Computational Intelligence - November 2017 - 34

3) Minimum Precision of Generated Predictions (MPGP)
In the field of predictive analytics, it is common to measure not
only the accuracy, but also the precision of a model in order to
validate whether all the possible predictable values are fairly
predicted. In the context of this work, using a DCMM with
high precision could lead to detect in time when the operator
is going to perform a "rare" interaction.
For this reason, the MPGP measure is created to evaluate
which of the possible K outputs of a DCMM is worst predicted in a test observation sequence. Predictions are generated
using the same method of the AGP measure, but here, instead
of scoring the expectation vector " t kt + 1 ,Kk=1, we just extract
the most expected observation from it and check whether it
matches the actual observation Yt + 1 . Counting these matchings
for every t ! 1, 2, f, T leads to a K × K confusion matrix
CM, where the precision for each possible observation symbol
is computed as:
Precision k = CM kk k ! " 1, f, K ,,
/ CM kj

transition matrix. A n is an M l # M l matrix, in which each row
represents the probability of moving to a new sequence of
states from a sequence of l previous states. Let ali (n) = max j A ijn
the maximum probability of each row of the transition matrix,
then the Coefficient of High Probability Hidden Transitions
(CHPHT) can be defined as:
Ml

1
CHPHT (n) = 1 l /
.
M i = 1 1 + e -l^ali (n) - TRT h

(9)

As it can be seen, we have used a logistic function to score the
values of ali (n) = max j A ijn , in order to favour more remarkably
the highest probabilities and penalize the lowest. The Transition
Relevancy Threshold (TRT) marks the sigmoid's midpoint of
the function and the order of the hidden chain, l, denotes the
steepness of the curve, so that the same value of ali (n) will be
scored better in a higher order model. This is because high probability transitions in higher order DCMMs usually offer more
informative and longer patterns.

(7)
V. Experimentation

j

assuming that rows in CM denote predictions and columns
denote actual values. Finally, the MPGP measure is computed
as the minimum value of Eq. (7) in the set " 1, f, K , .

In this section, the modelling strategy proposed in this work is
applied to model the interactions extracted from a multi-UAV
simulation environment. Below are described the simulation environment used, the experimental setup and the
results obtained.

B. Interpretability Measures

This family of evaluation measures intends to rate, automatically, different aspects around the interpretability of a DCMM,
once it has been fit to a specific training dataset.
1) Bayesian Information Criterion (BIC)
The Bayesian Information Criterion (BIC) has become very
popular in the field of statistical modelling, specially for comparing Markovian models [26]. It penalizes the likelihood of a
model by a complexity factor proportional to number of
parameters in the model and the number of training observations,
so it gives advantages to simple and general models and avoids
overfitting. It is defined as:
BIC = - 2 log L + p log x,

(8)

where L is the likelihood of the model, p is the number of
independent parameters and x is the number of components
in the likelihood, i.e., the number of observations used to train
the model. The less the BIC scores, the better the model is
considered. This measure makes a balance between the predictability and the interpretability of a DCMM, and thus, we
will include it in both perspectives for the model selection (See
Table 1).
2) Coefficient of High Probability Hidden
Transitions (CHPHT)
This measure favours those models in which every hidden state
(or sequence of hidden states for higher order chains) has a
clear leading transition. Let n (M, l, f ) be a DCMM and A n its

34

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2017

A. Drone Watch & Rescue

The simulation environment used as test-bed for this work is
called Drone Watch & Rescue (DWR), and its complete
description can be found in [10]. DWR gamifies the concept of
a multi-UAV mission, challenging the operator to capture all
mission targets consuming the minimum amount of resources,
while avoiding at the same time the possible incidents that may
occur during a mission (e.g., danger areas, sensor breakdowns).
To avoid these incidents, an operator in DWR can perform
multiple interactions to modify both the UAVs in the mission
and the waypoints composing their mission plan. These interactions are the following: 1) Change Control Mode (CCM);
2) Change UAV Path (CUP); 3) Modify Waypoints Table
(MWT); 4) Select UAV (SU); 5) Change UAV Speed (CUS);
and 6) Change Simulation Speed (CSS).
B. Experimental Setup

In this experiment, we will apply the process flow shown in Figure 2 to the data extracted from DWR, from the point of view
of an instructor who wants to obtain a behavioural model. Due
to the lack of space, we will not qualitatively analyse the best
DCMMs found, so we will focus in studying how changing the
importance balance between the predictability and interpretability affects to the hyperparameters of the best DCMM found.
Furthermore, we will check if the capabilities of DCMMs over
the currently used HMMs are worthwhile when it comes to
find better models.
The choice of the parameters involved in the process flow
described in the previous section is very important for the



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