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I am just getting started with Hidden Markov Models. In selecting my observable variables, there are some where I believe the recent change in the variable is potentially more predictive than its level. For example, in finance, the level of of an interest rate may not be as important as how much it has recently changed.

Given that HMM presumes that only the present state matters, am I violating a best practice if I used the delta of variables in this manner? Are there potential pitfalls of which I should be aware?

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Your question is whether you violate best practice if you use the delta in your HMM, and if there's any pitfalls.

I think to answer the first, and prevent the second, you could do what is often done in the case where people want to use a HMM whilst using information from more than one previous state: rewrite the latent chain, such that it becomes a 'standard' Markov chain again. This will probably require some wrangling, but prevents violating the Markov property and any potential pitfalls you're worried about.

Say your original latent chain is $X_t$. You can rewrite this to $Y_t=(X_t, X_{t-1})$.
This $Y_t$ then meets the fulfills the Limited Horizon assumption in the sense that it only depends on $Y_{t-1}$.

Depending on your specific use-case, make sure to also adjust the definition of the original state $Y_0$

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  • $\begingroup$ Excellent. So to see that I understand, if Sales is an observable, and I include say Sales - SalesT-10, if I just make the later its own observable, I haven't violated the Markov property? $\endgroup$
    – Tom
    Commented Jan 6, 2022 at 11:29
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    $\begingroup$ That is correct $\endgroup$
    – Tim J
    Commented Jan 6, 2022 at 12:38
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In machine learning, features need to have variance in order for the model to have the ability to learn to use them. If there is no variance in a feature then the feature can not contribute to making a prediction between target values. Hidden Markov Models (HMM) are a type of machine learning so feature variance is required.

Most machine learning models have the ability to learn which features are important for the task. Thus, providing the model with raw features tends to work. The raw observed variables should be provided to the HMM model and the model will learn to use them.

The question could also be answered through cross-validation. Provide the model with both the raw features and the delta versions and see which version has the most predictive ability.

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