Linear Gaussian Bayesian network manipulations

Description

General functions to generate, transform, display general and particular linear Gaussian Bayesian networks [/nbn/] are provided.
Specific /nbn/ are chain and crossed /nbn/s. Focus is given in getting joint and conditional probability distributions of the set of nodes.
rbmn stands for R'eseau Bay'esien MultiNormal.

Details

Some basic concepts:

• chain /nbn/s are /nbn/s where all nodes are connected with two other nodes, except the two ending nodes of the chain having only one connection. (This is not the usual terminology in graphical models but I didn't find a more appropriate word: suggestions are welcome.)

• crossed /nbn/s are /nbn/s having the node set defined as a Cartesian product of two series of items, and a DAG based on this structure. See the crossed4nbn1nbn function and/or Tian (2014) for details.

• An adjacency matrix is a matrix equivalent to the DAG associated to a /nbn/. Its rows as well as its columns are associated to the set of nodes. The (i,j) cell is one when there is an arc going from node i to j and zero otherwise.

Three equivalent ways can be used to represent the joint probability distribution of a set of nodes respectively associated to the structures /mn/, /nbn/ and /gema/:

• /mn/ (for multivariate normal) is just the list of the expectation ($mu) and the variance matrix ($gamma).

• /nbn/ (for normal Bayesian network) is a simple list, a component a node described with a list. The names are node names and each list associated to a node provides the conditional expectation and variance, the parent (if any) and the associated regression coefficients.

• /gema/ (for generating matrices) is a list of a vector (mu) and a matrix (li) such that the vector of the nodes can be defined by X = mu + li%*%E where E is a normal random vector with expectation zero and variance matrix unity.

• It is planned to add a fourth one under the name of /gbn/.

To relieve the memory effort, most names of the functions have been given a two (or more) components structure separated with a figure. This idea will be explained and exploited in a package to come named documair. The approximate meaning of the figures are:

• 0 (similar to 'o') rbmn0chain.01 to indicate an object example provided by rbmn.

• 1 (similar to an ~ and) ??? to link different objects or actions train1car for train and car.

• 2 (as usual but only one-to-one) nbn2gema means \"transforming a /nbn/ into a /gema/ objects\".

• 3 (remind the 'belong to' sign) form3repeat could be interpreted as "repeat action from the series of 'form' functions".

• 4 (associated to 'from') adja4nbn means "get the adjacency matrix from a /nbn/ object".

• 7 (upper bar of '7' similar to the hyphen) arc7nb4nbn means "get the arc-numbers from a /nbn/".

• 8 (similar to 'a') generate8nbn or print8nbn for \"generating or printing a /nbn/ object\".

A number of ancillary functions have not been exported to give a better access to the main function of /rbmn/. Nevertheless they are available in the ../rbmn/R/ directory, and with all their comments (equivalent to Rd files into ../rbmn/inst/original/ directory). Some of them are visible when defining the default arguments of some functions.

Projected evolution of /mn/

• Generalize the /mn/ object with a regression part like the output of function condi4joint when argument pour is not of length zero and argument x2 is not null. With such a structure, every node of a /nbn/ could be described with a /mn/ comprising a unique variable... Also the two arguments of function mn4joint1condi would be just two /mn/ objects... This is also the generalized /mn/ proposed in function simulate8gmn under the argument of loi... Of course almost all functions dealing with /nbn/ objects will be to rewrite!

• Introduce a new object gbn for Gaussian Bayesian network similar to the list provided by function nbn2rr.

TO DO list

• Systemize the existence of check8object functions

• Introduce their systematic use conditionned with a rbmn0check variable.

• Follow the main checking of every functions

• Give (and use) class attributes to the main objects.

• Introduce the main objects in this short presentation.

• Make a true small example in this short presentation.

• Make the function nbn4string7dag.

• Add the computation made with /bnlearn/ in the example of estimate8nbn.

• Check the topological order within nbn2nbn depending on rbmn0check value.

• Make a super transformation function from an object associated to a Bayesian network to any other type, including itself.

• Correct the ord option in order4chain.

• Check the topological order in rm8nd4adja.

• Think about removing all rmatrix transformations to the benefit of the to-come gbn object.

• Introduce a check of non-negativity of ma into cor4var.

• Add examples to all functions without any.

Author(s)

Original author: Jean-Baptiste Denis
Maintainer: Marco Scutari

References

Scutari M (2010). "Learning Bayesian Networks with the bnlearn R Package". Journal of Statistical Software, 35(3), 1-22.

