Package 'kstMatrix'

Title: Basic Functions in Knowledge Space Theory Using Matrix Representation
Description: Knowledge space theory by Doignon and Falmagne (1999) <doi:10.1007/978-3-642-58625-5> is a set- and order-theoretical framework, which proposes mathematical formalisms to operationalize knowledge structures in a particular domain. The 'kstMatrix' package provides basic functionalities to generate, handle, and manipulate knowledge structures and knowledge spaces. Opposed to the 'kst' package, 'kstMatrix' uses matrix representations for knowledge structures. Furthermore, 'kstMatrix' contains several knowledge spaces developed by the research group around Cornelia Dowling through querying experts.
Authors: Cord Hockemeyer [aut, cre], Peter Steiner [aut], Wai Wong [aut]
Maintainer: Cord Hockemeyer <[email protected]>
License: GPL-3
Version: 2.3-2
Built: 2026-05-26 06:16:55 UTC
Source: https://github.com/cran/kstMatrix

Help Index

Knowledge spaces on AutoCAD knowledge

Description

Bases of knowledge spaces on AutoCAD knowledge obtained from querying experts.

Usage

cad

Format

A list containing seven bases (cad1 to cad6, and cadmaj) in binary matrix form. Each matrix has 28 columns representing the different knowledge items and a varying number of rows containing the basis elements.

Details

Six experts were queried about prerequisite relationships between 28 AutoCAD knowledge items (Dowling, 1991; 1993). A seventh basis represents those prerequisite relationships on which the majority (4 out of 6) of the experts agree (Dowling & Hockemeyer, 1998).

References

Dowling, C. E. (1991). Constructing Knowledge Structures from the Judgements of Experts. Habilitationsschrift, Technische Universität Carolo-Wilhelmina, Braunschweig, Germany.

Dowling, C. E. (1993). Applying the basis of a knowledge space for controlling the questioning of an expert. Journal of Mathematical Psychology, 37, 21–48.

Dowling, C. E. & Hockemeyer, C. (1998). Computing the intersection of knowledge spaces using only their basis. In Cornelia E. Dowling, Fred S. Roberts, & Peter Theuns, editors, Recent Progress in Mathematical Psychology, pp. 133–141. Lawrence Erlbaum Associates Ltd., Mahwah, NJ.

See Also

Other Data: fractions, phsg, readwrite, xpl

Knowledge spaces on fractions

Description

Bases of knowledge spaces on fractions obtained from querying experts.

Usage

fractions

Format

A list containing four bases (frac1 to frac3, and fracmaj) in binary matrix form. Each matrix has 77 columns representing the different knowledge items and a varying number of rows containing the basis elements.

Details

Three experts were queried about prerequisite relationships between 77 items on fractions (Baumunk & Dowling, 1997). A forth basis represents those prerequisite relationships on which the majority of the experts agree (Dowling & Hockemeyer, 1998).

References

Baumunk, K. & Dowling, C. E. (1997). Validity of spaces for assessing knowledge about fractions. Journal of Mathematical Psychology, 41, 99–105.

Dowling, C. E. & Hockemeyer, C. (1998). Computing the intersection of knowledge spaces using only their basis. In Cornelia E. Dowling, Fred S. Roberts, & Peter Theuns, editors, Recent Progress in Mathematical Psychology, pp. 133–141. Lawrence Erlbaum Associates Ltd., Mahwah, NJ.

See Also

Other Data: cad, phsg, readwrite, xpl

Perform a probabilistic knowledge assessment

Description

kmassess performs a probabilistic knowledge assessment for a given response vector, knowledge structure, and BLIM parameters.

kmsassess performs a simplified probabilistic knowledge assessment for a given response vector, knowledge structure, and BLIM parameters. It assumes an equal probability distribution over the knowledge structure as starting point and identical beta and eta values for all items.

Usage

kmassess(
  r,
  pks,
  questioning,
  update,
  beta,
  eta,
  zeta0,
  zeta1,
  threshold,
  probdev = FALSE,
  directory = tempdir()
)

kmsassess(
  r,
  ks,
  questioning,
  update,
  beta,
  eta,
  zeta0,
  zeta1,
  threshold,
  probdev = FALSE,
  directory = NULL
)

Arguments

r

Response pattern (binary vector)
pks

Probabilistic knowledge structure: a data frame with a probability distribution in the first columns and the structure matrix in the subsequent columns.
questioning

Questioning rule ("halfsplit" o "informative")
update

Update rule ("Bayesian" or "multiplicative")
beta

Careless error probability
eta

Lucky guess probability
zeta0

Update parameter for wrong responses
zeta1

Update parameter for correct responses
threshold

Probability threshold for stopping criterion
probdev

Provide information on the probability development including Hasse diagrams stored in tempdir(). Defaults to FALSE.
directory

Where to store the Hasse diagrams.
ks

Knowledge structure: a binary matrix

Details

kmassess implements the stochastic assessment procedures according to Doignon & Falmagne, 1999, chapter 10.

kmassess stops if the number of questions has reached twice the number of items.

