spatial prediction of radioactivity using general regression neural ...

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=pfÒMMQ=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= pfÒMMQ= 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= pfÒMMQ= 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 →∞


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


⎛ 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==


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


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=pfÒMMQ=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=


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= pfÒMMQ= 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


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=


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


~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


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


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=


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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=Î= 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

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