# Granulated Tables with Frequency by Discretization and Their Application

Hiroshi Sakai, Zhiwen Jian

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Hiroshi Sakai, Zhiwen Jian. Granulated Tables with Frequency by Discretization and Their Application. 4th International Conference on Intelligence Science (ICIS), Feb 2021, Durgapur, India. pp.137-146, 10.1007/978-3-030-74826-5_12. hal-03741720

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# Granulated Tables with Frequency by Discretization and Their Application

Hiroshi Sakai and Zhiwen Jian

Graduate School of Engineering, Kyushu Institute of Technology,
Tobata, Kitakyushu 804-8550, Japan
sakai@mns.kyutech.ac.jp, zhiwen.jian389@mail.kyutech.jp

**Abstract.** We have coped with rule generation from tables with discrete attribute values and extended the Apriori algorithm to the DIS-Apriori algorithm and the NIS-Apriori algorithm. Two algorithms use table data characteristics, and the NIS-Apriori generates rules from tables with uncertainty. In this paper, we handle tables with continuous attribute values. We usually employ continuous data discretization, and we often had such a property that the different objects came to have the same attribute values. We define a granulated table with frequency by discretization and adjust the above two algorithms to granulated tables due to this property. The adjusted algorithms toward big data analysis improved the performance of rule generation. The obtained rules are also applied to rule-based reasoning, which gives one solution to the black-box problem in AI.

**Keywords:** rule generation, the Apriori algorithm, rule-based reasoning, big data analysis and machine learning

## 1 Introduction

We are applying rough sets [10, 13, 19] to rule generation from table data sets and are adjusting the Apriori algorithm for transaction data sets [1, 2] to table data sets. We proposed a framework termed "NIS-Apriori" [16-18] based on the combination of equivalence classes in rough sets and the effective enumeration of the candidates of rules in the Apriori algorithm.

We term such a table in Table 1 as a *Deterministic Information System* (DIS). Several rough-set based rule generation methods are proposed [5, 10, 13, 15, 19,

<table><caption>**Table 1.** An exemplary DIS $\psi$.</caption><thead><tr><th>Object</th><th>P1</th><th>P2</th><th>P3</th><th>Dec</th></tr></thead><tbody><tr><td>x1</td><td>c</td><td>1</td><td>b</td><td>d1</td></tr><tr><td>x2</td><td>b</td><td>2</td><td>b</td><td>d1</td></tr><tr><td>x3</td><td>a</td><td>2</td><td>b</td><td>d2</td></tr><tr><td>x4</td><td>a</td><td>3</td><td>c</td><td>d2</td></tr><tr><td>x5</td><td>c</td><td>2</td><td>c</td><td>d3</td></tr></tbody></table>
<table><caption>**Table 2.** An exemplary DIS $\psi$ with missing values.</caption><thead><tr><th>Object</th><th>P1</th><th>P2</th><th>P3</th><th>Dec</th></tr></thead><tbody><tr><td>x1</td><td>c</td><td>?</td><td>b</td><td>d1</td></tr><tr><td>x2</td><td>b</td><td>?</td><td>b</td><td>d1</td></tr><tr><td>x3</td><td>?</td><td>2</td><td>b</td><td>d2</td></tr><tr><td>x4</td><td>a</td><td>3</td><td>c</td><td>?</td></tr><tr><td>x5</td><td>c</td><td>2</td><td>?</td><td>d3</td></tr></tbody></table>

<table><caption>**Table 3.** An exemplary NIS $\Phi$. Each ? is changed to a set of all possible values.</caption><thead><tr><th>Object</th><th>P1</th><th>P2</th><th>P3</th><th>Dec</th></tr></thead><tbody><tr><td>x1</td><td>c</td><td>{1,2,3}</td><td>b</td><td>d1</td></tr><tr><td>x2</td><td>b</td><td>{1,2,3}</td><td>b</td><td>d1</td></tr><tr><td>x3</td><td>{a,b,c}</td><td>2</td><td>b</td><td>d2</td></tr><tr><td>x4</td><td>a</td><td>3</td><td>c</td><td>{d1,d2,d3}</td></tr><tr><td>x5</td><td>c</td><td>2</td><td>{a,b,c}</td><td>d3</td></tr></tbody></table>