Value

A list with the following elements:

state

Diagnosed knowledge state (binary vector)

probs

Resulting probability distribution. If probdev is set to TRUE, a list of probability distributions for each step is given instead.

queried

Sequence of items used in the assessment (list)

qtime

Average time for finding a question

utime

Average time for updating the probabilities

A list with the following elements:

state

Diagnosed knowledge state (binary vector)

probs

Resulting probability distribution. If probdev is set to TRUE, a list of probability distributions for each step is given instead.

queried

Sequence of items used in the assessment (list)

qtime

Average time for finding a question

utime

Average time for updating the probabilities

Background

Doignon & Falmagne (1985, 1999) proposed knowledge space theory originally with adaptive knowledge assessment in mind. The basic idea is to apply prerequisite relationships between items for reducing the number of problems to be posed to a learner in knowledge assessment.

Falmagne & Doignon (1988; Doignon & Falmange, 1999, chapte 10) proposed a class of stochastic procdures for such adaptive assessment which take into account that careless errors and lucky guesses may happen during the assessment by estimating a probability distribution over the knowledge structure. Such an assessment consists of three important parts

  • Question rule

  • Update rule

  • Stopping criterion

For the question rule, they propose the halfsplit and the infomrative rules, implemented in kmassesshalfslit and kmassessinfomrative.

For the update rule, they again propose two possibilities there the multiplicative rule is a generalisation of the (classical) Bayesian update rule implemented here in kmassessmultiplicative and kmassessbayesian, respectively.

As stopping criterion, usually a threshold for the maximal probability for one knowledge state is used. It is strongly recommended to keep this larger than 0.5 in order to have one unequivocal resulting state (see also Hockemeyer, 2002).

Framework of assessment functions within the kstMatrix package:

The founding stones are the four aforementioned functions for finding suitable questions and for updating the probability estimates, respectively. They could also be used in an interactive system, e.g. a Shiny app, for "real" adaptive assessment.

The remaining three assessment functions serve for mere simulation of adaptive assessment. kmassess takes, among others, a full response pattern as parameter and takes the responses for the selected questions from this vector. kmsassess is a simplified version where the update parameters (beta and eta for Bayesian or zeta0 and zeta1 for multiplicative update, respectively) are identical for all items whereas they are item-specific in kmassess. Finally, kmassesssimulation takes a whole data set, i.e. a collection of response patterns, and does an assessment for each of these patterns. Its result is a data frame which should be suitable for further statistical evaluation, especially if it is called several times with variant parameters (e.g., structures, update parameters, update and question rules).

Both, kmsassess and kmassesssimulation call kmassess.

Problems

In rare cases kmassess may flip forth and back between probability distributions resulting in an endless loop. Therefore, it stops after twice the number of items delivering a NULL result.

References

Doignon, J.-P. & Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machne-Studies, 23, 175-196. doi:10.1016/S0020-7373(85)80031-6.

Doignon, J.-P. & Falmagne, J.-C. (1999). Knowledge Spaces. Springer Verlag, Berlin. doi:10.1007/978-3-642-58625-5.

Falmagne, J.-C. & Doignon, J.-P. (1988). A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology, 41, 1-23. doi:10.1111/j.2044-8317.1988.tb00884.x.

Hoxkemeyer, C. (2002). A comparison of non-deterministic procedures for the adaptive assessment of knowledge. Psychlogische Beiträge, 44(4), 495-503.

See Also

Other Knowledge assessment: kmassessbayesian(), kmassesshalfsplit(), kmassessinformative(), kmassessmentsimulation(), kmassessmultiplicative()

Other Knowledge assessment: kmassessbayesian(), kmassesshalfsplit(), kmassessinformative(), kmassessmentsimulation(), kmassessmultiplicative()

Examples

kmassess(c(1, 1, 0, 0),
         cbind(as.data.frame(as.matrix(rep(1/9.0, 9), ncol=1)), xpl$space),
         "halfsplit",
         "Bayesian",
         rep(0.12, 4),
         rep(0.1, 4),
         NULL,
         NULL,
         0.55
        )

kmsassess(c(1,1,0,0), xpl$space, "halfsplit", "Bayesian", 0.1, 0.1, NULL, NULL, 0.55)

Update probability distribution applying Bayesian update

Description

kmassessbayesian updates a probability distribution over a knowledge structure according to the Bayesian update rule.

