PnodeLnAlphas: Access the combination function slope parameters for a Pnode
In ralmond/Peanut: Parameterized Bayesian Networks, Abstract Classes
PnodeLnAlphas R Documentation
Access the combination function slope parameters for a Pnode
Description
In constructing a conditional probability table using the discrete
partial credit framework (see calcDPCTable),
the effective thetas for each parent variable are combined into a
single effect theta using a combination rule. The expression
PnodeAlphas(node) accesses the slope parameters associated with
the combination function PnodeRules(node). The
expression PnodeLnAlphas(node) which is used in
mapDPC.
Usage
PnodeLnAlphas(node)
PnodeLnAlphas(node) <- value
PnodeAlphas(node)
PnodeAlphas(node) <- value
## Default S3 method:
PnodeAlphas(node)
## Default S3 replacement method:
PnodeAlphas(node) <- value
Arguments
node
A Pnode object.
value
A numeric vector of (log) slope parameters or a list of
such vectors (see details). The length of the vector depends on the
combination rules (see PnodeRules). If
a list, it should have length one less than the number of states in
node. For PnodeAlphas(node) <- value, value
should only contain positive numbers.
Details
Following the framework laid out in Almond (2015), the function
calcDPCTable calculates a conditional
probability table using the following steps:
Each set of parent variable states is converted to a set of
continuous values called effective thetas (see
PnodeParentTvals). These are built into an array,
eTheta, using expand.grid where each
column represents a parent variable and each row a possible
configuration of parents.
For each state of the node except the last,
the set of effective thetas is filtered using the local Q-matrix,
PnodeQ(node) = Q. Thus, the actual effect thetas
for state s is eTheta[,Q[s,]].
For each state of the node except the last, the
corresponding rule is applied to the effective thetas to get a
single effective theta for each row of the table. This step is
essentially calls the expression:
do.call(rules[[s]], list(eThetas[,Q[s,]]),
PnodeAlphas(node)[[s]], PnodeBetas(node)[[s]]).
The resulting set of effective thetas are converted into
conditional probabilities using the link function
PnodeLink(node).
The function PnodeRules accesses the function used in step 3.
It should should be the name of a function or a function with the
general signature of a combination function described in
Compensatory. The compensatory function is a
useful model for explaining the roles of the slope parameters,
alpha. Let theta_{i,j} be the effective theta value for the
jth parent variable on the ith row of the effective theta
table, and let alpha_{j} be the corresponding slope parameter.
Then the effective theta for that row is:
Z(theta_{i}) =
(alpha_1theta_{i,1}+\ldots+alpha_Jtheta_{J,1})/C - beta ,
where C=\sqrt(J) is a variance stabilization constant and
beta is a value derived from PnodeBetas. The
functions Conjunctive and
Disjunctive are similar replacing the sum with
a min or max respectively.
In general, when the rule is one of
Compensatory,
Conjunctive, or
Disjunctive, the the value of
PnodeAlphas(node) should be a vector of the same length as the
number of parents. As a special case, if it is a vector of length 1,
then a model with a common slope is used. This looks the same in
calcDPCTable but has a different implication
in mapDPC where the parameters are constrained
to be the same.
The rules OffsetConjunctive, and
OffsetDisjunctive, work somewhat differently,
in that they assume there is a single slope and multiple intercepts.
Thus, the OffsetConjunctive has equation:
Z(theta_{i}) = alpha min(theta_{i,1}-beta_1,
\ldots,theta_{J,1}-beta_{J}) .
In this case the assumption is that PnodeAlphas(node) will be a
scalar and PnodeBetas(node) will be a vector of length
equal to the number of parents.
If the value of PnodeLink is
partialCredit, then the link function can be
different for each state of the node. (If it is
gradedResponse then the curves need to be
parallel and the slopes should be the same.) If the value of
PnodeAlphas(node) is a list (note: list, not numeric vector or
matrix), then a different set of slopes is used for each state
transition. (This is true whether PnodeRules(node) is a
single function or a list of functions. Note that if there is a
different rule for each transition, they could require different
numbers of slope parameters.) The function
calcDPCTable assumes the states are ordered
from highest to lowest, and no transition is needed into the lowest
state.
Node that if the value of PnodeQ(node) is not a matrix
of all TRUE values, then the effective number of parents for
each state transition could be different. In this case the value of
PnodeAlphas(node) should be a list of vectors of different
lengths (corresponding to the number of true entries in each row of
PnodeQ(node)).
