Content

• Introduction
• S3 classes in kstMatrix
  • Constructors
• Different representations for knowledge spaces
• Working with knowledge structures
  • Creating trivial knowledge spaces
  • Determining variants
    • Closure operatorsa
    • Dealing with equivalent items
    • Combining knowledge structures
    • Reducing and expanding structures
• Working with data
  • Simulating response patterns
  • Generating knowledge structures from data
  • Validating knowledge structures
• Assessment
• Fringes and learning paths
• Plotting and pretty-printing
  • Creating color vectors
  • Pretty-printing
• Utilities
• Datasets provided by kstMatrix
  • Empirical structures
  • Example structures
• Removed and obsoleted functions and methods
• 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 most of which are in some hierarchy. A short explanation of the various classes:

• kmdata: A matrix of response patterns. These may be either empirical or simulated.
• kmfamset: A family of sets, i.e. of knowledge states. Due to the set character, any row may occur only once.
• kmbasis: Basis of a knowledge space; sub-class of kmfamset.
• kmstructure: Knowledge structure; a kmfamset which contains teh empty set and the full item set as states.
• kmspace: Knowledge space; a kmstructure which is closed under union.
• kmqspace: Quasi-ordinal knowledge space; a kmspace which is additionally closed under intersection.
• kmneidghbourhood: The neighbourhood of an individual knowledge state; sub-class of kmfamset.
• kmattributionrelation: Incidence matrix of an attribution relation; please note that here the rows contain the minimal states.
• kmsurmiserelation: Closure of a kmattributionrelation under reflexivity and transitivity.
• kmattributionfunction: Data frame describing an attribution function.
• kmsurmisefunction: Closure of a kmattributionfunction under extended reflexivity, transitivity, and incomparability.

The following classes are isolated from the general class hierarchy.

• kmlearningpath: A list of printed knowledge states building a gradation.
• kmlearningpaths: A list of kmlearningpath objects.
• kmlearningpathmatrix: A matrix whose rows describe the states within a gradation. Technically also a family of sets.
• kmlearningpathmatrices: A list of kmlearningpathmatrix objects.

Figure 1 shows the dependencies between the former group of classes, e.g., kmspace being a sub-class of kmstructure. Besides the kstMatrix classes in blue, also the underlying standard R classes are shown in green.

Figure 1: kstMatrix object 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.

Working with knowledge structures

Working with data

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 can then 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 and pretty-printing

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)

Figure 2: Example space

Figure 3: Example surmise relation

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="orange", vertexshape="circle")

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)

Figure 4: Neighbourhood

Alternatively, one can also plot the whole structure and highlight a certain state and its neighbourhood within the structure. Here, four colors are used.

plot(xpl$space, state=c(1,1,0,0))

Figure 5: Neighbourhood within a knowledge space

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. Please note that, for technical reasons, you should specify the arrowhead for horizontal but the arrowtail for vertical Hasse diagrams if you want arrows instead of plain lines.

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

Figure 6: Horizontal Hasse diagram

Figure 7: Using arrows instead of lines

kmprettyprint(xpl$basis)
#> [1] "{{a}, {b}, {a, c}, {b, c}, {a, b, d}}"
kmprettyprint(xpl$basis, simplified=TRUE)
#> [1] "{a, b, ac, bc, abd}"
kmprettyprint(xpl$sr,tuple=FALSE)
#> [1] "{a  ⪯ a, a  ⪯ d, b  ⪯ b, b  ⪯ d, c  ⪯ c, d  ⪯ d}"
cat(kmprettyprint(kmlearningpaths(xpl$space)))
#> ∅ →{a} →{a, b} →{a, b, c} →{a, b, c, d}, 
#> ∅ →{a} →{a, b} →{a, b, d} →{a, b, c, d}, 
#> ∅ →{a} →{a, c} →{a, b, c} →{a, b, c, d}, 
#> ∅ →{b} →{a, b} →{a, b, c} →{a, b, c, d}, 
#> ∅ →{b} →{a, b} →{a, b, d} →{a, b, c, d}, 
#> ∅ →{b} →{b, c} →{a, b, c} →{a, b, c, d}

Utilities

This group contains several smaller helper functions.

• binarymatrixproduct() computes the Boolean multiplaction of two binary matrices.
• kmsymmsetdiff(): Determine the symmetric set difference between two sets.
• kmsetdistance(): Determine the cardinality of the symmetric set difference
• kmsetiselement(): Test if an item is element of a state/set
• kmminimalfamset(): Determine the minimal states within a kmfamset
• kmdoubleequal(): Test if two doubles are (almost) equal thus avoiding rounding problems with the ‘==’ operator
• kmcolors(): Produce a color vector for plot() based on existing palettes
• kmtrivial(): Create a minimal or a maximal knowledge space for a given number of items
• kmheights(): Determine a table of items and their heights in the knowledge structure/space

Datasets provided by kstMatrix

Removed and obsoleted functions and methods

With the never-ending addition of new functions and methods, is sometimes becomes necessary to re-structure the interface.

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. 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.
• 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.