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Cocluster structure analysis is a basic technique for revealing intrinsic structural information from cooccurrence data among objects and items, in which coclusters are composed of mutually familiar pairs of objects and items. In many real applications, it is also the case that we have not only cooccurrence information among objects and items but also intrinsic relation among items and other ingredients. For example, in food preference analysis, users’ preferences on foods should be found considering not only user-food cooccurrences but also the implicit relation among users and cooking ingredients. In this paper, two FCM-type fuzzy coclustering models, that is, FCCM and Fuzzy CoDoK, are extended for revealing intrinsic cocluster structures from three-mode cooccurrence data, where the aggregation degree of three elements in each cocluster is maximized through iterative updating of three types of fuzzy memberships for objects, items, and ingredients. The characteristic features of the proposed methods are demonstrated through a numerical experiment.

In many web data analyses, we often have cooccurrence information among objects and items instead of multidimensional observations on objects. For example, web document summarization and web market purchase summarization are reduced to document-keyword cooccurrence analysis and customer-product basket analysis, respectively. FCM-type fuzzy coclustering is an extension of fuzzy

Besides their usefulness in many applications, it is also the case that the conventional fuzzy coclustering models cannot work well under severe influences of other intrinsic features. For example, in food preference analysis, users’ preferences on foods cannot be revealed considering only user-food cooccurrences but should be found considering implicit relation among users and cooking ingredients, which compose the foods. Then, when we have not only cooccurrence information among objects and items but also intrinsic relation among items and other ingredients; we can expect to find more useful cocluster structures in three-mode cooccurrence information data.

In this paper, two FCM-type fuzzy coclustering models are extended for analyzing three-mode cooccurrence information data, in which FCM-like alternative optimization schemes are performed considering cooccurrence relation among objects, items, and other ingredients. First, the FCCM algorithm is extended to the three-mode FCCM (3FCCM) algorithm by utilizing three types of fuzzy memberships for objects, items, and ingredients, where the aggregation degree of three features in each cocluster is maximized through iterative updating of memberships supported by the entropy-based fuzzification. Second, the 3FCCM algorithm is further extended to the three-mode Fuzzy CoDoK (3Fuzzy CoDoK) by introducing the quadratic regularization-based fuzzification. The characteristic features of the proposed methods are demonstrated through a numerical experiment.

The remainder of this paper is organized as follows: Section

Fuzzy

Assume that we have

Because of the linear nature with respect to

In [

Based on the alternative optimization principle,

Although the two updating rules are always fair under the constraints, they can be numerically unstable due to overflows because

As an alternative approach, Kummamuru et al. [

Based on the Lagrangian multiplier method, the updating rules are obtained as

The updating rules are more numerically stable than those of FCCM because their calculation ranges are in linear orders with respect to

Besides the usefulness of these fuzzy coclustering models in handling two-modes cooccurrence information, their cocluster structures may be influenced by other third elements. Specifically, if each item is related to some other ingredients, the partition quality is expected to be improved by considering the intrinsic relation among three-mode elements. In the following section, the FCM-type coclustering algorithms are extended for analyzing such three-mode cooccurrence information data.

Assume that we have

In order to extend the conventional FCCM and Fuzzy CoDoK algorithms to three-mode cocluster analysis, additional memberships

In the following parts of this section, the conventional FCCM and Fuzzy CoDoK algorithms are extended to their three-mode versions utilizing the above aggregation criterion.

First, the FCCM algorithm is extended by using the modified aggregation criterion of (

Here, it should be noted that we can adopt two different types of constraints to ingredient memberships

The clustering algorithm is an iterative process of updating

Next, Fuzzy CoDoK is extended to the three-mode coclustering model named three-mode Fuzzy CoDoK (3Fuzzy CoDoK). The objective function of (

The updating rules are given in the similar manner to the previous section as follows:

In a similar manner to Fuzzy CoDoK, the above updating rules are computationally more stable than 3FCCM because of the lack of

Following the above derivation, a sample algorithm is represented as follows:

Given

Update

Update

Update

In order to demonstrate the characteristics of the proposed algorithms, a numerical experiment was performed with an artificially generated three-mode data set, in which 40 objects

Artificial three-mode cooccurrence information data.

Intrinsic connection

Cooccurrence matrix

Cooccurrence matrix

The goal of this experiment is to extract the intrinsic cocluster structure among objects, items, and ingredients from cooccurrence matrices

First, the proposed 3FCCM and 3Fuzzy CoDoK algorithms were applied to

Derived memberships by proposed 3FCCM.

Object memberships

Ingredient memberships

Item memberships

Derived memberships by proposed 3Fuzzy CoDoK.

Object memberships

Ingredient memberships

Item memberships

Figures

By the way, Figure

These results imply that the 3FCCM algorithm is more suitable for clearly capturing the intrinsic connections although 3Fuzzy CoDoK has an advantage in computational stability.

Second, the above clustering results are compared with the conventional FCCM and Fuzzy CoDoK, which are designed only for two-mode cooccurrence information. Although the intrinsic connection

Estimated intrinsic connection matrix

The conventional FCCM and Fuzzy CoDoK were applied to

Derived memberships by conventional FCCM.

Object memberships

Ingredient memberships

Derived memberships by conventional Fuzzy CoDoK.

Object memberships

Ingredient memberships

Next, the robustness of the algorithms against random initialization is studied by comparing the frequencies of the plausible solutions. Table

Comparison of frequencies of plausible solutions in 100 trials.

Algorithm | Two-mode | Three-mode | ||
---|---|---|---|---|

FCCM | Fuzzy CoDoK | 3FCCM | 3Fuzzy CoDoK | |

Frequency | 78 | 47 | 90 | 100 |

Therefore, the proposed algorithms are useful in analyzing three-mode cooccurrence information, which simultaneously consider the typicality of three elements.

Finally, the partition characteristic of the proposed coclustering models is compared with the relational matrix decomposition method. Multiple correspondence analysis (MCA) [

2D Plots given by multicorresponding analysis.

The proposed algorithms have advantages in handling three-mode elements by emphasizing their contributions to each coclusters. Additionally, while the implicit fuzziness degree of MCA is fixed (unchangeable), the proposed coclustering model can improve the interpretability of cluster partition by tuning the fuzziness degrees.

In this paper, novel coclustering models were proposed for analyzing three-mode cooccurrence information with the goal being to improve the partition quality of the conventional two-modes analysis. The proposed 3FCCM and 3Fuzzy CoDoK algorithms extended the conventional FCCM and Fuzzy CoDoK algorithms by introducing an additional membership for ingredients into the aggregation degree of three elements: objects, items, and ingredients. A numerical experiment with an artificial data set demonstrated that 3FCCM is more useful in capturing the intrinsic connection among objects and ingredients while 3Fuzzy CoDoK is suitable for handling large data sets with its computational stability.

Besides the simplicity of FCM-type coclustering, FCCM and fuzzy CoDoK sometimes have the difficulty in tuning of fuzziness degrees. In the conventional two-modes coclustering, an MMMs-induced model [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported in part by Tateisi Science and Technology Foundation, Japan, under Research Grant 2017.