Tian S, Scutari M & Denis J-B (2014). "Predicting with Crossed Linear Gaussian Bayesian Networks". Journal de la Societe Francaise de Statistique, 155(3), 1-21.

Examples

library(rbmn)

## getting the data set
data(boco)
print(head(boco));

Arc matrix from an adjacency matrix

Description

returns the arc matrix from an adjacency matrix.

Usage

adja2arcs(adj)

Arguments

Value

a matrix with two columns ("from","to")

Examples

 adja2arcs(rbmn0adja.02)

creates a crossed-adjacency matrix from two ones

Description

Like crossed4nbn1nbn but at the level of adjacency matrices. Must be much efficient when regression coefficients are not needed.

Usage

adja2crossed(adj1, adj2, nona=as.vector(outer(dimnames(adj1)[[1]],
	  dimnames(adj2)[[1]], paste, sep="_")))

Arguments

Details

Just two Kronecker products of matrices.

Value

The resulting crossed adjacency matrix.

Examples

 print(adja2crossed(rbmn0adja.01, rbmn0adja.01));

standardized /nbn/ from an adjacency matrix

Description

returns a nbn object with O/1 regression coefficients having adja as adjacency matrix.

Usage

adja2nbn(adja)

Arguments

Value

The corresponding standardized nbn object.

Examples

 print8nbn(adja2nbn(adja4nbn(rbmn0nbn.03)));

adjacency matrix of a /nbn/

Description

returns a dimnamed matrix indicating with 1 an arc from row to column nodes (0 everywhere else); i.e. the adjacency matrix.

Usage

adja4nbn(nbn)

Arguments

Value

A dimnamed matrix

Examples

 adja4nbn(rbmn0nbn.04);

Adjacency matrices of DAGs having three nodes

Description

Returns the list of the 25 adjacency matrices associated to DAGs comprising three nodes. The first character of the name components gives the number of arcs in the DAG.

Usage

adja4three(nona=LETTERS[1:3])

Arguments

Details

Poor filling...

Value

a named list having 25 components, each being a 3x3 matrix.

returns the number(s) of arcs of a /nbn/

Description

returns the arc numbers of the node of /nbn/ object.

Usage

arc7nb4nbn(nbn, each=FALSE)

Arguments

Details

Parents associated with a zero regression coefficient are not excluded in the counting.

Value

Either a number or a named vector of numbers (names being the node names).

Examples

 arc7nb4nbn(rbmn0nbn.05);

returns the list of 'parallel' arcs of a crossed-nbn

Description

Returns a list of matrices with two columns (as needed by estimate8constrainednbn) indicating corresponding arcs for each arcs/nodes of nbn1 (or nbn2) of the crossed /nbn/ obtained when crossing /nbn1/ and /nbn2/ with node names given by nona.

Usage

arcs4nbn1nbn(nbn1, nbn2, type="a1", nona=as.vector(outer(names(nbn1),
	  names(nbn2), paste, sep="_")))

Arguments

Value

The resulting named (after node names) list of matrices.

Examples

 print(arcs4nbn1nbn(rbmn0nbn.01, rbmn0nbn.04));

transforms a /bn/ of /bnlearn/ package to a /nbn/

Description

returns a nbn object from a DAG (bn object) of /bnlearn/ package. O and 1 coefficients are introduced...

Usage

bn2nbn(bn)

Arguments

Value

A list following the nbn specification

transforms a /bn.fit/ of /bnlearn/ package to a /nbn/

Description

returns a nbn object from a Gaussian bn.fit object of /bnlearn/ package.

Usage

bnfit2nbn(bn.fit)

Arguments

Details

If bn.fit is not pertinent, a fatal error is issued.

Value

A list following the nbn specification

Body Composition Variables and Covariables

Description

Real-world data set extracted from the Nhanes data base comprising nine variables describing the body composition and five easy measurable covariables.

Usage

data(boco)

Format

The boco data set stored in variable boco comprises 100 individuals with the following variables:

• A the age in years

• H the height in cm

• W the weight in kg

• C the waist circumference in cm

• TF the trunk fat in kg

• LF the leg fat in kg

• AF the arm fat in kg

• TL the trunk lean in kg

• LL the leg lean in kg

• AL the arm lean in kg

• TB the trunk bone in kg

• LB the leg bone in kg

• AB the arm bone in kg

Source

Centers for Disease Control and Prevention. The 1999-2004 dual energy X-ray absorptiometry (DXA) multiple imputation data files and technical documentation.
Available from: http://www.cdc.gov/nchs/about/major/nhanes/dxx/dxa.html (accessed on 13_07_03).

