Jt~íëçå=hÉêåÉä=oÉÖêÉëëáçå=bëíáã~íçê=äççâë=äáâÉW=
⎛ x − xi 2 ⎞
⎟
y i exp⎜ −
∑
2
⎟
⎜
2
σ
i =1
⎠
⎝
fˆ ( x) =
2
N
⎛ x − xi ⎞
⎟
exp⎜ −
∑
⎜
2σ 2 ⎟
i =1
⎝
⎠
N
(17)
ïÜÉêÉ=íÜÉ=ÇÉåçãáå~íçê=íÉêã=êÉéêÉëÉåíë=íÜÉ=m~êòÉåJoçëÉåÄä~íí=ÇÉåëáíó=Éëíáã~íçê=EïáíÜçìí=
åçêã~äáò~íáçåF= ~ë= ~= ëìã= çÑ= k= ãìäíáî~êá~íÉ= d~ìëëá~å= ÇáëíêáÄìíáçåë= ÅÉåíÉêÉÇ= çå= íÜÉ= Ç~í~=
éçáåíë=ñi=EpéÉÅÜí=NVVNFK=qç=ÄÉ=ìåÄá~ëÉÇ=íÜáë=íÉêã=ëÜçìäÇ=ÄÉ=ÇáîáÇÉÇ=Äó=íÜÉ=åçêã~äáòáåÖ=
Åçåëí~åí=
(2πσ )
2
p
2
N K= qÜÉ= Ñçêã= ENTF= éêÉëÉåíë= íÜÉ= k~Ç~ê~ó~Jt~íëçå= d~ìëëá~å= hÉêåÉä=
oÉÖêÉëëáçå=bëíáã~íçêK=
qÜÉ=Ñçêã=ENTF=Å~å=ÄÉ=îáÉïÉÇ=~ë=~=åçêã~äáòÉÇ=Ñçêã=çÑ=íÜÉ=d~ìëëá~å=ê~Çá~ä=Ä~ëáë=ÑìåÅJ
íáçåë=E_áëÜçé=NVVRI=éK=NTTF=ÇÉÑáåÉÇ=áå=íÜÉ=áåéìí=ëé~ÅÉK=b~ÅÜ=Ä~ëáë=ÑìåÅíáçå=áë=ÅÉåíÉêÉÇ=çå=
~=Ç~í~=éçáåíI=~åÇ=íÜÉ=ÅçÉÑÑáÅáÉåíë=áå=íÜÉ=Éñé~åëáçå=~êÉ=ÖáîÉå=Äó=íÜÉ=í~êÖÉí=î~äìÉë=òiK=
2.2. NWKRE AS A NEURAL NETWORK: GRNN
qÜÉ=kthob=áå=íÜÉ=Ñçêã=ENTF=Å~å=ÄÉ=áãéäÉãÉåíÉÇ=áå=íÉêãë=çÑ=~=åÉìê~ä=åÉíïçêâ=EpéÉÅÜí=
NVVNFI=~åÇ=íÜÉ=ÅçêêÉëéçåÇáåÖ=åÉìê~ä=åÉíïçêâ=áë=Å~ääÉÇ=~=dÉåÉê~ä=oÉÖêÉëëáçå=kÉìê~ä=kÉíJ
ïçêâ= EdokkFK= cáÖìêÉ= N= áë= ~= Öê~éÜáÅ~ä= êÉéêÉëÉåí~íáçå= çÑ= dokk= áå= íÉêãë= çÑ= ~= åÉìê~ä=
åÉíïçêâK=
dokk=Åçåëáëíë=çÑ=Ñçìê=ä~óÉêëW=áåéìíI=é~ííÉêåI=ëìãã~íáçå=~åÇ=çìíéìíK=qÜÉ=áåéìí=ä~óÉê=
íê~åëÑÉêë= ~å= áåéìí= ëáÖå~ä= Ó= ~= éJÇáãÉåëáçå~ä= îÉÅíçê= ñI= áåíç= íÜÉ= åÉñí= é~ííÉêå= ä~óÉêK= få= íÜÉ=
ëé~íá~ä=ã~ééáåÖ=Å~ëÉ=éZOK=qÜÉ=åìãÄÉê=çÑ=åÉìêçåë=EâÉêåÉäëF=áå=íÜÉ=é~ííÉêå=ä~óÉê=áë=Éèì~ä=
íç=íÜÉ=åìãÄÉê=çÑ=íê~áåáåÖ=ë~ãéäÉë=EkFI=É~ÅÜ=åÉìêçå=ÅçêêÉëéçåÇáåÖ=íç=~=íê~áåáåÖ=ë~ãéäÉK=
cçê=~å=áåéìí=îÉÅíçê=~ää=é~ííÉêå=ä~óÉê=åÉìêçåë=ÅçãéìíÉ=íÜÉ=bìÅäáÇá~å=Çáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=
áåéìí= îÉÅíçê= ~åÇ= íÜÉ= ÅçêêÉëéçåÇáåÖ= åÉìêçå= äçÅ~íáçåK= qÜÉëÉ= Çáëí~åÅÉë= ~êÉ= ~ëëìãÉÇ= ~ë= ~å=
~Åíáî~íáçå=ÑìåÅíáçå=E~=d~ìëëá~å=âÉêåÉä=áå=çìê=Å~ëÉF=~åÇ=íê~åëÑÉêêÉÇ=íç=íÜÉ=ÑçääçïáåÖ=ä~óÉêK=
qÜÉ=ëìãã~íáçå=ä~óÉê=Åçåëáëíë=çÑ=íïç=åÉìêçåë=íÜ~í=Å~äÅìä~íÉ=íÜÉ=åçãáå~íçê=~åÇ=ÇÉåçãáJ
å~íçê=êÉëéÉÅíáîÉäó=áå=Éèì~íáçå=NTK=b~ÅÜ=çÑ=íÜÉëÉ=åÉìêçåë=ÅçãéìíÉë=~=ïÉáÖÜíÉÇ=ëìã=çÑ=íÜÉ=
çìíéìí=Ñêçã=~=éêÉîáçìë=ä~óÉêK=qÜÉ=ïÉáÖÜíë=ÅçêêÉëéçåÇ=íç=íÜÉ=äáåâë=ÄÉíïÉÉå=íÜÉ=åÉìêçåëK=
b~ÅÜ= äáåâ= Ñêçã= íÜÉ= é~ííÉêå= ä~óÉê= åÉìêçå= íç= íÜÉ= åçãáå~íçê= åÉìêçå= áë= ~= í~êÖÉí= î~äìÉ==
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-5
~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÅçêêÉëéçåÇáåÖ=åÉìêçå= äçÅ~íáçåK=^ää=äáåâë=ÄÉíïÉÉå=é~ííÉêå=ä~óÉê=åÉìJ
êçåë= ~åÇ= ~= ÇÉåçãáå~íçê= åÉìêçå= ~êÉ= Éèì~ä= íç= ~= ìåáíK= qÜÉ= çìíéìí= ä~óÉê= åÉìêçå= éÉêÑçêãë=
Çáîáëáçå=çéÉê~íáçå=~åÇ=íê~åëÑÉêë=íÜÉ=çìíéìí=î~äìÉK=
Figure 1
NWKRE as a general regression neural network (GRNN)
qÜÉ= çåäó= åçåJåÉìê~ä= çéÉê~íáçå= EÇáîáëáçåF= áå= íÜáë= ~êÅÜáíÉÅíìêÉ= áë= íÜ~í= çÑ= íÜÉ= çìíéìí=
ä~óÉêK= qÜÉêÉ= áë= ~= ÖÉåÉê~ä= EíÜçìÖÜ= åçí= ìåáîÉêë~äF= ~ÖêÉÉãÉåí= íÜ~í= ÄáçäçÖáÅ~ä= åÉìêçåë= ~êÉ=
áåÅ~é~ÄäÉ= çÑ= ÇáîáëáçåK= qÜáë= áë= ~= éçáåí= çÑ= ÅêáíáÅáëã= ~Ö~áåëí= áåíÉêéêÉíáåÖ= íÜÉ= kthob= ~ë= ~=
åÉìê~ä=åÉíïçêâK=píáääI=ëìÅÜ=~å=áåíÉêéêÉí~íáçå=áë=îÉêó=~ííê~ÅíáîÉ=ÄÉÅ~ìëÉ=çÑ=íÜÉ=ëáãéäÉ=ëÅÜÉJ
