#
Null Models for Formal Contexts^{ †}

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. FCA Basics and Problem Description

## 3. Related Work

## 4. Stochastic Modelling

#### 4.1. Coin-Toss—Direct Model

**Example**

**1.**

#### 4.2. Coin-Toss: Indirect Model

#### 4.3. Dirichlet Model

Algorithm 1: Dirichlet Approach |

## 5. Experiments

`Python 3`and all further computations, i.e., the I-PI coordinates, were done using

`conexp-clj`[14]. The generator code as well as the generated contexts can be found on GitHub (Online Repository containing Data, Results, Code: https://github.com/maximilian-felde/formal-context-generator).

#### 5.1. Observations

**A**. The results are shown in Figure 4. We can see that again many contexts have less than 100 pseudo-intents and the number of intents once again varies over the full possible range. There are 1909 contexts that contain a contranominal scale of size $\left|M\right|$. However, we notice that there is a not negligible number of contexts with over 100 and up to almost 252 pseudo-intents, which constitutes theoretical maximum [8]. Most of these gather around nearly vertical lines close to 75, 200, 380, 600 and 820 intents. Even though most of the contexts have an I-PI coordinate along one of those lines there are a few contexts in-between 100 and 175 pseudo-intents that do not fit this description. Looking at the histogram we can observe again that while the number of pseudo-intents increases the number of generated contexts to that pseudo-intent number decreases. This is in contrast to Example 1. This time, however, we can clearly see a peak at seven to ten pseudo-intents with 190 contexts having ten pseudo-intents. Apart from this we observed no other significant dips or peaks. We also tried randomizing the base measure $\mathit{\alpha}$ using Dirichlet distributions. However, this did not improve the results.

**B**the factor $c=0.1$ suitable, as we will explain in Section 5.2. A plot of the results can be found in Figure 5. We can see that most of the contexts have less than 150 pseudo-intents and the number of intents is between 1 and 1024. Furthermore, the quantity of contexts containing a contranominal scale of size $\left|M\right|$ is 1169. This number is about 700 lower than in variation

**A**, roughly 500 lower compared to the coin-tossing results in Example 1, and over 1200 lower than in the unaltered Dirichlet approach. We can again observe the same imaginary lines as mentioned for variation

**A**, with even more contrast. Finally, we observe that the space between these lines contains significantly more I-PI coordinates. Choosing even smaller values for c may result in less desirable sets of contexts. In particular, we found that lower values for c appear to increase the bias towards the imaginary lines.

**B**differs distinguishably to the one in Figure 4. The distribution of pseudo-intent numbers is more volatile and more evenly distributed. There is a first peak of 366 contexts with ten pseudo-intents, followed by a dip to eleven contexts with seventeen pseudo-intents and more relative peaks of 50 to 60 contexts each at 28, 36 and 45 pseudo-intents. After 62 pseudo-intents the number of contexts having this amount of pseudo-intents or more declines with the exception of the peak at 120 pseudo-intents.

**A**(orange dashed line) and variation

**B**(blue dotted line). In all three plots we recognize that there is a steep increase of distinct I-PI coordinates at the beginning followed by a fast decline in new I-PI coordinates, i.e., a slow increase in the total number of distinct I-PI coordinates, for all three random generation methods. The graphs remind of sublinear growth. For all three attribute set sizes we can observe that the graphs of variation

**A**and

**B**lie above the graph of the coin-toss. Hence, variation

**A**and

**B**generated more distinct contexts compared to the coin-toss. Exemplary for $\left|M\right|=7$ the coin-tossing approach resulted in 1963 distinct I-PI coordinates and reached them after around 99,000 generated contexts. Variation

**A**generated around 19,000 contexts until it hit 1963 distinct I-PI coordinates and reached a total of around 2450 after 100,000 contexts generated. Variation

**B**reached the same number of distinct I-PI values already at around 13,000 generated contexts and resulted in 2550 distinct I-PI coordinates.

