MACHINE LEARNING TECHNIQUES - LASA

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Figure 7-1: Schematic illustrating the concepts of a HMM. The process is assumed to be composed of 7
hidden states. All transitions across all states are possible (fully connected). Note that for simplicity, we
show only arrows across adjacent states. The schematic at the bottom shows a particular instance of
transition across states over five time steps. Transitiosn across each state leads to an observation of a
particular set of values for the system’s variables. As we see, the system rotates first across states 1 and 2
and then stays for two time steps on state 3.
Transitions across states are described as a stochastic finite automata process. The stochasticity
of the process is represented by computing a set of transition probabilities that determine the
likelihood to stay in a state or to jump to another state. Transition probabilities are encapsulated
in a N N
represent the probability of transiting from
× matrix A , whose elements { } ij
i , j=
1...
state i to state j , i.e. aij p( sj | si
)
a
N
= . The sum of all elements in each row of A equals 1. Each
state is associated an initial probability π
i, i= 1,.. Nthat represents the likelihood to be in that
state at any given point in time. In addition, one assigns to each state i a density ( ) i
b o , socalled
the emission probability that determines the probability of the observation to take a
particular value when in state S . Depending on whether the observables take discrete versus
i
continuous values, we talk about a discrete versus continuous HMM. When continuous, the
density may be estimated through a Gaussian Mixture Model. In this case, one associates a
GMM per state. HMM are used widely in speech processing. There, often, one uses very few
hidden states (at maximum 3!). The complexity of the speech is then embedded in the GMM
density modeling associated at each state.
© A.G.Billard 2004 – Last Update March 2011