ã~íáÅ= áääìëíê~íáçå= çÑ= íÜÉ= áåíÉêéçä~íáçå= éêçÅÉëë= EcáÖìêÉ= NF= ~åÇ= íÜÉ= éçëëáÄáäáíó= íç= ìëÉ=
ã~ÅÜáåÉ=äÉ~êåáåÖ=íÉêãáåçäçÖóK=
2.3. TRAINING OF THE GRNN
qÜÉ=çåäó=~Ç~éíáîÉ=EÑêÉÉF=é~ê~ãÉíÉê=áå=íÜÉ=dokk=ãçÇÉä=ïáíÜ=d~ìëëá~å=âÉêåÉä=EÉèì~íáçå=
NTF=áë= σI=íÜÉ=ïáÇíÜ=çÑ=íÜÉ=âÉêåÉäK=qç=ÇÉãçåëíê~íÉ=Üçï=~=dokk=ïçêâëI=äÉí=ìë=ÅçåëáÇÉê=~=
ëáãéäÉ=çåÉJÇáãÉåëáçå~ä=éêçÄäÉã=EcáÖìêÉ=OFK=^=ëáãéäÉ=ëáåÉ=ÑìåÅíáçå=êÉéêÉëÉåíë=~=íêìÉ=ÑìåÅJ
íáçå= çÑ= ~å= ìåÇÉêäóáåÖ= ëíêìÅíìêÉ= çÑ= ~= ÅçääÉÅíáçå= çÑ= ë~ãéäÉ= Ç~í~K= få= çêÇÉê= íç= ÖÉåÉê~íÉ= ~=
íê~áåáåÖ=ëÉíI=íÜáë=ëáåÉ=ÑìåÅíáçå=áë=ë~ãéäÉÇ=~í=~=ïáÇÉ=î~êáÉíó=çÑ=éçáåíëI=~åÇ=ê~åÇçã=åçáëÉ=áë=
~ÇÇÉÇK=cìêíÜÉêãçêÉI=çåÉ=ïáäÇ=éçáåí=EçìíäáÉêF=áë=áåÅäìÇÉÇK=qÜáë=íêìÉ=ÑìåÅíáçå=~åÇ=íê~áåáåÖ=
ë~ãéäÉë=~êÉ=Öê~éÜÉÇ=áå=cáÖìêÉ=O~K=
kçï= äÉí= ìë= Éñ~ãáåÉ= íÜÉ= ÉÑÑÉÅí= éêçÇìÅÉÇ= Äó= ~= î~êáÉíó= çÑ= σ= î~äìÉëI= íÜÉ= ëãççíÜáåÖ==
é~ê~ãÉíÉêK= få= cáÖìêÉ= OÄ= ïÉ= ëÉÉ= ïÜ~í= Ü~ééÉåë= ïÜÉå= ~= îÉêó= ëã~ää= î~äìÉ= áë= ìëÉÇK= qÜÉ=
dokk=Ñçääçïë=íÜÉ=íê~áåáåÖ=Ç~í~=ÅäçëÉäóI=~äãçëí=ãçîáåÖ=Ñêçã=éçáåí=íç=éçáåíK=fÑ=íÜÉ=Ç~í~=áë=
âåçïå=íç=ÄÉ=ÅäÉ~êI=íÜÉ=dokk=ã~âÉë=~å=ÉñÅÉääÉåí=áåíÉêéçä~íáçå=~äÖçêáíÜãI=~å~äçÖçìë=íç=
íÜÉ=åÉ~êÉëí=åÉáÖÜÄçê=ãÉíÜçÇK=kçíÉI=ÜçïÉîÉêI=íÜ~í=íÜáë=êÉëìäí=ïáää=ÄÉ=~ÅÅÉéí~ÄäÉ=çåäó=áÑ=íÜÉ=
ÇÉåëáíó=çÑ=íÜÉ=íê~áåáåÖ=Ç~í~=áë=ÜáÖÜ=ÉåçìÖÜK=få=çíÜÉê=Å~ëÉëI=~å=“çîÉêÑáííáåÖÒ=ÉÑÑÉÅíI=ïÜáÅÜ=
áë= ïÉääJâåçïå= áå= åÉìê~ä= åÉíïçêâëI= ã~ó= ~ééÉ~ê= ~åÇ= ëìÅÜ= ëçäìíáçåë= ïáää= åçí= ÄÉ= ~= Ñ~áäìêÉK=
pçI=ëáåÅÉ=ïÉ=âåçï=íÜ~í=íÜÉ=Ç~í~=~êÉ=ÇáëíçêíÉÇ=Äó=åçáëÉ=áå=ãçëí=Å~ëÉëI=ëíê~áÖÜíÑçêï~êÇ=áåJ
íÉêéçä~íáçå=áë=åçí=ïÜ~í=ïÉ=ï~åíK=
cáÖìêÉ=OÅ=áë=çÄí~áåÉÇ=Äó=ìëáåÖ=~=ãçÇÉê~íÉäó=ëã~ää=ëãççíÜáåÖ=é~ê~ãÉíÉêK=lÄëÉêîÉ=íÜ~í=
íÜÉ=ïáäÇ=éçáåí=E~å=çìíäáÉêF=ëìÅÅÉÉÇë=áå=éìääáåÖ=íÜÉ=ÑìåÅíáçå=~=Äáí=çìí=çÑ=íÜÉ=äáåÉK=fÑ=ïÉ=~êÉ=
ÅçåÑáÇÉåí=íÜ~í=~åó=ëìééçëÉÇ=ïáäÇ=éçáåí=áë=~Åíì~ääó=~=î~äáÇ=Ç~íìãI=íÜÉå=íÜáë=áë=àìëí=ïÜ~í=ïÉ=
ï~åíK=_ìí=ìëì~ääó=áí=áë=ÇìÉ=íç=ëçãÉ=âáåÇ=çÑ=Ç~í~=ÇáëíçêíáçåK=^=ëçãÉïÜ~í=ä~êÖÉê=ëãççíÜáåÖ=
é~ê~ãÉíÉê=ÖáîÉë=ìë=íÜÉ=êÉëìäí=ëÜçïå=áå=cáÖìêÉ=OÇI=ïÜáÅÜ=äççâë=áÇÉ~äK=
cáå~ääóI=ïÉ=ãìëí=ÄÉ=~ï~êÉ=çÑ=í~âáåÖ=~=ÖççÇ=íÜáåÖ=íçç=Ñ~êK=cáÖìêÉ=OÉ=áääìëíê~íÉë=íÜÉ=ÉÑJ
ÑÉÅí= çÑ= íçç= ä~êÖÉ= ~= ëãççíÜáåÖ= é~ê~ãÉíÉêK= qÜÉ= ÖäçÄ~ä= ëÜ~éÉ= çÑ= íÜÉ= íê~áåáåÖ= ëÉí= Ü~ë=
ëï~ãéÉÇ=çìí=~ää=íÜÉ=ÇÉí~áäëK=dê~åíÉÇI=ïÉ=ìëì~ääó=ï~åí=íÜÉ=ÑáåÉ=ÇÉí~áäëI=ïÜáÅÜ=~êÉ=ÇìÉ=íç=
åçáëÉI=íç=ÄÉ=Éäáãáå~íÉÇK=
19-6
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
qÜìë=ïÉ=Å~å=ã~âÉ=íÜÉ=ÅçåÅäìëáçå=íÜ~í=ÅÜççëáåÖ=íÜÉ=î~äìÉ=çÑ=íÜÉ=ïáÇíÜ=é~ê~ãÉíÉê= σ=áë=
îáí~ä=Ñçê=íÜÉ=dokk=ãçÇÉä=~åÇ=éêçÄäÉãJÇÉéÉåÇÉåíK=
Figure 2
A simple problem to illustrate the influence of the parameter on the GRNN
result: a) true function and noised training samples with an outlier; b) too
small σ; c) perhaps a bit small σ; d) a perfect σ; e) too large σ
qÜÉ=çêÇáå~êó= dokk=íê~áåáåÖ=éêçÅÉÇìêÉ=áë=~=ãÉ~åJëèì~êÉJÉêêçê=EjpbF=ãáåáãáò~íáçå=
éêçÅÉÇìêÉI= ~ÅÅçãéäáëÜÉÇ= ìëáåÖ= ~= ÅêçëëJî~äáÇ~íáçå= EâJÑçäÇI= äÉ~îÉJçåÉJçìíF= ~ééêç~ÅÜ=
Ee~óâáå= NVVVX= _áëÜçé= NVVRFK= ^= í~êÖÉí= ÑìåÅíáçå= EjpbF= áë= ÅçãéìíÉÇ= Ñêçã= êÉëìäíë= Ñçê= ~ää=