Examples

# load the data and build the correct network from the model string.
data(boco);
print(head(boco));
boco7dag <- "[H][W|H][TF|W;H]";
# to be finished

computes the correlation matrix of a chain

Description

returns the correlation matrix of a /chain/ object.

Usage

chain2correlation(chain)

Arguments

Value

The correlation matrix. It is not sorted to respect a topological order contrary to chain2mn function.

Examples

 chain2correlation(rbmn0chain.03);

transforms a /chain/ to a /gema/

Description

From a chain object returns the gema using a direct formulae.
Much precised than to use the /nbn/ way.

Usage

chain2gema(chain)

Arguments

Value

The corresponding gema object.

Examples

 identical(chain2gema(rbmn0chain.02)$mu, rbmn0gema.02$mu);
 print(chain2gema(rbmn0chain.02)$li-rbmn0gema.02$li);

computes the distribution of a chain

Description

returns the /mn/ object associated to a /chain/ object. Much better to use this function that the general function nbn2mn since exact formulae are applied.

Usage

chain2mn(chain, order=TRUE)

Arguments

Value

The resulting /mn/ object. Following the convention of mn objects, a topological order is given to it. This is necessary to retrieve the associate /nbn/.

Examples

 print8mn(chain2mn(rbmn0chain.01));

transforms a /chain/ to a /nbn/

Description

From a chain object returns the nbn translation.

Usage

chain2nbn(chain)

Arguments

Value

The corresponding nbn object.

Examples

 print8nbn(chain2nbn(rbmn0chain.02), ordering=names(rbmn0nbn.02));

computes the precision of a chain

Description

returns the precision matrix of a chain, that is the inverse of its variance (correlation) matrix. Much better to use this function that solve(chain2mn(chain)$gamma) since exact formulae are applied.

Usage

chain2pre(chain, corre=FALSE)

Arguments

Value

A dimnamed matrix

Examples

 chain2pre(rbmn0chain.02);

extracts a chain from a chain

Description

returns the chain obtained from chain retaining only nodes indicated by nodes and conditioned with nodes indicated in condi.

Usage

chain4chain(chain, nodes, condi=numeric(0), value=rep(0, length(condi)))

Arguments

Details

Integration is done for nodes not belonging to the extracted chain nor being in the conditioning subset. Then the distribution of the retained nodes is left identical to this in the initial chain.

Value

The resulting chain

Examples

 chain4chain(rbmn0chain.02, c("a", "d"), c("b"), 12);

checks a /chain/ object

Description

checks the consistency of chain as a /chain/ object issues a fatal error with some clues if inconsistent.

Usage

check8chain(chain)

Arguments

Details

Looking a the code of this function provides a way to know which are the requirements of a /chain/ object.

Value

TRUE or a character containing some clue about the discovered inconsistency.

Examples

 check8chain(rbmn0chain.01);
 res <- check8chain(rbmn0adja.01);
 if (is.na(as.logical(res))) { print(res);}

checks a /gema/ object

Description

checks the consistency of gema as a /gema/ object issues a fatal error with some clues if inconsistent.

Usage

check8gema(gema)

Arguments

Details

Looking a the code of this function provides a way to know which are the requirements of a /chain/ object.

Value

TRUE or a character containing some clue about the discovered inconsistency.

Examples

 check8gema(rbmn0gema.01);
 res <- check8gema(rbmn0adja.01);
 if (is.na(as.logical(res))) { print(res);}

checks a /nbn/ object

Description

checks the consistency of nbn as a /nbn/ object issues a fatal error with some clues if inconsistent.

Usage

check8nbn(nbn, sto = FALSE)

Arguments

Details

Looking a the code of this function provides a way to know which are the requirements of a /chain/ object.

Value

TRUE or a character containing some clue about the discovered inconsistency.

Examples

 check8nbn(rbmn0nbn.01);
 res <- check8nbn(rbmn0adja.01);
 if (is.na(as.logical(res))) { print(res);}

computes some conditional distribution of a multinormal vector

Description

returns the expectation and variance of a sub-vector conditionned with another (non overlapping) sub-vector from an initial random vector described by mn.

Usage

condi4joint(mn, par, pour, x2=NULL)

Arguments

Details

when no names are given to mn$mu, par and pour are supposed containing indices and default sequential names are provided.

Value

A list:
when x2 provides the values taken by the conditioning part, it is a /mn/ object with its two components: $mu for the expectation vector and $gamma for the variance matrix.
when x2 is NULL the list has got three components: $mu for the fixed part of the expectation vector, $b for the regression coefficients to be associated to the non precised x2 values, varying part of the expectation and $gamma for the variance matrix.