Usage

kmassessbayesian(probs, ks, beta, eta, question, response)

Arguments

probs

Probability distribution over the knowledge structure (vector)
ks

Binary matrix of the knowledge structure
beta

Vector of careless error probabilities
eta

Vector of lucky guess probabilities
question

Item that has been posed
response

Correctness of received response (0 or 1)

Value

Updated probability vector

See Also

Other Knowledge assessment: kmassess(), kmassesshalfsplit(), kmassessinformative(), kmassessmentsimulation(), kmassessmultiplicative()

Examples

kmassessbayesian(c(0.02, 0.1, 0.07, 0.01, 0.4, 0.17, 0.07, 0.08, 0.08),
                 xpl$space,
                 rep(0.2,4),
                 rep(0.1,4),
                 3,
                 1
                )

Determine next question for probabilistic knowledge assessment

Description

kmassesshalfsplit determines the next question in a probabilistic assessment according to the halfsplit rule.

Usage

kmassesshalfsplit(probs, ks)

Arguments

probs

Probability distribution over the knowledge structure (vector)
ks

Binary matrix of the knowledge structure

Value

Number of the selected question

See Also

Other Knowledge assessment: kmassess(), kmassessbayesian(), kmassessinformative(), kmassessmentsimulation(), kmassessmultiplicative()

Examples

kmassesshalfsplit(c(0.02, 0.1, 0.07, 0.01, 0.4, 0.17, 0.07, 0.08, 0.08),
                  xpl$space)

Determine next question for probabilistic knowledge assessment

Description

kmassessinfmrative determines the next question in a probabiliststic assessment according to the informative rule.

Usage

kmassessinformative(probs, ks, update, beta, eta, zeta0, zeta1)

Arguments

probs

Probability distribution over the knowledge structure (vector)
ks

Binary matrix of the knowledge structure
update

Update rule ("Bayesian" or "multiplicative")
beta

Careless error probabilities (vector)
eta

Lucky guess probabilities (vector)
zeta0

Vector of update parameters for wrong responses
zeta1

Vector of update parameters for correct responses

Value

Number of the selected question

See Also

Other Knowledge assessment: kmassess(), kmassessbayesian(), kmassesshalfsplit(), kmassessmentsimulation(), kmassessmultiplicative()

Examples

kmassessinformative(c(0.02, 0.1, 0.07, 0.01, 0.4, 0.17, 0.07, 0.08, 0.08),
                  xpl$space,
                  "Bayesian",
                  rep(0.3,4),
                  rep(0.2,4),
                  NULL,
                  NULL
                 )

Simulate assessments for a set of response patterns

Description

kmassessmentsimulation does a probabilistic knowledge assessment for each response pattern in a data matrix and stores information about the assessment.

Usage

kmassessmentsimulation(
  respdata,
  ks,
  questioning,
  update,
  beta,
  eta,
  zeta0,
  zeta1,
  threshold
)

Arguments

respdata

Data matrix
ks

Knowledge structure
questioning

Question rule
update

Updating rule
beta

Careless error probability
eta

Lucky guess probability
zeta0

Update parameter for wrong responses
zeta1

Update parameter for correct responses
threshold

Stopping criterion

Details

kmassessmentsimulation applies the kmsassess function.

Value

Assessment data as data frame

See Also

Other Knowledge assessment: kmassess(), kmassessbayesian(), kmassesshalfsplit(), kmassessinformative(), kmassessmultiplicative()

Examples

kmassessmentsimulation(
  xpl$data,
  xpl$space,
  "halfsplit",
  "multiplicative",
  NULL,
  NULL,
  5,
  5,
  0.55
)

Update probability distribution applying multiplicative rule

Description

kmassessmultiplicative updates a probability distribution on a knowledge structure according to the multiplicative rule.

Usage

kmassessmultiplicative(probs, ks, zeta0, zeta1, question, response)

Arguments

probs

Probability distribution over the knowledge structure (vector)
ks

Binary matrix of the knowledge structure
zeta0

Vector of update parameters for wrong responses
zeta1

Vector of update parameters for correct responses
question

Item that has been posed
response

Correctness of received response (0 or 1)

Value

Updated probability vector

See Also

Other Knowledge assessment: kmassess(), kmassessbayesian(), kmassesshalfsplit(), kmassessinformative(), kmassessmentsimulation()

Examples

kmassessmultiplicative(c(0.02, 0.1, 0.07, 0.01, 0.4, 0.17, 0.07, 0.08, 0.08),
                 xpl$space,
                 rep(1.2,4),
                 rep(2.1,4),
                 3,
                 1
                )

Compute the basis of a knowledge space

Description

kmbasis.matrix returns a matrix representing the basis of a knowledge space. If x is a knowledge structure or an arbitrary family of sets kmbasis returns the basis of the smallest knowledge space containing x.

kmbasis.kmsurmiserelation takes a surmise relation and returns the corresponding basis.

kmbasis.matrix returns a matrix representing the basis of a knowledge space. If x is a knowledge structure or an arbitrary family of sets kmbasis returns the basis of the smallest knowledge space containing x.

