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The question is about the equivalence between ARIMA models and hidden Markov models in the context of time series analysis/prediction. Specifically:

  1. Can any ARIMA(p,d,q) model bet represented by an equivalent HMM?
  2. Can any HMM be represented as ARIMA (or are HMMs a bigger class of models).

My impression is that the answers are 1. yes, and 2. no. However I am looking for a definitive answer. A reference to a reliable sources would be greatly appreciated as well.

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In the standard HMMs, the state spaces of hidden variables are discrete. On the other hand ARIMA models can be represented as a Kalman Filter which is a continuous state space model (underlying hidden variables are continuous). This is the main difference. However there should be HMM extensions to manage continuous state spaces.

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  • $\begingroup$ Thanks! It is not in the original question, but I am also wondering if or to what extent ARIMA and HMMs are made obsolete by LSTM neural networks? $\endgroup$
    – Roger V.
    Commented Aug 15, 2020 at 9:26
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    $\begingroup$ @Vadim, neural networks typically require longer series for training, while ARIMA can be fit quite effectively on just a handful of data points. This is not unique to time series; flexible machine learning methods in general have not made simple statistical models obsolete for the same reason. $\endgroup$
    – Richard Hardy
    Commented Aug 18, 2020 at 15:38
  • $\begingroup$ @RichardHardy this is a good point. But assuming that we have a sufficiently long sequence? I have encountered claims that there are situations where the models described by HMMs are poorly modeled by LSTMs. $\endgroup$
    – Roger V.
    Commented Aug 19, 2020 at 12:33
  • $\begingroup$ @Vadim, I do not know LSTMs well enough, but assuming they are flexible enough to approximate any HMM to a high degree and given a sufficient large training set, I do not see why they would be inferior. But perhaps the assumption I just made does not hold? $\endgroup$
    – Richard Hardy
    Commented Aug 19, 2020 at 12:36

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