Hidden Markov Models - Weight observations

Question

I am interested in using Hidden Markov Models in the context of multi-sensors sequences in vegetation remote sensing.

The observation labels are the same as the state labels.

Each observation at a given time $t$ is classified and I want to use HMM in order to reduce classification errors.

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For example, the states could be : FOREST, CROP, SOIL, WATER.

I have an associated transition matrix between those states.

Thanks to a sensor, I can observe a sequence of observations from which I want to infer the most likely sequence of hidden states.

Sequence of observations : FFFWFFFF

--> Sequence of states : FFFFFFFF

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Now, each observation comes from a given sensor (or maybe ground data) and I am able to assign a given weight or confidence in this observation.

As a limiting case, let's imagine I'm 100% positive that the fourth observation is water because I was on the ground, and that the transitions from/to water are null.

I would like the algorithm to output :

--> Sequence of states : WWWWWWWW

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I was thinking about using the following states:

• FOREST_SENSOR1, FOREST_SENSOR2,
• etc.

and computing a new transition matrix but this does not seem right.

Thank you for your help!

Answer 1

I think you can model your problem with the original states, where each state has multiple emissions (the different sensors), and there are missing emissions (where a sensor hasn't measured a particular state). Maybe you can find a software library that allows for both circumstances, or you might be able to implement this with a flexible probabilistic programming language such as rstan. If you already have the transition probabilities and measurement probabilities, your problem is only decoding, the Viterbi algorithm. A recent book on text processing has a good chapter on the Hidden Markov Model, you can find it here: https://web.stanford.edu/~jurafsky/slp3/9.pdf.