Triclustring Toolbox is a collection of triclustering methods consolidated into a single interface. It provides access to both box- and prime-based OAC (Object-Attribute-Condition) triclustering, Spectral triclustering and features implementations of DataPeeler and Trias. The application also contains algorithms for mining triclusters of similar values: NOAC and Tri-K-Means. Quality of triclusters is measured in terms of density, diversity, coverage, and variance, if applicable. Formats for input and output data of all the methods are universal, which makes comparison and interpretation of the results easier. The code is written in C# (.Net 4.5) and runs on Windows. Triclustring Toolbox was used to provide experimental results in several articles on triclustering.
Triadic Formal Concept Analysis (3-FCA) was introduced by Lehman and
Wille [
The remainder of the paper consists of three sections: Section 2 brie y introduces basic notions, Section 3 describes our Triclustering Toolbox, and Section 4 concludes the paper with future prospects of triclustering software development.
De nition 1. Let G, M and B be arbitrary sets. Subset of their Cartesian product de nes a triadic relation I G M B. The quadruple K = (G; M; B; I) is called a triadic formal context, or tricontext. The sets G, M and B are called set of objects, set of attributes, and set of conditions, respectively.
For each triple (g; m; b) I, where g 2 G, m 2 M , and b 2 B, it is said that \object g has attribute m under condition b". In case of numeric context we add a value function: V : I ! R.
Now we de ne a tricluster in its most general form: De nition 2. Let An k l be a three-dimensional binary matrix. Let sets G = fg1; g2; : : : ; gng, M = fm1; m2; : : : ; mkg, and B = fb1; b2; : : : ; blg be index sets of A. Then for some arbitrary sets X G, Y M and Z B submatrix AXY Z = faxyz j x 2 X; y 2 Y; z 2 Zg is called a tricluster. Sets X, Y and Z are called respectively extent, intent and modus of the tricluster.
Various constraints may be applied to triclusters. Usually they feature structure requirements, cardinality restrictions on extent, intent and modus, and limitations of other parameters. For example, the most common conditions, which eliminate small and meaningless structures from the output, are minimal support condition (jXj sX ; jY j sY ; jZj sZ ) and minimum density threshold: (AXY Z ) =
PjXj PjY j PjkZ=j1 axiyjzk i=1 j=1 jXjjY jjZj min:
In case of numeric triclusters we can also measure variance of the values of triples in the tricluster. Lower variance means that values are more similar. If we consider each tricluster T = (X; Y; Z) as an independent sample S(T ) = fV (g; m; b) j (g; m; b) 2 I \ X Y Zg, then unbiased estimate of the variance is: 2(S) =
Ps2S s jSj 2 1
S2 3
Triclustring Toolbox is a Windows Forms UI application. It was programmed in
C# using .Net Framework 4.5. The program takes an input le that contains a
triadic context as set of triples. Each triple is described in a separate line with
its tab-separated components, which are members of the context's respective
modalities. For numerical tricontexts lines also contain the triple's value. The
path to the le should be speci ed in the "Context le" eld of the interface.
Other parameters are:
{ Output folder, where the results and log les of the processing will be placed.
{ Limited context uploading. If only several rst triples of the context should
be processed, the number of triples should be speci ed.
{ Selection of triclustering method:
OAC-triclustering [
Tri-K-Means [
When all of the necessary options are set, user can press the \Start" button.
For example, let us look at speci c options for numerical triclustering algorithms. For NOAC it is the parameter , which is set to 0 by default. It also supports minimal density thresholds for extent, intent, and modus. Tri-K-Means requires the number of clusters k and parameter , which de nes degree of extending the tricluster over closeness of its values. User can also set upper bound for the number of steps the algorithm takes before terminating and the manner of initialization for centroids, which can be random or prede ned. The options are delimited by commas, decimals are separated with a dot. In case several experiments are planned, the sets of options are separated by semicolon.
The program writes two les in the output folder. The rst one contains a list of the extracted triclusters along with calculated measures. The rst string of the le contains a header, explaining the order of values: Density, Variance, Average Coverage, Objects coverage, Attributes coverage, Conditions coverage, Extent, Intent, and Modus. Then, in separate lines, the triclusters follow in the format described in the header. Name of the le is composed automatically, using names of the method, input le and the set of options. The second le is a log le. If it already exists in the folder, it will be appended by new execution information. Each line of the le corresponds to a separate experiment and contains the set of options, execution time, number of found triclusters, total coverage of the context by the tricluster set, as well as coverage of extent, intent and modus.
Figure 1 shows an example of Triclustering Toolbox application interface with all described controls. 4
With recent development of the C# programming language the application may be easily transferred to a cross-platform framework, namely .Net Core 2.0, but only as a console application. UI options are announced in upcoming .Net Core 3.0 release. We would also like to support more recent triclustering methods and provide extension possibilities for other developers.
Another possible important direction is using distributed and parallel
computing [
Acknowledgements We would like to thank Dmitry V. Gnatyshak for his valuable help with the initial development of the toolbox and Engelbert Mephu Nguifo for the inspiration. The work of Dmitry I. Ignatov (contributed to all the sections) was supported by the Russian Science Foundation under grant 17-11-01294 and performed at National Research University Higher School of Economics, Russia.