Gaussian mixture graphical model learning and inference

Description

This package provides a complete framework to deal with Gaussian mixture graphical models, which covers Bayesian networks and dynamic Bayesian networks (their temporal extension) whose local probability distributions are described by Gaussian mixture models. It includes a wide range of functions for:

• creating or modifying the structure of Gaussian mixture models (add_var, gmm, merge_comp, remove_var, rename_var, reorder, split_comp) or graphical models (add_arcs, add_nodes, gmbn, gmdbn, relevant, remove_arcs, remove_nodes, rename_nodes);

• describing or visualizing Gaussian mixture models (conditional, ellipses, summary.gmm) or graphical models (network, structure, summary.gmbn, summary.gmdbn);

• computing densities, expectations, or sampling Gaussian mixture models (density, expectation, sampling);

• computing scores of Gaussian mixture models (AIC.gmm, BIC.gmm, logLik.gmm) or graphical models (AIC.gmbn, AIC.gmdbn, BIC.gmbn, BIC.gmdbn, logLik.gmbn, logLik.gmdbn);

• learning the structure and/or the parameters of Gaussian mixture models (em, smem, stepwise) or graphical models (param_em, param_learn, struct_em, struct_learn);

• performing inference in Gaussian mixture graphical models (aggregation, filtering, inference, particles, prediction, propagation, smoothing).

Descriptions of these functions are provided in this manual with related references. Most of the algorithms are described in the PhD thesis of Roos (2018, in french). To better handle this package, two real-world datasets are provided (data_air, data_body) with examples of Gaussian mixture models and graphical models (gmbn_body, gmdbn_air, gmm_body).

References

Roos, J. (2018). Prévision a court terme des flux de voyageurs : une approche par les réseaux bayésiens. PhD thesis, University of Lyon.

Compute the Akaike Information Criterion (AIC) of a Gaussian mixture model or graphical model

Description

This function computes the Akaike Information Criterion (AIC) of a Gaussian mixture model or graphical model:

AIC = logLik - n_{par}

where logLik is the log-likelihood and n_{par} the number of free parameters.

Usage

## S3 method for class 'gmm'
AIC(object, data, y = NULL, regul = 0.01, ...)

## S3 method for class 'gmbn'
AIC(object, data, col_seq = NULL, ...)

## S3 method for class 'gmdbn'
AIC(object, data, col_seq = NULL, ...)

Arguments

Value

If object is a gmm object, a numeric value corresponding to the AIC.

If object is a gmbn or gmdbn object, a list with elements:

See Also

BIC, logLik

Examples

data(gmm_body, data_body)
aic_1 <- AIC(gmm_body, data_body)
aic_2 <- AIC(gmm_body, data_body, y = "WAIST")

data(gmbn_body, data_body)
aic_3 <- AIC(gmbn_body, data_body)

data(gmdbn_air, data_air)
aic_4 <- AIC(gmdbn_air, data_air, col_seq = "DATE")

Compute the Bayesian Information Criterion (BIC) of a Gaussian mixture model or graphical model

Description

This function computes the Bayesian Information Criterion (BIC) of a Gaussian mixture model or graphical model:

BIC = logLik - \frac{\log(n_{obs})}{2} n_{par}

where logLik is the log-likelihood, n_{obs} the number of observations in the data and n_{par} the number of free parameters.

Usage

## S3 method for class 'gmm'
BIC(object, data, y = NULL, regul = 0.01, ...)

## S3 method for class 'gmbn'
BIC(object, data, col_seq = NULL, ...)

## S3 method for class 'gmdbn'
BIC(object, data, col_seq = NULL, ...)

Arguments

Value

If object is a gmm object, a numeric value corresponding to the BIC.

If object is a gmbn or gmdbn object, a list with elements:

See Also

AIC, logLik

Examples

data(gmm_body, data_body)
bic_1 <- BIC(gmm_body, data_body)
bic_2 <- BIC(gmm_body, data_body, y = "WAIST")

data(gmbn_body, data_body)
bic_3 <- BIC(gmbn_body, data_body)

data(gmdbn_air, data_air)
bic_4 <- BIC(gmdbn_air, data_air, col_seq = "DATE")

Add arcs to a Gaussian mixture graphical model

Description

This function adds arcs to a Gaussian mixture graphical model. For each added arc, a variable related to the start node is added to the Gaussian mixture model describing the local distribution over the end node and its parents, with mean 0 and variance 1 for each mixture component.

Usage

add_arcs(gmgm, arcs)

Arguments

Value

The gmbn or gmdbn object after adding the arcs.

See Also

add_nodes, relevant, remove_arcs, remove_nodes, rename_nodes

Examples

data(gmbn_body)
gmbn_1 <- add_arcs(gmbn_body,
                   data.frame(from = c("GENDER", "AGE"),
                              to = c("GLYCO", "WEIGHT")))

data(gmdbn_air)
gmdbn_1 <- add_arcs(gmdbn_air,
                    list(b_2 = data.frame(from = "WIND", to = "NO2", lag = 1),
                         b_13 = data.frame(from = c("NO2", "NO2"),
                                           to = c("O3", "O3"), lag = c(0, 1))))

Add nodes to a Gaussian mixture graphical model

Description

This function adds nodes to a Gaussian mixture graphical model. If this model is a dynamic Bayesian network, the nodes are added to each of its transition models. For each added node, a one-component univariate Gaussian mixture model is created with mean 0 and variance 1.

Usage

add_nodes(gmgm, nodes)

Arguments

Value

The gmbn or gmdbn object after adding the nodes.

See Also

add_arcs, relevant, remove_arcs, remove_nodes, rename_nodes

Examples

data(gmbn_body)
gmbn_1 <- add_nodes(gmbn_body, c("CHOL", "TRIGLY"))

data(gmdbn_air)
gmdbn_1 <- add_nodes(gmdbn_air, "PM10")

Add variables to a Gaussian mixture model

Description

This function adds variables to a Gaussian mixture model.

Usage

add_var(gmm, var)

Arguments

Value

The gmm object after adding the variables.

See Also

remove_var, rename_var

Examples

data(gmm_body, data_body)
gmm_1 <- add_var(gmm_body, "GENDER")
gmm_2 <- add_var(gmm_body, data_body[, "GENDER"])

Aggregate particles to obtain inferred values

Description

This function aggregates particles to obtain inferred values. Assuming that the particles have been propagated to a given time slice t, the weighted average of the samples is computed to estimate the state of the system at t or at previous time slices (Koller and Friedman, 2009).