Examples

 print8mn(condi4joint(rbmn0mn.04, c("1.1", "2.2", "1.2", "2.1"), NULL));
 print8mn(condi4joint(rbmn0mn.04, c("1.1", "2.2", "1.2", "2.1"), "C", 0));
 print(condi4joint(rbmn0mn.04, c("1.1", "2.2", "1.2", "2.1"), "C", NULL));

returns the correlation matrix from the variance

Description

returns the correlation matrix from the variance preserving possible variable names

Usage

cor4var(ma)

Arguments

Details

Zero variances are detected and accepted (all associated correlation coefficients are forced to be zero.>>

Value

The correlation matrix

Examples

 cor4var(rbmn0mn.04$gamma);

creates a crossed-nbn from two /nbn/s

Description

A crossed /nbn/ is a /nbn/ obtained when replacing each node of the first /nbn/ by the second /nbn/ and vice-versa.
Let nn1/nn2 and na1/na2 be the node and arc numbers of the two nbns, the node number of the crossed nbn is nn1*nn2 and its arc number is nn1*na2+nn2*na1.
The regression coefficients attributed to the crossed nbn are the products of the weights (we1/we2) and the regression coefficients of the initial nbn.

Usage

crossed4nbn1nbn(nbn1, nbn2, we1=rep(1, length(nbn1)), we2=rep(1, length(nbn2)),
	  nona=as.vector(outer(names(nbn1), names(nbn2), paste,
	  sep="_")))

Arguments

Details

The mu coefficient is the sum of the two corresponding mus of the generating nbn.
The sigma coefficient is the product of the two corresponding sigmas of the generating nbn.
The regression coefficient are directed inherited from the nbn which is duplicated with this arc.

Value

The resulting crossed nbn object.

Examples

 print8nbn(crossed4nbn1nbn(rbmn0nbn.01, rbmn0nbn.04));

Computes the deviance for a sample of multinormal vector

Description

From the n observed values of a vector of size p (Y), their expectations (EY) and the variance matrix (VY) supposed identical for all vectors, returns the deviance, i.e. -2*log(p(Y)).

Usage

dev4mn(Y, EY, VY)

Arguments

Details

When EY is a vector with length ncol(Y) this supposes that all observations have the same expectation.

Value

A scalar

Examples

 dev4mn(matrix(runif(3), 1), t(rbmn0mn.01$mu), rbmn0mn.01$gamma);

returns a score of the difference between two /nbn/s

Description

Returns a positive scalar value measuring, in some way, the difference existing within two /nbn/s sharing the same structure.

Usage

diff8nbn(nbn1, nbn2, type=1, scalar=TRUE)

Arguments

Details

For type==1 it is the canonical euclidian difference between all parameters, including the sigma. The score to use to measure the differences between two successive estimations is not well established (see the code).

Value

Either a scalar or a named vector (according to scalar).

Examples

 diff8nbn(rbmn0nbn.01, rbmn0nbn.01);
 diff8nbn(rbmn0nbn.01, rbmn0nbn.01, scalar=FALSE);

estimates the parameters of a nbn with equality constraints

Description

Estimations of the parameters of a /nbn/ is done when there are some equality constraints onto the regression coefficients.
Constant terms (mu) and conditional standard deviations (sigma) are supposed independent (that is not constrained with equalities).
Equality constraints are given by sarc, a list of matrices with two columns, indicating each the series of arcs having the same regression coefficient.

Usage

estimate8constrainednbn(nbn, sarc, data, imp=0, nite=10, eps=10^-5)

Arguments

Details

Not linked regression coefficients doesn't require to be included in sarc, the function do it by itself.
The score to use to measure the differences between two successive estimations is not well established (see the code).

Value

The resulting /nbn/ object with the estimated parameters.

Examples

 data(boco);
 print8nbn(rbmn0nbn.05);
 print8nbn(estimate8nbn(rbmn0nbn.05, boco));
 print8nbn(estimate8constrainednbn(rbmn0nbn.05, rbmn0crarc.05, boco));

estimating the /nbn/ parameters

Description

From a /nbn/ to describe the DAG, and a data.frame containing the necessary observations, returns the /nbn/ with all its parameters newly estimated.

Usage

estimate8nbn(nbn, data)

Arguments

Details

No constraints are put on the parameters.

Value

The resulting /nbn/ with the estimated parameters.

