Los valores inclusivos.

This method is a hybrid that combines the recency and establishment effects. It dynamically estimates the weights of each of these effects for each meeting and corrects itself over time. We refer to this method as Kalman combination of Re-

cency and Establishment (K2RE). It utilizes the Kalman filter[73] to estimate the Recency and Establishment weights over time. By measuring the meeting behav- ior of a group in terms of establishment and recency, K2RE adapts itself to data of each group. In the following, we first present a general overview of K2RE and then elaborate on its details. Figure 7.1 illustrates the components of the K2RE method. This method integrates scores from the recency and the establishment effects, described in Sections 7.2.2 and 7.2.3, using a linear interpolation.

The linear interpolation for a meeting at time slice t is defined as:

K2RE_t = w_E,t∗ Scoreest a bl ishment+ wR,t∗ Scorer ecenc y (7.5)

where Scor eest a bl ishment and Scor er ecenc y are computed by the establishment and

the recency effects, respectively. Furthermore, w_E,t and w_R,t are establishment weights and recency weights computed by the Kalman filter at time t, such that:

121 7.2 Methods

Figure 7.1. K2RE Method’s Architecture

w_E,t+ wR,t= 1 (7.6)

This means that at each time slice t each of the two effects will be given a weight, either equally or if one effect is assigned a higher weight, the other will receive a lower weight.

The Kalman filter is always initialized by assigning equal probability of 0.5 to both w_E,t and w_R,t.

f(n) =

¨

XG= AGf_t−1+ "_tG t = 2, . . . , T z_t = HGf_t+ w_tG t = 1, . . . , T

where f_t is the system state at time t, AG _{denotes the transition of the dynamic}

system from t − 1 to t, HG describes how to map state ft to to an observation

(i.e., measurement) zt and both"tGand wtGare mutually independent Gaussian

noise variables with co-variances R_tand Q_t respectively. The superscript G in the system of equations explains that we compute these equations per each group. The dynamic system, then evolves over time and updates itself proportional to the Kalman gain.

Now that we explained the Kalman filter module and how it computes the weights for recency and establishment effects, we explain the topic linking mod- ule and describe how the measurement process explained in the Kalman filter

122 7.2 Methods

equations works.

In Chapter 4, we introduced a topic model for tracking the evolution of in- termittent topics over time[23]. In other words, this model tracks the evolution of topics that may occur discretely over time, such that a topic does not need to be necessarily present over all time slices. In the topic linking module shown in Figure 7.1 we use a component of the mentioned topic model. This component links together similar topics over time. As explained, the linking of a topic may be discrete or continuous under this model (i.e., a topic may be present over all time slices or it may skip some time slices). We briefly explain the model in the following. The general idea is to form a Gaussian random walk in a Markovian state space model. The Markov assumption enforces probabilities of a hidden state at time t to be computed merely dependent on the previous time slice and not all the previous states. We utilized this assumption to compute topic chains that capture the evolution of a topic discretely over time. If two topics over two different time slices are similar according to the following criterion, they will be linked.

βt,k|βt−m,1..k∼ N (βt−1,σ2I) (7.7)

where β_t,k is topic k at time slice t, m ∈ (1, .., n) with n being the number of

previous time slices (meetings) andσ is the maximum variance allowed from the mean of a topic in the previous time slice. By assigning a very small value to σ, the model links two topics that are highly similar. Furthermore, the Baum-Welch [28] algorithm learns the forward and backward probabilities of the transitions among the topics. We used this model as a component of K2RE for linking the topics that are similar across different meetings. The model takes as input the topics from the first n meetings whose continuation in the(n + 1)thmeeting is to

be computed.

After linking similar topics over every two consecutive meetings, the topic linking module computes the recency rate of the topics for meeting n by comput- ing the number of topics from the meeting n− 1 that have been present in the meeting n divided by the total number of topics in the same meeting. This mea- surement is given as the observation matrix to the Kalman filter for each meeting which we explained above. Subsequently, the Kalman filter computes the evo- lution of recency and establishment weights using the Kalman filter equations presented above. Furthermore, using Equation 7.6, similarly to the recency and establishment methods, a K2RE reference vector is generated.

Moreover, analogously to the other three previous methods, all steps for com- puting word vectors and computation of Pearson correlation as an energy func-

123 7.3 Evaluation

tion are performed.