Usage

aggregation(part, nodes, col_seq = NULL, col_weight = "weight", lag = 0)

Arguments

Value

If lag has one element, a data frame (tibble) containing the aggregated values of the inferred nodes and their observation sequences (if col_seq is not NULL). If lag has two or more elements, a list of data frames (tibbles) containing these values for each time lag.

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

aggregation, particles

Examples

library(dplyr)
set.seed(0)
data(gmdbn_air, data_air)
evid <- data_air %>%
  group_by(DATE) %>%
  slice(1:3) %>%
  ungroup()
evid$NO2[sample.int(150, 30)] <- NA
evid$O3[sample.int(150, 30)] <- NA
evid$TEMP[sample.int(150, 30)] <- NA
evid$WIND[sample.int(150, 30)] <- NA
aggreg <- particles(data.frame(DATE = unique(evid$DATE))) %>%
  propagation(gmdbn_air, evid, col_seq = "DATE", n_times = 3) %>%
  aggregation(c("NO2", "O3", "TEMP", "WIND"), col_seq = "DATE", lag = c(0, 1))

Conditionalize a Gaussian mixture model

Description

This function conditionalizes a Gaussian mixture model (Sun et al., 2006).

Usage

conditional(gmm, y = rownames(gmm$mu)[1])

Arguments

Value

A list with elements:

References

Sun, S., Zhang, C. and Yu, G. (2006). A Bayesian Network Approach to Traffic Flow Forecasting. IEEE Transactions on Intelligent Transportation Systems, 7(1):124–132.

Examples

data(gmm_body)
cond <- conditional(gmm_body)

Beijing air quality dataset

Description

This dataset includes hourly air pollutants and weather data measured at the Dongsi air quality monitoring site in Beijing (China) for 320 complete days of the year 2015. These data are taken from the Beijing Multi-Site Air Quality Dataset published in the UCI Machine Learning Repository: https://archive.ics.uci.edu/ml/datasets/Beijing+Multi-Site+Air-Quality+Data (Zhang et al., 2017).

Usage

data_air

Format

A data frame (tibble) with 7680 rows and 6 columns:

• DATE: day's date;

• HOUR: hour of the day;

• NO2: nitrogen dioxide concentration (μg/m³);

• O3: ozone concentration (μg/m³);

• TEMP: temperature (°C);

• WIND: wind speed (m/s).

References

Zhang, S., Guo, B., Dong, A., He, J., Xu, Z. and Chen, S. X. (2017). Cautionary Tales on Air-Quality Improvement in Beijing. Proceedings of the Royal Society A, 473.

See Also

data_body, gmbn_body, gmdbn_air, gmm_body

NHANES body composition dataset

Description

This dataset includes body composition data measured in 2148 adults aged 20 to 59 years in the United States. These data are taken from the National Health and Nutrition Examination Survey (NHANES) 2017-2018: https://wwwn.cdc.gov/nchs/nhanes/continuousnhanes/default.aspx?BeginYear=2017 (Centers for Disease Control and Prevention, 2020).

Usage

data_body

Format

A data frame (tibble) with 2148 rows and 8 columns:

• ID: respondent identifier;

• GENDER: gender (0: male, 1: female);

• AGE: age (years);

• HEIGHT: height (cm);

• WEIGHT: weight (kg);

• FAT: body fat (%);

• WAIST: waist circumference (cm);

• GLYCO: glycohemoglobin (%).

References

Centers for Disease Control and Prevention (2020). National Health and Nutrition Examination Survey Data.

See Also

data_air, gmbn_body, gmdbn_air, gmm_body

Compute densities of a Gaussian mixture model

Description

This function computes densities of a Gaussian mixture model.

Usage

density(gmm, data, y = NULL, log = FALSE)

Arguments

Value

A numeric vector containing the (log-)densities.

See Also

expectation, sampling

Examples

data(gmm_body, data_body)
dens_1 <- density(gmm_body, data_body, log = TRUE)
dens_2 <- density(gmm_body, data_body, y = "WAIST", log = TRUE)

Display the mixture components of a Gaussian mixture model

Description

This function displays the mixture components of a Gaussian mixture model. For each pair of variables, the covariance matrices are represented by confidence ellipses.

Usage

ellipses(
  gmm,
  data = NULL,
  y = rownames(gmm$mu),
  x = rownames(gmm$mu),
  level = 0.95
)

Arguments

Value

A ggplot object displaying the mixture components (and the data if required).

Examples

set.seed(0)
data(gmm_body)
ellipses(gmm_body, sampling(gmm_body, n = 500))

Estimate the parameters of a Gaussian mixture model

Description

This function estimates the parameters of a Gaussian mixture model using the expectation-maximization (EM) algorithm. Given an initial model, this algorithm iteratively updates the parameters, monotonically increasing the log-likelihood until convergence to a local maximum (Bilmes, 1998). A Bayesian regularization is applied by default to prevent that a mixture component comes down to a single point and leads to a zero covariance matrix (Ormoneit and Tresp, 1996). Although the EM algorithm only applies to the joint model, good parameters can be found for a derived conditional model. However, care should be taken as the monotonic increase of the conditional log-likelihood is not guaranteed.

Usage

em(
  gmm,
  data,
  regul = 0.01,
  epsilon = 1e-06,
  max_iter_em = 100,
  verbose = FALSE
)

Arguments

Value

A list with elements:

References

Bilmes, J. A. (1998). A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. Technical report, International Computer Science Institute.

Ormoneit, D. and Tresp, V. (1996). Improved Gaussian Mixture Density Estimates Using Bayesian Penalty Terms and Network Averaging. In Advances in Neural Information Processing Systems 8, pages 542–548.

See Also

smem, stepwise

Examples

data(data_body)
gmm_1 <- split_comp(add_var(NULL,
                            data_body[, c("WAIST", "AGE", "FAT", "HEIGHT",
                                          "WEIGHT")]),
                    n_sub = 3)
res_em <- em(gmm_1, data_body, verbose = TRUE)

Compute expectations of a Gaussian mixture model

Description

This function computes expectations of a Gaussian mixture model.

Usage

expectation(gmm, data_x = NULL)

Arguments

Value

A numeric matrix containing the expectations.