Examples

 data(boco);
 print8nbn(rbmn0nbn.05);
 print8nbn(estimate8nbn(rbmn0nbn.05, boco));

computes a /mn/ from a /gema/

Description

from a /gema/ object defining a normal Bayesian network, computes the expectation and variance matrix.

Usage

gema2mn(gema)

Arguments

Value

a list with the following components: mu and gamma.

Examples

 print8mn(gema2mn(rbmn0gema.04));

computes a /nbn/ from a /gema/

Description

from a /gema/ object defining a normal Bayesian network, computes more standard /nbn/ where each node is defined from its parents.

Usage

gema2nbn(gema)

Arguments

Details

using general formulae rather a sequential algorithm as done in the original gema2nbn implementation.

Value

the corresponding /nbn/.

Examples

 print8nbn(gema2nbn(rbmn0gema.02));

generation of a /chain/ /nbn/

Description

[randomly] generates a /chain/ /nbn/.

Usage

generate8chain(rnn=c(3, 7), proo=0.5, rcor=c(-1, 1), rmu=c(0, 0), rsig=c(0, 1),
	  nona=r.form3names(max(rnn)))

Arguments

Details

Proposed ranges can be a unique value, implying no randomness in the value.
Roots are placed according to proo probabilities, then collider are placed in between with uniform probability on the possibles nodes.

Value

A /chain/ coding list is returned.

Examples

 set.seed(1234);
 print8chain(generate8chain());
 print8chain(generate8chain());
 print8chain(generate8chain(rnn=10, rcor=0.5));
 print8chain(generate8chain(rnn=10, rcor=0.5));

returns a randomly built /nbn/ object.

Description

To obtain systematic results, you have to call set.seed before hands.

Usage

generate8nbn(rnn=c(3, 7), ppar=0.5, rreg=c(-1, 1), rmu=c(0, 0), rsig=c(0, 1),
	  nona=r.form3names(max(rnn)))

Arguments

Details

Node numbers are uniformly drawn. Parent numbers are independently drawn from all ancestors with the probability associated to the considered node. Regression coefficient are uniformly drawn. Conditional expectations and standard deviations are uniformly drawn.
All range arguments can be given one value instead of two, to precise the unique value to use.

Value

a /nbn/ object, with nodes in topological order.

Examples

 set.seed(1234)
 print8nbn(generate8nbn());
 print8nbn(generate8nbn());

reduces a chain to its inputs and outputs

Description

From a chain returns the reduced chain comprising only inputs (that is root nodes) and outputs (that is colliders and ends which are not roots)

Usage

inout4chain(chain)

Arguments

Value

The resulting chain

Examples

 print8chain(inout4chain(rbmn0chain.02));

Checks if a given /nbn/ is a /chain/

Description

returns TRUE [the order] or FALSE [NULL] according that nbn is a chain of not [according to order].

Usage

is8nbn8chain(nbn, order=FALSE)

Arguments

Value

A logical(1) when order si TRUE if not the resulting chain order versus NULL.

Examples

is8nbn8chain(rbmn0nbn.01);
is8nbn8chain(rbmn0nbn.04);

returns marginal expectations and standard deviations of a chain

Description

From a chain object returns a list with two components: $mu and $sigma vectors of marginal expectations and standard deviations.

Usage

marginal4chain(chain)

Arguments

Value

a list with the two components $mu and $sigma.

Examples

 marginal4chain(rbmn0chain.02);

computes a /gema/ from a /mn/

Description

proposes generating matrices of a Bayesian network from a /mn/ object defining a multinormal distribution by expectation and variance, under the assumption that the nodes are in topological order.

Usage

mn2gema(mn)

Arguments

Value

a list with the /gema/ components $mu and $li.

Examples

 print8gema(mn2gema(rbmn0mn.04));

computes a joint distribution from a marginal and a conditional one for multinormal distributions

Description

returns the expectation and variance of the multinormal normal distribution defined through a marginal subcomponent and a conditional distribution.

Usage

mn4joint1condi(lmar, lcon)

Arguments

Details

The conditional distribution is defined with a list with elements a for the constant part of the expectation; b for the regression coefficient part of the expectation; and S for the residual variance matrix.

Value

A list with elements:

that is a /mn/ object.

Examples

 lcon <- list(a=c(D=2, E=4),
 b=matrix(1:6, 2, dimnames=list(LETTERS[4:5],
 LETTERS[1:3])),
 S=matrix(c(1, 1, 1, 2), 2));

 print8mn(mn4joint1condi(rbmn0mn.01, lcon));

number of Bayesian networks

Description

returns the number of different Bayesian networks having n labelled or not nodes. Non labelled nodes means that nodes are exchangeable: A -> B is identical to A <- B.

