Consider events represented by spatio-temporal-textual data, a tuple consists of time, location, and text. And we model the sequence of spatio-temporal-textual events using a multivariate Hawkes point process, called spatio-temporal-textual point process. Spatio-temporal-textual point process is essentially a marked multivariate Hawkes process, where each component is discretized location, and text is mark. By using an adapted kernel function, as well as text embedding techniques, our proposed spatio-temporal-textual point process is able to incorporate the text similarity as part of the influence between events. The intensity function of the point process is shown below.

Equiped with the conditional intensity, we explicitly denote the dependence of the likelihood function on the spatio-temporal coefficients in the presence of spatio-temporal-textual data. The log-likelihood function is shown as below.

We then construct the linkage between events by introducing auxiliary variables that indicates the probability i-th event is linked to j-th event. Moreover, an EM algorithm for learning the parameters is presented.

Below is an simple example for initialization and fitting of the model.

# init MPPEM object
# - t: a sequence of time
# - u: a sequence of discretized locations (indices)
# - l: a sequence of labels (optional)
# - l: a sequence of marks
# - d: dimension of components of the point process (number of discretized locations)
mppem = MPPEM(seq_t=t, seq_u=u, seq_l=l, seq_m=m, d=len(u_set))
# init A
distance_matrix = utils.calculate_beats_pairwise_distance(u_set, csv_filename)
mppem.init_A(distance_matrix, gamma=gamma)
# init Mu
mppem.init_Mu(gamma=gamma)
# fit model
# - ps:  a list of precisions over iterations
# - rs:  a list of recalls over iterations
# - lls: a list of loglikelihoods over iterations
# - lbs: a list of lower bounds over iterations
# - t, T:  start and end of the time window for fitting
# - iters: number of iterations
ps, rs, lls, lbs = em.fit(T=t[-1], tau=t[0], iters=iters)
# results
print(em.P)
print(em.A)

• S. Zhu and Y. Xie. "Crime Linkage Detection by Spatial-Temporal Text Point Processes"