ë~ãéäÉë=í~âÉå=çìíI=çåÉ=Äó=çåÉI=~åÇ=Éëíáã~íÉÇ=Ä~ëÉÇ=çå=íÜÉ=çíÜÉê=ë~ãéäÉë=~åÇ=~=ÖáîÉå= σ=
î~äìÉK=qÜÉ=î~äìÉë=çÑ=σ=ÅÜ~åÖÉI=í~âáåÖ=î~äìÉë=Ñêçã=~å=áåíÉêî~ä=xσlowI=σhighzW=
σ i = σ low + (i − 1)
σ high − σ low
M
(18)
ïÜÉêÉ=j=áë=íÜÉ=åìãÄÉê=çÑ=ëíÉéë=íç=éÉêÑçêãK=qÜÉ=áåáíá~ä=áåíÉêî~ä=çÑ=éçëëáÄäÉ= σ=î~äìÉë=~åÇ=
íÜÉ= åìãÄÉê= çÑ= ëíÉéë= ãìëí= ÄÉ= ÇÉÑáåÉÇK= qÜÉ= áåíÉêî~ä= ~åÇ= íÜÉ= åìãÄÉê= çÑ= ëíÉéë= Ü~îÉ= íç= ÄÉ=
ÅçåëáëíÉåí= áå= çêÇÉê= íç= Å~íÅÜ= íÜÉ= ÉñéÉÅíÉÇ= çéíáã~ä= î~äìÉK= qç= ã~âÉ= íÜÉ= áåíÉêî~ä= ëÉäÉÅíáçå=
ãçêÉ=ÖÉåÉê~äI=áí=áë=ìëÉÑìä=íç=íê~åëÑçêã=íÜÉ=áåéìí=ëé~ÅÉ=î~äìÉë=íç=íÜÉ=xJNI=Nz=áåíÉêî~ä=ìëáåÖI=
Ñçê= Éñ~ãéäÉI= ~= ëáãéäÉ= äáåÉ~ê= íê~åëÑçêãK= ^ÑíÉê= íÜ~í= ~ää= éçëëáÄäÉ= ïáÇíÜë= ~êÉ= ïáíÜáå= ~= ìåáíK=
qÜìëI=íÜÉ=ëí~êíáåÖ=áåíÉêî~ä=~åÇ=íÜÉ=åìãÄÉê=çÑ=ëíÉéë=~êÉ=åçí=îÉêó=áãéçêí~åíK=`ÜççëáåÖ=íçç=
ïáÇÉ=~å=áåíÉêî~ä=~åÇLçê=~=ä~êÖÉ=åìãÄÉê=çÑ=ëíÉéë=çåäó=äÉ~Ç=íç=ëéÉåÇáåÖ=ãçêÉ=íáãÉ=çå=Å~äÅìJ
ä~íáçåëK=_ìí=ïáíÜ=ãçÇÉêå=ÅçãéìíÉêë= ~åÇ=~=Ç~í~=ëÉí=ïáíÜ=~Äçìí=OMM=éçáåíë=íÜáë=Ñ~Åíçê=áë=
åçí=îÉêó=ëáÖåáÑáÅ~åíK=
^= î~äìÉ= çÑ= σ= ïÜáÅÜ= ãáåáãáòÉë= íÜÉ= jpb= Å~å= ÄÉ= ~ÇçéíÉÇ= ~ë= ~= ëçäìíáçåK= cìêíÜÉêãçêÉI=
~åó=Öê~ÇáÉåí=ëÉ~êÅÜ=ãÉíÜçÇ=Å~å=ÄÉ=ìëÉÇ=Ñçê=íìåáåÖ=íÜáë=î~äìÉK=táíÜ=~å=ÉñÅÉääÉåí=ëí~êíáåÖ=
éçáåí= Ó= ~= ïÉääJÇÉÑáåÉÇ= ãáåáãìãI= ~åÇ= ~= ëã~ää= ëÉ~êÅÜ= ~êÉ~= Ó= íÜÉ= Öê~ÇáÉåí= ëÉ~êÅÜ= ãÉíÜçÇ=
ÑáåÇë=íÜÉ=ëçäìíáçå=èìáÅâäó=~åÇ=É~ëáäóK=bîÉå=áÑ=~å=çéíáã~ä=î~äìÉ=çÑ= σ=áë=çìíëáÇÉ=íÜÉ=áåáíá~ä=
áåíÉêî~äI=áí=áë=åçí=~=éêçÄäÉã=Ñçê=íÜÉ=Öê~ÇáÉåí=ëÉ~êÅÜ=ãÉíÜçÇ=íç=Çç=ëçãÉ=ëíÉéë=çìíI=~ë=áí=áë=
åçí=äáãáíÉÇ=Äó=~åó=ÄçìåÇ~êáÉëK=
qÜÉ=pf`OMMQ=ÉñÉêÅáëÉ=ÇÉ~äë=ïáíÜ=íïçJÇáãÉåëáçå~ä=Ç~í~=áå=íÜÉ=EñIóF=ÅççêÇáå~íÉ=ëé~ÅÉK=
få= íÜáë= Å~ëÉI= ~å= ~Çî~åÅÉÇ= dokk= êÉ~äáò~íáçå= ã~ó= ÄÉ= ~ééäáÉÇI= ìëáåÖ= ~å= ~åáëçíêçéáÅ= Oa=
é~ê~ãÉíÉê= σZEσxIσyFK= `çåëÉèìÉåíäóI= ~= Oa= EêÉÅí~åÖìä~êF= áåíÉêî~ä= áë= ìëÉÇ= Ñçê= σ= çéíáãáòáåÖ=
íÜÉ=~äÖçêáíÜã=~å~äçÖçìë=íç=çåÉ=ÇÉëÅêáÄÉÇ=~ÄçîÉ=Ñçê=~å=áëçíêçéáÅ=Å~ëÉK=låäó=íÜÉêÉ=ïçìäÇ=
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-7
ÄÉ=jxGjó=î~äìÉë=Ñçê= σ=íç=ÅÜÉÅâK=rëì~ääó=íÜáë=Oa=Éêêçê=ÑìåÅíáçå=Ü~ë=çåÉ=ëãççíÜÉÇI=ïÉääJ
ÇÉÑáåÉÇ=ãáåáãìãK=qÜÉ=î~äìÉ=çÑ=σ=ÖáîáåÖ=íÜÉ=ãáåáãìã=Éêêçê=ã~ó=ÄÉ=~ÇçéíÉÇ=~ë=~=ëçäìíáçå=
çê=íìåÉÇ=ìëáåÖ=~=Öê~ÇáÉåí=ëÉ~êÅÜ=ãÉíÜçÇK=få=íÜÉ=ÅìêêÉåí=ëíìÇó=~=ÅçåàìÖ~íÉ=Öê~ÇáÉåí=ëÉ~êÅÜ=
ãÉíÜçÇ=ï~ë=ìëÉÇ=Ñçê=Ñáå~ä=íìåáåÖ=çÑ=σK=qÜáë=~äÖçêáíÜã=áë=~ÄëçäìíÉäó=~ìíçã~íáÅI=åçí=êÉèìáêJ
áåÖ= ~åó= çéíáãáò~íáçå= é~ê~ãÉíÉêë= ~åÇ= ~äëç= îÉêó= Ñ~ëí= ~åÇ= ÉÑÑÉÅíáîÉK= fíë= ëçìêÅÉ= ÅçÇÉ= ~åÇ=
~ééäáÅ~íáçå=ÇÉí~áäë=Å~å=ÄÉ=ÑçìåÇ=áå=j~ëíÉêë=ENVVRFK=^ééäóáåÖ=~=Öê~ÇáÉåí=ëÉ~êÅÜ=Ñçê=Ñáå~ä=
é~ê~ãÉíÉê=íìåáåÖ=~ääçïÉÇ=ìë=íç=Å~íÅÜ=~=î~äìÉ=Ñçê= σ=ÖççÇ=ÉåçìÖÜ=Ñçê=íÜÉ=ëÉÅçåÇ=E“àçâÉêÒF=
Ç~í~=ëÉíI=ïÜáÅÜ=~ééÉ~êÉÇ=íç=Ñ~ää=çìíëáÇÉ=íÜÉ=áåáíá~ä=áåíÉêî~äK=
cáå~ääóI= íÜÉ= éêçÅÉÇìêÉ= Ñçê= ~ééäóáåÖ= dokk= Ñçê= íÜÉ= pf`OMMQ= ÉñÉêÅáëÉ= ï~ë= íÜÉ==
ÑçääçïáåÖW=
NK kçêã~äáòÉ=íÜÉ=áåéìí=ëé~ÅÉ=íç=íÜÉ=áåíÉêî~ä=xJNI=Nz=EÄó=äáåÉ~ê=íê~åëÑçêã~íáçåFK=
OK aÉÑáåÉ=σlow=~ë=MKMP=~åÇ=σhigh=~ë=MKQ=Ñçê=ÄçíÜ=ÇáêÉÅíáçåë=Eáå=íê~åëÑçêãÉÇ=ÅççêÇáå~íÉëFK=
qÜÉ= ÄçìåÇë= ïÉêÉ= ëÉäÉÅíÉÇ= Ä~ëÉÇ= çå= ~å~äóëáë= çÑ= íÜÉ= éêáçê= Ç~í~K= qÜÉ= åìãÄÉê= çÑ==
