kstMatrix

Cord Hockemeyer

r Sys.Date()

Content

Introduction

Knowledge space theory applies prerequisite relationships between items of knowledge within a given domain for efficient adaptive assessment and training (Doignon & Falmagne, 1999). The kstMatrix package implements some basic functions for working with knowledge space. Furthermore, it provides several empirically obtained knowledge spaces in form of their bases.

There is a certain overlap in functionality between the kstand kstMatrix packages, however the former uses a set representation and the latter a matrix representation. The packages are to be seen as complementary, not as a replacement for each other.

This document gives a short overview on the kstMatrix package. For more details on the specific functions, please have a look at the manual.

Subsequently, we start with a introduction into the S3 classes used in kstMatrix. Afterwards, the various functions are presented groupd by topic.

S3 classes in kstMatrix

With version 2, S3 classes were introduced in kstMatrix. The following classes are used:

The following figure shows the dependencies between these classes, e.g., kmspace being a sub-class of kmstructure.

Constructors

There exist four constructors

which create the respective objects from a given matrix.

Different representations for knowledge spaces

There exist various ways to represent knowledge spaces: the whole space itself, its basis, and the corresponding surmise function; in case of quasi-ordinal spaces also the surmise relation. The subsequent functions allow to map between these representations.

kmbasis()

The kmbasis() S3 method allows to ap different representations to bases. There are implementations for the classes kmsurmiserelation, kmsurmisefunction, and matrix, the latter aiming at kmfamsets including kmstructures and kmspaces.

kmsf2basis()

This function determines a basis from a surmise function.

kmSR2basis()

This function creates a basis from a surmise relation.

kmsurmisefunction()

This function takes an arbitrary kmfamset (family of sets) and determines the surmise function of the smallest knowledge space containing this family of sets, i.e. its closure under union.

kmsurmiserelation()

This function determines the surmise relation for a quasi-ordinal knowledge space. If the parameter is an arbitrary kmfamset, the result is the surmise relation for the smalest quasi-ordinal knowledge space containing the family of sets, i.e. its closure under union and intersection.

kmunionclosure()

This function returns the closure under union (i.e. a knowledge space) of a family of sets.

Properties of knowledge spaces

There are kmeqreduction() and kmnotions() for dealing with equivalent items. The former reduces a famset to the structure on its notions, and the latter gives the notions itself.

Another point is the weelgradedness - kmiswellgraded()tests if a structure fulfils this condition.

Creating trivial knowledge spaces

For a given item number, there are two trivial knowledge spaces, the maximal knowledge space representing absolutely no prerequisite relationships (the knowledge space is the power set of the item set and the basis matrix is the diagonal matrix), and the minimal knowledge space representing equivalence of all items (the knowledge space contains just the empty set and the full item set, and the basis matrix contains one line full of ‘1’s).

Simulating response patterns

Response patterns are simulated by kmsimulate() applying the BLIM (Basic Local Independence Model). It assumes identical \(\beta\) and \(\eta\) values for all items.

More elaborated functions may follow.

Validating knowledge structures

There exist two core functions for validation. kmvalidate() determines validity indices based on the states in the knowledge structure, and kmSRvalidate()determines indices based on the surmise relation of a quasi-ordinal knowledge space. Furthermore, there is a helpüer function kmdist() ehich returns a distance distribution.

Assessment

There exist several functions in the assessment context. They are all based on a paper by Falmagne & Doignon (1988, see also Doignon & Falmagne, 1999, Chapter 10).

The core functiuon is kmassess(). It does a complete assessmednt for a given response vector and a given knowledge structure. A simplified version is kmsassess() where the originally item-specific parameters are identical for all items.

Both these functions rely on alternative question and update rules realised by kmassesshalfsplit() and kmassessinfomrative() and by kmassessbayesian() and kmassessmultipöicatiuve(), respectively. The separation of these functions allows also for a use in an interactive system, e.g., a Shiny app.

Finally, there is kmassessmentsimulation() which does assessments for a whole dataset of response patterns and produces a table which could be further evaluated statistically.

Fringes and Learning Paths

Well-graded learning paths are an important concept in knowledge space theory. With kmlearningpaths(), we obtain a collection of all learning paths in a knowledge structure, with kmgradations() a collection of all gradations between two states in a structure. In this context also kmiswellgraded() should be mentioned which tests if a structure is well-graded.

In this context, we may also need fringes and neighbourhoods. The former can be determined with kmfringe(), kminnerfringe(), and kmouterfringe(). For the latter, there are the functions kmneighbourhood() (determining the 1-neighbourhood of a state) and kmnneighbourhood() (determining an arbitrary n-neighbourhood).

There exists also an alternative implementation with the functions kmbasisfringe(), kmbasisinnerfringe(), and kmbasisouterfringe(), as well as kmbasisneighbourhood(). These functions work on the basis of a knowledge space and may, thus, be faster, especially for large knowledge spaces (see Hockemeyer, 1997). Please note that, as a consequence of using the basis, these functions do not apply for general knowledge structures but only for knowledge spaces. Furthermore, due to complexity, there is no kmbasisnneighbourhood()(yet).

Plotting

With version 2.0 of kstMatrix, plotting functionality is provided through the S3 plot method. The method is available for the kmfamset, kmsurmiserelation, and kmneighbourhood classes.

plot(xpl$space)
plot(xpl$sr)

plot uses the DiagrammeR package which in turn uses the Graphviz software. The default parameters are slightly different for kmfamsets and kmsurmiserelations.

plot(xpl$sr, colors="#aaffaa", vertexshape="box")

For plotting neighbourhoods, by default three different colors are used.

plot(kmneighbourhood(c(1,1,0,0), xpl$space, include=TRUE), state=c(1,1,0,0), edgelabel=TRUE)

By default, Hasse diagrams are drawn vertically with just lines as edges. However, on finds also horizontal diagrams in which case the arrows are set as arrows from left to right.

plot(xpl$space, horizontal=TRUE, arrowtail="open")
plot(xpl$space, arrowhead="empty")

Utilities

This group contains several smaller helper functions.

Datasets provided by kstMatrix

Empirical structures

The provided empirical datasets were obtained by the research group around Cornelia Dowling through querying experts in the respective fields.

cad

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

fractions

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

readwrite

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

Example structures

phsg

A small knowledge space on a set of seven items on linear functions. This example is used in a manuscript by Steiner et al.

xpl

This is just a small fictitious 4-item-example used for the examples in the documentation. Please note that this knowledge space is not quasi-ordinal.

References