21] in DISs. In Table 1, we have an implication $\tau$: [P1,c] $\Rightarrow$ [Dec,d1] from object x1. It occurs 1 time for 5 objects, namely $support(\tau)$ (a ratio of occurrence) =1/5. It occurs 1 time for 2 objects with [P1,c], namely $accuracy(\tau)$ (a ratio of consistency)=1/2. We term such formulas like [P1,c] and [Dec,d1] descriptors. We usually specify constraints $support(\tau) \ge \alpha$ and $accuracy(\tau) \ge \beta$ for 0 < $\alpha, \beta \le 1$, and we see each implication satisfying the constraints as a rule.

To cope with information incompleteness in DISs, missing values ‘?’ [6] (Table 2) and a *Non-deterministic Information System* (NIS) [11, 12] (Table 3) were also investigated. In NIS, some attribute values are given as a set of possible attribute values. In Table 3, we interpret {1, 2, 3} in x2 as that one of 1, 2, and 3 is the actual value, but there is not enough information to decide it due to information incompleteness.

If we replace each set of attributes in NIS $\Phi$ with one element of the set, we have one possible DIS. In Table 3, there are 243 ($=3^5$) possible DISs and let $DD(\Phi)$ denote a set of possible DISs. We considered the certain rules and the possible rules from NIS below:

* An implication $\tau$ is a certain rule, if $\tau$ is a rule in each $\psi \in DD(\Phi)$.

* An implication $\tau$ is a possible rule, if $\tau$ is a rule in at least one $\psi \in DD(\Phi)$. This definition seems natural, but the $|DD(\Phi)|$ increases exponentially. However, we proved some properties and developed the NIS-Apriori algorithm, which does not depend upon the number of $|DD(\Phi)|$ [17, 18].

This paper extends the previous frameworks and considers rule generation from tables with continuous attribute values. Furthermore, this paper applies the obtained rules to decision making. The reasoning based on the obtained rules will recover the black-box problem in AI.

This paper's organization is as follows: Section 2 clarifies the discretization of the continuous attribute values and considers an example of the Iris data set [4]. Section 3 proposes granulated tables $\Gamma$ with frequency by discretization and extends the previous algorithms for NIS to that for $\Gamma$. Section 4 applies the extended algorithms to some data sets and shows the improvement of rule generation performance. An application of the obtained rules to decision making is considered. Section 5 concludes this paper.
# 2 Tables with Continuous Attribute Values

This section briefly examines the discretization of continuous attribute values and considers the case of the Iris data set [4].

## 2.1 Rules and Discretization of Continuous Attribute Values

To generate rules from tables with continuous attribute values, we need to discretize tables because there may be too many descriptors. There seem to be several methods for discretization [7]. Most of them focus on the optimal discretization specified by the constraints like minimal entropy, equal-interval width, equal-interval frequency, etc. In these researches on discretization, descriptors seem to be obtained as a side effect.

However, our research purpose is to generate rules by specified descriptors. We at first specify descriptors and their intervals, then generate rules by them. Thus, we can have our rules for our specifications.

## 2.2 An Example of the Iris Data Set

The Iris data set consists of 150 objects, four condition attributes {$spl, spw, pel, pew$} (each attribute value of them is continuous), one decision attribute class whose attribute value is one of setosa, versicolor, and virginica. Fig. 1 is a part of the Iris data set.