Å~äÅìä~íáçåë=áë=NM=Ñçê=ÄçíÜ=ÇáêÉÅíáçåëK=pç=ïÉ=Ü~îÉ=NMGNMZNMM=ÇáÑÑÉêÉåí=î~äìÉë=çÑ=σK=
PK açïåäç~Ç=Ç~í~=ëÉíë=Ñêçã=íÜÉ=ïÉÄ=ëáíÉK=pí~êí=íÜÉ=ÅçãéÉíáíáçåK=
QK pí~êí=íÜÉ=äÉ~îÉJçåÉJçìí=ÅêçëëJî~äáÇ~íáçå=êçìíáåÉ=Ñçê=É~ÅÜ=NMM=î~äìÉë=çÑ= σK=`~äÅìä~íÉ=
íÜÉ=ÅêçëëJî~äáÇ~íáçå=Éêêçê=Ñçê=É~ÅÜ=î~äìÉK=cáÖìêÉ=P=éêÉëÉåíë=ÅêçëëJî~äáÇ~íáçå=Éêêçêë=~ë=
~=ÑìåÅíáçå=çÑ=σK=qÜÉ=äçïÉê=áë=íÜÉ=jpbI=íÜÉ=äáÖÜíÉê=áë=íÜÉ=ÅçäçêK=qÜÉ=ëÅ~äÉ=Ñçê=Éêêçêë=áë=
åçí=áãéçêí~åí=ÄÉÅ~ìëÉ=ïÉ=ëÉ~êÅÜ=Ñçê=ãáåáãìã=çåäó=~åÇ=Çç=åçí=é~ó=~åó=~ííÉåíáçå=íç=
íÜÉ=î~äìÉë=íÜÉãëÉäîÉëK=
RK pÉäÉÅí=íÜÉ= σ=î~äìÉ=íÜ~í=ãáåáãáòÉë=íÜÉ=ÅêçëëJî~äáÇ~íáçå=ÉêêçêK=rëÉ=íÜáë=î~äìÉ=~ë=~=ëí~êíJ
áåÖ=éçáåí=Ñçê=íÜÉ=ÅçåàìÖ~íÉÇ=Öê~ÇáÉåí=íìåáåÖ=éêçÅÉÇìêÉK=
SK rëáåÖ= íÜÉ= íê~åëÑçêã~íáçå= é~ê~ãÉíÉêë= Ñêçã= ëíÉé= NI= Çç= Ä~Åâ= íê~åëÑçêã~íáçå= Ñçê= íÜÉ=
çéíáã~ä= σ=î~äìÉK=q~ÄäÉ=N=éêÉëÉåí=íÜÉ=ÅçêêÉëéçåÇáåÖ=êÉëìäíëK=`çãé~êÉ=ïáíÜ=q~ÄäÉ=P=
Ñçê=êÉëìäíë=çÑ=σ=íìåáåÖ=Ñçê=éêáçê=Ç~í~K=
TK mÉêÑçêã=éêÉÇáÅíáçå=Ñçê=UMU=éçáåíë=ìëáåÖ=íÜÉ=çéíáã~ä=σ=î~äìÉK=
UK pÉåÇ=êÉëìäíë=íç=íÜÉ=ÉñÉêÅáëÉ=çêÖ~åáòÉêëK=båÇ=çÑ=íÜÉ=ÅçãéÉíáíáçåK=
Figure 3
Contour lines of the cross-validation errors (MSE) for the 1st (left) and the 2nd
data set (right). Crosses indicate locations of the
values from the initial
interval
σ
σx
Data sets
First data set
Second data set
Original
27 287
21 861
σy
Transformed
0.16
0.10
Original
34 285
6 959
Transformed
0.13
0.02
Table 1
Optimal anisotropic σ=(σx,σy) values after step 6
19-8
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
2.4. USE OF PRIOR INFORMATION
qÜÉ= dokk= íê~áåáåÖ= éêçÅÉÇìêÉ= ïáíÜ= ~å= áåáíá~ä= áåíÉêî~ä= çÑ= xMKMNI= MKSz= ï~ë= ~ééäáÉÇ= íç= ~ää=
éêáçê=Ç~í~=ëÉíëK=q~ÄäÉ=O=éêÉëÉåíë=íÜÉ=êççíJãÉ~åJëèì~êÉ=Éêêçê=EojpbF=~åÇ=mÉ~êëçåÛë=ÅçêêÉJ
ä~íáçå= ÅçÉÑÑáÅáÉåí= ê= Ñçê= ~ää= NM= Ç~í~= ëÉíëK= léíáã~ä= ~åáëçíêçéáÅ= σ= î~äìÉë= ~êÉ= éêÉëÉåíÉÇ= áå=
q~ÄäÉ=PK=
Prior
day
Std.
Dev.
1
2
3
4
5
6
7
8
9
10
Mean
RMSE
8.85
8.87
8.41
8.52
8.37
7.99
8.34
8.39
8.72
8.70
8.52
0.26
r
0.89
0.89
0.89
0.87
0.87
0.87
0.86
0.87
0.88
0.87
0.88
0.01
Table 2
Training errors for all prior data sets
Prior
Std.
Dev.
1
2
3
4
5
6
7
8
9
10
Mean
σx
0.16
0.16
0.16
0.15
0.17
0.17
0.20
0.16
0.16
0.14
0.16
0.01
σy
0.09
0.09
0.09
0.10
0.10
0.09
0.08
0.10
0.10
0.11
0.10
0.01
Table 3
Optimal σ values for all prior data sets (in transformed coordinates)
qÜÉ=ã~áå=éìêéçëÉ=çÑ=íÜÉ=éêáçê=Ç~í~=~å~äóëáë=ï~ë=íç=ÇÉÑáåÉ=íÜÉ=áåáíá~ä=áåíÉêî~ä=Ñçê=íÜÉ=
çéíáãáë~íáçå= éêçÅÉÇìêÉK= qÜÉ= áåíÉêî~ä= ï~ë= ëÉäÉÅíÉÇ= ~ë= xMKMPI= MKQz= Eáå= íÜÉ= íê~åëÑçêãÉÇ= ÅçJ
çêÇáå~íÉëFI= íÜÉ= ë~ãÉ= Ñçê= ÄçíÜ= ÇáêÉÅíáçåëK= qÜÉ= ÉñéÉÅíÉÇ= çéíáã~ä= âÉêåÉä= ïáÇíÜ= ï~ë= ~Äçìí=
MKNS=Ñçê=σñ=~åÇ=~Äçìí=MKN=Ñçê=σóK=fí=ï~ë=ÅçãéäÉíÉäó=ÅçåÑáêãÉÇ=Ñçê=íÜÉ=Ñáêëí=Ç~í~=ëÉíI=ëáãáä~ê=
áå=ÑÉ~íìêÉë=íç=íÜÉ=éêáçê=Ç~í~=ëÉíëK=
cçê=íÜÉ=ëÉÅçåÇ=Ç~í~=ëÉí=íÜÉ=çéíáã~ä= σ=î~äìÉë=ÇáÑÑÉê=ëáÖåáÑáÅ~åíäóW=MKNP=~åÇ=MKMOK=fí=áë=~=
ÖççÇ= áåÇáÅ~íçê= íÜ~í= íÜÉêÉ= áë= “ëçãÉíÜáåÖ= ïêçåÖÒ= ïáíÜ= íÜáë= Ç~í~= ëÉíK= _ìí= ~ìíçã~íáÅ= ÅêçëëJ
î~äáÇ~íáçå= ÅçãÄáåÉÇ= ïáíÜ= ~= Öê~ÇáÉåí= íìåáåÖ= éêçÅÉÇìêÉ= ~ääçïÉÇ= ìë= íç= ÅÜççëÉ= ÖççÇ==
