# Create a Gaussian mixture dynamic Bayesian network 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: [REMOVE_ME]$$ p(X^{(1)}, \dots , X^{(T)}) = p(X^{(1)})\prod_{t = 2}^T p(X^{(t)} | X^{(t - 1)}, \dots , X^{(1)}) [REMOVE_ME_2]$$ 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). ```r gmdbn(...) ``` ## 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). ## Returns A list of class `gmdbn` containing the `gmbn` objects passed as arguments. ## 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). ## Examples ```r 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) ) ) ``` ## 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. ## See Also `gmbn`, `gmm`

gmdbn function