Sample observations from the ordinal MRF

This function samples states from the ordinal MRF using a Gibbs sampler. The Gibbs sampler is initiated with random values from the response options, after which it proceeds by simulating states for each variable from a logistic model using the other variable states as predictor variables.

Usage

mrfSampler(
  no_states,
  no_variables,
  no_categories,
  interactions,
  thresholds,
  variable_type = "ordinal",
  reference_category,
  iter = 1000
)

Arguments

no_states: The number of states of the ordinal MRF to be generated.
no_variables: The number of variables in the ordinal MRF.
no_categories: Either a positive integer or a vector of positive integers of length no_variables. The number of response categories on top of the base category: no_categories = 1 generates binary states.
interactions: A symmetric no_variables by no_variables matrix of pairwise interactions. Only its off-diagonal elements are used.
thresholds: A no_variables by max(no_categories) matrix of category thresholds. The elements in row i indicate the thresholds of variable i. If no_categories is a vector, only the first no_categories[i] elements are used in row i. If the Blume-Capel model is used for the category thresholds for variable i, then row i requires two values (details below); the first is $\alpha$, the linear contribution of the Blume-Capel model and the second is $\beta$, the quadratic contribution.
variable_type: What kind of variables are simulated? Can be a single character string specifying the variable type of all p variables at once or a vector of character strings of length p specifying the type for each variable separately. Currently, bgm supports “ordinal” and “blume-capel”. Binary variables are automatically treated as “ordinal’’. Defaults to variable_type = "ordinal".
reference_category: An integer vector of length no_variables specifying the reference_category category that is used for the Blume-Capel model (details below). Can be any integer value between 0 and no_categories (or no_categories[i]).
iter: The number of iterations used by the Gibbs sampler. The function provides the last state of the Gibbs sampler as output. By default set to 1e3.

Value

A no_states by no_variables matrix of simulated states of the ordinal MRF.

Details

There are two modeling options for the category thresholds. The default option assumes that the category thresholds are free, except that the first threshold is set to zero for identification. The user then only needs to specify the thresholds for the remaining response categories. This option is useful for any type of ordinal variable and gives the user the most freedom in specifying their model.

The Blume-Capel option is specifically designed for ordinal variables that have a special type of reference_category category, such as the neutral category in a Likert scale. The Blume-Capel model specifies the following quadratic model for the threshold parameters: $$\mu_{\text{c}} = \alpha \times \text{c} + \beta \times (\text{c} - \text{r})^2,$$ where $\mu_{\text{c}}$ is the threshold for category c (which now includes zero), $\alpha$ offers a linear trend across categories (increasing threshold values if $\alpha > 0$ and decreasing threshold values if $\alpha <0$), if $\beta < 0$, it offers an increasing penalty for responding in a category further away from the reference_category category r, while $\beta > 0$ suggests a preference for responding in the reference_category category.

Examples

# Generate responses from a network of five binary and ordinal variables.
no_variables = 5
no_categories = sample(1:5, size = no_variables, replace = TRUE)

Interactions = matrix(0, nrow = no_variables, ncol = no_variables)
Interactions[2, 1] = Interactions[4, 1] = Interactions[3, 2] =
  Interactions[5, 2] = Interactions[5, 4] = .25
Interactions = Interactions + t(Interactions)
Thresholds = matrix(0, nrow = no_variables, ncol = max(no_categories))

x = mrfSampler(no_states = 1e3,
               no_variables = no_variables,
               no_categories = no_categories,
               interactions = Interactions,
               thresholds = Thresholds)
#> Warning: The matrix ``thresholds'' contains numeric values for variable 1 for category 
#> (categories, i.e., columns) exceding the maximum of 1. These values will 
#> be ignored.
#> Warning: The matrix ``thresholds'' contains numeric values for variable 2 for category 
#> (categories, i.e., columns) exceding the maximum of 1. These values will 
#> be ignored.
#> Warning: The matrix ``thresholds'' contains numeric values for variable 4 for category 
#> (categories, i.e., columns) exceding the maximum of 1. These values will 
#> be ignored.
#> Warning: The matrix ``thresholds'' contains numeric values for variable 5 for category 
#> (categories, i.e., columns) exceding the maximum of 1. These values will 
#> be ignored.

# Generate responses from a network of 2 ordinal and 3 Blume-Capel variables.
no_variables = 5
no_categories = 4

Interactions = matrix(0, nrow = no_variables, ncol = no_variables)
Interactions[2, 1] = Interactions[4, 1] = Interactions[3, 2] =
  Interactions[5, 2] = Interactions[5, 4] = .25
Interactions = Interactions + t(Interactions)

Thresholds = matrix(NA, no_variables, no_categories)
Thresholds[, 1] = -1
Thresholds[, 2] = -1
Thresholds[3, ] = sort(-abs(rnorm(4)), decreasing = TRUE)
Thresholds[5, ] = sort(-abs(rnorm(4)), decreasing = TRUE)

x = mrfSampler(no_states = 1e3,
               no_variables = no_variables,
               no_categories = no_categories,
               interactions = Interactions,
               thresholds = Thresholds,
               variable_type = c("b","b","o","b","o"),
               reference_category = 2)