#### 5.2. Discussion

**A**and 1200 for variation

**B**. One reason for the huge number of contranominal scales generated by the base version of the Dirichlet approach is that most of the realizations of the Dirichlet distribution (Algorithm 1, Line 6) are inner points of the probability simplex, i.e., they lie near the center of the simplex. These points or probability vectors result in almost balanced categorical distributions (Algorithm 1, Line 8), i.e., every category is drawn at least a few times for a fixed number of draws. This fact may explain the frequent occurrence of contranominal scales. The expected number of objects with $\left|M\right|-1$ attributes that need to be generated for a context to contain a contranominal scale is low. In more detail, we only need to hit the $\left|M\right|$ equally likely distinct objects, having $\left|M\right|-1$ attributes during the generation process. To be more precise, the mean ${\mu}_{N}$ and the standard deviation ${\sigma}_{N}$ of the number of required objects with $\left|M\right|-1$ attributes can easily be computed via ${\mu}_{N}=N{\sum}_{k=1}^{N}\frac{1}{k}$ and ${\sigma}_{N}^{2}=N{\sum}_{k=1}^{N}\frac{N-k}{{k}^{2}}$ with $N:=\left(\genfrac{}{}{0pt}{}{\left|M\right|}{\left|M\right|-1}\right)=\left|M\right|$, cf. [15], as this is an instance of the so-called Coupon Collector Problem. For example for a context with ten attributes we get ${\mu}_{10}\approx 29.3$ and ${\sigma}_{10}\approx 11.2$, hence we need to generate on average around 30 objects with nine attributes to create a contranominal scale. While, there is already a high probability of obtaining a contranominal scale after generating around 18 objects. This means if we generate a context with $\left|G\right|=300$ objects and the probability for the category with nine attributes is around $10\%$ we can expect the context to contain a contranominal scale.

**A**and

**B**. This due to the fact that a fixed row-density context with density $8/10$ that contains all possible objects has exactly ten pseudo-intents, cf. ([8] Prop. 1). This is again related to the Coupon Collector Problem. The solution to this problem yields the expected number of objects that we need in order to hit every possible combination. In particular for the case of the peak at ten pseudo-intents, $N=\left(\genfrac{}{}{0pt}{}{10}{8}\right)=45$, ${\mu}_{45}\approx 198$ and ${\sigma}_{45}\approx 56$, meaning if we generate a fixed row-density context with around 200 objects we can expect it to contain all possible combinations and therefore have ten pseudo-intents. This fits well with the observed 366 contexts with ten pseudo-intents in variation

**B**. Consider the case that we only generate fixed row-density contexts containing all possible attribute combinations and all densities are equally likely. The expected number of contexts with eight attributes and therefore ten pseudo-intents for 5000 generated contexts then is $5000/11\approx 455$. Naturally, variation

**B**does not predominantly generate fixed row-density contexts or even fixed row-density contexts with all possible attribute combinations. Hence, the before mentioned 366 observed contexts with ten pseudo-intents seem reasonable.

#### 5.3. The Problem with Contranominal Scales

## 6. Applications

#### 6.1. Null Models for Formal Contexts

#### The Dirichlet Null Model for Formal Contexts

#### 6.2. Evaluation of the Dirichlet Approach for Null Model Generation

`conexp-clj`software [14]. Short summaries of the used contexts and their properties are depicted in Table 2 and Table 3.

#### 6.2.1. Observations

#### 6.2.2. Evaluation

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A2.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A3.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A4.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A5.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A6.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A7.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A8.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A9.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure A10.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

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**Figure 3.**Dirichlet generated contexts with $\mathit{\alpha}=(\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1},\dots ,\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1})$, $\beta =\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1$.

**Figure 4.**Dirichlet generated contexts with $\mathit{\alpha}=(\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1},\dots ,\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1})$, $\beta \sim \mathrm{Uniform}(0,|M|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1)$.

**Figure 5.**Dirichlet generated contexts with $\mathbf{\alpha}=(\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1},\dots ,\frac{1}{\left|M\right|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1})$, $\beta =0.1\left(\right|M|\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1)$.

**Figure 6.**Number of distinct I-PI coordinates for up to 100,000 randomly generated contexts with 6, 7 and 8 attributes.

**Figure 7.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach. The first three charts show the resulting densities, numbers of intents and numbers of pseudo-intents as box-whiskers plots with outliers. The chart in the top right is an I-PI-plot of all generated contexts. The four other charts depict the mean and standard deviation of the frequencies of the numbers of attributes, i.e., of the row sum distributions. The red line indicates the row sum distribution of the original context for which the null models were generated.

**Figure 8.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure 9.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Figure 10.**Results of generating null models consisting of 100 random contexts each using resampling, coin-tossing and the Dirichlet approach, c.f. Figure 7.

**Table 1.**Non-exhaustive list of methods for simple null models under the indicated constraints. We write G-dist for the row sum distribution and M-dist for the column sum distribution of a formal context. Further we write $\mathbb{E}$(G-dist) and $\mathbb{E}$(M-dist) for the expected row (column) sum distribution.