Usage

nb8bn(n, label=FALSE)

Arguments

Details

When not labelled nodes, the results were proposed by Sloane in 'the on line encyclopedy of integer sequences' (http://oeis.org/A003087). For labelled nodes, just the application of the recursive formula of Robinson.

Value

Number of Bayesian networks

Examples

 nb8bn(5)
 nb8bn(5, TRUE);

transforms a /nbn/ to a /bn.fit/ of /bnlearn/ package

Description

returns a bn.fit object from a Gaussian nbn object of /rbmn/ package.

Usage

nbn2bnfit(nbn, onlydag=FALSE)

Arguments

Value

The resulting bn.fit (or bn) object.

transforms a /nbn/ into a /chain/

Description

returns the chain obtained from nbn which is supposed to a chain. If it is not a chain, an error is issued.

Usage

nbn2chain(nbn)

Arguments

Details

It is advised to use is8nbn8chain before calling this function.

Value

The resulting chain

Examples

 print8chain(nbn2chain(rbmn0nbn.02));

computes a /gema/ from a /nbn/

Description

from a /nbn/ object defining a normal Bayesian network, computes the vector mu and the matrix li such that if the vector E is a vector of i.i.d. centred and standardized normal, then mu + li * E has the same distribution as the input /nbn/.

Usage

nbn2gema(nbn)

Arguments

Value

a list with the two following components: mu and li.

Examples

 identical(nbn2gema(rbmn0nbn.02), rbmn0gema.02);

computes the joint distribution of a /nbn/

Description

Computes the joint distribution of a /nbn/ with three possible algorithms according to algo.

Usage

nbn2mn(nbn, algo=3)

Arguments

Details

To be explained if it works

Value

the resulting /mn/ object

Examples

 print8mn(nbn2mn(rbmn0nbn.05));

computes the /nbn/ changing its topological order

Description

returns the proposed /nbn/ with a new topological order without modifying the joint distribution of all variables.
This allows to directly find regression formulae within the Gaussian Bayesian networks.

Usage

nbn2nbn(nbn, norder)

Arguments

Details

BE aware that for the moment, no check is made about the topological order and if it is not, the result is FALSE!

Value

The resulting /nbn/.

Examples

 print8mn(nbn2mn(rbmn0nbn.01, algo=1));
 print8mn(nbn2mn(rbmn0nbn.01, algo=2));
 print8mn(nbn2mn(rbmn0nbn.01, algo=3));
 print8mn(nbn2mn(nbn2nbn(rbmn0nbn.02, c(1, 2, 4, 5, 3))));
 print8mn(nbn2mn(nbn2nbn(rbmn0nbn.02, c(4, 1, 2, 3, 5))));

computes standard matrices from a /nbn/

Description

from a /nbn/ object defining a normal Bayesian network, returns a list comprising (i) mm the vector of the mean of the different nodes when the parents are nought, (ii) ss the vector of the conditional standard deviations and (iii) rr the matrix of the regression coefficients of the direct parents (rr[i,j] contains the regression coefficient of the node j for its parents i or zero when i is not a parent of j.

Usage

nbn2rr(nbn)

Arguments

Value

the resulting list with the three components: mm, ss and rr.

Examples

 nbn2rr(rbmn0nbn.01);

From a /nbn/ computes the associated nbn1

Description

returns a /nbn/ object with the same structure as nbn but all $mu are put to zero, all $sigma to one as well as $regcof.

Usage

nbn4nbn(nbn)

Arguments

Details

These coefficient values allows the easy study of the /nbn/ structure.

Value

The resulting nbn.

Examples

 print8nbn(nbn4nbn(rbmn0nbn.04));

a /nbn/ from a regression matrix

Description

reverse of rmatrix4nbn but the standard deviations must be included.

Usage

nbn4rmatrix(rmatrix)

Arguments

Details

mus are put to nought

Value

A /nbn/ object

Examples

 print8nbn(nbn4rmatrix(rmatrix4nbn(rbmn0nbn.02)));

normalizes a /nbn/

Description

returns a nbn with a given expectation and variance through an transformation leaving the correlation unchanged.

Usage

normalize8nbn(nbn, mu=0, sigma=1)

Arguments

Value

The transformed nbn.

Examples

 print8nbn(normalize8nbn(rbmn0nbn.01));

returns a topological order of a /chain/ or checks a proposed order.

