Package {kstMatrix}

Compute a binary matrix product

Description

binary_matrix_product expects two binary matrices and computes their Boolean product.

Usage

binary_matrix_product(m, n)

Arguments

Value

Boolean matrix product of m and n

See Also

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

Examples

binary_matrix_product(xpl$sr, kmsurmiserelation(kmmaximalspace(4)))
binary_matrix_product(xpl$sr, kmsurmiserelation(kmminimalspace(4)))

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.

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.

DEPRECATED: kmSF2basis() has been replaced by kmbasis().

Description

DEPRECATED: kmSF2basis() has been replaced by kmbasis().

Usage

kmSF2basis(x)

Arguments

Value

Corresponding basis

See Also

Other Deprecated functions and methods: kmSR2basis(), kmSRdiagram(), kmbasisdiagram(), kmhasse()

DEPRECATED: kmSR2basis() has been replaced by kmbasis().

Description

DEPRECATED: kmSR2basis() has been replaced by kmbasis().

Usage

kmSR2basis(x)

Arguments

Value

Corresponding basis

See Also

Other Deprecated functions and methods: kmSF2basis(), kmSRdiagram(), kmbasisdiagram(), kmhasse()

DEPRECATED: kmSRdiagram() has been replaced by a plot() method.

Description

DEPRECATED: kmSRdiagram() has been replaced by a plot() method.

Usage

kmSRdiagram(x, horizontal = FALSE, colors = NULL)

Arguments

See Also

Other Deprecated functions and methods: kmSF2basis(), kmSR2basis(), kmbasisdiagram(), kmhasse()

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

Value

A list with two elements:

gamma

Goodman & Kruskal's gamma index

vc

Violational Coefficient

See Also

Other Functions for working with data: kmdist(), kmgenerate(), kmiita2SR(), kmsimulate(), kmvalidate()

Examples

kmSRvalidate(xpl$data, xpl$sr)

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

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 functions: 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

Value

Updated probability vector

See Also

Other Knowledge assessment functions: 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

Value

Number of the selected question

See Also

Other Knowledge assessment functions: 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

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

Usage

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

Arguments

Value

Number of the selected question

See Also

Other Knowledge assessment functions: 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

Details

kmassessmentsimulation applies the kmsassess function.

Value

Assessment data as data frame

See Also

Other Knowledge assessment functions: 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

Value

Updated probability vector

See Also

Other Knowledge assessment functions: 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.kmsurmisefunction takes a surmise function and returns the corresponding basis, i.e. effectively the collection of all clauses in the surmise function.

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

Value

Binary matrix representing the basis of the corresponding knowledge space.

See Also

Other Different representations for knowledge structures: kmspace(), kmsurmisefunction(), kmsurmiserelation()

Examples

kmbasis(xpl$sf)


kmbasis(xpl$sr)

kmbasis(xpl$space)

DEPRECATED: kmbasisdiagram() has been replaced by a plot() method.

Description

DEPRECATED: kmbasisdiagram() has been replaced by a plot() method.

Usage

kmbasisdiagram(x, horizontal = TRUE, colors = NULL)

Arguments

See Also

Other Deprecated functions and methods: kmSF2basis(), kmSR2basis(), kmSRdiagram(), kmhasse()

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

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 Functions for fringes & neighbourhoods: kmbasisneighbourhood(), kmfringe(), kmneighbourhood()

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

Value

Matrix containing the neighbouring states, one per row

See Also

Other Functions for fringes & neighbourhoods: kmbasisfringe(), kmfringe(), kmneighbourhood()

Examples

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

Determine the closure of KST objects

Description

kmclosure() is an S3 method closing kstMatrix objects.

Usage

kmclosure(x, ...)

## S3 method for class 'kmattributionfunction'
kmclosure(x, ...)

## S3 method for class 'kmattributionrelation'
kmclosure(x, ...)

## S3 method for class 'kmdata'
kmclosure(x, ...)

## S3 method for class 'kmfamset'
kmclosure(x, ..., closure = "union")

Arguments

Value

Corresponding closed KST object

An attribution relation or attribution function is closed to the corresponding surmise relation or surmise function, respectively.

A family of sets (including bases, knowledge structure, and knowledge space) is closed to a knowledge space or quasi-ordinal knowledge space.