Usage

kmbasis(x)

## S3 method for class 'kmsurmisefunction'
kmbasis(x)

## S3 method for class 'kmsurmiserelation'
kmbasis(x)

## S3 method for class 'matrix'
kmbasis(x)

Arguments

x

Binary matrix representing a knowledge space

Value

Binary matrix representing the basis of the knowledge space.

Basis

Basis

Binary matrix representing the basis of the knowledge space.

See Also

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Examples

kmbasis(xpl$space)

kmbasis(xpl$space)

Compute the fringe of a state within a knowledge structure using its basis

Description

kmbasisfringe computes the fringe of a state within a knowledge structure, i.e. the set of items by which the state differs from its neighbours.

Usage

kmbasisfringe(state, basis)

kmbasisinnerfringe(state, basis)

kmbasisouterfringe(state, basis)

Arguments

state

Binary vector representing a knowledge state
basis

kmbasis object

Value

Binary vector representing the fringe

References

Hockemeyer C (1997). Using the Basis of a Knowledge Space for Determining the Fringe of a Knowledge State. Journal of Mathematical Psychology, 41, 275–279.

See Also

Other Fringes & learning paths: kmbasisneighbourhood(), kmfringe(), kmgradations(), kmlearningpaths(), kmneighbourhood(), kmnneighbourhood()

Examples

kmbasisfringe(c(1,0,0,0), xpl$basis)

Compute the neighbourhod of a state within a knowledge structure using its basis

Description

kmbasisneighbourhood computes the neighbourhood of a state within a knowledge structure, i.e. the family of all other states with a symmetric set difference of 1.

Usage

kmbasisneighbourhood(state, basis, include = FALSE)

Arguments

state

Binary vector representing a knowledge state
basis

kmbasis object
include

Boolean whether the original state should be included in the result (default FALSE)

Value

Matrix containing the neighbouring states, one per row

See Also

Other Fringes & learning paths: kmbasisfringe(), kmfringe(), kmgradations(), kmlearningpaths(), kmneighbourhood(), kmnneighbourhood()

Examples

kmbasisneighbourhood(c(1,1,0,0), xpl$basis)

Determine a color vector based on probabilities

Description

kmcolors takes a probabilty vector and a color palette and creates a color vector to be used with kstMatrix::plot.

Usage

kmcolors(prob, palette = cm.colors)

Arguments

prob

Probability vector
palette

Color palette (default = cm.colors)

See Also

Other Plotting knowledge structures: plot()

Other Utilities: kmdoubleequal(), kmsetiselement(), kmsymmsetdiff()

Compute the distance between a data set and a knowledge structure

Description

kmdist returns a named vector with the frequencies of distances between a set of response patterns and a knowledge structure. This vector can be used to compute, e.g., the Discrepancy Index (DI) or the Distance Agreement Coefficient (DA).

Usage

kmdist(data, struct)

Arguments

data

Binary matrix representing a set of response patterns
struct

Binary matrix representing a knowledge structure

Value

Distance distribution vector

See Also

Other Validating knowledge spaces: kmSRvalidate(), kmvalidate()

Examples

kmdist(xpl$data, xpl$space)

Test two double numbers on equity with a certain tolerance

Description

Test two double numbers on equity with a certain tolerance

Usage

kmdoubleequal(x, y, tol = sqrt(.Machine$double.eps))

Arguments

x

First double to compare
y

Second double to compare
tol

Tolerance optional)

Value

Boolean for (approximate) equity

See Also

Other Utilities: kmcolors(), kmsetiselement(), kmsymmsetdiff()

Examples

kmdoubleequal(0.5+0.5, 1)

Reduce a family of knowledge states with respect to item equivalence

Description

kmeqreduction takes a family of knowledge states and returns its reduction to non-equivalent items.

Usage

kmeqreduction(x)

Arguments

x

Binary matrix

Value

Binary matrix reduced by equivalences

See Also

Other Properties of knowledge structures: kmiswellgraded(), kmnotions()

Examples

kmeqreduction(xpl$space)

Convert a binary matrix to a kmfamset object (family of sets)

Description

kmfamset returns a kmfamset object after checking that the passed object is a binary matrix with all different rows. If the passe object inherits the kmfamset property, nothing else is changed.

Usage

kmfamset(x)

Arguments

x

Binary matrix representing a family of sets

Value

kmfamset object

See Also

Other Constructors: kmspace(), kmstructure()

Examples

m <- as.matrix(c(1,0,0,0,1,0,1,1,1), nrow=3, byrow=TRUE)
kmfamset(m)

Compute the fringe of a state within a knowledge structure

Description

kmfringe computes the fringe of a state within a knowledge structure, i.e. the set of items by which the state differs from its neighbours.