See Also

density, sampling

Examples

data(gmm_body, data_body)
expect_1 <- expectation(gmm_body)
expect_2 <- expectation(gmm_body,
                        data_body[, c("WEIGHT", "FAT", "HEIGHT", "AGE")])

Perform filtering inference in a Gaussian mixture dynamic Bayesian network

Description

This function performs filtering inference in a Gaussian mixture dynamic Bayesian network. For a sequence of T time slices, this task consists in estimating the state of the system at each time slice t (for 1 \le t \le T) given all the data (the evidence) collected up to t. This function is also designed to perform fixed-lag smoothing inference, which consists in defining a time lag l such that at each time slice t (for l + 1 \le t \le T), the state at t - l is estimated given the evidence collected up to t (Murphy, 2002). Filtering and fixed-lag smoothing inference are performed by sequential importance resampling, which is a particle-based approximate method (Koller and Friedman, 2009).

Usage

filtering(
  gmdbn,
  evid,
  nodes = names(gmdbn$b_1),
  col_seq = NULL,
  lag = 0,
  n_part = 1000,
  max_part_sim = 1e+06,
  min_ess = 1,
  verbose = FALSE
)

Arguments

Value

If lag has one element, a data frame (tibble) with a structure similar to evid containing the estimated values of the inferred nodes and their observation sequences (if col_seq is not NULL). If lag has two or more elements, a list of data frames (tibbles) containing these values for each time lag.

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

Murphy, K. (2002). Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, University of California.

See Also

inference, prediction, smoothing

Examples

set.seed(0)
data(gmdbn_air, data_air)
evid <- data_air
evid$NO2[sample.int(7680, 1536)] <- NA
evid$O3[sample.int(7680, 1536)] <- NA
evid$TEMP[sample.int(7680, 1536)] <- NA
evid$WIND[sample.int(7680, 1536)] <- NA
filt <- filtering(gmdbn_air, evid, col_seq = "DATE", lag = c(0, 1),
                  verbose = TRUE)

Create a Gaussian mixture Bayesian network

Description

This function creates a Gaussian mixture Bayesian network as an object of S3 class gmbn. A Bayesian network is a probabilistic graphical model that represents the conditional dependencies and independencies between random variables by a directed acyclic graph. It encodes a global joint distribution over the nodes, which decomposes into a product of local conditional distributions:

p(X_1, \dots , X_n) = \prod_{i = 1}^n p(X_i | Pa(X_i))

where Pa(X_i) is the set of parents of X_i in the graph. In a Gaussian mixture Bayesian network, each local joint distribution over a node and its parents is described by a Gaussian mixture model, which means that the global distribution is a product of local conditional Gaussian mixture models (Davies and Moore, 2000). The gmbn class can be extended to the time factor by regarding the nodes as the state of the system at a given time slice t (denoted by X^{(t)}) and allowing them to have parents at previous time slices. This makes it possible to create a (k + 1)-slice temporal Bayesian network that encodes the transition distribution p(X^{(t)} | X^{(t - 1)}, \dots , X^{(t - k)}) (Hulst, 2006). Finally, note that a Gaussian mixture Bayesian network can be created with functions add_nodes (by passing NULL as argument gmgm) and add_arcs, which allows to quickly initialize a gmbn object that can be passed to a learning function.

Usage

gmbn(...)

Arguments

Value

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

References

Davies, S. and Moore, A. (2000). Mix-nets: Factored Mixtures of Gaussians in Bayesian Networks with Mixed Continuous And Discrete Variables. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, pages 168–175, Stanford, CA, USA.

Hulst, J. (2006). Modeling physiological processes with dynamic Bayesian networks. Master's thesis, Delft University of Technology.

See Also

gmdbn, gmm

Examples

data(data_body)
gmbn_1 <- gmbn(
  AGE = split_comp(add_var(NULL, data_body[, "AGE"]), n_sub = 3),
  FAT = split_comp(add_var(NULL,
                           data_body[, c("FAT", "GENDER", "HEIGHT", "WEIGHT")]),
                   n_sub = 2),
  GENDER = split_comp(add_var(NULL, data_body[, "GENDER"]), n_sub = 2),
  GLYCO = split_comp(add_var(NULL, data_body[, c("GLYCO", "AGE", "WAIST")]),
                     n_sub = 2),
  HEIGHT = split_comp(add_var(NULL, data_body[, c("HEIGHT", "GENDER")])),
  WAIST = split_comp(add_var(NULL,
                             data_body[, c("WAIST", "AGE", "FAT", "HEIGHT",
                                           "WEIGHT")]),
                     n_sub = 3),
  WEIGHT = split_comp(add_var(NULL, data_body[, c("WEIGHT", "HEIGHT")]),
                      n_sub = 2)
)

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()
gmbn_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)
)

Gaussian mixture Bayesian network learned from the NHANES body composition dataset

Description

This Gaussian mixture dynamic Bayesian network is learned from the NHANES body composition dataset, following the example provided in the documentation page of function struct_learn.

Usage

gmbn_body

Format

A gmbn object.

See Also

data_air, data_body, gmdbn_air, gmm_body

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(...)

Arguments

Value

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

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

Examples

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)
  )
)

Gaussian mixture dynamic Bayesian network learned from the Beijing air quality dataset

Description

This Gaussian mixture dynamic Bayesian network is learned from the Beijing air quality dataset, following the example provided in the documentation page of function struct_learn.

Usage

gmdbn_air

Format

A gmdbn object.

See Also

data_air, data_body, gmbn_body, gmm_body

Create a Gaussian mixture model

Description

This function creates a Gaussian mixture model as an object of S3 class gmm. A Gaussian mixture model is a weighted sum of multivariate Gaussian distributions:

p(x) = \sum_{i = 1}^M \alpha_i \mathcal{N}(x | \mu_i, \Sigma_i)

where \alpha_i is the ith mixture proportion such that \alpha_i > 0 and \sum_{i = 1}^M \alpha_i = 1, \mu_i the mean vector and \Sigma_i the covariance matrix of the ith mixture component (Bilmes, 1998). Since conditional distributions can be derived from joint distributions, the gmm class is also used to work with conditional Gaussian mixture models (see function conditional to explicit their parameters). Finally, note that a one-component Gaussian mixture model can be created with function add_var (by passing NULL as argument gmm), which allows to quickly initialize a gmm object that can be passed to a learning function.