See Also

Other Closure operators: kmintersectionclosure(), kmunionclosure()

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

See Also

Other Functions for plotting and printing knowledge structures: kmprettyprint(), plot

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

Value

Distance distribution vector

See Also

Other Functions for working with data: kmSRvalidate(), kmgenerate(), kmiita2SR(), kmsimulate(), 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

Value

Boolean for (approximate) equity

See Also

Other Utilities: binary_matrix_product(), kmminimalfamset(), kmsetiselement(), kmsymmsetdiff(), kmtrivial

Examples

kmdoubleequal(0.5+0.5, 1)
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

Value

Binary matrix reduced by equivalences

See Also

Other Functions for working with knowledge structures: kmexpand(), kmintersection(), kmmesh(), kmnotions(), kmsubstructure(), kmtrivial, kmunion()

Examples

kmeqreduction(xpl$space)

Expand an attribution function to a larger set of items and close it to a surmise function

Description

kmexpand() expands an attribution function by additional items and closes it to a surmise function.

Usage

kmexpand(af, ni)

Arguments

Value

Expanded surmise function

See Also

Other Functions for working with knowledge structures: kmeqreduction(), kmintersection(), kmmesh(), kmnotions(), kmsubstructure(), kmtrivial, kmunion()

Examples

xpl$sf
kmexpand(xpl$sf, c("a", "b", "c", "d", "e", "f"))

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

Value

kmfamset object

See Also

Other Constructors: kmqspace(), 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

Value

Binary vector representing the fringe

See Also

Other Functions for fringes & neighbourhoods: kmbasisfringe(), kmbasisneighbourhood(), kmneighbourhood()

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

Value

Binary matrix representing the generated knowledge structure

See Also

Other Functions for working with data: kmSRvalidate(), kmdist(), kmiita2SR(), kmsimulate(), kmvalidate()

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

Value

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

See Also

Other Functions for wellgradedness & learning paths: kmiswellgraded(), kmlearningpathmatrices(), kmlearningpaths()

Examples

kmgradations(xpl$space)

DEPRECATED: kmhasse() has been replaced by a plot() method.

Description

DEPRECATED: kmhasse() has been replaced by a plot() method.

Usage

kmhasse(x, horizontal = TRUE, colors = NULL)

Arguments

See Also

Other Deprecated functions and methods: kmSF2basis(), kmSR2basis(), kmSRdiagram(), kmbasisdiagram()

Compute the heights of items in a knowledge space specified through its basis

Description

The height of an item in the knowledge space is defined as the minimal cardinality of the states containing it minus 1, i.e. the minimal number of prerequisites.

Usage

kmheights(b)

Arguments

Value

Data.frame with a table of heights.

See Also

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

Examples

kmheights(xpl$basis)

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

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 Functions for working with data: kmSRvalidate(), kmdist(), kmgenerate(), kmsimulate(), kmvalidate()

Compute the "intersection" of two KST objects

Description

Both objects must be of the same kstMatrix class, the result will be of the same class, i.e. a respective closure operator will be executed automatically.

Please note that the intersection of surmise relations is automatically a surmise relation without any closure needs.

Usage

kmintersection(x, y)

## S3 method for class 'kmqspace'
kmintersection(x, y)

## S3 method for class 'kmspace'
kmintersection(x, y)

## S3 method for class 'kmsurmisefunction'
kmintersection(x, y)

## S3 method for class 'kmsurmiserelation'
kmintersection(x, y)

Arguments

Value

Intersection of x and y

See Also

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmmesh(), kmnotions(), kmsubstructure(), kmtrivial, kmunion()

Examples

sf <- kmsurmisefunction(kmbasis(xpl$sr))
kmintersection(xpl$sf, sf)

Close a knowledge space under intersection

Description

Close a knowledge space under intersection

Usage

kmintersectionclosure(x)

Arguments

Value

Quasi–ordinal knowledge space

See Also

Other Closure operators: kmclosure(), kmunionclosure()

Examples

kmintersectionclosure(xpl$space)

Check for wellgradedness of a knowledge structure

Description

kmiswellgraded() returns whether a knowledge space (or its basis) is well-graded.

Usage

kmiswellgraded(x)

Arguments

Value

Logical value specifying whether x is well-graded

References

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

See Also

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

Other Functions for wellgradedness & learning paths: kmgradations(), kmlearningpathmatrices(), kmlearningpaths()

Examples

kmiswellgraded(xpl$space)

Determine all learning paths in a knowledge structure

Description

Determine all learning paths in a knowledge structure

Usage

kmlearningpathmatrices(structure, through = NULL, from = NULL, to = NULL)

Arguments

Value

A list of learning paths where each learning path matrix

If through is specified, only those learning paths are listed which go through that state (binary vector). If from and to are specified, the respective sub paths are returned.

See Also

Other Functions for wellgradedness & learning paths: kmgradations(), kmiswellgraded(), kmlearningpaths()

Examples

kmlearningpathmatrices(xpl$space)

Determine all learning paths in a knowledge structure

Description

Determine all learning paths in a knowledge structure

Usage

kmlearningpaths(
  structure,
  through = NULL,
  from = NULL,
  to = NULL,
  simplified = FALSE
)

Arguments

Value

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

If through is specified, only those learning paths are listed which go through that state (binary vector). If from and to are specified, the respective sub paths are returned.

See Also

Other Functions for wellgradedness & learning paths: kmgradations(), kmiswellgraded(), kmlearningpathmatrices()

Examples

kmlearningpaths(xpl$space)

Determine the maximal mesh of two knowledge structures

Description

If the two structures are not compatible, NULL is returned.

Usage

kmmesh(x, y)

Arguments

Value

Maximal mesh

References

Falmagne J, Doignon J (1998). “Meshing Knowledge Structures.” In Dowling CE, Roberts FS, Theuns P (eds.), Recent Progress in Mathematical Psychology, 143–153. Lawrence Erlbaum Associates Ltd., Mahwah, NJ.

See Also

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmintersection(), kmnotions(), kmsubstructure(), kmtrivial, kmunion()

Determine subfamily of minimal states in a kmfamset

Description

Determine subfamily of minimal states in a kmfamset

Usage

kmminimalfamset(x)

Arguments

Value

kmfamset of minimal states

kmminimalfamset() determines the sub family of minimal sets within a famset. If x is a knowledge structure, the result will be a family containing (only) the empty set because the empty set is a state in any knowledge structure and is its minimal state.

The concept of the minimal state of a family of sets should not be confused with the concept of knowledge states minimal for some item.

See Also

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

Examples

m <- matrix(c(1,0,0,0,1,0,1,0,1), ncol=3, byrow=TRUE)
m
kmminimalfamset(kmfamset(m))

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.

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

kmneighbourhood(state, struct, include = FALSE)

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

Arguments

Value

Matrix containing the neighbouring states, one per row

Matrix containing the neighbouring states, one per row

See Also

Other Functions for fringes & neighbourhoods: kmbasisfringe(), kmbasisneighbourhood(), kmfringe()

Examples

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

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

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: kmheights(), kmiswellgraded()

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmintersection(), kmmesh(), kmsubstructure(), kmtrivial, kmunion()

Examples

kmnotions(xpl$space)

Print KST objects in set notation

Description

kmprettyprint() formats KST objects in set notation for printing. For further details, see below.

Usage

kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "{",
  close = "}",
  inneropen = "{",
  innerclose = "}"
)

## S3 method for class 'numeric'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "{",
  close = "}",
  itemsep = ", "
)

## S3 method for class 'kmfamset'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "{",
  close = "}",
  inneropen = "{",
  innerclose = "}"
)

## S3 method for class 'kmattributionfunction'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "[",
  close = "]",
  inneropen = "{",
  innerclose = "}",
  mapsto = NULL
)

## S3 method for class 'kmattributionrelation'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "{",
  close = "}",
  inneropen = "(",
  innerclose = ")",
  tuple = TRUE,
  relation = NULL
)

## S3 method for class 'kmdata'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "(",
  close = ")",
  inneropen = "{",
  innerclose = "}"
)

## S3 method for class 'kmlearningpathlist'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "",
  close = "",
  inneropen = "{",
  innerclose = "}",
  path = NULL
)

## S3 method for class 'kmlearningpathmatrices'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "",
  close = "",
  inneropen = "{",
  innerclose = "}",
  path = NULL
)

## S3 method for class 'kmlearningpathmatrix'
kmprettyprint(
  x,
  simplified = FALSE,
  ...,
  open = "",
  close = "",
  inneropen = "{",
  innerclose = "}",
  path = NULL
)

Arguments

Details

If simplified is TRUE, a state '{a, b, c}' is printed as 'abc'. This should only be used if the item names are single letters.

For Families of sets (kmfamset and sub classes), the states are ordered by cardinality.

inneropen and innerclose are ignored for numeric vectors (i.e. states/plain sets).

mapsto defines the 'mapsto' symbol for pretty-printing a surmise or attribution function. It may contain more than one character, as in the default.

For surmise and attribution relations, there are two special parameters. tuple sets whether the relation shall be printed as set of ordered pairs or whether it shall be printed as tuple of pairs with relation symbol which can be set with the relation parameter.

For kmlearningpaths lists, thre result should be printed through cat() instead of print() because the latter does not execute the line breaks.

Please note that the methods can also be used for sub classes, e.g. the kmprettyprinting.kmfamset() method is also applicable for bases, knowledge structures, and knowledge spaces.

kmprettyprint() was decidedly not implemented as print() method. This way, one can print KST objects in matrix or data frame notation or prettyprint them in set notation etc.

See Also

Other Functions for plotting and printing knowledge structures: kmcolors(), plot

Examples

kmprettyprint(xpl$basis)
kmprettyprint(xpl$basis, simplified=TRUE)
kmprettyprint(xpl$sr)
kmprettyprint(xpl$sf)

Obtain a quasi-ordinal knowledge space

Description

kmqspace() computes the quasi-ordinal space corresponding to a given KST object.

kmqspace() computes the smallest quasi-ordinal space containing a given family of sets.

kmqspace.kmsurmisefunction() takes a surmise function and determines the quasi-ordinal knowledge space which is the closure under intersection of the knowledge space delineated by this surmise function.

kmqspace.kmsurmiserelation() computes the quasi-ordinal knowledge space delineated by a surmise relation

Usage

kmqspace(x)

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

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

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

Arguments

Value

Corresponding kmqspace object

See Also

Other Constructors: kmfamset(), kmspace(), kmstructure()

Examples

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

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

sr <- kmsurmiserelation(xpl$space)
kmqspace(sr)

Refine a famset by dissolving a notion through a basis on the notion

Description

Refine a famset by dissolving a notion through a basis on the notion

Usage

kmrefine(x, b)

Arguments

Value

Refined family of sets

Examples

x <- kmfamset(matrix(c(1,0,0,0,0,1,0,0,1,1,1,1), ncol=4, byrow=TRUE))
colnames(x) <- c("a", "b", "c", "d")
x
b <- kmbasis(kmfamset(matrix(c(1,0,1,1), ncol=2, byrow=TRUE)))
colnames(b) <- c("c", "d")
b
kmrefine(x,b)

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

Value

Boolean is s is contained in f

See Also

Other Utilities: binary_matrix_product(), kmdoubleequal(), kmminimalfamset(), kmsymmsetdiff(), kmtrivial

Examples

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

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

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.

See Also

Other Functions for working with data: kmSRvalidate(), kmdist(), kmgenerate(), kmiita2SR(), kmvalidate()

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)

kmspace constructor method

Description

kmspace() determines the knowledge space corresponding to a KST object

Please note that a knowledge space delineated by a surmise relation is quasi-ordinal. Therefore, the result is automatically marked as kmqspace object.

Usage

kmspace(x)

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

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

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

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

Arguments

Value

Corresponding knowledge space

See Also

Other Constructors: kmfamset(), kmqspace(), kmstructure()

Other Different representations for knowledge structures: kmbasis(), kmsurmisefunction(), kmsurmiserelation()

Examples

kmspace(xpl$basis)
kmspace(xpl$sf)

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

Value

kmstructure object

See Also

Other Constructors: kmfamset(), kmqspace(), kmspace()

Examples

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

Reduce a KST object to a subset of items

Description

Reduce a KST object to a subset of items

Usage

kmsubstructure(x, items)

## S3 method for class 'kmdata'
kmsubstructure(x, items)

## S3 method for class 'kmbasis'
kmsubstructure(x, items)

## S3 method for class 'kmstructure'
kmsubstructure(x, items)

## S3 method for class 'kmspace'
kmsubstructure(x, items)

## S3 method for class 'kmsurmiserelation'
kmsubstructure(x, items)

## S3 method for class 'kmsurmisefunction'
kmsubstructure(x, items)

Arguments

Value

Sub structure KST object (the kstMatrix class and its properties are preserved)

kmsubstructure() takes a KST object and determines a sub-structure based on the list of items.

The items can be specified through their item names or through the respective column numbers. At least two items must be specified.

See Also

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmintersection(), kmmesh(), kmnotions(), kmtrivial, kmunion()

Examples

phsg$basis
kmsubstructure(phsg$basis, c("I1", "I2", "I5"))

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, itemname = "Item")

Arguments

Value

Data frame representing the surmise unction of x.

See Also

Other Different representations for knowledge structures: kmbasis(), kmspace(), kmsurmiserelation()

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

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 structures: kmbasis(), kmspace(), kmsurmisefunction()

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

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: binary_matrix_product(), kmdoubleequal(), kmminimalfamset(), kmsetiselement(), kmtrivial

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

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

See Also

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

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmintersection(), kmmesh(), kmnotions(), kmsubstructure(), kmunion()

Examples

kmminimalspace(3)
kmmaximalspace(3)

Compute the "union" of two KST objects

Description

Both objects must be of the same kstMatrix class, the result will be of the same class, i.e. a respective closure operator will be executed automatically.

Usage

kmunion(x, y)

## S3 method for class 'kmfamset'
kmunion(x, y)

## S3 method for class 'kmbasis'
kmunion(x, y)

## S3 method for class 'kmspace'
kmunion(x, y)

## S3 method for class 'kmstructure'
kmunion(x, y)

## S3 method for class 'kmsurmisefunction'
kmunion(x, y)

## S3 method for class 'kmsurmiserelation'
kmunion(x, y)

Arguments

Details

For surmise functions, the algorithm by Dowling & Hockemeyer (1998) is used.

Please note that, for families of sets, there are the functions kmunionclosure() and kmintersectionclosure().

Value

Union of x and y

References

Dowling CE, Hockemeyer C (1998). “Computing the Intersection of Knowledge Spaces Using only their Basis.” In Dowling CE, Roberts FS, Theuns P (eds.), Recent Progress in Mathematical Psychology, 133–141. Lawrence Erlbaum Associates Ltd., Mahwah, NJ.

See Also

Other Functions for working with knowledge structures: kmeqreduction(), kmexpand(), kmintersection(), kmmesh(), kmnotions(), kmsubstructure(), kmtrivial

Close a family of sets under union, i.e. determine the smallest knowledge space containing it

Description

Close a family of sets under union, i.e. determine the smallest knowledge space containing it

Usage

kmunionclosure(x)

Arguments

Value

Knowledge space

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 Closure operators: kmclosure(), kmintersectionclosure()

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

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 Functions for working with data: kmSRvalidate(), kmdist(), kmgenerate(), kmiita2SR(), kmsimulate()

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

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,
  ...,
  state = NULL,
  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 = NULL,
  verbose = 0
)

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

## S3 method for class 'kmlearningpathmatrices'
plot(
  x,
  ...,
  horizontal = FALSE,
  colors = NULL,
  keepNames = TRUE,
  itemsep = ",",
  braces = TRUE,
  vertexshape = "oval",
  arrowhead = "none",
  arrowtail = "none",
  edgelabel = FALSE,
  verbose = 0,
  structure = NULL,
  pathcolors = c("red", "blue", "gold", "deeppink", "brown", "violet", "green")
)

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

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

Arguments

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.

If the KST object to be plotted is a family of sets (kmfamset) and a state is specified, the colors vector must have four elements, (i) the color for the state, (ii) the color for its upper neighbours, (iii) the color for its lower neighbours, and (iv) the color for all other states.

See Also

Other Functions for plotting and printing knowledge structures: kmcolors(), kmprettyprint()

Examples

## Not run: 
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")

## End(Not run)

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.

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.