Finally, note that if we want the conditional probability table
associated with node to be monotonic, then the
PnodeAlphas(node) must be positive. To ensure this,
mapDPC works with the log of the slopes, not
the raw slopes. Similarly, calcDPCTable
expects the log slope parameters as its lnAlphas argument, not
the raw slopes. For that reason PnodeLnAlphas(node) is
considered the primary function and a default method for
PnodeAlphas(node) which simply takes exponents (or logs in the
setter) is provided. Note that a sensible range for the slope
parameters is usually between 1/2 and 2, with 1 (0 on the log scale) as
a sensible first pass value.
Value
A list of numeric vectors giving the slopes for the combination
function of each state transition. The vectors may be of different
lengths depending on the value of PnodeRules(node) and
PnodeQ(node). If the slopes are the same for all
transitions (as is required with the
gradedResponse link function) then a single
numeric vector instead of a list is returned.
Note that the setter form may destructively modify the Pnode object
(this depends on the implementation).
Note
The functions PnodeLnAlphas and PnodeLnAlphas<- are
abstract generic functions, and need specific implementations.
The default methods for the functions PnodeAlphas and
PnodeAlphas<-. Depend on PnodeLnAlphas and
PnodeLnAlphas<-, respectively. See the
PNetica-package for an example.
The values of PnodeLink, PnodeRules,
PnodeQ, PnodeParentTvals,
PnodeLnAlphas, and PnodeBetas all need to
be consistent for this to work correctly, but no error checking is
done on any of the setter methods.
Author(s)
Russell Almond
References
Almond, R. G. (2015) An IRT-based Parameterization for Conditional
Probability Tables. Paper presented at the 2015 Bayesian Application
Workshop at the Uncertainty in Artificial Intelligence Conference.
Almond, R.G., Mislevy, R.J., Steinberg, L.S., Williamson, D.M. and
Yan, D. (2015) Bayesian Networks in Educational Assessment.
Springer. Chapter 8.
See Also
Pnode, PnodeQ,
PnodeRules, PnodeLink,
PnodeBetas, BuildTable,
PnodeParentTvals, maxCPTParam
calcDPCTable, mapDPC
Compensatory,
OffsetConjunctive
Examples
## Not run:
library(PNetica) ## Requires implementation
sess <- NeticaSession()
startSession(sess)
tNet <- CreateNetwork("TestNet",sess)
theta1 <- NewDiscreteNode(tNet,"theta1",
c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta1) <- effectiveThetas(PnodeNumStates(theta1))
PnodeProbs(theta1) <- rep(1/PnodeNumStates(theta1),PnodeNumStates(theta1))
theta2 <- NewDiscreteNode(tNet,"theta2",
c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta2) <- effectiveThetas(PnodeNumStates(theta2))
PnodeProbs(theta2) <- rep(1/PnodeNumStates(theta1),PnodeNumStates(theta2))
partial3 <- NewDiscreteNode(tNet,"partial3",
c("FullCredit","PartialCredit","NoCredit"))
PnodeParents(partial3) <- list(theta1,theta2)
## Usual way to set rules is in constructor
partial3 <- Pnode(partial3,rules="Compensatory", link="partialCredit")
PnodePriorWeight(partial3) <- 10
BuildTable(partial3)
## slopes of 1 for both transitions
PnodeLnAlphas(partial3) <- c(0,0)
BuildTable(partial3)
## log slope 0 = slope 1
stopifnot(
all(abs(PnodeAlphas(partial3) -1) <.0001)
)
## Make Skill 1 more important than Skill 2
PnodeLnAlphas(partial3) <- c(.25,-.25)
BuildTable(partial3)
## Make Skill 1 more important for the transition to ParitalCredit
## And Skill 2 more important for the transition to FullCredit
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
PartialCredit=c(.25,-.25))
BuildTable(partial3)
## Set up so that first skill only needed for first transition, second
## skill for second transition; Adjust alphas to match
PnodeQ(partial3) <- matrix(c(TRUE,TRUE,
TRUE,FALSE), 2,2, byrow=TRUE)
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
PartialCredit=0)
BuildTable(partial3)
## Using OffsetConjunctive rule requires single slope
PnodeRules(partial3) <- "OffsetConjunctive"
## Single slope parameter for each transition
PnodeLnAlphas(partial3) <- 0
PnodeQ(partial3) <- TRUE
PnodeBetas(partial3) <- c(0,1)
BuildTable(partial3)
## Separate slope parameter for each transition;
## Note this will only different from the previous transition when
## mapDPC is called. In the former case, it will learn a single slope
## parameter, in the latter, it will learn a different slope for each
## transition.
PnodeLnAlphas(partial3) <- list(0,0)
BuildTable(partial3)
DeleteNetwork(tNet)
## End(Not run)
ralmond/Peanut documentation built on Sept. 19, 2023, 8:27 a.m.