Description

From a chain object returns one of the possible topological orders, through a permutation when is.null(ord). If not ord must be a proposed order to be checked given as a permutation if is.numeric(ord) or a vector of ordered names if is.character(ord).

Usage

order4chain(chain, ord=NULL)

Arguments

Details

For the moment the ord option is bad and an error message is returned when used.

Value

a permutation vector of the nodes of the /nbn/ or a named character with the nodes not having their parents before them; when it is of length zero this means that the check was successful.

Examples

 order4chain(rbmn0chain.02);
 order4chain(rbmn0chain.02, order4chain(rbmn0chain.02));

topological order of a /gema/

Description

returns one of the orders of the nodes such as the parents of any node are less ranked than it when is.null(ord). If not check that the proposed order is either a right permutation (is.numeric(ord)) or a vector of node names providing a topological order (is.character(ord)).

Usage

order4gema(gema, ord=NULL)

Arguments

Details

When !is.null(ord) the order must be an order, if not an error is issued.

Value

a permutation vector of the nodes of the /gema/ or a named list with the nodes not having their parents before them. That is a topological order.

Examples

 names(rbmn0gema.04$mu)[order4gema(rbmn0gema.04)];

topological order of a /nbn/

Description

returns one of the orders of the nodes such as the parents of any node are less ranked than it when is.null(ord). If not check that the proposed order is either a right permutation (is.numeric(ord)) or a vector of node names providing a topological order (is.character(ord)).

Usage

order4nbn(nbn, ord=NULL)

Arguments

Details

When !is.null(ord) the order must be an order, if not an error is issued.

Value

a permutation vector of the nodes of the /nbn/ or a named list with the nodes not having their parents before them.

Examples

 names(rbmn0nbn.04)[order4nbn(rbmn0nbn.04)];

prints a /chain/ object

Description

prints a /chain/ object.

Usage

print8chain(chain, digits=3)

Arguments

Details

See nbn2chain code for some details about the definition of a /chain/.

Value

nothing but something is printed

Examples

 print8chain(rbmn0chain.01);
 print8chain(rbmn0chain.02);
 print8chain(rbmn0chain.03);

standard print function for a /gema/ object.

Description

prints a /gema/ object completely or a part of it according to what specification.

Usage

print8gema(gema, what="ml", ordering=NULL, digits=3, printed=TRUE)

Arguments

Value

The gema is printed or a matrix having nn x ? is returned binding which elements are precised in the argument what.

Examples

 print8gema(rbmn0gema.01);
 print8gema(rbmn0gema.02, "m");
 print8gema(rbmn0gema.03, "l", digit=1);
 print8gema(rbmn0gema.04, printed=FALSE);

standard print function for a /mn/ object.

Description

prints a /mn/ object completely or a part of it.

Usage

print8mn(mn, what="msC", ordering=NULL, digits=3, printed=TRUE)

Arguments

Value

The mn is printed or a matrix having nn x ? is returned binding which elements precised in the argument what.

Examples

 print8mn(rbmn0mn.01);

print function for a /nbn/ object.

Description

prints a /nbn/ object.

Usage

print8nbn(nbn, what="pr", digits=3, ordering=NULL, chk=TRUE)

Arguments

Value

Nothing but but nbn is printed.

Examples

 print8nbn(rbmn0nbn.01);
 print8nbn(rbmn0nbn.03, "pm", order=1:2)

Some examplifying structures

Description

Small examples of adjacency matrices, /nbn/, /chain/, /gema/ and /mn/ objects.

Usage

       rbmn0chain.01
       rbmn0chain.02
       rbmn0chain.03
       rbmn0nbn.01
       rbmn0nbn.02
       rbmn0nbn.03
       rbmn0nbn.04
       rbmn0adja.01
       rbmn0adja.02
       rbmn0adja.03
       rbmn0adja.04
       rbmn0mn.01
       rbmn0mn.02
       rbmn0mn.03
       rbmn0mn.04
       rbmn0gema.01
       rbmn0gema.02
       rbmn0gema.03
       rbmn0gema.04

Details

• rbmn0chain.# objects are chain /nbn/ objects

• rbmn0nbn.# objects are general /nbn/ objects

• rbmn0adja.# objects are adjacency matrices

• rbmn0mn.# objects are /mn/ distributions

• rbmn0gema.# objects are /gema/ generating matrices

Every last numbers (#) refer to the same Gaussian Bayesian networks.