é~ê~ãÉíÉêë=ÉîÉå=áÑ=íÜÉó=~ééÉ~êÉÇ=íç=ÄÉ=çìíëáÇÉ=íÜÉ=áåáíá~ä=áåíÉêî~äK=lìê=ÖìÉëë=áë=íÜ~í=é~êíáÅáJ
é~åíë=ïÜç=ìëÉÇ=çåäó=íÜÉ=éêáçê=áåÑçêã~íáçå=Ñçê=íê~áåáåÖ=~åÇ=ÇáÇ=åçí=éÉêÑçêã=~åó=~ìíçã~íáÅ=
íìåáåÖ=~äÖçêáíÜãë=ïçìäÇ=åçí=çÄí~áå=îÉêó=ÖççÇ=êÉëìäíë=Ñêçã=íÜÉ=“àçâÉêÒ=Ç~í~=ëÉíK=
2.5. MEASUREMENT OF THE UNCERTAINTY
råÇÉê= íÜÉ= ~ëëìãéíáçå= çÑ= áåÇÉéÉåÇÉåíI= áÇÉåíáÅ~ääó= ÇáëíêáÄìíÉÇ= EáKáKÇKF= ê~åÇçã= î~êá~ÄäÉë=
çîÉê=íÜÉ=ïÜçäÉ=áåéìí=ëé~ÅÉ=Eáå=çìê=Å~ëÉ=~=åçêã~ä=Ed~ìëëá~åF=ÇáëíêáÄìíáçåFI=~=dokk=ÉëíáJ
ã~íÉ=EwGF=Å~å=ÄÉ=íêÉ~íÉÇ=~ë=~=ãÉ~å=çÑ=~=ê~åÇçã=î~êá~ÄäÉK=s~êá~íáçå=Å~å=ÄÉ=ÅçåëáÇÉêÉÇ=~ë=
O
íÜÉ=ëèì~êÉÇ=çÑ=íÜÉ=êÉëáÇì~äë=Ó=EwGJwF K=qÜìëI=ìåÅÉêí~áåíó=Eî~êá~íáçåF=áë=ÅçãéìíÉÇ=Ñêçã=íÜÉ=
êÉëáÇì~äë=~í=íÜÉ=íê~áåáåÖ=äçÅ~íáçåë=EïÜÉêÉ=íÜÉ=êÉ~ä=î~äìÉë=~êÉ=âåçïåF=Äó=áåíÉêéçä~íáåÖ=çîÉê=
íÜÉ=ïÜçäÉ=ëé~ÅÉK=
fåíÉêéçä~íáçå= Å~å= ÄÉ= éÉêÑçêãÉÇ= ïáíÜ= íÜÉ= ÜÉäé= çÑ= ~å= ~äêÉ~Çó= íê~áåÉÇ= dokk= Eσ= = é~J
ê~ãÉíÉê=íìåÉÇ=Ñçê=Ç~í~=éêÉÇáÅíáçåFK=fí=áë=åÉÅÉëë~êó=íç=âÉÉé=áå=ãáåÇ=íÜ~í=ëìÅÜ=~å=Éëíáã~íáçå=
çÑ= î~êá~íáçåëI= ÑáêëíI= ÇÉéÉåÇë= çå= íÜÉ= Ç~í~= éçáåíëÛ= ÇáëíêáÄìíáçå= EÇÉåëáíóFI= ~åÇI= ëÉÅçåÇI= áë==
Äá~ëÉÇ=E~ë=Ä~ëÉÇ=çå=íÜÉ=Äá~ëÉÇ=ãÉ~å=Éëíáã~íáçåFK=qÜÉ=áKáKÇK=~ëëìãéíáçå=áë=íçç=ëíêçåÖI=~åÇ=
áí=~ääçïë=ìë=íç=çÄí~áå=çåäó=êìÇÉI=~ééêçñáã~íÉ=Éëíáã~íÉë=çÑ=íÜÉ=êÉ~ä=ìåÅÉêí~áåíóK=få=êÉ~äáíó=
áí=çåäó=~ääçïë=ìë=íç=ÇÉíÉÅí=~êÉ~ë=ïÜÉêÉ=ìåÉñéÉÅíÉÇ=EÑ~ê=Ñêçã=éêÉÇáÅíÉÇF=î~äìÉë=ã~ó=çÅÅìê=
ïáíÜ=éçëáíáîÉ=éêçÄ~ÄáäáíóK=
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-9
qÜÉ= çíÜÉê= éçëëáÄäÉ= ~ééêç~ÅÜ= áë= íç= ÄìáäÇ= ~= åÉï= ëéÉÅá~ä= dokk= íç= éêÉÇáÅí= íÜÉ= ìåÅÉêJ
í~áåíóI=~ë=ï~ë=ÇçåÉ=áå=íÜÉ=ÅìêêÉåí=Å~ëÉK=qÜÉ=éêçÅÉëë=áë=Ñìääó=~ìíçã~íáÅ=E~ë=ïÉää=~ë=íÜÉ=ã~áå=
êçìíáåÉFI= ~åÇ= áí= ï~ë= áãéäÉãÉåíÉÇ= ~ë= é~êí= çÑ= íÜÉ= ~ìíçã~íáÅ= ÇÉÅáëáçå= ã~âáåÖ= ëóëíÉã= íçJ
ÖÉíÜÉê=ïáíÜ=~=dokk=áíëÉäÑK=fí=çåäó=~ÇÇë=~å=~ÇÇáíáçå~ä=íê~áåáåÖ=éêçÅÉÇìêÉ=ïÜÉêÉ=ëèì~êÉÇ=
êÉëáÇì~äë=~í=íÜÉ=íê~áåáåÖ=éçáåíë=~êÉ=ìëÉÇ=~ë=~=í~êÖÉí=ÑìåÅíáçåK=qÜÉ=çéíáã~ä=î~äìÉ=çÑ= σ=Ñçê=
íÜáë=ìåÅÉêí~áåíó=dokk=áë=ÑçìåÇ=Äó=íÜÉ=ë~ãÉ=éêçÅÉÇìêÉ=~ë=ÇÉëÅêáÄÉÇ=~ÄçîÉK=qÜÉ=íê~áåÉÇ=
ãçÇÉä=Å~å=ÄÉ=ìëÉÇ=íç=Éëíáã~íÉ=íÜÉ=ìåÅÉêí~áåíó=~í=íÜÉ=UMU=ìåâåçïå=äçÅ~íáçåë=çê=~åóïÜÉêÉ=
ÉäëÉK=
3. RESULTS
3.1. OVERALL RESULTS
q~ÄäÉ= Q= éêÉëÉåíë= íÜÉ= Ä~ëáÅ= ëí~íáëíáÅë= EãáåáãìãI= ã~ñáãìãI= ãÉ~åI= ãÉÇá~å= ~åÇ= ëí~åÇ~êÇ=
ÇÉîá~íáçåF=çÑ=íÜÉ=UMU=Éëíáã~íÉÇ=~åÇ=çÄëÉêîÉÇ=î~äìÉë=Ñçê=íÜÉ=íïç=Ç~í~=ëÉíëK=
N = 808
Observed (first data set)
Estimates (first data set)
Observed (second data set)
Estimates (second data set)
Min.
57.0
69.9
57.0
58.7
Max.
180.0
125.9
1528.2
1330.5
Mean
98.0
96.8
105.4
104.9
Median
98.8
98.8
99.0
100.3
Std. Dev.
20.0
13.9
83.7
71.5
Table 4
Comparison of the estimated and measured values (nSv/h)
qÜÉ=ãÉ~å=~ÄëçäìíÉ=Éêêçê=Ej^bFI=íÜÉ=Äá~ë=Eçê=ãÉ~å=ÉêêçêI=jbFI=íÜÉ=êççíJãÉ~åJëèì~êÉÇ=
Éêêçê= EojpbFI= mÉ~êëçåÛë= ÅçêêÉä~íáçå= ÅçÉÑÑáÅáÉåí= ê= ÄÉíïÉÉå= íÜÉ= Éëíáã~íÉÇ= ~åÇ= íêìÉ= î~äìÉë=
~åÇ= åçêã~äáòÉÇ= ojpb= EkojpbF= Ñçê= íÜÉ= éêÉÇáÅíáçåë= ~í= íÜÉ= å= Z= UMU= äçÅ~íáçåë= ~êÉ= éêÉJ
ëÉåíÉÇ=áå=q~ÄäÉ=RK=kojpb=áë=ÅçãéìíÉÇ=Ñêçã=
NRMSE=RMSE / Std.Dev.
