Hidden Markov Models: Motivation
Biotelemetry enables humans to remotely monitor various energetic status or track both animals and humans. It is the preferred tool to study the movement of species and understand how animals migrate (Nigel Hussey (2015), K Whoriskey et al. (2019)).
|
Hidden Markov Models: Framework
In the hidden markov models' framework, the observation is related to a discrete underlying real-world state that changes over time. This latent process is a Markov chain. At each time-step, the observation gives information about the hidden state with some level of confidence. The idea of a hidden Markov model is to use this observation to infer the state at every time-step. For example, Hmms have been used with animals tracks to extract information about the underlying behaviour states. In this context, Dr. Ron Togunov was able to identify three key behaviors in polar bears: a passive drift state, an olfactory search state, and an area restricted search state. You can see his publications here.
Example of a Markov chain
A sequence of discrete random variable \(C_t\) is a discrete-time Markov chain if, for all:
1) \( \mathbb{P}(C_{t+1}|C_{1},\ldots,C_{t})=\mathbb{P}(C_{t+1}|C_{t}), \)
2) Conditioning on the history of the process up to time is equivalent to conditioning only on the most recent value of \(C_t\).
\[\Gamma= \begin{bmatrix}1/4 & 3/4 \\ 1/5 & 4/5 \end{bmatrix}. \]
This state dependent distribution represents the state-specific probability of observation \(x\).
Hidden Markov Models: Key Parameters
The key parameters of a hidden Makov model are:
1) The number of states: m,
2) State probability: \(\mathbb{P}(C_t|C_{t-1})\),
3) The state-dependent distribution: \(\mathbb{P}(x_t=x|C_{t}=i)=p_{i}(x)\),
4) \(u_t=(\mathbb{P}(C_t=1),\ldots,\mathbb{P}(C_t=m)) \),
5) For \(P(x)\) the matrix with zeros and diagonal term of index \(i\), \((P(x))_{ii}=p_{i}(x)\):
\(u_tP(x)1^{\top}=\mathbb{P}(x_t = x)\).