<table><caption>**Fig. 1.** A part of the Iris data set.</caption><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th></tr></thead><tbody><tr><td>1</td><td>object</td><td>spl</td><td>spw</td><td>pel</td><td>pew</td><td>class</td></tr><tr><td>2</td><td>1</td><td>5.1</td><td>3.5</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><td>3</td><td>2</td><td>4.9</td><td>3</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><td>100</td><td>99</td><td>5.1</td><td>2.5</td><td>3</td><td>1.1</td><td>versicolor</td></tr><tr><td>101</td><td>100</td><td>5.7</td><td>2.8</td><td>4.1</td><td>1.3</td><td>versicolor</td></tr><tr><td>150</td><td>149</td><td>6.2</td><td>3.4</td><td>5.4</td><td>2.3</td><td>virginica</td></tr><tr><td>151</td><td>150</td><td>5.9</td><td>3</td><td>5.1</td><td>1.8</td><td>virginica</td></tr></tbody></table>

<table><caption>**Fig. 2.** A definition of the discretization of continuous attribute values.</caption><thead><tr><th></th><th>small</th><th>medium</th><th>large</th></tr></thead><tbody><tr><td>spl</td><td>5.5&lt;</td><td>5.5&lt;= &lt; 6.7</td><td>6.7&lt;=</td></tr><tr><td>spw</td><td>3&lt;</td><td>3&lt;= &lt; 4</td><td>4&lt;=</td></tr><tr><td>pel</td><td>2.5&lt;</td><td>2.5&lt;= &lt; 5</td><td>5&lt;=</td></tr><tr><td>pew</td><td>1&lt;</td><td>1&lt;= &lt; 2</td><td>2&lt;=</td></tr></tbody></table>

Fig. 2 shows the discretization specification; namely, we want to generate rules using the attribute values small, medium, and large for each attribute. Here, every object is identified as one element of the Cartesian product {$small, medium, large$}⁴ × {$setosa, versicolor, virginica$}, whose number of elements is 243 (=3⁵). On the other hand, the number of objects is 150. Thus, there exist many such elements of the Cartesian product that do not correspond to objects.
<figure><img src="image_2.png" /><figcaption>Fig. 3. Discretized table with frequency.</figcaption></figure>

<table><caption>**Table 4.** A relationship between data sets and the discretized data sets. Here, type I: tables with continuous values, type II: tables with discrete values, type III: tables with missing values, #object: the number of objects, #con: the number of condition attributes, #Cartesian: the number of elements of the Cartesian product, #object_discre: the number of objects after discretization.</caption><thead><tr><th>data sets</th><th>type</th><th>#object</th><th>#con</th><th>#Cartesian</th><th>#object_discre</th></tr></thead><tbody><tr><td>Iris</td><td>I</td><td>150</td><td>4</td><td>243 (=3<sup>4</sup> × 3)</td><td>25</td></tr><tr><td>Wine quality [4]</td><td>I</td><td>4898</td><td>11</td><td>1240029 (=3<sup>11</sup> × 7)</td><td>564</td></tr><tr><td>Htru2</td><td>I</td><td>17898</td><td>8</td><td>13122 (=3<sup>8</sup> × 2)</td><td>134</td></tr><tr><td>Phishing [4]</td><td>II</td><td>1353</td><td>9</td><td>59049 (=3<sup>9</sup> × 3)</td><td>724</td></tr><tr><td>Car Evaluation [4]</td><td>II</td><td>1728</td><td>6</td><td>27648 (=4<sup>4</sup> × 3<sup>3</sup>)</td><td>1728</td></tr><tr><td>Suspicious Network [3]</td><td>II</td><td>39427</td><td>51</td><td>-</td><td>39427</td></tr><tr><td>Mammographic [4]</td><td>III</td><td>961</td><td>5</td><td>3200 (=4<sup>3</sup> × 5<sup>2</sup> × 2)</td><td>301</td></tr><tr><td>Congress Voting [4]</td><td>III</td><td>435</td><td>16</td><td>131072 (=2<sup>17</sup>)</td><td>342</td></tr></tbody></table>

We examined the relationship between the Iris data set and the elements of the Cartesian product. Fig. 3 shows lists of the sequential number, the Cartesian product element, and the duplicated number of objects. For example, the 10th list represents 42 objects for 150 objects. We may say that 42 objects are granulated to one granule represented by the 10th Cartesian product element. Thus, we have a discretized table with re-numbered 25 objects from a table with 150 objects. This phenomenon seems interesting because we can handle re-numbered 25 objects for a total of 150 objects.