ïÜÉêÉ=píÇKaÉîK=áë=íÜÉ=ëí~åÇ~êÇ=ÇÉîá~íáçå=çÑ=íÜÉ=çÄëÉêîÉÇ=î~äìÉëK=qÜÉ=kojpb=ëÜçïë=íÜÉ=
êÉä~íáçå=ÄÉíïÉÉå=íÜÉ=ojpb=çÑ=íÜÉ=Éëíáã~íÉÇ=î~äìÉë=~åÇ=íÜÉ=ëí~åÇ~êÇ=ÇÉîá~íáçå=çÑ=íÜÉ=çÄJ
ëÉêîÉÇ=î~äìÉëK=qÜìë=çåäó=î~äìÉë=çÑ=kojpb=äÉëë=íÜ~å=N=ã~ó=ÄÉ=~ÅÅÉéíÉÇ=~ë=ë~íáëÑ~ÅíçêóK=
líÜÉêïáëÉI= áÑ= íÜÉ= éêÉÇáÅíçê= éêÉëÉåíë= íÜÉ= ãÉ~å= çÑ= íÜÉ= çÄëÉêîÉÇ= î~äìÉë= ~ë= íÜÉ= êÉëìäíI= íÜÉå=
kojpbZNKM=áë=çÄí~áåÉÇK=
Data sets
MAE
First data set
Second data set
ME
RMSE
Pearson’s r
NRMSE
9.40
– 1.25
12.59
0.78
0.63
14.85
– 0.51
45.46
0.84
0.54
Table 5
Comparison of errors
qÜÉ=Ñáêëí=íÜêÉÉ=Éêêçê=ÅÜ~ê~ÅíÉêáëíáÅë=Ej^bI=jbI=ojpbF=ÇÉéÉåÇ=çå=î~äìÉë=çÑ=íÜÉ=Ç~í~=
ëÉíë=éêÉëÉåíÉÇK=pç=áí=áë=ÇáÑÑáÅìäí=íç=Åçãé~êÉ=íÜÉ=èì~äáíó=çÑ=éêÉÇáÅíáçå=Ñçê=ÇáÑÑÉêÉåí=Ç~í~=ëÉíë=
ïáíÜ=íÜÉëÉ=ÉêêçêëK=^åçíÜÉê=íïç=Éêêçêë=EêI=kojpbF=~êÉ=êÉä~íáîÉ=~åÇ=Çç=åçí=ÇÉéÉåÇ=çå=íÜÉ=
ê~åÖÉ=çÑ=íÜÉ=Ç~í~K=s~äìÉë=çÑ=j^b=~åÇ=ojpb=Ñçê=íÜÉ=ëÉÅçåÇ=Ç~í~=ëÉí=~êÉ=ïçêëÉI=Äìí=íÜÉ=
ÅçêêÉä~íáçå= ÅçÉÑÑáÅáÉåí= ~åÇ= kojpb= ~êÉ= ãìÅÜ= ÄÉííÉêK= qÜìëI= íÜÉ= ëÉÅçåÇ= Ç~í~= ëÉí= áë= éêÉJ
ÇáÅíÉÇ=ÄÉííÉê=íÜ~å=íÜÉ=Ñáêëí=çåÉK=kÉÖ~íáîÉ=î~äìÉë=çÑ=íÜÉ=jb=áåÇáÅ~íÉ=íÜ~í=dokkI=çå=íÜÉ=
19-10
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
~îÉê~ÖÉI= ãçêÉ= íÉåÇë= íç= êÉÇìÅÉ= EìåÇÉêÉëíáã~íÉF= ÜáÖÜ= î~äìÉë= íÜ~å= áåÅêÉ~ëÉ= EçîÉêÉëíáã~íÉF=
äçï=çåÉëK=fí=áë=íÜÉ=Å~ëÉ=Ñçê=ÄçíÜ=Ç~í~=ëÉíëK=
qÜêÉÉ=ëÉíë=çÑ=ã~éë=ïáíÜ=áëçäáåÉë=~êÉ=éêÉëÉåíÉÇW=íÜÉó=ëÜçï=íÜÉ=Éëíáã~íÉÇ=î~äìÉë=EcáÖìêÉ=
QF=~åÇ=íÜÉ=~ëëçÅá~íÉÇ=ìåÅÉêí~áåíáÉë=EcáÖìêÉë=R=~åÇ=SFK=cáÖìêÉ=S=éêÉëÉåíë=íÜÉ=ìåÅÉêí~áåíáÉë=
Ñêçã=íÜÉ=Ää~Åâ=ëéçí=çÑ=cáÖìêÉ=R=EêáÖÜíFI=ïÜáÅÜ=~êÉ=åçí=ÇáëíáåÖìáëÜ~ÄäÉ=áå=íÜÉ=çêáÖáå~ä=ëÅ~äÉK=
få=~ää=íÜÉëÉ=ÑáÖìêÉë=ÅêçëëÉë=áåÇáÅ~íÉ=éçáåí=äçÅ~íáçåë=çÑ=íÜÉ=Éëíáã~íÉÇ=î~äìÉë=~åÇ=ÄçñÉë=ÇÉJ
éáÅí=äçÅ~íáçåë=çÑ=íÜÉ=áåéìí=î~äìÉëK=
Figure 4
Isoline levels (nSv/h) for the 1st set (left) and the 2nd set (right)
Figure 5
Isoline levels showing the uncertainty associated to the estimations obtained
st
nd
for the 1 set (left) and the 2 set (right) ) in the same colour scale for comparison
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-11
Figure 6
Isoline levels showing the uncertainty for the black spot of Figure 5 (right)
3.2. DETECTING ANOMALIES AND OUTLIERS
Figure 7
3D map showing extreme values found in the 2nd set by GRNN
åÇ
cáÖìêÉ=T=éêÉëÉåíë=íÜÉ=éÉ~â=ÑçìåÇ=áå=íÜÉ=O =Ç~í~=ëÉí=Äó=íÜÉ=dokk=~ééêç~ÅÜK=qÜÉ=ëãççíÜJ
áåÖ=ÅÜ~ê~ÅíÉêáëíáÅ=Ñçê=dokk=äçïÉêÉÇ=íÜÉ=ÜÉáÖÜí=çÑ=íÜÉ=éÉ~âK=_ìí=îáëì~ääó=áí=ÇçÉë=åçí=äççâ=
ëãççíÜÉÇ= Ó= ~= ÅçääÉÅíáçå= çÑ= ëÉîÉê~ä= ê~íÜÉê= ëÜ~êé= íçéëK= ^ìíçã~íáÅ= dokk= ~ääçïë= ìë= íç=
~Ç~éí=èìáÅâäó=íç=ÅçãéäÉíÉäó=åÉï=Ç~í~=~åÇ=íç=ã~âÉ=îÉêó=ÖççÇ=éêÉÇáÅíáçåëK=
4. DISCUSSION
^= dÉåÉê~ä= oÉÖêÉëëáçå= kÉìê~ä= kÉíïçêâ= EdokkF= áë= ~å= áãéäÉãÉåí~íáçå= çÑ= ~= åçåJ
é~ê~ãÉíêáÅ=âÉêåÉä=Éëíáã~íçêK=få=íÜÉ=ÅìêêÉåí=ïçêâ=áí=áë=íêÉ~íÉÇ=~ë=~å=~ìíçã~íáÅ=ëé~íá~ä=éêÉJ
ÇáÅíçê= çÑ= ê~Çáç~ÅíáîáíóK= fí= ï~ë= ëìÅÅÉëëÑìääó= ~ééäáÉÇ= íç= íïç= ÇáÑÑÉêÉåí= Ç~í~= ëÉíë= Ó= ìëì~ä=
êçìíáåÉ=Ç~í~=~åÇ=ìåìëì~ä=ÉãÉêÖÉåÅó=Ç~í~K=dokk=áë=îÉêó=Ñ~ëí=Ó=ê~ï=íê~áåáåÖ=~åÇ=éêÉÇáÅJ
íáçå=Å~äÅìä~íáçåë=íççâ=~Äçìí=PM=ëÉÅçåÇë=Ñçê=É~ÅÜ=Ç~í~=ëÉí=çå=~=mÉåíáìã=fff=RMM=jeò=m`K=
