Omega2Pnet: Constructs a parameterized network from an Omega matrix.

In ralmond/Peanut: Parameterized Bayesian Networks, Abstract Classes

Constructs a parameterized network from an Omega matrix.

Description

An Omega matrix (represented as a data frame) is a structure which describes a Bayesian network as a series of regressions from the parent nodes to the child nodes. It actually contains two matrixes, one giving the structure and the other the regression coefficients. A skeleton matrix can be constructed through the function Pnet2Omega.

Usage

Omega2Pnet(OmegaMat, pn, nodewarehouse, defaultRule = "Compensatory",
defaultLink = "normalLink", defaultAlpha = 1, defaultBeta = 0,
defaultLinkScale = 1, defaultPriorWeight=10, debug = FALSE, override =FALSE,
addTvals = TRUE)

Arguments

OmegaMat	A data frame containing an Omega matrix (see values section of Pnet2Omega).
pn	A (possible empty) Pnet object. This will be modified by the function.
nodewarehouse	A Node Warehouse which contains instructions for building nodes referenced in the Omega matrix but not in the network.
defaultRule	This should be a character scalar giving the name of a CPTtools combination rule (see Compensatory). With the regression model assumed in the algorithm, currently “Compensatory” is the only value that makes sense.
defaultLink	This should be a character scalar giving the name of a CPTtools link function (see normalLink). With the regression model assumed in the algorithm, currently “normalLink” is the only value that makes sense.
defaultAlpha	A numeric scalar giving the default value for slope parameters.
defaultBeta	A numeric scalar giving the default value for difficulty (negative intercept) parameters.
defaultLinkScale	A positive number which gives the default value for the link scale parameter.
defaultPriorWeight	A positive number which gives the default value for the node prior weight hyper-parameter.
debug	A logical scalar. If true then recover is called after an error, so that the node in question can be inspected.
override	A logical value. If false, differences between any exsiting structure in the graph and the Omega matrix will raise an error. If true, the graph will be modified to conform to the matrix.
addTvals	A logical value. If true, nodes which do not have state values set, will have those state values set using the function effectiveThetas.

Details

Whittaker (1990) noted that a normal Bayesian network (one in which all nodes followed a standard normal distribution) could be described using the inverse of the covariance matrix, often denoted Omega. In particular, zeros in the inverse covariance matrix represented variables which were conditionally independent, and therefore reducing the matrix to one with positive and zero values could provide the structure for a graphical model. Almond (2010) proposed using this as the basis for specifying discrete Bayesian networks for the proficiency model in educational assessments (especially as correlation matrixes among latent variables are a possible output of a factor analysis).

The Omega matrix is represented with a data.frame object which contains two square submatrixes and a couple of auxiliary columns. The first column should be named “Node” and contains the names of the nodes. This defines a collection of nodes which are defined in the Omega matrix. Let J be the number of nodes (rows in the data frame). The next J columns should have the names the nodes. Their values give the structural component of the matrix. The following two columns are “Link” and “Rules” these give the name of the combination rule and link function to use for this row. Next follows another series J “A” columns, each should have a name of the form “A.node”. This defines a matrix A containing regression coefficients. Finally, there should be two additional columns, “Intercept” and “PriorWeight”.

Let Q be the logical matrix formed by the J columns after the first and let A be the matrix of coefficients. The matrix Q gives the structure of the graph with Q[i,j] being true when Node j is a parent of node i. By convention, Q[j,j]=1. Note that unlike the inverse covariance matrix from which it gets its name, this matrix is not symmetric. It instead reflects the (possibly arbitrary) directions assigned to the edges. Except for the main diagonal, Q[i,j] and Q[j,i] will not both be 1. Note also, that A[i,j] should be positive only when Q[i,j]=1. This provides an additional check that structures were correctly entered if the Omega matrix is being used for data entry.

When the link function is set to normalLink and the rules is set of Compensatory the model is described as a series of regressions. Consider Node j which has K parents. Let \theta_j be a real value corresponding to that node and let \theta_k be a real (standard normal) value representing Parent Node k a_k represent the corresponding coefficient from the A-table. Let \sigma_j = a_{j,j} that is the diagonal element of the A-table corresponding to the variable under consideration. Let b_j be the value of the intercept column for Node j. Then the model specifies that theta_j has a normal distribution with mean

\frac{1}{\sqrt{K}}\sum a_k\theta_k + b_j,

and standard deviation \sigma_j. The regression is discretized to calculate the conditional probability table (see normalLink for details).

Note that the parameters are deliberately chosen to look like a regression model. In particular, b_j is a normal intercept and not a difficulty parameter, so that in general PnodeBetas applied to the corresponding node will have the opposite sign. The 1/\sqrt{K} term is a variance stabilization parameter so that the variance of \theta_j will not be affected by number of parents of Node j. The multiple R-squared for the regression model is

\frac{1/K \sum a_k^2}{ 1/K \sum a_k^2 + \sigma_j^2} .

This is often a more convenient parameter to elicit than \sigma_j.

The function Omega2Pnet attempts to make adjustments to its pnet argument, which should be a Pnet, so that it conforms to the information given in the Omega matrix. Nodes are created as necessary using information in the nodewarehouse argument, which should be a Warehouse object whose manifest includes instructions for building the nodes in the network. The warehouse supply function should either return an existing node in pnet or create a new node in pnet. The structure of the graph is adjusted to correspond to the Q-matrix (structural part of the data frame). If the value of the override argument is false, an error is raised if there is existing structure with a different topology. If override is true, then the pnet is destructively altered to conform to the structural information in the Omega matrix.