Usage

gmm(alpha, mu, sigma, var = rownames(mu))

Arguments

Value

A list of class gmm containing the elements alpha, mu and sigma passed as arguments (completed with the variable names passed as argument var).

References

Bilmes, J. A. (1998). A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. Technical report, International Computer Science Institute.

See Also

gmbn, gmdbn

Examples

gmm_1 <- gmm(alpha = c(0.2, 0.5, 0.3),
             mu = matrix(c(109, 91, 44, 160, 41, 99, 87, 27, 173, 40, 86, 65,
                           35, 161, 40),
                         nrow = 5),
             sigma = list(matrix(c(208, 240, 32, 17, -6, 240, 378, 40, 55, -38,
                                   32, 40, 15, -2, 1, 17, 55, -2, 47, -13, -6,
                                   -38, 1, -13, 127),
                                 nrow = 5),
                          matrix(c(242, 270, 82, 10, 49, 270, 363, 83, 44, 19,
                                   82, 83, 38, -2, 15, 10, 44, -2, 45, -7, 49,
                                   19, 15, -7, 137),
                                 nrow = 5),
                          matrix(c(109, 102, 41, 11, 29, 102, 128, 34, 38, 10,
                                   41, 34, 36, -9, 16, 11, 38, -9, 56, -5, 29,
                                   10, 16, -5, 138),
                                 nrow = 5)),
             var = c("WAIST", "WEIGHT", "FAT", "HEIGHT", "AGE"))

Gaussian mixture model learned from the NHANES body composition dataset

Description

This Gaussian mixture model is learned from the NHANES body composition dataset, following the example provided in the documentation page of function stepwise.

Usage

gmm_body

Format

A gmm object.

See Also

data_air, data_body, gmbn_body, gmdbn_air

Perform inference in a Gaussian mixture Bayesian network

Description

This function performs inference in a (non-temporal) Gaussian mixture Bayesian network. This task consists in estimating the state of the system given partial observations of it (the evidence). Inference is performed by likelihood weighting, which is a particle-based approximate method (Koller and Friedman, 2009).

Usage

inference(
  gmbn,
  evid,
  nodes = names(gmbn),
  n_part = 1000,
  max_part_sim = 1e+06,
  verbose = FALSE
)

Arguments

Value

A data frame (tibble) with a structure similar to evid containing the estimated values of the inferred nodes.

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

filtering, prediction, smoothing

Examples

set.seed(0)
data(gmbn_body, data_body)
evid <- data_body
evid$GENDER[sample.int(2148, 430)] <- NA
evid$AGE[sample.int(2148, 430)] <- NA
evid$HEIGHT[sample.int(2148, 430)] <- NA
evid$WEIGHT[sample.int(2148, 430)] <- NA
evid$FAT[sample.int(2148, 430)] <- NA
evid$WAIST[sample.int(2148, 430)] <- NA
evid$GLYCO[sample.int(2148, 430)] <- NA
infer <- inference(gmbn_body, evid, verbose = TRUE)

Compute the log-likelihood of a Gaussian mixture model or graphical model

Description

This function computes the log-likelihood of a Gaussian mixture model or graphical model.

Usage

## S3 method for class 'gmm'
logLik(object, data, y = NULL, regul = 0.01, ...)

## S3 method for class 'gmbn'
logLik(object, data, col_seq = NULL, ...)

## S3 method for class 'gmdbn'
logLik(object, data, col_seq = NULL, ...)

Arguments

Value

If object is a gmm object, a numeric value corresponding to the log-likelihood.

If object is a gmbn or gmdbn object, a list with elements:

See Also

AIC, BIC

Examples

data(gmm_body, data_body)
loglik_1 <- logLik(gmm_body, data_body)
loglik_2 <- logLik(gmm_body, data_body, y = "WAIST")

data(gmbn_body, data_body)
loglik_3 <- logLik(gmbn_body, data_body)

data(gmdbn_air, data_air)
loglik_4 <- logLik(gmdbn_air, data_air, col_seq = "DATE")

Merge mixture components of a Gaussian mixture model

Description

This function merges mixture components of a Gaussian mixture model (Zhang et al., 2003).

Usage

merge_comp(gmm, comp = seq_along(gmm$alpha))

Arguments

Value

The gmm object after merging the mixture components.

References

Zhang, Z., Chen, C., Sun, J. and Chan, K. L. (2003). EM algorithms for Gaussian mixtures with split-and-merge operation. Pattern Recognition, 36(9):1973–1983.

See Also

split_comp

Examples

data(gmm_body)
gmm_1 <- merge_comp(gmm_body, c(1, 2))

Display the graphical structure of a Gaussian mixture Bayesian network

Description

This function displays the graphical structure of a Gaussian mixture Bayesian network.

Usage

network(gmbn)

Arguments

Value

A visNetwork object displaying the graphical structure.

Examples

data(gmbn_body)
network(gmbn_body)

data(gmdbn_air)
network(gmdbn_air$b_2)

Learn the parameters of a Gaussian mixture graphical model with incomplete data

Description

This function learns the parameters of a Gaussian mixture graphical model with incomplete data using the parametric EM algorithm. At each iteration, inference (smoothing inference for a dynamic Bayesian network) is performed to complete the data given the current estimate of the parameters (E step). The completed data are then used to update the parameters (M step), and so on. Each iteration is guaranteed to increase the log-likelihood until convergence to a local maximum (Koller and Friedman, 2009). In practice, due to the sampling process inherent in particle-based inference, it may happen that the monotonic increase no longer occurs when approaching the local maximum, resulting in an earlier termination of the algorithm.