Usage

kmfringe(state, struct)

kminnerfringe(state, struct)

kmouterfringe(state, struct)

Arguments

state

Binary vector representing a knowledge state
struct

Binary matrix representing a knowledge structure

Value

Binary vector representing the fringe

See Also

Other Fringes & learning paths: kmbasisfringe(), kmbasisneighbourhood(), kmgradations(), kmlearningpaths(), kmneighbourhood(), kmnneighbourhood()

Examples

kmfringe(c(1,0,0,0), xpl$space)

Generate a knowledge structure from a set of response patterns

Description

kmgenerate returns a matrix representing a knowledge structure generated from data. It uses a simplistic approach: patterns with a frequency above a specified threshold are considered as knowledge states. If the specified threshold is 0 (default) a real threshold is computed as N (number of response patterns) divided by 2^|Q|. Please note that the number of response patterns should be much higher than the size of the power set of the item set Q. A factor of art least 10 is recommended. Currently, the number of items is limited to the number of bits in a C long minus one (i.e. 31 under Windows and 63 otherwise). But we would probably run into memory problems way earlier anyway.

Usage

kmgenerate(x, threshold = 0)

Arguments

x

Binary matrix representing a data set
threshold

Threshold for taking response patterns as knowledge states (default 0)

Value

Binary matrix representing the generated knowledge structure

See Also

Other Generating knowledge spaces: kmiita2SR()

Examples

kmgenerate(xpl$sim, 15)

Determine all gradations between two states

Description

Determine all gradations between two states

Usage

kmgradations(structure, from = NULL, to = NULL)

Arguments

structure

Knowledge structure
from

Starting state (if NULL (default), it is the empty set)
to

Goal state (if NULL (default), it is the full item set)

Value

A list of gradations where each gradation is a list of states

See Also

Other Fringes & learning paths: kmbasisfringe(), kmbasisneighbourhood(), kmfringe(), kmlearningpaths(), kmneighbourhood(), kmnneighbourhood()

Examples

kmgradations(xpl$space)

Convert an IITA result into a surmise relation matrix

Description

kmiita2SR takes the result of a DAKS::iita() call and delivers the matrix of the computed surmise relation.

Usage

kmiita2SR(ii, names = NULL, items = 0)

Arguments

ii

iita() result
names

Vector of item names (default NULL)
items

Minimal number of items (default 0)

Value

Surmise relation matrix

The iita() function looses information on the item names and uses consecutive numbers instead. The implications part of its result does not give any hint on isolated items, i.e. items which# neither have a prerequisite nor are prerequisite of any other item. Therefore, a minimal number of items can be passed to kmiita2SR(). If the highest item number within implications is higher, this items parameter is ignored.

See Also

Other Generating knowledge spaces: kmgenerate()

Check for wellgradedness of a knowledge structure

Description

kmiswellgraded returns whether a knowledge structure is wellgraded.

Usage

kmiswellgraded(x)

Arguments

x

Binary matrix representing a knowledge space

Value

Logical value specifying whether x is wellgraded

References

Doignon, J.-P. & Falmagne, J.-C. (1999). Knowledge Spaces. Springer–Verlag, Berlin.

See Also

Other Properties of knowledge structures: kmeqreduction(), kmnotions()

Examples

kmiswellgraded(xpl$space)

Determine all learning paths in a knowledge structure

Description

Determine all learning paths in a knowledge structure

Usage

kmlearningpaths(structure)

Arguments

structure

Knowledge structure

Value

A list of learning paths where each learning path is a list of states

See Also

Other Fringes & learning paths: kmbasisfringe(), kmbasisneighbourhood(), kmfringe(), kmgradations(), kmneighbourhood(), kmnneighbourhood()

Examples

kmlearningpaths(xpl$space)

Compute the neighbourhod of a state within a knowledge structure

Description

kmneighbourhood computes the neighbourhood of a state within a knowledge structure, i.e. the family of all other states with a symmetric set difference of 1.

Usage

kmneighbourhood(state, struct, include = FALSE)

Arguments

state

Binary vector representing a knowledge state
struct

Binary matrix representing a knowledge structure
include

Boolean whether the original state should be included in the result (default FALSE)

Value

Matrix containing the neighbouring states, one per row

See Also

Other Fringes & learning paths: kmbasisfringe(), kmbasisneighbourhood(), kmfringe(), kmgradations(), kmlearningpaths(), kmnneighbourhood()

Examples

kmneighbourhood(c(1,1,0,0), xpl$space)

Compute the n-neighbourhod of a state within a knowledge structure

Description

kmnneighbourhood computes the n-neighbourhood of a state within a knowledge structure, i.e. the family of all other states with a symmetric set difference maximal n.

