Content

• Introduction
• S3 classes in kstMatrix
• Different representations for knowledge spaces
• Propertiesd of knowledge spaces
• Creating trivial knowledge spaces
• Simulating response patterns
• Validating knowledge structures
• Assessment
• Fringes and paths
• Plotting
• Utilities
• Datasets provided by kstMatrix
• References

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:

• kmfamset
• kmbasis
• kmstructure
• kmspace
• kmneighbourhood
• kmsurmiserelation
• kmsurmisefunction

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

Structure of kstMatrix classes

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.

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 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 mthis 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).

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 kmneidghbourhood 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)

Utilities

This group contains several smaller helper functions.

• kmsymmsetdiff()
• kmsetdistance()
• kmsetiselement()
• kmdoubleequal()
• kmcolors()

Datasets provided by kstMatrix

References

• Baumunk, K. & Dowling, C. E. (1997). Validity of spaces for assessing knowledge about fractions. Journal of Mathematical Psychology, 41, 99–105.
• Doignon, J.-P. & Falmagne, J.-C. (1999). Knowledge Spaces. Springer–Verlag, Berlin.
• Dowling, C. E. (1991). Constructing Knowledge Structures from the Judgements of Experts. Habilitationsschrift, Technische Universität Carolo-Wilhelmina, Braunschweig, Germany.
• Dowling, C. E. (1993a). Applying the basis of a knowledge space for controlling the questioning of an expert. Journal of Mathematical Psychology, 37, 21–48.
• Dowling, C. E. (1993b). On the irredundant construction of knowledge spaces. Journal of Mathematical Psychology, 37, 49–62.
• 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.
• 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.