Usage

param_em(
  gmgm,
  data,
  nodes = structure(gmgm)$nodes,
  col_seq = NULL,
  n_part = 1000,
  max_part_sim = 1e+06,
  min_ess = 1,
  max_iter_pem = 5,
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

param_learn, struct_em, struct_learn

Examples

set.seed(0)
data(data_body)
data_1 <- data_body
data_1$GENDER[sample.int(2148, 430)] <- NA
data_1$AGE[sample.int(2148, 430)] <- NA
data_1$HEIGHT[sample.int(2148, 430)] <- NA
data_1$WEIGHT[sample.int(2148, 430)] <- NA
data_1$FAT[sample.int(2148, 430)] <- NA
data_1$WAIST[sample.int(2148, 430)] <- NA
data_1$GLYCO[sample.int(2148, 430)] <- NA
gmbn_1 <- gmbn(
  AGE = split_comp(add_var(NULL, data_1[, "AGE"]), n_sub = 3),
  FAT = split_comp(add_var(NULL,
                           data_1[, c("FAT", "GENDER", "HEIGHT", "WEIGHT")]),
                   n_sub = 2),
  GENDER = split_comp(add_var(NULL, data_1[, "GENDER"]), n_sub = 2),
  GLYCO = split_comp(add_var(NULL, data_1[, c("GLYCO", "AGE", "WAIST")]),
                     n_sub = 2),
  HEIGHT = split_comp(add_var(NULL, data_1[, c("HEIGHT", "GENDER")])),
  WAIST = split_comp(add_var(NULL,
                             data_1[, c("WAIST", "AGE", "FAT", "HEIGHT",
                                        "WEIGHT")]),
                     n_sub = 3),
  WEIGHT = split_comp(add_var(NULL, data_1[, c("WEIGHT", "HEIGHT")]), n_sub = 2)
)
res_learn_1 <- param_em(gmbn_1, data_1, verbose = TRUE)

library(dplyr)
set.seed(0)
data(data_air)
data_2 <- data_air
data_2$NO2[sample.int(7680, 1536)] <- NA
data_2$O3[sample.int(7680, 1536)] <- NA
data_2$TEMP[sample.int(7680, 1536)] <- NA
data_2$WIND[sample.int(7680, 1536)] <- NA
data_3 <- data_2 %>%
  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_3[, c("NO2", "NO2.1", "WIND")]),
                     n_sub = 3),
    O3 = split_comp(add_var(NULL,
                            data_3[, c("O3", "NO2", "NO2.1", "O3.1", "TEMP",
                                       "TEMP.1")]),
                    n_sub = 3),
    TEMP = split_comp(add_var(NULL, data_3[, c("TEMP", "TEMP.1")]), n_sub = 3),
    WIND = split_comp(add_var(NULL, data_3[, c("WIND", "WIND.1")]), n_sub = 3)
  ),
  b_13 = gmbn(
    NO2 = split_comp(add_var(NULL, data_3[, c("NO2", "NO2.1", "WIND")]),
                     n_sub = 3),
    O3 = split_comp(add_var(NULL,
                            data_3[, c("O3", "O3.1", "TEMP", "TEMP.1",
                                       "WIND")]),
                    n_sub = 3),
    TEMP = split_comp(add_var(NULL, data_3[, c("TEMP", "TEMP.1")]), n_sub = 3),
    WIND = split_comp(add_var(NULL, data_3[, c("WIND", "WIND.1")]), n_sub = 3)
  )
)
res_learn_2 <- param_em(gmdbn_1, data_2, col_seq = "DATE", verbose = TRUE)

Learn the parameters of a Gaussian mixture graphical model

Description

This function learns the parameters of a Gaussian mixture graphical model. Using the local decomposability of the log-likelihood, this task consists in learning each local conditional model independently with the EM algorithm (Koller and Friedman, 2009).

Usage

param_learn(
  gmgm,
  data,
  nodes = structure(gmgm)$nodes,
  col_seq = NULL,
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

param_em, struct_em, struct_learn

Examples

data(data_body)
gmbn_1 <- gmbn(
  AGE = split_comp(add_var(NULL, data_body[, "AGE"]), n_sub = 3),
  FAT = split_comp(add_var(NULL,
                           data_body[, c("FAT", "GENDER", "HEIGHT", "WEIGHT")]),
                   n_sub = 2),
  GENDER = split_comp(add_var(NULL, data_body[, "GENDER"]), n_sub = 2),
  GLYCO = split_comp(add_var(NULL, data_body[, c("GLYCO", "AGE", "WAIST")]),
                     n_sub = 2),
  HEIGHT = split_comp(add_var(NULL, data_body[, c("HEIGHT", "GENDER")])),
  WAIST = split_comp(add_var(NULL,
                             data_body[, c("WAIST", "AGE", "FAT", "HEIGHT",
                                           "WEIGHT")]),
                     n_sub = 3),
  WEIGHT = split_comp(add_var(NULL, data_body[, c("WEIGHT", "HEIGHT")]),
                      n_sub = 2)
)
res_learn_1 <- param_learn(gmbn_1, data_body, verbose = TRUE)

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)
  )
)
res_learn_2 <- param_learn(gmdbn_1, data_air, col_seq = "DATE", verbose = TRUE)

Initialize particles to perform inference in a Gaussian mixture graphical model

Description

This function initializes particles to perform (approximate) inference in a Gaussian mixture graphical model. Particles consist in weighted sample sequences propagated forward in time by sampling the model and aggregated to obtain the inferred values (Koller and Friedman, 2009).

Usage

particles(seq = NULL, col_weight = "weight", n_part = 1000)

Arguments

Value

A data frame (tibble) containing the initial particles.

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

aggregation, propagation

Examples

data(data_air)
part <- particles(data.frame(DATE = unique(data_air$DATE)))

Perform predictive inference in a Gaussian mixture dynamic Bayesian network

Description

This function performs predictive inference in a Gaussian mixture dynamic Bayesian network. For a sequence of T time slices, this task consists in defining a time horizon h such that at each time slice t (for 0 \le t \le T - h), the state of the system at t + h is estimated given all the data (the evidence) collected up to t. Although the states at t + 1, \dots , t + h are observed in the future, some information about them can be known a priori (such as contextual information or features controlled by the user). This "predicted" evidence can be taken into account when propagating the particles from t to t + h in order to improve the predictions. Predictive inference is performed by sequential importance resampling, which is a particle-based approximate method (Koller and Friedman, 2009).

Usage

prediction(
  gmdbn,
  evid,
  evid_pred = NULL,
  nodes = names(gmdbn$b_1),
  col_seq = NULL,
  horizon = 1,
  n_part = 1000,
  max_part_sim = 1e+06,
  min_ess = 1,
  verbose = FALSE
)

Arguments

Value

If horizon has one element, a data frame with a structure similar to evid containing the predicted values of the inferred nodes and their observation sequences (if col_seq is not NULL). If horizon has two or more elements, a list of data frames (tibbles) containing these values for each time horizon.