We dealt with this phenomenon for other data sets with continuous attribute values. Table 4 shows the results. For example, there are 17898 objects in the
<figure><img src="image_3.png" /><figcaption>Fig. 4. Previous rule generation and new rule generation.</figcaption></figure>

Htru2 data set [4]. We employed a similar discretization, which divides each attribute into three classes, as Fig. 2. Then, each object is identified with an element of the Cartesian product with 13122 elements. Furthermore, 17898 objects are granulated to 134 elements in the Cartesian product.

*Remark 1.* In tables with continuous attribute values, we specify descriptors for discretization. If the number of discretized objects is much smaller than that of the original data sets, we will reduce the execution time of rule generation. Of course, we have the same rules as that of the original data set. It will be meaningful to consider the new rule generation in Fig. 4.

The strategy in Fig. 4 will be related to researches on rough sets (coarse classes) [13, 19], the Infobright technology [20], granular computing [14], and the zoom out operation [21]. As for the Car Evaluation and Suspicious Network data sets (type II) in Table 4, each tuple was different, and we cannot reduce the number of objects. Remark 1 will be applied to tables with continuous attribute values, and it may not be useful to tables with discrete attribute values.

## 3 Rule Generation from Discretized Tables with Frequency

This section defines a granulated table $\Gamma$ with frequency.

**Definition 1.** A discretized table $\Gamma$ with frequency (from a DIS $\psi$) consists of the following:

1. A finite set $AT$ of attributes,

2. A finite set $DESC_A$ of descriptors for $A \in AT$,

3. A pair $(LIST, freq)$ ($LIST \in \Pi_{A \in AT} DESC_A$: the Cartesian product of sets of descriptors, and $freq (> 0)$: the number of objects in $\psi$ satisfying $LIST$),

4. We assign a number $num$ to each pair and see a tuple $(num, LIST, freq)$ as an object in $\Gamma$.
**Input:** Table data set DIS $\psi$, decision attribute Dec, threshold values $\alpha, \beta$.

**Output:** A set $\mathbf{Rule}(\psi)$ of minimal rules.

1: $\mathbf{Rule}(\psi) \leftarrow \{\}; i \leftarrow 1$;

2: create a set $\mathbf{CAN}_1$ of candidates of rules with 1 condition;

3: **while** ($|\mathbf{CAN}_i| \ge 1$) **do**

4: &nbsp;&nbsp;&nbsp;&nbsp;Rest$_i \leftarrow \{\}; \mathbf{Rule}_i \leftarrow \{\}$;

5: &nbsp;&nbsp;&nbsp;&nbsp;**for all** $\tau_{i,j} \in \mathbf{CAN}_i$ **do**

6: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**if** $\text{support}(\tau_{i,j}) \ge \alpha$ **then**

7: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**if** $\text{accuracy}(\tau_{i,j}) \ge \beta$ **then** add $\tau_{i,j}$ to $\mathbf{Rule}_i$; **else** add $\tau_{i,j}$ to $\mathbf{Rest}_i$;

8: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end if**

9: &nbsp;&nbsp;&nbsp;&nbsp;**end if**

10: **end for**

11: $i \leftarrow i + 1$;

12: create $\mathbf{CAN}_i$ (candidates of rules with $i$-th conditions) from $\mathbf{Rest}_{i-1}$ and $\mathbf{Rest}_1$;

13: **end while**

14: **return** $\mathbf{Rule}(\psi) = \bigcup_{k<i} \mathbf{Rule}_k$

<caption>**Fig. 5.** The DIS-Apriori algorithm adjusted to table data set DIS $\psi$ [9].</caption>

Fig. 3 is an example of $\Gamma$. Now, we consider rule generation from $\Gamma$. Previously, we have investigated rules and rule generators [17, 18] in the following.

