[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

238 H. Josiński et al.
2.3 Hidden Markov Model
A Hidden Markov Model (HMM) is a statistical tool used for modeling generative
sequences characterized by a set of observable sequences [16].
An HMM of a system is defined as a triple λ = (π, A, B), where A is the state
transition probability matrix, B is the observation probability matrix and π is the
initial state distribution [2]. The system has N states (s 1 , s 2 , …, s N ), M observation
symbols (v 1 , v 2 , …, v M ), q t denotes the state of the system at time t and O t
represents the observation symbol of the system at time t.
The initial state distribution is described by π = {π i }, where π i = P(q 1 = s i ), 1 ≤ i
≤ N. The state transition probability matrix is defined as A = {a ij }, where a ij =
P(q t+1 = s j | q t = s i ), 1 ≤ i, j ≤ N. The observation matrix is specified as B = {b j (k)},
where b j (k) = P(O t = v k | q t = s j ), 1 ≤ j ≤ N, 1 ≤ k ≤ M.
Modeling human gait using HMM essentially involves identifying the N
characteristic stances (poses) of an individual, which correspond to the N states of
the system, and modeling the dynamics between the N such stances [2].
The “parallel-oriented” sequences from the dataset A with reduced
dimensionality were divided evenly into training set and test set according to the
schema presented in Fig. 2. During the training phase separate model λ j was
constructed for every actor j, 1 ≤ j ≤ 20 for the given number N of states.
Subsequently, in the testing phase the probability P(O | λ j ), that the considered
sequence O from the test set was produced by the individual j, was computed for
all estimated models λ j . Finally, the unknown actor X in the sequence O was
determined by the largest value of the probability P(O | λ j ):
X = arg max log( P( O | λ )) .
j
j
(5)
Fig. 2. Phases of the HMM-based classification