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

filtering, inference, smoothing

Examples

set.seed(0)
data(gmdbn_air, data_air)
evid <- data_air
evid$NO2[sample.int(7680, 1536)] <- NA
evid$O3[sample.int(7680, 1536)] <- NA
pred <- prediction(gmdbn_air, evid, evid[, c("DATE", "TEMP", "WIND")],
                   nodes = c("NO2", "O3"), col_seq = "DATE",
                   horizon = c(1, 2), verbose = TRUE)

Propagate particles forward in time

Description

This function propagates particles forward in time. Assuming that the particles have been propagated to a given time slice t, the aim is to propagate them to a later time slice t + k according to the Gaussian mixture graphical model and to the evidence collected over time. At first, a renewal step is performed if the effective sample size (ESS) is below a given threshold (Doucet and Johansen, 2009). This step consists in randomly selecting new particles among the old ones proportionately to their current weights. Upon receiving the data (the evidence) of t + 1, each particle is used to generate samples for the unknown values. Its weight is then updated to the likelihood for the observed values. The higher this likelihood, the more likely the particle is selected at the next renewal step for propagation to t + 2, and so on (Koller and Friedman, 2009).

Usage

propagation(
  part,
  gmgm,
  evid = NULL,
  col_seq = NULL,
  col_weight = "weight",
  n_times = 1,
  min_ess = 1
)

Arguments

Value

A data frame (tibble) containing the particles supplemented with the samples of time slices t + 1, \dots , t + k.

References

Doucet, A. and Johansen, A. M. (2009). A Tutorial on Particle Filtering and Smoothing: Fifteen years later. Handbook of nonlinear filtering, 12:656–704.

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

aggregation, particles

Examples

library(dplyr)
set.seed(0)
data(gmdbn_air, data_air)
evid <- data_air %>%
  group_by(DATE) %>%
  slice(1:3) %>%
  ungroup()
evid$NO2[sample.int(150, 30)] <- NA
evid$O3[sample.int(150, 30)] <- NA
evid$TEMP[sample.int(150, 30)] <- NA
evid$WIND[sample.int(150, 30)] <- NA
part <- particles(data.frame(DATE = unique(evid$DATE))) %>%
  propagation(gmdbn_air, evid, col_seq = "DATE", n_times = 3)

Extract the minimal sub-Gaussian mixture graphical model required to infer a subset of nodes

Description

This function extracts the minimal sub-Gaussian mixture graphical model required to infer a subset of nodes (i.e. the sub-model relevant to these nodes). The nodes that do not contribute to inference are removed, which includes those that are d-separated from the inferred ones by the nodes whose values are observed, as well as the barren nodes (Druzdzel and Suermondt, 1994).

Usage

relevant(gmgm, nodes, nodes_obs = NULL, nodes_miss = NULL)

Arguments

Value

The gmbn or gmdbn object relevant to the subset of nodes.

References

Druzdzel, M. J. and Suermondt, H. J. (1994). Relevance in Probabilistic Models: "Backyards" in a "Small World". In Working Notes of the AAAI 1994 Fall Symposium Series: Relevance, pages 60–63, New Orleans, LA, USA.

See Also

add_arcs, add_nodes, remove_arcs, remove_nodes, rename_nodes

Examples

data(gmbn_body)
gmbn_1 <- relevant(gmbn_body, "AGE",
                   nodes_obs = c("FAT", "HEIGHT", "WEIGHT"),
                   nodes_miss = "GLYCO")

data(gmdbn_air)
gmdbn_1 <- do.call("gmdbn", gmdbn_air[c("b_1", "b_2")])
gmdbn_2 <- relevant(gmdbn_1, "O3", nodes_obs = "NO2")

Remove arcs from a Gaussian mixture graphical model

Description

This function removes arcs from a Gaussian mixture graphical model.

Usage

remove_arcs(gmgm, arcs)

Arguments

Value

The gmbn or gmdbn object after removing the arcs.

See Also

add_arcs, add_nodes, relevant, remove_nodes, rename_nodes

Examples

data(gmbn_body)
gmbn_1 <- remove_arcs(gmbn_body,
                      data.frame(from = c("HEIGHT", "AGE"),
                                 to = c("FAT", "WAIST")))

data(gmdbn_air)
gmdbn_1 <- remove_arcs(gmdbn_air,
                       list(b_2 = data.frame(from = c("NO2", "TEMP"),
                                             to = c("O3", "O3"), lag = c(1, 1)),
                            b_13 = data.frame(from = "TEMP", to = "O3",
                                              lag = 1)))

Remove nodes from a Gaussian mixture graphical model

Description

This function removes nodes from a Gaussian mixture graphical model. If this model is a dynamic Bayesian network, the nodes are removed from each of its transition models.

Usage

remove_nodes(gmgm, nodes)

Arguments

Value

The gmbn or gmdbn object after removing the nodes.

See Also

add_arcs, add_nodes, relevant, remove_arcs, rename_nodes

Examples

data(gmbn_body)
gmbn_1 <- remove_nodes(gmbn_body, c("FAT", "GLYCO"))

data(gmdbn_air)
gmdbn_1 <- remove_nodes(gmdbn_air, "TEMP")

Remove variables from a Gaussian mixture model

Description

This function removes variables from a Gaussian mixture model.

Usage

remove_var(gmm, var)

Arguments

Value

The gmm object after removing the variables.

See Also

add_var, rename_var

Examples

data(gmm_body)
gmm_1 <- remove_var(gmm_body, "FAT")

Rename nodes of a Gaussian mixture graphical model

Description

This function renames nodes of a Gaussian mixture graphical model. If this model is a dynamic Bayesian network, the nodes are renamed for each of its transition models.

Usage

rename_nodes(gmgm, nodes, names)

Arguments

Value

The gmbn or gmdbn object after renaming the nodes.

See Also

add_arcs, add_nodes, relevant, remove_arcs, remove_nodes

Examples

data(gmbn_body)
gmbn_1 <- rename_nodes(gmbn_body, c("FAT", "GLYCO"),
                       c("BODY_FAT", "GLYCOHEMOGLOBIN"))

data(gmdbn_air)
gmdbn_1 <- rename_nodes(gmdbn_air, "TEMP", "TEMPERATURE")

Rename variables of a Gaussian mixture model

Description

This function renames variables of a Gaussian mixture model.

Usage

rename_var(gmm, var, names)

Arguments

Value

The gmm object after renaming the variables.

See Also

add_var, remove_var

Examples

data(gmm_body)
gmm_1 <- rename_var(gmm_body, "FAT", "BODY_FAT")

Reorder the variables and the mixture components of a Gaussian mixture model

Description

This function reorders the variables and the mixture components of a Gaussian mixture model.