Author(s)

Jean-Baptiste Denis

reverses the nodes of a chain

Description

returns the chain obtained after reversing its node order

Usage

reverse8chain(chain)

Arguments

Value

The resulting chain

Examples

 print8chain(rbmn0chain.02);
 print8chain(reverse8chain(rbmn0chain.02));

removes somes nodes from an adjacency matrix

Description

Eliminates from the adjacency matrix (adja)all nodes not breaking the existing links.
Important: the node order in adja must be topological.

Usage

rm8nd4adja(adja, nodes)

Arguments

Details

When a node is removed, all its parents become parent of its children.

Value

The reduced adjacency matrix.

Examples

 rm8nd4adja(rbmn0adja.04, "1.1");

removes some nodes from a /nbn/

Description

returns a /nbn/ object deduced from an original /nbn/ by integrating on a given subset of nodes.

Usage

rm8nd4nbn(nbn, nodes)

Arguments

Details

The transformation is made through the associated joint distributions for the probabilities and with the help of the function rm8nd4adja for the relationships.

Value

The resulting nbn.

Examples

 rm8nd4nbn(rbmn0nbn.04, "1.1"); 

regression matrix of a /nbn/

Description

returns a dimnamed matrix indicating with rho an arc from row to column nodes (0 everywhere else) where rho is the regression coefficient. Also conditional standard deviations can be introduced as diagonal elements but mu coefficient are lost... It is advisable to normalize the /nbn/ first.

Usage

rmatrix4nbn(nbn, stdev=TRUE)

Arguments

Value

A dimnamed matrix

Examples

 rmatrix4nbn(rbmn0nbn.02);
 (rmatrix4nbn(rbmn0nbn.02, FALSE)>0)*1;

simulates from a /gema/ object

Description

returns a matrix of simulated values with the variable in columns and the simulations in rows.

Usage

simulate8gema(gema, nbs)

Arguments

Details

Just the application of the standard formula to a white noise. Variables names are taken from those of gema$mu, when these does not exist, standard ones are provided.

Value

A matrix of size : nbs x length(gema$mu)

Examples

 simulate8gema(rbmn0gema.01, 10);

simulates a multinormal vector with varying expectation

Description

returns a matrix of simulated values with the variable in columns and the simulations in rows.

Usage

simulate8gmn(loi, cova, nbs, tol=1e-7)

Arguments

Details

Just a call to the function simulate8mn, adding the terms to the expectation due to the regression...

Value

A matrix of size : nbs x length(loi$mu)

Examples

 loi <- list(mu=c(D=2, E=4), 
 rho=matrix(1:6, 2, dimnames=list(LETTERS[4:5], 
 LETTERS[1:3])), 
 gamma=matrix(c(1, 1, 1, 2), 2));
 cova <- matrix(runif(36), 12, dimnames=list(NULL, LETTERS[1:3]));
 print(simulate8gmn(loi, cova, 12));

simulates a multinormal vector

Description

returns a matrix of simulated values with the variable in columns and the simulations in rows.

Usage

simulate8mn(mn, nbs, tol=1e-7)

Arguments

Details

Just a call to the basic function mvrnorm. Names of the variables are taken from those of mn$mu, when these does not exist, standard ones are provided.

Value

A matrix/data frame of size : nbs x length(mn$mu)

Examples

 print(simulate8mn(rbmn0mn.01, 12));

simulates from a /nbn/ object

Description

returns a matrix of simulated values with the variable in columns and the simulations in rows.

Usage

simulate8nbn(nbn, nbs)

Arguments

Details

Just the sequential simulations of the nodes

Value

A matrix of size : nbs x length(nbn)

Examples

 simulate8nbn(rbmn0nbn.01, 10);

returns the states of each node of a chain

Description

From a chain object returns a named character precising the role of each node: "r" for root, "c" for collider, "t" for transmitter and "l" for leaf.

Usage

state4chain(chain)

Arguments

Value

a character of the states named with node names.

Examples

 state4chain(rbmn0chain.01);
 state4chain(rbmn0chain.03);

provides so-called string model of a /nbn/

Description

returns a character(1) describing the dag of the nbn under the string form.

Usage

string7dag4nbn(nbn, sep=";")

Arguments

Value

A character(1).

Examples

 string7dag4nbn(rbmn0nbn.01);
 string7dag4nbn(rbmn0nbn.04, sep=", ");

returns the precision matrix from the variance

Description

returns the precision matrix from the variance preserving possible variable names

Usage

var2pre(ma)

Arguments

Details

Non full rank matrices are accepted, a generalized inverse is returned and a warning is issued.

Value

The precision matrix

Examples

 var2pre(rbmn0mn.04$gamma);