Constraint | Randomization Method(s) for Null Models |
---|---|

keep G-dist and M-dist | pairwise swapping of incidences |

keep G-dist or M-dist | shuffling of rows or columns |

keep $\mathbb{E}$(G-dist) or $\mathbb{E}$(M-dist) | Dirichlet approach based on the row sum distribution as base measure and a high precision parameter. |

keep $\mathbb{E}$(density) | coin-toss based on density, Dirichlet approach |

keep all implications | resampling of objects |

Context | Source | Description |
---|---|---|

Bird-Diet | [14] | A context of birds and what they eat. |

Brunson-Club | [20,21] | Membership information of corporate executive officers in social organisations. |

Diagnosis | [22,23] | The data was created by a medical expert as a data set to test the expert system, which will perform the presumptive diagnosis of two diseases of the urinary system. The temperature attribute is interval-scaled. |

Dolphins | [20,24] | A formal context created from a directed social network of bottlenose dolphins living in a fjord in New Zealand. A relation indicates frequent association based on observations between 1994 and 2001. |

Forum-Romanum | [2] | A context based on ratings of monuments on the Forum Romanum in different travel guides and scaled ordinally. This context can be found in the standard work on FCA. |

Living-Beings-and-Water | [2] | The first formal context in the standard work on FCA (the yellow book) by Ganter and Wille. |

Olympic-Disciplines | [25] | This context is about the disciplines of the Summer Olympic Games 2020. |

Seasoning-Planner | [14] | This context contains foods that are related to recommended seasonings based on a chart published by the spice company Fuchs Group. |

Southern-Woman | [20,26] | Participation of 18 white women in 14 social events over a nine-month period, collected in the Southern United States of America in the 1930s. |

Wood-Properties | [14] | A context about properties of different kinds of wood. |

Cointoss-1 | artificial | Artificially generated with the coin-toss approach. |

Cointoss-2 | artificial | Artificially generated with the coin-toss approach. |

Dirichlet-1 | artificial | Artificially generated with the Dirichlet approach. |

Dirichlet-2 | artificial | Artificially generated with the Dirichlet approach. |

**Table 3.**This table contains some basic properties of the formal contexts used in the experiment where each method of random generation was used to generate a null model consisting of 100 contexts. The aggregated results for density, number of intents and number of pseudo-intents are shown.

Context | Method | #Attributes | #Objects | ($\mathit{\mu}$)-Density | $\mathit{\sigma}$-Density | ($\mathit{\mu}$)-#Intents | $\mathit{\sigma}$-#Intents | ($\mathit{\mu}$)-#Pseudo-Intents | $\mathit{\sigma}$-#Pseudo-Intents |
---|---|---|---|---|---|---|---|---|---|