Usage

kmnneighbourhood(state, struct, distance, include = FALSE)

Arguments

state

Binary vector representing a knowledge state
struct

Binary matrix representing a knowledge structure
distance

Size of the n-neighbourhood
include

Boolean whether the original state should be included (defaul FALSE)

Value

Matrix containing the neighbouring states, one per row

See Also

Other Fringes & learning paths: kmbasisfringe(), kmbasisneighbourhood(), kmfringe(), kmgradations(), kmlearningpaths(), kmneighbourhood()

Examples

kmnneighbourhood(c(1,1,0,0), xpl$space, 2)

Determine the notions of a knowledge structure

Description

kmnotions returns a matrix representing the notions of a knowledge structure.

Usage

kmnotions(x)

Arguments

x

Binary matrix representing a knowledge structure

Value

Binary matrix representing notions in the knowledge structure

The matrix has a '1' in row 'i' and column 'j' if 'i' and 'j' belong to the same notion (i.e. are equivalent). It is a symmetric matrix with '1's in the main diagonal.

See Also

Other Properties of knowledge structures: kmeqreduction(), kmiswellgraded()

Examples

kmnotions(xpl$space)

Test if a state is contained in a family of states

Description

Test if a state is contained in a family of states

Usage

kmsetiselement(s, f)

Arguments

s

State
f

Family of sets

Value

Boolean is s is contained in f

See Also

Other Utilities: kmcolors(), kmdoubleequal(), kmsymmsetdiff()

Examples

kmsetiselement(c(1,1,1,0), xpl$space)

Derive a basis from a surmise function

Description

kmSF2basis expects a surmise function data frame and returns the corresponding basis.

Usage

kmSF2basis(sf)

Arguments

sf

Surmise function

Value

Matrix representing the basis.

See Also

Other Different representations for knowledge spaces: kmSR2basis(), kmbasis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Simulate a set of response patterns according to the BLIM

Description

kmsimulate returns a data set of n simulated response patterns based on the knowledge structure x given as a binary matrix. The simulation follows the BLIM (Basic Local Independence Model; see Doigon & Falmagne, 1999).

Usage

kmsimulate(x, n, beta, eta)

Arguments

x

Binary matrix representing a knowledge space
n

Number of simulated response patterns
beta

Careless error probability value or vector
eta

Lucky guess probability value or vector

Details

The beta and eta parameters must be either single numericals or vectors with a length identical to the number of rows in the x matrix. A mixture is possible.

The sample function used by kmsimulate might work inaccurately for knowledge structures x with 2^31 or more states.

Value

Binary matrix representing the simulated data set

References

Doignon, J.-P. & Falmagne, J.-C. (1999). Knowledge Spaces. Springer–Verlag, Berlin.

Examples

kmsimulate(xpl$space, 50, 0.2, 0.1)
kmsimulate(xpl$space, 50, c(0.2, 0.25, 0.15, 0.2), c(0.1, 0.15, 0.05, 0.1))
kmsimulate(xpl$space, 50, c(0.2, 0.25, 0.15, 0.2), 0)

Convert a binary matrix to a kmspace object

Description

kmspace() returns a kmspace object for a binary matrix.

Usage

kmspace(x)

Arguments

x

Binary matrix representing a family of sets

Value

kmspace object

See Also

Other Constructors: kmfamset(), kmstructure()

Examples

m <- matrix(c(1,0,0,0,1,0,1,1,1), nrow=3, byrow=TRUE)
kmspace(m)

Determine the basis of a knowledge space from a surmise relation

Description

kmSR2basis takes a surmise relation and returns the corresponding basis.

Usage

kmSR2basis(sr)

Arguments

sr

Surmise relation

Value

Basis

See Also

Other Different representations for knowledge spaces: kmSF2basis(), kmbasis(), kmsurmisefunction(), kmsurmiserelation(), kmunionclosure()

Validate a surmise relation against a data set

Description

kmSRvalidate returns a list with two elements, Goodman & Kruskal's gamma value and the violational coefficient (VC).

Usage

kmSRvalidate(data, sr)

Arguments

data

Binary matrix representing a set of response patterns
sr

Binary matrix representing a surmise relation

Value

A list with two elements:

gamma

Goodman & Kruskal's gamma index

vc

Violational Coefficient

See Also

Other Validating knowledge spaces: kmdist(), kmvalidate()

Examples

kmSRvalidate(xpl$data, xpl$sr)

Convert a binary matrix to a kmstructure object

Description

kmstructure() returns a kmstructure object after checking that the passed object is a binary matrix without double rows. The empty set and the full item set are added if missing.