1. Rules in DIS $\psi$ and the DIS-Apriori rule generator,

2. Certain rules and possible rules in NIS $\Phi$ and the NIS-Apriori rule generator,

3. Decision-making tool based on the obtained rules.

We identify each descriptor [attribute,value] as an item and adjusted the Apriori algorithm for transaction data sets to that of table data sets. The overview of the adjusted DIS-Apriori algorithm is in Fig. 5. There is usually one decision attribute in every table, and we can see one itemset defines one implication. Furthermore, in the line 12 in Fig. 5, we have proved that $\mathbf{CAN}_i$ can be generated from $\mathbf{Rest}_{i-1}$ and $\mathbf{Rest}_1$ [9]. Due to these characteristics, we reduced the number of candidates of rules and the execution time of rule generation.

*Remark 2.* The following properties are related to the DIS-Apriori algorithm.

1. We replace DIS $\psi$ with NIS $\Phi$, support and accuracy values with minsupp and minacc values [17, 18], respectively. Then, this algorithm generates all minimal certain rules.

2. We replace DIS $\psi$ with NIS $\Phi$, support and accuracy values with maxsupp and maxacc values [17, 18], respectively. Then, this algorithm generates all minimal possible rules.

3. We term the above two algorithms the NIS-Apriori algorithm.

4. Both DIS-Apriori and NIS-Apriori algorithms are logically sound and complete for rules. They generate rules without excess and deficiency.

To handle a discretized table $\Gamma$ with frequency, we revise the calculation of support and accuracy values in Fig. 5. To calculate them, we are handling
equivalence classes defined by the concept of rough sets. For example, in Table 1, we have equivalence classes {$x1, x5$} for [P1,c] and {$x1, x2$} for [Dec,d1], respectively. We can easily know that $\tau$: [P1,c]$\Rightarrow$ [Dec,d1] is supported by {$x1$} ($=\{x1, x5\} \cap \{x1, x2\}$). Here, the occurrence of $\tau$ is 1, however in $\Gamma$, we need to count the frequency $freq(x1)$ of $x1$. Due to this consideration, we have the next Remark 3 for handling $\Gamma$.

*Remark 3.* In a DIS $\psi$, if an implication $\tau$ is supported by an equivalence class {$x1, x2, \dots, x_n$}, $\tau$ is supported by $n$ objects. However, in $\Gamma$, $\tau$ is supported by $freq(x1)+freq(x2)+\dots+freq(x_n)$ objects (Here, $freq(x_i)$ means the frequency of $x_i$ in $\Gamma$). If we replace the number of occurrence (i.e., 1) of one object $x_i$ with $freq(x_i)$, we can have the DIS-Apriori algorithm for handling $\Gamma$.

# 4 An Apriori-based Rule Generator for $\Gamma$ and Some Experiments

We revised rule generation programs, the DIS-Apriori algorithm for $\Gamma$ and the NIS-Apriori algorithm for $\Gamma$, in Python based on Remarks 2-3. For simplicity, we omit the details of the NIS-Apriori algorithm for $\Gamma$ and show the execution time in Table 5.