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| PnodeLnAlphas | R Documentation |
Access the combination function slope parameters for a Pnode
Description
In constructing a conditional probability table using the discrete
partial credit framework (see calcDPCTable),
the effective thetas for each parent variable are combined into a
single effect theta using a combination rule. The expression
PnodeAlphas(node) accesses the slope parameters associated with
the combination function PnodeRules(node). The
expression PnodeLnAlphas(node) which is used in
mapDPC.
Usage
PnodeLnAlphas(node)
PnodeLnAlphas(node) <- value
PnodeAlphas(node)
PnodeAlphas(node) <- value
## Default S3 method:
PnodeAlphas(node)
## Default S3 replacement method:
PnodeAlphas(node) <- value
Arguments
node |
A |
value |
A numeric vector of (log) slope parameters or a list of
such vectors (see details). The length of the vector depends on the
combination rules (see |
Details
Following the framework laid out in Almond (2015), the function
calcDPCTable calculates a conditional
probability table using the following steps:
Each set of parent variable states is converted to a set of continuous values called effective thetas (see
PnodeParentTvals). These are built into an array,eTheta, usingexpand.gridwhere each column represents a parent variable and each row a possible configuration of parents.For each state of the
nodeexcept the last, the set of effective thetas is filtered using the local Q-matrix,PnodeQ(node) = Q. Thus, the actual effect thetas for statesiseTheta[,Q[s,]].For each state of the
nodeexcept the last, the corresponding rule is applied to the effective thetas to get a single effective theta for each row of the table. This step is essentially calls the expression:do.call(rules[[s]],list(eThetas[,Q[s,]]),PnodeAlphas(node)[[s]],PnodeBetas(node)[[s]]).The resulting set of effective thetas are converted into conditional probabilities using the link function
PnodeLink(node).
The function PnodeRules accesses the function used in step 3.
It should should be the name of a function or a function with the
general signature of a combination function described in
Compensatory. The compensatory function is a
useful model for explaining the roles of the slope parameters,
alpha. Let theta_{i,j} be the effective theta value for the
jth parent variable on the ith row of the effective theta
table, and let alpha_{j} be the corresponding slope parameter.
Then the effective theta for that row is:
Z(theta_{i}) =
(alpha_1theta_{i,1}+\ldots+alpha_Jtheta_{J,1})/C - beta ,
where C=\sqrt(J) is a variance stabilization constant and
beta is a value derived from PnodeBetas. The
functions Conjunctive and
Disjunctive are similar replacing the sum with
a min or max respectively.
In general, when the rule is one of
Compensatory,
Conjunctive, or
Disjunctive, the the value of
PnodeAlphas(node) should be a vector of the same length as the
number of parents. As a special case, if it is a vector of length 1,
then a model with a common slope is used. This looks the same in
calcDPCTable but has a different implication
in mapDPC where the parameters are constrained
to be the same.
The rules OffsetConjunctive, and
OffsetDisjunctive, work somewhat differently,
in that they assume there is a single slope and multiple intercepts.
Thus, the OffsetConjunctive has equation:
Z(theta_{i}) = alpha min(theta_{i,1}-beta_1,
\ldots,theta_{J,1}-beta_{J}) .
In this case the assumption is that PnodeAlphas(node) will be a
scalar and PnodeBetas(node) will be a vector of length
equal to the number of parents.
If the value of PnodeLink is
partialCredit, then the link function can be
different for each state of the node. (If it is
gradedResponse then the curves need to be
parallel and the slopes should be the same.) If the value of
PnodeAlphas(node) is a list (note: list, not numeric vector or
matrix), then a different set of slopes is used for each state
transition. (This is true whether PnodeRules(node) is a
single function or a list of functions. Note that if there is a
different rule for each transition, they could require different
numbers of slope parameters.) The function
calcDPCTable assumes the states are ordered
from highest to lowest, and no transition is needed into the lowest
state.
Node that if the value of PnodeQ(node) is not a matrix
of all TRUE values, then the effective number of parents for
each state transition could be different. In this case the value of
PnodeAlphas(node) should be a list of vectors of different
lengths (corresponding to the number of true entries in each row of
PnodeQ(node)).