Usage

kmstructure(x)

Arguments

x

Binary matrix representing a family of sets

Value

kmstructure object

See Also

Other Constructors: kmfamset(), kmspace()

Examples

m <- matrix(c(1,0,0,0,1,0,1,1,1), nrow=3, byrow=TRUE)
kmstructure(m)

Compute the surmise function for a knowledge space or basis

Description

kmsurmisefunction returns a data frame representing the surmise function for a knowledge space or basis. The rows of the data frame are ordered by item name.

Usage

kmsurmisefunction(x)

Arguments

x

Binary matrix representing a knowledge space or basis

Value

Data frame representing the surmise unction of x.

See Also

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmbasis(), kmsurmiserelation(), kmunionclosure()

Examples

kmsurmisefunction(xpl$space)

Compute the surmise relation of a quasi-ordinal knowledge space

Description

kmsurmiserelation returns a matrix representing the surmise relation of a quasi-ordinal knowledge space. If x is a general knowledge space, a knowledge structure or an arbitrary family of sets, kmsurmiserelation returns the surmise relation of the smallest quasi-ordinal knowledge space containing x.

Usage

kmsurmiserelation(x)

Arguments

x

Binary matrix representing a family of sets (class kmfamset)

Value

Binary matrix representing the surmise relation

Note: The columns of the surmise relation matrix describe the minimal state for the respective item in the quasi-ordinal knowledge space.

See Also

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmbasis(), kmsurmisefunction(), kmunionclosure()

Examples

kmsurmiserelation(xpl$space)

Compute the symmetric set difference between two sets

Description

Compute the symmetric set difference between two sets

Usage

kmsymmsetdiff(x, y)

kmsetdistance(x, y)

Arguments

x

Binary vector representing a set
y

Binary vector representing a set

Value

kmsymmsetdiff: Symmetric set difference between 'x' and 'y'

kmsetdistance: Distance between the sets 'x' and 'y', i.e. the cardinality of the symmetric set difference

See Also

Other Utilities: kmcolors(), kmdoubleequal(), kmsetiselement()

Other Utilities: kmcolors(), kmdoubleequal(), kmsetiselement()

Examples

kmsymmsetdiff(c(1,0,0), c(1,1,0))

kmsetdistance(c(1,0,0), c(1,1,0))

Create trivial knowledge spaces

Description

These functions create trivial knowledge spaces of a given item number. The minimal space contains just the empty set and the full item set while the maximal space is equal to the power set.

Usage

kmminimalspace(noi)

kmmaximalspace(noi)

Arguments

noi

Number of items

Details

Please note that the computation time for creating large power sets can grow quite large easily.

Value

A binary matrix representing the respective knowledge space

Examples

kmminimalspace(5)
kmmaximalspace(5)

Close a family of sets under union

Description

kmunionclosure returns a matrix representing a knowledge space. Please note that it may take quite some time for computing larger knowledge spaces.

Usage

kmunionclosure(x)

## S3 method for class 'kmdata'
kmunionclosure(x)

## S3 method for class 'kmfamset'
kmunionclosure(x)

## S3 method for class 'kmstructure'
kmunionclosure(x)

Arguments

x

Binary matrix representing a family of sets

Value

Binary matrix representing the corresponding knowledge space, i.e. the closure of the family under union including the empty set and the full set.

kmunionclosure implements the irredundant algorithm developed by Dowling (1993).

References

Dowling, C. E. (1993). On the irredundant construction of knowledge spaces. Journal of Mathematical Psychology, 37, 49–62.

See Also

Other Different representations for knowledge spaces: kmSF2basis(), kmSR2basis(), kmbasis(), kmsurmisefunction(), kmsurmiserelation()

Examples

kmunionclosure(xpl$basis)

Validate a knowledge structure against a data set

Description

kmvalidate returns a list with three elements, a named vector (dist) with the frequencies of distances between a set of response patterns and a knowledge structure, the Discrepancy Index (DI), and the Distance Agreement Coefficient (DA).

Usage

kmvalidate(data, struct)

Arguments

data

Binary matrix representing a set of response patterns
struct

Binary matrix representing a knowledge structure

Value

A list with three elements:

dist

Distance distribution vector

DI

Discrepancy Index

DA

Distance Agreement Coefficient

Warning

The DA computation can take quite some time for larger item sets as the power set has to be computed. For item sets with around 30 items or more, it may even crash the system due to huge memory requests.

See Also

Other Validating knowledge spaces: kmSRvalidate(), kmdist()

Examples

kmvalidate(xpl$data, xpl$space)

Knowledge space on linear functions

Description

Basis of a small knowledge space and list of items on linear functions used in a manuscript by Steiner et al.

Usage

phsg

Format

A list containing the basis and the list of items

See Also

Other Data: cad, fractions, readwrite, xpl

Plot a Hasse diagram

Description

plot draws a Hasse diagram for a family of sets or a surmise relation.