<table><caption>**Table 5.** A Comparison of the execution time: the original tables and the granulated tables. As for Iris, we coped with four cases. We duplicated the original table by ten times, 100 times, and 1000 times.</caption><thead><tr><th rowspan="2">Table</th><th colspan="2">support accuracy</th><th colspan="2">Original</th><th colspan="2">Granulated</th></tr><tr><th colspan="2"></th><th colspan="2">#object exec (sec)</th><th colspan="2">exec (sec)</th></tr></thead><tbody><tr><td>Iris</td><td>0.01</td><td>0.9</td><td>150</td><td>0.022</td><td>25</td><td>0.018</td></tr><tr><td>Iris1500</td><td>0.01</td><td>0.9</td><td>1500</td><td>0.030</td><td>25</td><td>0.020</td></tr><tr><td>Iris15000</td><td>0.01</td><td>0.9</td><td>15000</td><td>0.206</td><td>25</td><td>0.022</td></tr><tr><td>Iris150000</td><td>0.01</td><td>0.9</td><td>150000</td><td>2.162</td><td>25</td><td>0.022</td></tr><tr><td>Wine quality</td><td>0.001</td><td>0.5</td><td>4898</td><td>13.365</td><td>564</td><td>10.962</td></tr><tr><td>Htru2</td><td>0</td><td>0.7</td><td>17898</td><td>3.167</td><td>134</td><td>0.175</td></tr><tr><td>Mammographic<br>(certain rule)</td><td>0</td><td>0.8</td><td>961</td><td>0.343</td><td>299</td><td>0.297</td></tr><tr><td>(possible rule)</td><td>0</td><td>0.8</td><td>961</td><td>0.375</td><td>299</td><td>0.349</td></tr><tr><td>Congress Voting<br>(certain rule)</td><td>0.1</td><td>0.7</td><td>435</td><td>0.434</td><td>342</td><td>0.417</td></tr><tr><td>(possible rule)</td><td>0.1</td><td>0.7</td><td>435</td><td>0.448</td><td>342</td><td>0.419</td></tr></tbody></table>

Due to Table 5, we know the new rule generator is more effective in three cases of Iris15000, Iris150000, and Htru2. In the granulated tables $\Gamma$ from Iris15000 and Iris150000, the number of objects is the same, and only the frequency is changed. The rule generation process is the same as that of Iris, and the calculation of *support* and *accuracy* is slightly changed. Thus, the execution time of rule
<figure><img src="image_4.png" /><figcaption>**Fig. 6.** Obtained rules ($support(\tau) \ge 0.01$, $accuracy(\tau) \ge 0.9$) and $Rest_1$, $Rest_2$ from Iris150000.</figcaption></figure>

generation is almost constant. We think that to consider $\Gamma$ will be meaningful for big data analysis. Fig. 6 shows the obtained rules (the left hand side) and $Rest_1$, $Rest_2$, ..., we can reduce the number of candidates of rules. This reduction causes to reduce the execution time of rule generation.

<figure><img src="image_5.png" /><figcaption>**Fig. 7.** A chart of rule-based reasoning.</figcaption></figure>

Now, we consider the application of the obtained rules. We quickly think the rule-based decision making in Fig. 7. We have coped with ① and have developed the environment of rule generation. We also need to manage ②, i.e., decision making by the obtained rules. The decision making in Fig. 7 is based on the
applied rules, and the applied rule supports the reasoning. Thus, the reason is
apparent. This strategy will recover the black-box problem in AI. We need to
clarify the following subjects:

1. How do we select one rule if there are some applicable rules?

2. How do we have a decision if there is no candidate for a rule?

3. How do we think that the different results may be concluded?

For the Suspicious Network data set in Table 4, we employed the *lift* value for selecting one rule. We employed three-cross validation for 39427 objects and applied the obtained rules. This procedure is based on Fig. 7, and the averaged 94% correct estimation was obtained [8]. The research on Fig. 7 is in progress now.

5 Concluding Remarks

We considered discretization for tables with continuous attribute values and proposed granulated tables with frequency. In some cases, we can reduce the number of objects. This property causes to reduce the execution time of rule generation. We implemented a new rule generator and examined that the adjusted algorithms for big data analysis improved rule generation performance. The obtained rules are also applied to rule-based reasoning, which gives one solution to the black-box problem in AI. We need to improve much more comparative analysis and sensitive analysis comprehensively.

**Acknowledgment.** The authors would be grateful to the anonymous referees for their useful comments. This work is supported by JSPS (Japan Society for the Promotion of Science) KAKENHI Grant Number JP20K11954.

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