Finally, note that if we want the conditional probability table
associated with node to be monotonic, then the
PnodeAlphas(node) must be positive. To ensure this,
mapDPC works with the log of the slopes, not
the raw slopes. Similarly, calcDPCTable
expects the log slope parameters as its lnAlphas argument, not
the raw slopes. For that reason PnodeLnAlphas(node) is
considered the primary function and a default method for
PnodeAlphas(node) which simply takes exponents (or logs in the
setter) is provided. Note that a sensible range for the slope
parameters is usually between 1/2 and 2, with 1 (0 on the log scale) as
a sensible first pass value.
Value
A list of numeric vectors giving the slopes for the combination
function of each state transition. The vectors may be of different
lengths depending on the value of PnodeRules(node) and
PnodeQ(node). If the slopes are the same for all
transitions (as is required with the
gradedResponse link function) then a single
numeric vector instead of a list is returned.
Note that the setter form may destructively modify the Pnode object (this depends on the implementation).
Note
The functions PnodeLnAlphas and PnodeLnAlphas<- are
abstract generic functions, and need specific implementations.
The default methods for the functions PnodeAlphas and
PnodeAlphas<-. Depend on PnodeLnAlphas and
PnodeLnAlphas<-, respectively. See the
PNetica-package for an example.
The values of PnodeLink, PnodeRules,
PnodeQ, PnodeParentTvals,
PnodeLnAlphas, and PnodeBetas all need to
be consistent for this to work correctly, but no error checking is
done on any of the setter methods.
Author(s)
Russell Almond
References
Almond, R. G. (2015) An IRT-based Parameterization for Conditional Probability Tables. Paper presented at the 2015 Bayesian Application Workshop at the Uncertainty in Artificial Intelligence Conference.
Almond, R.G., Mislevy, R.J., Steinberg, L.S., Williamson, D.M. and Yan, D. (2015) Bayesian Networks in Educational Assessment. Springer. Chapter 8.
See Also
Pnode, PnodeQ,
PnodeRules, PnodeLink,
PnodeBetas, BuildTable,
PnodeParentTvals, maxCPTParam
calcDPCTable, mapDPC
Compensatory,
OffsetConjunctive
Examples
## Not run:
library(PNetica) ## Requires implementation
sess <- NeticaSession()
startSession(sess)
tNet <- CreateNetwork("TestNet",sess)
theta1 <- NewDiscreteNode(tNet,"theta1",
c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta1) <- effectiveThetas(PnodeNumStates(theta1))
PnodeProbs(theta1) <- rep(1/PnodeNumStates(theta1),PnodeNumStates(theta1))
theta2 <- NewDiscreteNode(tNet,"theta2",
c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta2) <- effectiveThetas(PnodeNumStates(theta2))
PnodeProbs(theta2) <- rep(1/PnodeNumStates(theta1),PnodeNumStates(theta2))
partial3 <- NewDiscreteNode(tNet,"partial3",
c("FullCredit","PartialCredit","NoCredit"))
PnodeParents(partial3) <- list(theta1,theta2)
## Usual way to set rules is in constructor
partial3 <- Pnode(partial3,rules="Compensatory", link="partialCredit")
PnodePriorWeight(partial3) <- 10
BuildTable(partial3)
## slopes of 1 for both transitions
PnodeLnAlphas(partial3) <- c(0,0)
BuildTable(partial3)
## log slope 0 = slope 1
stopifnot(
all(abs(PnodeAlphas(partial3) -1) <.0001)
)
## Make Skill 1 more important than Skill 2
PnodeLnAlphas(partial3) <- c(.25,-.25)
BuildTable(partial3)
## Make Skill 1 more important for the transition to ParitalCredit
## And Skill 2 more important for the transition to FullCredit
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
PartialCredit=c(.25,-.25))
BuildTable(partial3)
## Set up so that first skill only needed for first transition, second
## skill for second transition; Adjust alphas to match
PnodeQ(partial3) <- matrix(c(TRUE,TRUE,
TRUE,FALSE), 2,2, byrow=TRUE)
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
PartialCredit=0)
BuildTable(partial3)
## Using OffsetConjunctive rule requires single slope
PnodeRules(partial3) <- "OffsetConjunctive"
## Single slope parameter for each transition
PnodeLnAlphas(partial3) <- 0
PnodeQ(partial3) <- TRUE
PnodeBetas(partial3) <- c(0,1)
BuildTable(partial3)
## Separate slope parameter for each transition;
## Note this will only different from the previous transition when
## mapDPC is called. In the former case, it will learn a single slope
## parameter, in the latter, it will learn a different slope for each
## transition.
PnodeLnAlphas(partial3) <- list(0,0)
BuildTable(partial3)
DeleteNetwork(tNet)
## End(Not run)
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