Layered hidden Markov model: Difference between revisions

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{{Short description|Multilevel, non-directly observable 'probability engine'}}

The layered [[hidden Markov model]] (LHMM) is a [[statistical model]] derived from the hidden Markov model (HMM).

AThe '''layered hidden Markov model''' ('''LHMM''') consistsis ofa <math>N</math>[[statistical levelsmodel]] ofderived HMMs wherefrom the HMMs[[hidden onMarkov levelmodel]] <math>N+1</math>(HMM). corresponds to observation symbols or probability generators at level <math>N</math>.

A layered hidden Markov model consists of ''N'' levels of HMMs, where the HMMs on level ''i''&nbsp;+&nbsp;1 correspond to observation symbols or probability generators at level ''i''.

Every level <math>i</math> of the LHMM consists of <math>K_i</math> HMMs running in parallel.

Every level ''i'' of the LHMM consists of ''K''_''i'' HMMs running in parallel.<ref>N. Oliver, A. Garg and E. Horvitz, "Layered Representations for Learning and Inferring Office Activity from Multiple Sensory Channels", Computer Vision and Image Understanding, vol. 96, p. 163-&ndash;180, 2004.

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== Background ==

ItLHMMs isare sometimes useful to use HMMs in specific structures inbecause orderthey tocan facilitate learning and generalization. For example, even though a fully connected HMM could always be used if enough [[Training, validation, and test data sets|training data]] iswere available, it is often useful to constrain the model by not allowing arbitrary state transitions. In the same way it can be beneficial to embed the HMM intoin a greaterlayered structure; which, theoretically, may not be able to solve any other problems than the basic HMM cannot, but can solve some problems more efficiently whenbecause itless comestraining todata theis amount of required training dataneeded.

== The Layeredlayered Hiddenhidden Markov Modelmodel ==

A layered hidden Markov model (LHMM) consists of <math>N</math> levels of HMMs where the HMMs on level <math>N+1</math> corresponds to observation symbols or probability generators at level <math>N</math>.

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At any given level <math>L</math> in the LHMM a sequence of <math>T_L</math> observation symbols

<math>\mathbf{o}_L=\{o_1, o_2, ...\dots, o_{T_L}\}</math> can be used to classify the input into one of <math>K_L</math> classes, where each class corresponds to each of the <math>K_L</math> HMMs at level <math>L</math>. This classification can then be used to generate a new observation for the level <math>L-1</math> HMMs. At the lowest layer, i.e. level <math>N</math>, primitive observation symbols <math>\mathbf{o}_p=\{o_1, o_2, ...\dots, o_{T_p}\}</math> would be generated directly from observations of the modeled process. For example, in a trajectory tracking task the primitive observation symbols would originate from the quantized sensor values. Thus at each layer in the LHMM the observations originate from the classification of the underlying layer, except for the lowest layer where the observation symbols originate from measurements of the observed process.

It should be noted here that it is not necessary to run all levels at the same time granularity. For example, it is possible to use windowing at any level in the structure so that the classification takes the average of several classifications into consideration before passing the results up the layers of the LHMM.<ref>D. Aarno and D. Kragic "Evaluation of Layered HMM for Motion Intention Recognition", IEEE International Conference on Advanced Robotics, 2007</ref>

Instead of simply using the winning HMM at level <math>L+1</math> as an input symbol for the HMM at level <math>L</math> it is possible to use it as a [[probability generator]] by passing the complete [[probability distribution]] up the layers of the LHMM. Thus instead of having a "winner takes all" strategy where the most probable HMM is selected as an observation symbol, the likelihood <math>L(i)</math> of observing the <math>i</math>th HMM can be used in the recursion formula of the level <math>L</math> HMM to account for the uncertainty in the classification of the HMMs at level <math>L+1</math>. Thus, if the classification of the HMMs at level <math>n+1</math> is uncertain, it is possible to pay more attention to the a-priori information encoded in the HMM at level <math>L</math>.

It should be noted here that aA LHMM could in practice be transformed into a single layered HMM where all the different models are concatenated together.<ref>D. Aarno and D. Kragic: "Layered HMM for Motion Intention Recognition", IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006.</ref> Some of the advantages that may be expected from using the LHMM over a large single layer HMM is that the LHMM is less likely to suffer from over-fitting[[overfitting]] since the individual sub-components are trained independently on smaller amounts of data. A consequence of this is that a significantly smaller amount of training data is required for the LHMM to achieve a performance comparable of the HMM. Another advantage is that the layers at the bottom of the LHMM, which are more sensitive to changes in the environment such as the type of sensors, sampling rate etc,. can be retrained separately without altering the higher layers of the LHMM.

== See Also ==

== See Also also==

*[[Hierarchical hidden Markov model]]

== References ==

{{Reflist}}

[[Category:Hidden Markov models]]

N. Oliver, A. Garg and E. Horvitz, "Layered Representations for Learning and Inferring Office Activity from Multiple Sensory Channels", Computer Vision and Image Understanding, vol. 96, p. 163-180, 2004.