# Random field

Luncat ka: pituduh, sungsi

Harti dasar random field nyaeta daftar wilangan acak numana nileyna dipetakeun kana rohangan (dimensi-n). Nilai dina random field ilahar pakait sacara spatial antara hiji niley jeung nu sejenna, dina harti dasarna bisa oge niley ieu teu pati beda jeung niley saterusna. Contona keur kasus struktur covariance, numana sababaraha tipe kovarian nu beda ieu bisa dimodelkeun make random field.

## Sacara Matematika

Dina probability theory, anggap S = {X1, ..., Xn}, numana Xi dina {0, 1, ..., G − 1}, disusun salaku variabel acak di jero sampel ruang Ω = {0, 1, ..., G − 1}n. Ukuran probabiliti π nyaeta random field lamun

$\pi(\omega)>0\,$

keur sakabeh ω dina Ω. Sababaraha tipe random fields nu ilahar, diantara Markov random fields (MRF), Gibbs random fields (GRF), conditional random fields(CRF), sarta Gaussian random fields. MRF nembongkeun pasipatan Markovian

$\pi (X_i=x_i|X_j=x_j, i\neq j) = \pi (X_i=x_i|\partial_i), \,$

dimana $\partial_i$ nyaeta susunan pangdeukeutna tina variable acak Xi. Dina kalimah sejen, probabiliti variabel acak dianggap niley nu gumantung kana variabel acak sejenna ngaliwatan nilai pangdeukeutna nu kapanggih saanggeusna. Probabiliti variabel acak dina MRF ditembongkeun ku persamaan 1, Ω' sarua jeung niley real Ω, iwal ti keur variabel acak Xi. Gampang ditempo yen hese diitung gedena ieu niley migunakeun persamaan di luhur. Solusi keur ieu masalah diusulkeun ku Besag dina 1974, numana manehna nyieun hubungan antara MRF jeung GRF.

$\pi (X_i=x_i|\partial_i) = \frac{\pi(\omega)}{\sum_{\omega'}\pi(\omega')} \;\;\;\;(1)$

## Pamakean

Random field nu geus ilahar dipake keur nalungtik proses alam nyaeta Monte Carlo method, numana random field pakait jeung sifat spatial alami, saperti permeabilitas taneuh dina skala meter atawa kuat beton dina skala sentimeter.

## Rujukan

• Besag, J. E. "Spatial Interaction and the Statistical Analysis of Lattice Systems", Journal of Royal Statistical Society: Series B 36, 2 (May 1974), 192-236.