Answers.Body: <p>Yes, you can fit any time series (with or without external variables) using HMM, but there are some constraints:</p>
  
  <ol>
  <li>It should follow the Markov property.  </li>
  <li>There is some variance that other models are not able to capture (in other words, the system is partially observable).</li>
  </ol>
  
  <p>Adding to point 1, for HMM, it should hold true, but the way <a href="https://en.wikipedia.org/wiki/Baum%E2%80%93Welch_algorithm" rel="nofollow noreferrer">Baum Welch algorithm</a> works, indirectly it considers the values of more than the previous state for HMM (order-1). The state <span class="math-container">$t-1$</span>  depends on <span class="math-container">$t-2$</span>, which in turn depends on <span class="math-container">$t-3$</span>. The calculation of parameters (transition, emission, starting probabilities) happens over multiple iterations and it finds parameters in such a way that holds Markov property true.  </p>
  
  <p>I think that, when they say 'any', they mean even when you don't have all variables needed to forecast future values. </p>
  
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Answers.Body: <p>HMMs can be used to model sequential data, which is composed of discrete tokens, and it should generally follow the Markov property, which is the assumption that the probability of a class/label given observation depends only on the preceding class/label (rather than on some longer sequence).</p>
  
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Answers.Text: I'm sure this is not a valid answer.
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Answers.Text: I think the intention of asking this question is get a conceptual definition of what kind of time series can be modeled.
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Answers.Text: But time series is not Markov. Your answer doesn't answer the question wherther HMM can be used for non-Markov process.
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Answers.Text: A process with the Markov property is a Markov process. The type of time series itself does not define whether it can be modeled with a Markov process. That is a hypothesis/assumption that is leveraged.
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Answers.Text: @drwiner is right. My intention is to understand the topic at a conceptual level. So the statement I quoted is wrong?
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Answers.Text: @vinjk I think it's misleading, but also insufficient as described to be completely wrong. If you have a continuous time series with interval-based labels, the discrete tokens could be calculated from intervals, say by chunking the time series into bins and mapping to tokens based on some function whose input is the time series in that bin.  Consider human speech, where the audio can be mapped to phonemes and modeled with an HMM.
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Body: <p>Can HMMs be used to model <strong>any</strong> time series data? Or does the data have to be that of a Markov process?</p>
  
  <p>In <a href="http://www.ee.columbia.edu/~dpwe/LabROSA/doc/HTKBook21/node3.html" rel="nofollow noreferrer">HTK documentation</a>, I see that the first few lines state that it can model any time series </p>
  
  <blockquote>
    <p>HTK is a toolkit for building Hidden Markov Models (HMMs). HMMs can be used to model any time series and the core of HTK is similarly general-purpose. </p>
  </blockquote>
  
Tags: <machine-learning><applications><time-series><hidden-markov-model>
Title: Can hidden Markov models be used to model any time series data?
UserDisplayName: vinjk
ThreadText: # Can hidden Markov models be used to model any time series data?
  
  (Asked by: username_0 on 2017-08-03)
  
  Can HMMs be used to model **any** time series data? Or does the data have to be that of a Markov process?
  
  In [HTK documentation](http://www.ee.columbia.edu/~dpwe/LabROSA/doc/HTKBook21/node3.html), I see that the first few lines state that it can model any time series
  
  > HTK is a toolkit for building Hidden Markov Models (HMMs). HMMs can be used to model any time series and the core of HTK is similarly general-purpose.
  
  **Tags:** machine-learning, applications, time-series, hidden-markov-model
  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_1 on 2017-08-15. Score: 1 | Accepted answer
  
  Yes, you can fit any time series (with or without external variables) using HMM, but there are some constraints:
  
  1. It should follow the Markov property.
  2. There is some variance that other models are not able to capture (in other words, the system is partially observable).
  
  Adding to point 1, for HMM, it should hold true, but the way [Baum Welch algorithm](https://en.wikipedia.org/wiki/Baum%E2%80%93Welch_algorithm) works, indirectly it considers the values of more than the previous state for HMM (order-1). The state $t-1$ depends on $t-2$, which in turn depends on $t-3$. The calculation of parameters (transition, emission, starting probabilities) happens over multiple iterations and it finds parameters in such a way that holds Markov property true.
  
  I think that, when they say 'any', they mean even when you don't have all variables needed to forecast future values.
  
  
  ---
  
  ### Answer by username_2 on 2017-08-05. Score: 0
  
  HMMs can be used to model sequential data, which is composed of discrete tokens, and it should generally follow the Markov property, which is the assumption that the probability of a class/label given observation depends only on the preceding class/label (rather than on some longer sequence).
  
  > **Comments:**
  > **username_3**: I'm sure this is not a valid answer.  
  > **username_2**: I think the intention of asking this question is get a conceptual definition of what kind of time series can be modeled.  
  > **username_3**: But time series is not Markov. Your answer doesn't answer the question wherther HMM can be used for non-Markov process.  
  > **username_2**: A process with the Markov property is a Markov process. The type of time series itself does not define whether it can be modeled with a Markov process. That is a hypothesis/assumption that is leveraged.  
  > **username_0**: @drwiner is right. My intention is to understand the topic at a conceptual level. So the statement I quoted is wrong?  
  > **username_2**: @username_0 I think it's misleading, but also insufficient as described to be completely wrong. If you have a continuous time series with interval-based labels, the discrete tokens could be calculated from intervals, say by chunking the time series into bins and mapping to tokens based on some function whose input is the time series in that bin.  Consider human speech, where the audio can be mapped to phonemes and modeled with an HMM.  
  
  ---
  
  