The “Link” and “Rules” columns are used to set the values of PnodeLink(node) and PnodeRules(node). The off-diagonal elements of the A-matrix are used to set PnodeAlphas(node) and the diagonal elements to set PnodeLinkScale(node). The values in the “Intercept” column are the negatives of the values PnodeBetas(node). Finally, the values in the “PriorWeight” column correspond to the values of PnodePriorWeight(node). In any of these cases, if the value in the Omega matrix is missing, then the default value will be supplied instead.

One challenge is setting up a matrix with the correct structure. If the nodes have been defined, the the Pnet2Omega can be used to create a blank matrix with the proper format which can then be edited.

Value

The network pnet is returned. Note that it is destructively modified by the commands to conform to the Omega matrix.

Omega Matrix Structure

An Omega Matrix should be an object of class data.frame with number of rows equal to the number of nodes. Throughout let node stand for the name of a node.

Node

The name of the node described in this column.

node

One column for each node. The value in this column should be 1 if the node in the column is regarded as a parent of the node referenced in the row.

Link

The name of a link function. Currently, “normalLink” is the only value supported.

Rules

The name of the combination rule to use. Currently, “Compensatory” is recommended.

A.node

One column for each node. This should be a positive value if the corresponding node column has a 1. This gives the regression coefficient. If node corresponds to the current row, this is the residual standard deviation rather than a regression coefficient. See details.

Intercept

A numeric value giving the change in prevalence for the two variables (see details).

PriorWeight

The amount of weight which should be given to the current values when learning conditional probability tables. See PnodePriorWeight.

Logging and Debug Mode

As of version 0.6-2, the meaning of the debug argument is changed. In the new version, the flog.logger mechanism is used for progress reports, and error reporting. In particular, setting flog.threshold(DEBUG) (or TRACE) will cause progress reports to be sent to the logging output.

The debug argument has been repurposed. It now call recover when the error occurs, so that the problem can be debugged.

Side Effects

This function destructively modifies pnet and nodes referenced in the Qmat and supplied by the warehouses.

Note that unlike typical R implementations, this is not necessarily safe. In particular, if the Qmat references 10 node, and an error is raised when trying to modify the 5th node, the first 4 nodes will be modified, the last 5 will not be and the 5th node may be partially modified. This is different from most R functions where changes are not committed unless the function returns successfully.

Note

While the Omega matrix allows the user to specify both link function and combination rule, the description of the Bayesian network as a series of regressions only really makes sense when the link function is normalLink and the combination rule is Compensatory. These are included for future exapnsion.

The representation, using a single row of the data frame for each node in the graph, only works well with the normal link function. In particular, both the partial credit and graded response links require the ability to specify different intercepts for different states of the variable, something which is not supported in the Omega matrix. Furthermore, the OffsetConjunctive rule requires multiple intercepts. Presumable the Conjunctive rule could be used, but the interpretation of the slope parameters is then unclear. If the variables need a model other than the compensatory normal model, it might be better to use a Q-matrix (see Pnet2Qmat to describe the variable.

Author(s)

Russell Almond

References

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley.

Almond, R. G. (2010). ‘I can name that Bayesian network in two matrixes.’ International Journal of Approximate Reasoning. 51, 167-178.

Almond, R. G. (presented 2017, August). Tabular views of Bayesian networks. In John-Mark Agosta and Tomas Singlair (Chair), Bayeisan Modeling Application Workshop 2017. Symposium conducted at the meeting of Association for Uncertainty in Artificial Intelligence, Sydney, Australia. (International) Retrieved from http://bmaw2017.azurewebsites.net/

See Also

The inverse operation is Pnet2Omega.

See Warehouse for description of the node warehouse argument.

See normalLink and Compensatory for more information about the mathematical model.

The node attributes set from the Omega matrix include: PnodeParents(node), PnodeLink(node), PnodeLinkScale(node), PnodeRules(node), PnodeAlphas(node), PnodeBetas(node), and PnodePriorWeight(node)

Examples

## Sample Omega matrix.
omegamat <- read.csv(system.file("auxdata", "miniPP-omega.csv",
                                package="Peanut"),
                     row.names=1,stringsAsFactors=FALSE)

## Not run: 
library(PNetica) ## Needs PNetica
sess <- NeticaSession()
startSession(sess)

curd <- getwd()

netman1 <- read.csv(system.file("auxdata", "Mini-PP-Nets.csv", 
                                package="Peanut"),
                    row.names=1,stringsAsFactors=FALSE)

nodeman1 <- read.csv(system.file("auxdata", "Mini-PP-Nodes.csv", 
                                package="Peanut"),
                     stringsAsFactors=FALSE)

## Insures we are building nets from scratch
setwd(tempdir())
## Network and node warehouse, to create networks and nodes on demand.
Nethouse <- BNWarehouse(manifest=netman1,session=sess,key="Name")

Nodehouse <- NNWarehouse(manifest=nodeman1,
                         key=c("Model","NodeName"),
                         session=sess)
CM <- WarehouseSupply(Nethouse,"miniPP_CM")
CM1 <- Omega2Pnet(omegamat,CM,Nodehouse,override=TRUE,debug=TRUE)

Om2 <- Pnet2Omega(CM1,NetworkAllNodes(CM1))

DeleteNetwork(CM)
stopSession(sess)
setwd(curd)


## End(Not run)

ralmond/Peanut documentation built on Sept. 19, 2023, 8:27 a.m.