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I understand that Markov Chains are very important in modeling phenomena such as intergenerational socio-economic status, weather, random walks, memory-less board games, etc... But I'm struggling to find real, empirical data that satisfies a Markov Chain.

For example, I see a lot of..."imagine that if it rained yesterday, then it will rain today with probability 0.8". Where is this 0.8 from? I can't find anything that references a study, or some dataset upon which these figures are based. Not just weather, but any Markov Chain based model. Anyone have resources or suggestions on where to locate these? I searched the footnotes of a lot of these examples but they haven't revealed anything.

Edit Just because some people have provided answers to this, let me clarify and ask it in a different way: Are there processes in nature, and accompanying datasets, that satisfy the Markov Property to a "high" degree, rather than just toy examples that illustrate it?

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    $\begingroup$ All models are wrong! $\endgroup$
    – Xi'an
    Commented Sep 12, 2015 at 8:31
  • $\begingroup$ look at www.amstat.org/publications/jse/v18n3/rotondi.pdf $\endgroup$
    – KFkf
    Commented Jun 10, 2016 at 6:28
  • $\begingroup$ you can find some other papers by googling "markov chain real data" . but they were too advanced for me so i did not post them here . $\endgroup$
    – KFkf
    Commented Jun 10, 2016 at 6:37

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Usually these kinds of models are trained using some empirical data. In your example, if you have data about the weather for several years, you can estimate the probability of a rainy day given that previous day was rainy, or given that the previous day was sunny, etc.

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  • $\begingroup$ Right. Can you point me to where such a study exists? I can't seem to find one. $\endgroup$
    – ilanman
    Commented Sep 12, 2015 at 11:46
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I would agree on Xi'an's comment All Models Are Wrong.

However, these models (assumptions) enable us to have "describe data effectively". Without these (strong) assumptions, it is impossible to compute anything.

Let's use Markov Chain as an example. Suppose we want to model a length $100$ sequence, $(X_1, X_2, \cdots, X_{100})$, and you have say $5000$ sequences. And let's assume all symbols in the sequence is binary.

Without the Markov assumption, how can we describe the joint distribution $P(X_1, X_2, \cdots, X_{100})$? If you think about using a table, that is $2^{100}$ rows, and $2^{100}-1$ free parameters (needs to sum to $1.0$). There is no way we can do that with computer from computational perspective. On the other hand, to fit those number of parameters, we need much much more data. So, $5000$ sequences seems to be nothing...

With the Markov assumption,

$$P(X_1, X_2, \cdots, X_{100})=P(X_1)\prod_{n=2}^{100}P(X_n|X_{n-1})$$

We only have very few parameters:

  • Initial distribution: 1 free parameter
  • Transition matrix 2 free parameters

Such assumption (constrain) enables us to have a joint in a traceable way.


To your edit, even all the real world data does not satisfy Markov property in a reasonable degree, we still need reasonable strong assumptions. Because it is the assumption make the modeling work doable.

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