not a very good instrument for predicting extreme and out1ier va1ues, but it a11ows the user to detect their presence. Resu1ts from the second (âjokerâ) data set ...
INTRODUCTION TO THE SPATIAL INTERTICLES THE DA
ARTICLES
SPATIAL PREDICTION OF RADIOACTIVITY USING GENERAL REGRESSION NEURAL NETWORK
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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=
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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=
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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
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SPATIAL PREDICTION OF RADIOACTIVITY ARTICLES