Usage

reorder(gmm, var = NULL, comp = NULL)

Arguments

Value

The reordered gmm object.

Examples

data(gmm_body)
gmm_1 <- reorder(gmm_body, var = c("WAIST", "AGE", "FAT", "HEIGHT", "WEIGHT"),
                 comp = c(2, 1, 3))

Sample a Gaussian mixture model

Description

This function samples a Gaussian mixture model.

Usage

sampling(gmm, data_x = NULL, n = 1)

Arguments

Value

A numeric matrix containing the samples.

See Also

density, expectation

Examples

set.seed(0)
data(gmm_body, data_body)
sampl_1 <- sampling(gmm_body, n = 500)
sampl_2 <- sampling(gmm_body,
                    data_body[, c("WEIGHT", "FAT", "HEIGHT", "AGE")])

Select the number of mixture components and estimate the parameters of a Gaussian mixture model

Description

This function selects the number of mixture components and estimates the parameters of a Gaussian mixture model using a split-and-merge EM (SMEM) algorithm. At the first iteration, the classic EM algorithm is performed to update the parameters of the initial model. Then each following iteration consists in splitting a component into two or merging two components, before re-estimating the parameters with the EM algorithm. The selected split or merge operation is the one that maximizes a scoring function (after the re-estimation process). To avoid testing all possible operations, the split and merge candidates are initially ranked according to relevant criteria (Zhang et al., 2003). At first, the top-ranked split and top-ranked merge operations are tested. If neither of them increases the score, the second-ranked ones are considered, and so on. The SMEM algorithm stops if a given maximum rank is reached without improving the score.

Usage

smem(
  gmm,
  data,
  y = NULL,
  score = "bic",
  split = TRUE,
  merge = TRUE,
  min_comp = 1,
  max_comp = Inf,
  space = 0.5,
  max_rank = 1,
  max_iter_smem = 10,
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

References

Zhang, Z., Chen, C., Sun, J. and Chan, K. L. (2003). EM algorithms for Gaussian mixtures with split-and-merge operation. Pattern Recognition, 36(9):1973–1983.

See Also

em, stepwise

Examples

data(data_body)
gmm_1 <- add_var(NULL, c("WAIST", "AGE", "FAT", "HEIGHT", "WEIGHT"))
res_smem <- smem(gmm_1, data_body, max_comp = 3, verbose = TRUE)

Perform smoothing inference in a Gaussian mixture dynamic Bayesian network

Description

This function performs smoothing inference in a Gaussian mixture dynamic Bayesian network. For a sequence of T time slices, this task consists in estimating the state of the system at each time slice t (for 1 \le t \le T) given all the data (the evidence) collected up to T. Smoothing inference is performed by sequential importance resampling, which is a particle-based approximate method (Koller and Friedman, 2009).

Usage

smoothing(
  gmdbn,
  evid,
  nodes = names(gmdbn$b_1),
  col_seq = NULL,
  n_part = 1000,
  max_part_sim = 1e+06,
  min_ess = 1,
  verbose = FALSE
)

Arguments

Value

A data frame (tibble) with a structure similar to evid containing the estimated values of the inferred nodes and their observation sequences (if col_seq is not NULL).

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

filtering, inference, prediction

Examples

set.seed(0)
data(gmdbn_air, data_air)
evid <- data_air
evid$NO2[sample.int(7680, 1536)] <- NA
evid$O3[sample.int(7680, 1536)] <- NA
evid$TEMP[sample.int(7680, 1536)] <- NA
evid$WIND[sample.int(7680, 1536)] <- NA
smooth <- smoothing(gmdbn_air, evid, col_seq = "DATE", verbose = TRUE)

Split a mixture component of a Gaussian mixture model

Description

This function splits a mixture component of a Gaussian mixture model using the singular value decomposition of the covariance matrix (Zhang et al., 2003).

Usage

split_comp(gmm, comp = 1, n_sub = 2, space = 0.5)

Arguments

Value

The gmm object after splitting the mixture component.

References

Zhang, Z., Chen, C., Sun, J. and Chan, K. L. (2003). EM algorithms for Gaussian mixtures with split-and-merge operation. Pattern Recognition, 36(9):1973–1983.

See Also

merge_comp

Examples

data(gmm_body)
gmm_1 <- split_comp(gmm_body, n_sub = 3)

Select the explanatory variables, the number of mixture components and estimate the parameters of a conditional Gaussian mixture model

Description

This function selects the explanatory variables, the number of mixture components and estimates the parameters of a conditional Gaussian mixture model using a stepwise algorithm. At the first iteration, the SMEM algorithm is performed to update the number of components and the parameters of the initial model. Then each following iteration consists in adding or removing a candidate explanatory variable, before re-estimating the model with the SMEM algorithm. The selected add or remove operation is the one that maximizes a conditional scoring function (after the re-estimation process). The stepwise algorithm stops if none of the candidate operations improves the score.

Usage

stepwise(
  gmm,
  data,
  y = rownames(gmm$mu)[1],
  x_cand = setdiff(colnames(data), y),
  score = "bic",
  add = TRUE,
  remove = TRUE,
  min_x = 0,
  max_x = Inf,
  max_iter_step = 10,
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

See Also

em, smem

Examples

data(data_body)
gmm_1 <- add_var(NULL, "WAIST")
res_step <- stepwise(gmm_1, data_body, verbose = TRUE, max_comp = 3)

Learn the structure and the parameters of a Gaussian mixture graphical model with incomplete data

Description

This function learns the structure and the parameters of a Gaussian mixture graphical model with incomplete data using the structural EM algorithm. At each iteration, the parametric EM algorithm is performed to complete the data and update the parameters (E step). The completed data are then used to update the structure (M step), and so on. Each iteration is guaranteed to increase the scoring function until convergence to a local maximum (Koller and Friedman, 2009). In practice, due to the sampling process inherent in particle-based inference, it may happen that the monotonic increase no longer occurs when approaching the local maximum, resulting in an earlier termination of the algorithm.

