gmdbn: Create a Gaussian mixture dynamic Bayesian network

Description

This function creates a Gaussian mixture dynamic Bayesian network as an object of S3 class gmdbn. Assuming that the system evolves over time (possibly non-stationary) and denoting by $X^{(t)}$ its state at time slice $t$, a dynamic Bayesian network is a probabilistic graphical model that encodes the joint distribution over any finite time sequence: $$p(X^{(1)}, \dots , X^{(T)}) = p(X^{(1)}) \prod_{t = 2}^T p(X^{(t)} | X^{(t - 1)}, \dots , X^{(1)})$$ It is defined by a sequence of transition models $\mathcal{B}_1, \mathcal{B}_2, \dots , \mathcal{B}_N$ associated with transition time slices $t_1 = 1 < t_2 < \dots < t_N$, where:

$\mathcal{B}_1$ is a Bayesian network that encodes the distribution $p(X^{(t)})$ for $1 \le t \le t_2 - 1$, assuming that the states at these time slices do not depend on previous states;
for each $i \ge 2$, $\mathcal{B}_i$ is a ($k_i + 1$)-slice temporal Bayesian network (where $k_i < t_i$) that encodes the transition distribution $p(X^{(t)} | X^{(t - 1)}, \dots , X^{(t - k_i)})$ for $t_i \le t \le t_{i + 1} - 1$ (or $t \ge t_i$ if $i = N$), assuming that the states at these time slices only depend on the $k_i$ previous states (Hourbracq et al., 2017).

In a Gaussian mixture dynamic Bayesian network, these transition models are Gaussian mixture Bayesian networks (Roos et al., 2017).

Usage

gmdbn(...)

Value

A list of class gmdbn containing the gmbn objects passed as arguments.

Arguments

...: Objects of class gmbn corresponding to the transition models. Each gmbn object must be named with the prefix b_ followed by its associated transition time slice (e.g. a transition model whose transition time slice is 8 is represented by the gmbn object b_8). If the first gmbn object (chronologically) is associated with a transition time slice $t \ge 2$ (i.e. b_1 is not specified), it is duplicated to create transition models associated with $1, \dots , t - 1$ (removing the arcs whose time lags exceed the maximum temporal depths of these models).

References

Hourbracq, M., Wuillemin, P.-H., Gonzales, C. and Baumard, P. (2017). Learning and Selection of Dynamic Bayesian Networks for Non-Stationary Processes in Real Time. In Proceedings of the 30th International Flairs Conference, pages 742--747, Marco Island, FL, USA.

Roos, J., Bonnevay, S. and Gavin, G. (2017). Dynamic Bayesian Networks with Gaussian Mixture Models for Short-Term Passenger Flow Forecasting. In Proceedings of the 12th International Conference on Intelligent Systems and Knowledge Engineering, Nanjing, China.

Examples

Run this code

library(dplyr)
data(data_air)
data <- data_air %>%
  group_by(DATE) %>%
  mutate(NO2.1 = lag(NO2), O3.1 = lag(O3), TEMP.1 = lag(TEMP),
         WIND.1 = lag(WIND)) %>%
  ungroup()
gmdbn_1 <- gmdbn(
  b_2 = gmbn(
    NO2 = split_comp(add_var(NULL, data[, c("NO2", "NO2.1", "WIND")]),
                     n_sub = 3),
    O3 = split_comp(add_var(NULL,
                            data[, c("O3", "NO2", "NO2.1", "O3.1", "TEMP",
                                     "TEMP.1")]),
                    n_sub = 3),
    TEMP = split_comp(add_var(NULL, data[, c("TEMP", "TEMP.1")]), n_sub = 3),
    WIND = split_comp(add_var(NULL, data[, c("WIND", "WIND.1")]), n_sub = 3)
  ),
  b_13 = gmbn(
    NO2 = split_comp(add_var(NULL, data[, c("NO2", "NO2.1", "WIND")]),
                     n_sub = 3),
    O3 = split_comp(add_var(NULL,
                            data[, c("O3", "O3.1", "TEMP", "TEMP.1", "WIND")]),
                    n_sub = 3),
    TEMP = split_comp(add_var(NULL, data[, c("TEMP", "TEMP.1")]), n_sub = 3),
    WIND = split_comp(add_var(NULL, data[, c("WIND", "WIND.1")]), n_sub = 3)
  )
)