Bird-Diet | True Context | 8 | 10 | 0.30 | 16 | 15 | |||

Cointoss | 0.30 | 0.05 | 18 | 3.87 | 15 | 2.85 | |||

Dirichlet | 0.30 | 0.04 | 18 | 3.50 | 15 | 2.49 | |||

Resample | 0.30 | 0.05 | 11 | 2.01 | 11 | 1.81 | |||

Brunson-Club | True Context | 15 | 25 | 0.25 | 62 | 73 | |||

Cointoss | 0.25 | 0.02 | 84 | 14.18 | 88 | 7.41 | |||

Dirichlet | 0.25 | 0.02 | 79 | 10.86 | 86 | 7.17 | |||

Resample | 0.26 | 0.02 | 39 | 5.73 | 48 | 8.94 | |||

Diagnosis | True Context | 17 | 120 | 0.47 | 88 | 43 | |||

Cointoss | 0.47 | 0.01 | 5749 | 779.98 | 1422 | 100.19 | |||

Dirichlet | 0.47 | 0.00 | 3677 | 55.04 | 1420 | 38.12 | |||

Resample | 0.47 | 0.00 | 87 | 1.94 | 43 | 0.74 | |||

Dolphins | True Context | 62 | 62 | 0.08 | 282 | 1077 | |||

Cointoss | 0.08 | 0.00 | 227 | 21.15 | 1611 | 97.30 | |||

Dirichlet | 0.08 | 0.01 | 231 | 29.73 | 1580 | 117.76 | |||

Resample | 0.08 | 0.01 | 146 | 18.26 | 685 | 97.01 | |||

Forum-Romanum | True Context | 7 | 14 | 0.45 | 19 | 8 | |||

Cointoss | 0.45 | 0.05 | 33 | 7.29 | 13 | 1.92 | |||

Dirichlet | 0.45 | 0.08 | 27 | 10.84 | 12 | 2.79 | |||

Resample | 0.46 | 0.09 | 13 | 2.48 | 8 | 1.10 | |||

Living-Beings-and-Water | True Context | 9 | 8 | 0.47 | 19 | 10 | |||

Cointoss | 0.46 | 0.06 | 28 | 6.71 | 18 | 3.23 | |||

Dirichlet | 0.47 | 0.02 | 29 | 4.34 | 19 | 3.57 | |||

Resample | 0.47 | 0.03 | 12 | 2.47 | 10 | 0.78 | |||

Olympic-Disciplines | True Context | 19 | 50 | 0.46 | 529 | 86 | |||

Cointoss | 0.46 | 0.02 | 2178 | 380.38 | 831 | 83.09 | |||

Dirichlet | 0.46 | 0.02 | 2414 | 674.88 | 773 | 114.78 | |||

Resample | 0.46 | 0.02 | 301 | 47.00 | 65 | 6.51 | |||

Seasoning-Planner | True Context | 37 | 56 | 0.20 | 532 | 553 | |||

Cointoss | 0.20 | 0.01 | 631 | 83.08 | 1045 | 133.45 | |||

Dirichlet | 0.20 | 0.01 | 688 | 131.33 | 1044 | 172.95 | |||

Resample | 0.20 | 0.01 | 260 | 52.03 | 331 | 49.12 | |||

Southern-Woman | True Context | 14 | 18 | 0.35 | 65 | 23 | |||

Cointoss | 0.35 | 0.03 | 94 | 18.59 | 75 | 8.57 | |||

Dirichlet | 0.36 | 0.04 | 99 | 25.07 | 73 | 10.09 | |||

Resample | 0.35 | 0.04 | 36 | 9.00 | 21 | 2.15 | |||

Wood-Properties | True Context | 28 | 29 | 0.28 | 315 | 275 | |||

Cointoss | 0.28 | 0.01 | 362 | 54.30 | 432 | 42.88 | |||

Dirichlet | 0.28 | 0.02 | 361 | 61.43 | 427 | 45.90 | |||

Resample | 0.29 | 0.02 | 154 | 31.96 | 153 | 31.68 | |||

Cointoss-1 | True Context | 10 | 793 | 0.42 | 913 | 34 | |||

Cointoss | 0.42 | 0.01 | 866 | 27.40 | 45 | 8.39 | |||

Dirichlet | 0.42 | 0.01 | 880 | 29.42 | 42 | 8.97 | |||

Resample | 0.42 | 0.01 | 808 | 28.68 | 49 | 6.40 | |||

Cointoss-2 | True Context | 15 | 200 | 0.21 | 411 | 312 | |||

Cointoss | 0.21 | 0.01 | 434 | 39.18 | 315 | 12.16 | |||

Dirichlet | 0.21 | 0.01 | 408 | 40.04 | 312 | 11.37 | |||

Resample | 0.21 | 0.01 | 278 | 19.39 | 294 | 18.59 | |||

Dirichlet-1 | True Context | 10 | 198 | 0.39 | 308 | 101 | |||

Cointoss | 0.39 | 0.01 | 467 | 42.57 | 80 | 6.24 | |||

Dirichlet | 0.39 | 0.01 | 307 | 7.04 | 96 | 4.40 | |||

Resample | 0.39 | 0.01 | 259 | 7.21 | 84 | 6.61 | |||

Dirichlet-2 | True Context | 15 | 200 | 0.57 | 18,166 | 564 | |||

Cointoss | 0.57 | 0.01 | 12,451 | 1053.37 | 989 | 80.60 | |||

Dirichlet | 0.57 | 0.01 | 17,894 | 1342.92 | 625 | 68.66 | |||

Resample | 0.57 | 0.01 | 12,018 | 976.22 | 552 | 51.81 |

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Felde, M.; Hanika, T.; Stumme, G.
Null Models for Formal Contexts. *Information* **2020**, *11*, 135.
https://doi.org/10.3390/info11030135

**AMA Style**

Felde M, Hanika T, Stumme G.
Null Models for Formal Contexts. *Information*. 2020; 11(3):135.
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**Chicago/Turabian Style**

Felde, Maximilian, Tom Hanika, and Gerd Stumme.
2020. "Null Models for Formal Contexts" *Information* 11, no. 3: 135.
https://doi.org/10.3390/info11030135