19-12
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
qÜÉ=ãçëí=áãéçêí~åí=éêçÄäÉã=ïáíÜ=dokk=éêÉÇáÅíçê=áë=ëãççíÜáåÖK=qÜÉ=ëãççíÜáåÖ=ÇÉJ
ÖêÉÉ= ÇÉéÉåÇë= çå= dokk= é~ê~ãÉíÉêë= íìåÉÇ= Ñçê= íÜÉ= Ç~í~K= qÜÉ= éêÉëÉåÅÉ= çÑ= ëãççíÜáåÖ= Ñçê=
íÜÉ=ÅìêêÉåí=í~ëâë=Å~å=ÄÉ=ëÉÉå=áå=q~ÄäÉ=QK=jáåáã~=~êÉ=çîÉêÉëíáã~íÉÇI=ã~ñáã~=~êÉ=ìåÇÉêÉëJ
íáã~íÉÇI=~åÇ=ëí~åÇ~êÇ=ÇÉîá~íáçåë=Ñçê=íÜÉ=éêÉÇáÅíÉÇ=ÇáëíêáÄìíáçåë=~êÉ=ãìÅÜ=äçïÉê=íÜ~å=íÜÉ=
çåÉë=çÑ=íÜÉ=êÉ~ä=ÑáÉäÇëK=_ìí=íÜÉ=ãÉ~åë=~åÇ=ãÉÇá~åë=~êÉ=êÉéêçÇìÅÉÇ=îÉêó=ïÉääK=fí=ãÉ~åë=íÜ~í=
dokk=éêçîáÇÉÇ=~=ÖäçÄ~ääó=ìåÄá~ëÉÇI=ëãççíÜÉÇ=Éëíáã~íÉK=
qÜÉ=ã~áå=éêçÄäÉãë=ïáíÜ=~ééäóáåÖ=dokk=~êÉ=ïÜÉíÜÉê=ëìÅÜ=~=ÇÉÖêÉÉ=çÑ=ëãççíÜáåÖ=áë=
ë~íáëÑ~Åíçêó=~åÇ=ïÜÉíÜÉê=íÜÉ=êÉëìäíë=Ñêçã=íÜÉ=ãçÇÉäáåÖ=~êÉ=~ÅÅÉéí~ÄäÉ=Ñçê=ÇÉÅáëáçå=ã~âJ
áåÖK=fí=ÉîáÇÉåíäó=ÇÉéÉåÇë=çå=íÜÉ=ã~áå=Öç~ä=çÑ=íÜÉ=ëíìÇóW=áÑ=ïÉ=~êÉ=áåíÉêÉëíÉÇ=áå=éêÉÇáÅíáåÖ=
ÉñíêÉãÉ=î~äìÉë=éêÉÅáëÉäóI=íÜÉå=dokk=áë=åçí=ïÜ~í=ïÉ=åÉÉÇK=_ìí=áÑ=ïÉ=åÉÉÇ=~=Ñ~ëíI=É~ëáäó=
íê~áåÉÇ=íççä=Ñçê=êÉÅçêÇáåÖ=~=Ñäìñ=~ë=~å=ÉîÉåíI=íÜÉå=dokk=áë=ÖççÇ=ÉåçìÖÜK=dokk=ÇÉÑáåÉÇ=
íÜÉ=éêÉëÉåÅÉ=çÑ=~=Üçí=ëéçí=~åÇ=éçëáíáçåÉÇ=áí=ïÉääK=qç=ÄÉ=ãçêÉ=ÅÉêí~áå=~Äçìí=áíë=~ééäáÅ~ÄáäJ
áíóI=ïÉ=ëÜçìäÇ=íÉëí=íÜÉ=ãçÇÉä=çå=çíÜÉê=í~ëâë=áåîçäîáåÖ=ÉãÉêÖÉåÅó=ëáíì~íáçåëK=
iÉí= ìë= êÉã~êâ= ÜÉêÉ= íÜ~í= áí= áë= åçí= ~= éêçÄäÉã= íç= ÄìáäÇ= ~= dokk= EÇÉÑáåÉ= ~= σF= ïáíÜ= ~ää=
íê~áåáåÖ=Éêêçêë=ÄÉáåÖ= òÉêç=~åÇ=~=ÅçêêÉä~íáçå=ÅçÉÑÑáÅáÉåí=çÑ=NK=fí=áë= ÉåçìÖÜ=íç=ìëÉσ=î~äìÉë=
ÅäçëÉ=íç=òÉêçX=~=ëáãáä~ê=Éñ~ãéäÉ=ï~ë=ÇáëÅìëëÉÇ=Ñçê=~=ëáãéäÉ=éêçÄäÉã=EcáÖìêÉ=OÄFK=_ìí=ãçëí=
éêçÄ~Ääó=áí=ïçìäÇ=áãéäó=~å=çîÉêÑáííáåÖI=ÄÉÅ~ìëÉ=ëìÅÜ=~=dokk=ÅçìäÇ=åçí=ÄÉ=ÖÉåÉê~äáòÉÇI=
áKÉK=ìëÉÇ=íç=éêÉÇáÅí=î~äìÉë=Ñçê=åÉïI=ìåâåçïå=éçáåíëK=qÜÉ=ÖÉåÉê~äáò~íáçå=~Äáäáíó=áë=íÜÉ=ãçëí=
áãéçêí~åí=ÑÉ~íìêÉ=çÑ=~åó=éêÉÇáÅíáåÖ=ãçÇÉäK=
_ìí=~=ëã~ää=σ=ÇçÉë=åçí=~äï~óë=áãéäó=çîÉêÑáííáåÖK=cçê=íÜÉ=ëÉÅçåÇ=Ç~í~=ëÉí=íÜÉ=çéíáã~ä= σ=
áë=ãìÅÜ=ëã~ääÉê=íÜ~å=Ñçê=íÜÉ=Ñáêëí=Ç~í~=ëÉí=E~åÇ=Ñçê=~ää=éêáçê=Ç~í~=ëÉíëFK=_ìí=ëíáää=íÜÉ=dokk=
éêÉëÉåíë=îÉêó=ÖççÇ=ÖÉåÉê~äáò~íáçå=~ÄáäáíáÉëI=ïÜáÅÜ=Å~å=ÄÉ=çÄëÉêîÉÇ=íÜêçìÖÜ=~=ÜáÖÜ=ÅçêêÉä~J
íáçå= ÅçÉÑÑáÅáÉåí= ÄÉíïÉÉå= êÉ~ä= ~åÇ= Éëíáã~íÉÇ= î~äìÉë= Eq~ÄäÉ= RFK= ^äëç= íÜÉ= Ä~ëáÅ= ëí~íáëíáÅë=
Eq~ÄäÉ=QF=ëÜçïë=ÉîÉå=ÄÉííÉê=êÉéêçÇìÅíáçå=çÑ=ÖäçÄ~ä=ÇáëíêáÄìíáçå=íÜ~å=Ñçê=íÜÉ=Ñáêëí=Ç~í~=ëÉíK=
qÜÉêÉ=áë=ëçãÉ=íêáÅâ=ïáíÜ=ìåÅÉêí~áåíó=Éëíáã~íáçå=ïÜáäÉ=ïçêâáåÖ=ïáíÜ=~=dokk=Ó=ìåÅÉêJ
í~áåíó= áë= Éëíáã~íÉÇ= ~ë= ~= êÉÖêÉëëáçå= çÑ= âåçïå= ëèì~êÉÇ= êÉëáÇì~äëK= mÉêÜ~éë= ~= ãçêÉ=
íÜÉçêÉíáÅ~ääó=éêçîÉå=~ééêç~ÅÜ=áë=íç=é~ëë=íç=~=éêçÄ~ÄáäáëíáÅ=íêÉ~íãÉåí=çÑ=íÜÉ=ìåÅÉêí~áåíó=Ó=
íç= Éëíáã~íÉ= íÜÉ= éêçÄ~Äáäáíó= çÑ= ÉñÅÉÉÇáåÖ= ~= ëÉí= çÑ= ÅêáíáÅ~ä= äÉîÉäëK= fí= ãÉ~åë= íç= ÅÜ~åÖÉ= ~= êÉJ
ÖêÉëëáçå=éêçÄäÉã=íç=~=Åä~ëëáÑáÅ~íáçå=çåÉK=^=êÉèìáêÉÇ=Åä~ëëáÑáÉê=Å~å=ÄÉ=ÅçåëíêìÅíÉÇ=Ä~ëÉÇ=çå=
íÜÉ=ë~ãÉ=ã~íÜÉã~íáÅ~ä=éêáåÅáéäÉë=Ó=m~êòÉåÛë=åçåJé~ê~ãÉíêáÅ=ÇÉåëáíó=Éëíáã~íáçåK=^=éêçÄJ
~ÄáäáëíáÅ= ÇÉëÅêáéíáçå= çÑ= ìåÅÉêí~áåíó= áë= ãçêÉ= áåíÉêéêÉí~ÄäÉ= ~åÇI= íÜìëI= Å~å= ÄÉ= ÉîÉå= ãçêÉ=
~ííê~ÅíáîÉ=Ñçê=ÇÉÅáëáçå=ã~âÉêë=íÜ~å=ÇáêÉÅí=î~äìÉë=çÑ=ìåÅÉêí~áåíóK=
5. CONCLUSIONS
qÜÉ= ÅìêêÉåí= ëíìÇó= äÉÇ= íç= íÜÉ= ã~áå= ÅçåÅäìëáçå= íÜ~í= dokk= áë= ~= éêçãáëáåÖ= íççä= Ñçê= ~ìíçJ
