Violation of the Markov property

[carllacan]

Suppose we have a methereological model which describes the weather change as a Markov chain: we assume that the weather on each day depends only on the weather of the previous day. Suppose further that we extended this model so as to consider season change. defining a different transition table for each season. Would this violate the Markov property of the season change?

I think it would not, since the weather would still depend just on the previous day, no matter if it was another season. Am I right?

 

[Stephen Tashi]

the weather would still depend just on the previous day, no matter if it was another season. Am I right?

You haven't given a definition of what a "state" is in your model. Whether it is Markov or non-Markov depends on that definition.

The various pieces of information in a model or a real life process don't, by themselves determine, whether the model is a Markov process. To have a Markov process, the next "state" must depend only on the previous "state", so the Markov-ness or non_Markov-ness of the model depends on how you define "state".

If you define "state" to be the single variable W that represents "the weather for the day" then your model is not Markov since W for the next day depends on both W for the previous day and the season. A variable representing the season is not present in that definition of "state".

On way to make your model Markov is to define a "state" to be a vector of 3 pieces of information (J,S,W) where J is the julian date (1,2,...365), S is the season (summer, fall, winter,spring) and W is the weather (however you care to classify it).

(The Julian date is needed in order to define how the seasons change. For example to define when Spring begins, you need transition rules that say if the previous state is ( 78,winter,W= anything) then the next state has the form (79,spring,W = something).

 

[carllacan]

You haven't given a definition of what a "state" is in your model. Whether it is Markov or non-Markov depends on that definition.

The various pieces of information in a model or a real life process don't, by themselves determine, whether the model is a Markov process. To have a Markov process, the next "state" must depend only on the previous "state", so the Markov-ness or non_Markov-ness of the model depends on how you define "state".

If you define "state" to be the single variable W that represents "the weather for the day" then your model is not Markov since W for the next day depends on both W for the previous day and the season. A variable representing the season is not present in that definition of "state".

On way to make your model Markov is to define a "state" to be a vector of 3 pieces of information (J,S,W) where J is the julian date (1,2,...365), S is the season (summer, fall, winter,spring) and W is the weather (however you care to classify it).

(The Julian date is needed in order to define how the seasons change. For example to define when Spring begins, you need transition rules that say if the previous state is ( 78,winter,W= anything) then the next state has the form (79,spring,W = something).

Thank you, great explanation.