, ▁which ▁extends ▁a ▁single ▁Mark ov ▁chain ▁into ▁a ▁multi - dimensional ▁random ▁field ▁for ▁ge ost at ist ical ▁model ing . ▁A ▁Mark ov ▁chain ▁random ▁field ▁is ▁still ▁a ▁single ▁spatial ▁Mark ov ▁chain . ▁The ▁spatial ▁Mark ov ▁chain ▁moves ▁or ▁j umps ▁in ▁a ▁space ▁and ▁dec ides ▁its ▁state ▁at ▁any ▁un ob ser ved ▁location ▁through ▁interactions ▁with ▁its ▁nearest ▁known ▁neighb ors ▁in ▁different ▁directions . ▁The ▁data ▁interaction ▁process ▁can ▁be ▁well ▁explained ▁as ▁a ▁local ▁sequ ential ▁Bay esian ▁updating ▁process ▁within ▁a ▁neighborhood . ▁Because ▁single - step ▁transition ▁probability ▁matrices ▁are ▁difficult ▁to ▁estimate ▁from ▁sparse ▁sample ▁data ▁and ▁are ▁imp ract ical ▁in ▁representing ▁the ▁complex ▁spatial ▁heter ogene ity ▁of ▁states , ▁the ▁trans i ogram , ▁which ▁is ▁defined ▁as ▁a ▁transition ▁probability ▁function ▁over ▁the ▁distance ▁lag , ▁is ▁proposed ▁as ▁the ▁accomp any ing ▁spatial ▁measure ▁of ▁Mark ov ▁chain ▁random ▁fields . ▁ ▁References ▁▁▁ ▁Li , ▁W . ▁ 2 0 0 7 . ▁Mark ov ▁chain ▁random ▁fields ▁for ▁estimation ▁of ▁categor ical ▁variables . ▁Math . ▁Ge ol ., ▁ 3 9 ( 3 ): ▁ 3 2 1 – 3 3 5 . ▁ ▁Li , ▁W . ▁et ▁al . ▁ 2 0 1 5 . ▁Bay esian ▁Mark ov ▁chain ▁random ▁field ▁cos im ulation ▁for ▁impro ving ▁land ▁cover ▁classification ▁accuracy . ▁Math . ▁Ge osci ., ▁ 4 7 ( 2 ): ▁ 1 2 3