Probability; Tucker tensor decomposition; Likert scale; Fuzzy sets; Fuzzy transform

The research reported in this paper is motivated by the application of probabilistic graphical models in the social sciences. In this field, models are often learned from data collected in survey questionnaires. A Likert scale is typically used to collect respondents' opinions. For instance, in the seven point Likert scale, the possible responses are (1) strongly disagree, (2) disagree, (3) somewhat disagree, (4) neutral, (5) somewhat agree, (6) agree, and (7) strongly agree.Naturally, the dividing lines between these categories are vague, so it is reasonable to interpret them using fuzzy numbers instead of integers. This approach yields a new representation of the probability distributions learned from the collected data. The original state space of each measured variable is transformed into a new, typically reduced, state space. A function describes the correspondence to the original state space. For this function to have a natural interpretation, it must be unimodal, meaning it either increases to a peak and then decreases or decreases to a trough and then increases. Since each probability distribution can be viewed as a tensor, the problem of finding a new, compact representation can be formalized as a unimodal Tucker decomposition. We find these decompositions by combining the alternating least squares algorithm with projections of the columns of the factor matrices onto unimodal vectors. We tested the algorithm's performance using data from the Dividing Lines in Czech Society survey.

The research reported in this paper is motivated by the application of probabilistic graphical models in the social sciences. In this field, models are often learned from data collected in survey questionnaires. A Likert scale is typically used to collect respondents' opinions. For instance, in the seven point Likert scale, the possible responses are (1) strongly disagree, (2) disagree, (3) somewhat disagree, (4) neutral, (5) somewhat agree, (6) agree, and (7) strongly agree.Naturally, the dividing lines between these categories are vague, so it is reasonable to interpret them using fuzzy numbers instead of integers. This approach yields a new representation of the probability distributions learned from the collected data. The original state space of each measured variable is transformed into a new, typically reduced, state space. A function describes the correspondence to the original state space. For this function to have a natural interpretation, it must be unimodal, meaning it either increases to a peak and then decreases or decreases to a trough and then increases. Since each probability distribution can be viewed as a tensor, the problem of finding a new, compact representation can be formalized as a unimodal Tucker decomposition. We find these decompositions by combining the alternating least squares algorithm with projections of the columns of the factor matrices onto unimodal vectors. We tested the algorithm's performance using data from the Dividing Lines in Czech Society survey.