Usage

## S3 method for class 'kmfamset'
plot(
  x,
  ...,
  horizontal = FALSE,
  colors = NULL,
  keepNames = TRUE,
  itemsep = ",",
  braces = TRUE,
  vertexshape = "oval",
  arrowhead = "none",
  arrowtail = "none",
  edgelabel = FALSE,
  verbose = 0
)

## S3 method for class 'kmneighbourhood'
plot(
  x,
  ...,
  horizontal = FALSE,
  colors = c("#eeee00", "#aaccff", "#bbffbb"),
  keepNames = TRUE,
  itemsep = ",",
  braces = TRUE,
  vertexshape = "oval",
  arrowhead = "none",
  arrowtail = "none",
  edgelabel = FALSE,
  state,
  verbose = 0
)

## S3 method for class 'kmsurmiserelation'
plot(
  x,
  ...,
  horizontal = FALSE,
  colors = NULL,
  keepNames = TRUE,
  vertexshape = "circle",
  arrowhead = "none",
  arrowtail = "none",
  verbose = 0
)

Arguments

x

Binary matrix representing a family of sets
...

Optional inherited parameters
horizontal

Boolean defining orientation of the graph, default FALSE
colors

Color value or vector (default NULL).
keepNames

Keep item names (default TRUE)
itemsep

Item separator in sets (default ','; only for families of states)
braces

Put braces around vertices (default TRUE; only for families of states)
vertexshape

Shape of the vertex objects, e.g. circle, oval, box, or none. See Graphviz Node Shapes for a complete list of possible values.
arrowhead

Form of the arrow head, e.g. vee or none (default). See Graphviz Arrow Types for a complete list of possible values. This may be used for vertical graphs although none is the standard there.
arrowtail

Form of the arrow tail, e.g. vee or none (default). See Graphviz Arrow Types for a complete list of possible values. This should be used for horizontal graphs.
edgelabel

Boolean whether to label the edges of the diagram (default FALSE)
verbose

Verbosity level (0 (default), 1, or 2)
state

Knowledge state whose neighbourhood is to be pictured

Details

plot takes a matrix representing a family of sets (knowledge states) or a surmise relation and a color vector, and draws a Hasse diagram. If the color vector is NULL the states are drawn in green, the items in the relation are drawn in orange.

For a surmise relation, if there are equivalent items, they are contracted into one vertex labelled 'a ~ b ~ ...' for equivalent items a, b, ...

If the plot is to be used within a Shiny app, it must be included with grVizOutput() and renderGrViz() from the DiagrammeR package (plotOutput() and renderPlot() do not work).

Please note that, for equivalent items in a relation plot, the prerequisites are lost.

See Also

Other Plotting knowledge structures: kmcolors()

Examples

plot(phsg$basis)

sp <- kmunionclosure(phsg$basis)
n <- kmneighbourhood(phsg$basis[3,], sp, include=TRUE)
plot(n, state=phsg$basis[3,])

m <- matrix(c(1,0,0,1,1,1,1,1,1), ncol=3, byrow=FALSE)
class(m) <- unique(c("kmsurmiserelation", class(m)))
plot(m, vertexshape="oval")

Knowledge spaces on reading and writing abilities

Description

Bases of knowledge spaces on reading/writing abilities obtained from querying experts.

Usage

readwrite

Format

A list containing four bases (rw1 to rw3, and rwmaj) in binary matrix form. Each matrix has 48 columns representing the different knowledge items and a varying number of rows containing the basis elements.

Details

Three experts were queried about prerequisite relationships between 48 items on reading and writing abilities (Dowling, 1991; 1993). A forth basis represents those prerequisite relationships on which the majority of the experts agree (Dowling & Hockemeyer, 1998).

References

Dowling, C. E. (1991). Constructing Knowledge Structures from the Judgements of Experts. Habilitationsschrift, Technische Universität Carolo-Wilhelmina, Braunschweig, Germany.

Dowling, C. E. (1993). Applying the basis of a knowledge space for controlling the questioning of an expert. Journal of Mathematical Psychology, 37, 21–48.

Dowling, C. E. & Hockemeyer, C. (1998). Computing the intersection of knowledge spaces using only their basis. In Cornelia E. Dowling, Fred S. Roberts, & Peter Theuns, editors, Recent Progress in Mathematical Psychology, pp. 133–141. Lawrence Erlbaum Associates Ltd., Mahwah, NJ.

See Also

Other Data: cad, fractions, phsg, xpl

Small example knowledge space

Description

Basis and space matrix, surmise relation and surmise function of a small fictional knowledge space, and two data sets (data (7 patterns) and sim (500 patterns)to be used in examples. The latter was produced from the space with kmsimulate() with beta and eta values of 0.1.

Usage

xpl

Format

A list containing the basis, the space, the surmise relation, the surmise function, and the two data matrices data and sim.

See Also

Other Data: cad, fractions, phsg, readwrite