ã~íáÅ= ëé~íá~ä= éêÉÇáÅíáçå= çÑ= ê~Çáç~Åíáîáíó= áå= ÄçíÜ= êçìíáåÉ= ~åÇ= ÉãÉêÖÉåÅó= ëáíì~íáçåëK= fí= áë=
îÉêó= Ñ~ëíI= É~ëó= íç= ~ééäó= ~åÇ= áåíÉêéêÉíK= fí= éêçîáÇÉë= êÉ~ëçå~ÄäÉ= êÉëìäíëI= ÉîÉå= Ñçê= ÅçãéäÉñI=
ìåÉñéÉÅíÉÇ=Ç~í~K=qÜÉ=ëãççíÜáåÖ=ÉÑÑÉÅí=áë=~=îáëáÄäÉ=Çê~ïÄ~ÅâK=_ìí=áí=Å~å=ÄÉ=áÖåçêÉÇ=áÑ=ïÉ=
~êÉ=åçí=íçç=ãìÅÜ=áåíÉêÉëíÉÇ=éêÉÇáÅíáåÖ=~ÅÅìê~íÉäó=ÜáÖÜ=~åÇ=äçï=î~äìÉëI=Äìí=ï~åí=íç=ÇÉíÉÅí=
íÜÉ=éêÉëÉåÅÉ=çÑ=~=Üçí=ëéçí=~åÇ=áåÇáÅ~íÉ=íÜ~í=~=ÅêáíáÅ~ä=äÉîÉä=áë=ÉñÅÉÉÇÉÇK=tÉ=çåäó=åÉÉÇ=íç=
âÉÉé=áå=ãáåÇ=íÜ~í=ã~ñáãìãë=ïçìäÇ=~äï~óë=ÄÉ=ìåÇÉêÉëíáã~íÉÇ=Ó=ëãççíÜÉÇK=
qÜÉ=ÇáêÉÅí=Éëíáã~íáçå=çÑ=íÜÉ=éêÉÇáÅíáçå=ìåÅÉêí~áåíó=Ü~ë=ëçãÉ=íÜÉçêÉíáÅ~ä=éêçÄäÉãëK=fåáJ
íá~ääó=íÜÉ=ãÉíÜçÇ=ÇçÉë=åçí=éêçîáÇÉ=~=ãÉ~ëìêÉ=çÑ=ìåÅÉêí~áåíóX=áí=áë=Å~äÅìä~íÉÇ=ìåÇÉê=ê~íÜÉê=
ëíêçåÖ= ~ëëìãéíáçåëK= _ìí= íÜÉêÉ= ~êÉ= ëçãÉ= ï~óëI= ~äëç= ÇáëÅìëëÉÇ= áå= íÜáë= ïçêâI= íç= áåíêçÇìÅÉ=
ìåÅÉêí~áåíó= Éëíáã~íáçå= áå= íÜÉ= éêçÅÉëëK= qÜÉ= çíÜÉê= éçëëáÄáäáíó= áë= íç= ãçîÉ= íç= ~= éêçÄ~Äáäáíó=
íêÉ~íãÉåí=çÑ=ìåÅÉêí~áåíóK=
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES
19-13
CODES
qÜÉ= ëçìêÅÉ= ÅçÇÉë= çÑ= ~ää= dokkJêÉä~íÉÇ= ~äÖçêáíÜãë= ~åÇ= éêçÅÉÇìêÉë= ïÉêÉ= í~âÉå= Ñêçã==
j~ëíÉêë= ENVVRFK= qÜÉó= ïÉêÉ= ~ëëáãáä~íÉÇ= áå= ~= ëéÉÅá~äI= ~ìíçã~íáÅ= dokk= íê~áåáåÖ= ~åÇ==
éêÉÇáÅíáçå=ëóëíÉãK=
REFERENCES
_áëÜçéI=`KjKX=kÉìê~ä=kÉíïçêâë=Ñçê=m~ííÉêå=oÉÅçÖåáíáçåK=lñÑçêÇW=`ä~êÉåÇçå=mêÉëëX=NVVRK=
e~óâáåI=pKX=kÉìê~ä=kÉíïçêâW=^=`çãéêÉÜÉåëáîÉ=cçìåÇ~íáçåK=pÉÅçåÇ=bÇáíáçåK=kÉï=vçêâW==
j~Åãáääá~å=mìÄäáëÜáåÖ=`çãé~åóX=NVVVK=
h~åÉîëâóI=jKX=Dpé~íá~ä=mêÉÇáÅíáçåë=çÑ=pçáä=`çåí~ãáå~íáçå=rëáåÖ=dÉåÉê~ä=oÉÖêÉëëáçå=kÉìê~ä==
kÉíïçêâëDK=póëíÉãë=oÉëÉ~êÅÜ=~åÇ=fåÑçK=póëíÉãëI=NVVU=j~êX=UW=éé=OQNJORSK=
h~åÉîëâóI=jKX=D^êíáÑáÅá~ä=kÉìê~ä=kÉíïçêâë=~åÇ=pé~íá~ä=fåíÉêéçä~íáçåëK=`~ëÉ=ëíìÇóW=`ÜÉêåçÄóä==
Ñ~ääçìíDK=mêÉéêáåí=VRJM=jçëÅçïW=kìÅäÉ~ê=p~ÑÉíó=fåëíáíìíÉX=NVVRK=
j~ëíÉêëI=qKX=^Çî~åÅÉÇ=^äÖçêáíÜãë=Ñçê=kÉìê~ä=kÉíïçêâëK=^=`HH=pçìêÅÉÄççâK=qçêçåíçW=gçÜå=táäÉó=
C=pçåëI=fåÅX=NVVRK=
k~Ç~ê~ó~I=bK^KX=Dlå=Éëíáã~íáåÖ=êÉÖêÉëëáçåDK=qÜÉçêó=çÑ=mêçÄ~Äáäáíó=~åÇ=áíë=^ééäáÅ~íáçåëI=NVSQX=VW=ééK=
NQNJNQOK=
m~êòÉåI=bKX=Dlå=Éëíáã~íáçå=çÑ=~=éêçÄ~Äáäáíó=ÇÉåëáíó=ÑìåÅíáçå=~åÇ=ãçÇÉDK=^åå~äë=çÑ=j~íÜÉã~íáÅ~ä=
pí~íáëíáÅëX=NVSOX=PPW=ééK=NMSRJNMTSK=
oçëÉåÄä~ííI=jKX=DaÉåëáíó=Éëíáã~íÉë=~åÇ=j~êâçî=ëÉèìÉåÅÉëDK=fåW=mìêáI=jKI=ÉÇáíçêK=kçåé~ê~ãÉíêáÅ=
qÉÅÜåáèìÉë=áå=pí~íáëíáÅ~ä=fåÑÉêÉåÅÉK=içåÇçåW=`~ãÄêáÇÖÉ=råáîÉêëáíó=mêÉëëX=NVTMK=ééK=NVVJONP=
oçëÉåÄä~ííI=jKX=DoÉã~êâë=çå=ëçãÉ=åçåé~ê~ãÉíêáÅ=Éëíáã~íÉë=çÑ=~=ÇÉåëáíó=ÑìåÅíáçåDK=^åå~äë=çÑ=
j~íÜÉã~íáÅ~ä=pí~íáëíáÅëI=NVRSX=OTW=ééK=UPOJUPTK=
péÉÅÜíI=aKbKX=D^=dÉåÉê~ä=oÉÖêÉëëáçå=kÉìê~ä=kÉíïçêâDK=fbbb=qê~åë~Åíáçåë=çå=kÉìê~ä=kÉíïçêâëI=
NVVNX=OW=ééK=RSUJRTSK=
t~íëçåI=dKpKX=DpãççíÜ=êÉÖêÉëëáçå=~å~äóëáëDK=p~åâÜó~W=qÜÉ=fåÇá~å=gçìêå~ä=çÑ=pí~íáëíáÅëI=pÉêáÉë=^I=
NVSQX=OSW=ééK=PRVJPTOK=
Cite this article as: Timonin, Vadim; Savelieva, Elena. ‘Spatial prediction of radioactivity using general
regression neural network’. Applied GIS, Vol 1, No 2, 2005. pp. 19-01 to 19-14. DOI: 10.2104/ag050019
Copyright © 2005 Vadim Timonin and Elena Savelieva
19-14
SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES