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, 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, readwrite, xpl

Plot the Hasse diagram of a basis stored as a matrix

Description

kmSRdiagram takes a matrix representing a surmise relation and a color vector and draws a Hasse diagram. If the color vector is NULL the states are drawn in green.

Usage

kmSRdiagram(structure, horizontal = TRUE, colors = NULL)

Arguments

See Also

Other Plotting knowledge structures: 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

Viola<tional Coefficient

See Also

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

Examples

kmSRvalidate(xpl$data, xpl$sr)

Compute the basis of a knowledge space

Description

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

Usage

kmbasis(x)

Arguments

Value

Binary matrix representing the basis of the knowledge space.

See Also

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

Examples

kmbasis(xpl$space)

Plot the Hasse diagram of a basis stored as a matrix

Description

kmbasisdiagram takes a matrix representing a basis and a color vector and draws a Hasse diagram. If the color vector is NULL the states are drawn in green.

Usage

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

Arguments

See Also

Other Plotting knowledge structures: kmSRdiagram(), kmhasse()

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 kmhasse.

Usage

kmcolors(prob, palette = cm.colors)

Arguments

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 Validating knowledge spaces: kmSRvalidate(), kmvalidate()

Examples

kmdist(xpl$data, xpl$space)

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

Examples

kmeqreduction(xpl$space)

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)

Arguments

Value

Binary vector representing the fringe

See Also

Other Neighbourhood & fringe: 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

Examples

kmgenerate(xpl$sim, 15)

Plot the Hasse diagram of a knowledge structure stored as a matrix

Description

kmhasse takes a matrix representing a knowledge structure and a color vector and draws a Hasse diagram. If the color vector is NULL the states are drawn in green.

Usage

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

Arguments

See Also

Other Plotting knowledge structures: kmSRdiagram(), kmbasisdiagram()

Check for wellgradedness of a knowledge structure

Description

kmiswellgraded returns whether a knowledge structure is wellgraded.

Usage

kmiswellgraded(x)

Arguments

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)

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)

Arguments

Value

Matrix containing the neighbouring states, one per row

See Also

Other Neighbourhood & fringe: kmfringe()

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)

Arguments

Value

Matrix containing the neighbouring states, one per row

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

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)

Derive a basis from a surmise function

Description

kmsf2basis expects a surmise function data frame and returns the corresponding basis.

Usage

kmsf2basis(sf)

Arguments

Value

Matrix representing the basis.

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.

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)

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

Value

Data frame representing the surmise unction of x.

See Also

Other Different representations for knowledge spaces: 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

Value

Binary matrix representing the surmise relation of the corresponding quasi-ordinal knowledge space

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: 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

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

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

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)

Arguments

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: 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

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 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, 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, readwrite