Usage

struct_em(
  gmgm,
  data,
  nodes = structure(gmgm)$nodes,
  arcs_cand = tibble(lag = 0),
  col_seq = NULL,
  score = "bic",
  n_part = 1000,
  max_part_sim = 1e+06,
  min_ess = 1,
  max_iter_sem = 5,
  max_iter_pem = 5,
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

param_em, param_learn, struct_learn

Examples

set.seed(0)
data(data_body)
data_1 <- data_body
data_1$GENDER[sample.int(2148, 430)] <- NA
data_1$AGE[sample.int(2148, 430)] <- NA
data_1$HEIGHT[sample.int(2148, 430)] <- NA
data_1$WEIGHT[sample.int(2148, 430)] <- NA
data_1$FAT[sample.int(2148, 430)] <- NA
data_1$WAIST[sample.int(2148, 430)] <- NA
data_1$GLYCO[sample.int(2148, 430)] <- NA
gmbn_1 <- add_nodes(NULL,
                    c("AGE", "FAT", "GENDER", "GLYCO", "HEIGHT", "WAIST",
                      "WEIGHT"))
arcs_cand_1 <- data.frame(from = c("AGE", "GENDER", "HEIGHT", "WEIGHT", NA,
                                   "AGE", "GENDER", "AGE", "FAT", "GENDER",
                                   "HEIGHT", "WEIGHT", "AGE", "GENDER",
                                   "HEIGHT"),
                          to = c("FAT", "FAT", "FAT", "FAT", "GLYCO", "HEIGHT",
                                 "HEIGHT", "WAIST", "WAIST", "WAIST", "WAIST",
                                 "WAIST", "WEIGHT", "WEIGHT", "WEIGHT"))
res_learn_1 <- struct_em(gmbn_1, data_1, arcs_cand = arcs_cand_1,
                         verbose = TRUE, max_comp = 3)

set.seed(0)
data(data_air)
data_2 <- data_air
data_2$NO2[sample.int(7680, 1536)] <- NA
data_2$O3[sample.int(7680, 1536)] <- NA
data_2$TEMP[sample.int(7680, 1536)] <- NA
data_2$WIND[sample.int(7680, 1536)] <- NA
gmdbn_1 <- gmdbn(b_2 = add_nodes(NULL, c("NO2", "O3", "TEMP", "WIND")),
                 b_13 = add_nodes(NULL, c("NO2", "O3", "TEMP", "WIND")))
arcs_cand_2 <- data.frame(from = c("NO2", "NO2", "NO2", "O3", "TEMP", "TEMP",
                                   "WIND", "WIND"),
                          to = c("NO2", "O3", "O3", "O3", NA, NA, NA, NA),
                          lag = c(1, 0, 1, 1, 0, 1, 0, 1))
res_learn_2 <- struct_em(gmdbn_1, data_2, arcs_cand = arcs_cand_2,
                         col_seq = "DATE", verbose = TRUE, max_comp = 3)

Learn the structure and the parameters of a Gaussian mixture graphical model

Description

This function learns the (graphical and mixture) structure and the parameters of a Gaussian mixture graphical model. Using the local decomposability of the scoring function, this task consists in learning each local conditional model independently with the stepwise algorithm (Koller and Friedman, 2009). Note that some candidate arcs may be discarded to avoid that the global graphical structure contains cycles. To limit recourse to this action, the learning process is performed sequentially. The more arcs of a local model are likely to be part of cycles (considering the worst case where all the candidate arcs are selected), the later this local model is processed. By gradually taking into account the local structures learned over time, the number of possible cycles decreases and, with it, the number of candidate arcs to discard.

Usage

struct_learn(
  gmgm,
  data,
  nodes = structure(gmgm)$nodes,
  arcs_cand = tibble(lag = 0),
  col_seq = NULL,
  score = "bic",
  verbose = FALSE,
  ...
)

Arguments

Value

A list with elements:

References

Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. The MIT Press.

See Also

param_em, param_learn, struct_em

Examples

data(data_body)
gmbn_1 <- add_nodes(NULL,
                    c("AGE", "FAT", "GENDER", "GLYCO", "HEIGHT", "WAIST",
                      "WEIGHT"))
arcs_cand_1 <- data.frame(from = c("AGE", "GENDER", "HEIGHT", "WEIGHT", NA,
                                   "AGE", "GENDER", "AGE", "FAT", "GENDER",
                                   "HEIGHT", "WEIGHT", "AGE", "GENDER",
                                   "HEIGHT"),
                          to = c("FAT", "FAT", "FAT", "FAT", "GLYCO", "HEIGHT",
                                 "HEIGHT", "WAIST", "WAIST", "WAIST", "WAIST",
                                 "WAIST", "WEIGHT", "WEIGHT", "WEIGHT"))
res_learn_1 <- struct_learn(gmbn_1, data_body, arcs_cand = arcs_cand_1,
                            verbose = TRUE, max_comp = 3)

data(data_air)
gmdbn_1 <- gmdbn(b_2 = add_nodes(NULL, c("NO2", "O3", "TEMP", "WIND")),
                 b_13 = add_nodes(NULL, c("NO2", "O3", "TEMP", "WIND")))
arcs_cand_2 <- data.frame(from = c("NO2", "NO2", "NO2", "O3", "TEMP", "TEMP",
                                   "WIND", "WIND"),
                          to = c("NO2", "O3", "O3", "O3", NA, NA, NA, NA),
                          lag = c(1, 0, 1, 1, 0, 1, 0, 1))
res_learn_2 <- struct_learn(gmdbn_1, data_air, arcs_cand = arcs_cand_2,
                            col_seq = "DATE", verbose = TRUE, max_comp = 3)

Provide the graphical structure of a Gaussian mixture graphical model

Description

This function provides the graphical structure of a Gaussian mixture graphical model.

Usage

structure(gmgm)

Arguments

Value

A list with elements:

Examples

data(gmbn_body)
struct_1 <- structure(gmbn_body)

data(gmdbn_air)
struct_2 <- structure(gmdbn_air)

Summarize a Gaussian mixture model or graphical model

Description

This function summarizes a Gaussian mixture model or graphical model.

Usage

## S3 method for class 'gmm'
summary(object, ...)

## S3 method for class 'gmbn'
summary(object, ...)

## S3 method for class 'gmdbn'
summary(object, ...)

Arguments

Value

If object is a gmm object, an integer vector containing the number of variables, mixture components and free parameters.

If object is a gmbn or gmdbn object, a list with elements:

Examples

data(gmm_body)
summ_1 <- summary(gmm_body)

data(gmbn_body)
summ_2 <- summary(gmbn_body)

data(gmdbn_air)
summ_3 <- summary(gmdbn_air)