A Linear Model for Three-Way Analysis of Facial Similarity Daryl H. Hepting [0000-0002-3138-3521] , Hadeel Hatim Bin Amer [0000-0003-2515-2329] , and Yiyu Yao [0000-0001-6502-6226] Department of Computer Science University of Regina, Regina, SK, S4S 0A2, Canada hepting@cs.uregina.ca, binamerh@uregina.ca, yyao@cs.uregina.ca Abstract. Card sorting was used to gather information about facial similarity judgments. A group of raters put a set of facial photos into an unrestricted number of different piles according to each rater’s judgment of similarity. This paper proposes a linear model for 3-way analysis of similarity. An overall rating function is a weighted linear combination of ratings from individual raters. A pair of photos is considered to be similar, dissimilar, or divided, respectively, if the overall rating function is greater than or equal to a certain threshold, is less than or equal to another threshold, or is between the two thresholds. The proposed framework for 3-way analysis of similarity is complementary to studies of similarity based on features of photos. Keywords: similarity, three-way decision, card sorting, linear model 1 Introduction A basic idea of three-way decisions (3WD) is thinking and problem solving in threes [15]. According to a trisecting-and-acting model of 3WD, we divide a whole into three parts and devise strategies to process the three parts [14, 15]. While each part captures a particular aspect of the whole or consists of elements of particular interest, their integration reflects the whole. By thinking in threes, 3WD may provide a simplification of processing the whole through processing three parts. The theory of 3WD has been applied in many fields [6–9, 13, 16]. In a previous paper [3], we presented some preliminary results on applying the 3WD theory to a card sorting problem. Card sorting has been successfully applied to gain insight about the structure of information in different contexts [1, 2, 10, 11]. For our card sorting problem, we have a set of facial photos and a group of raters. Each rater was instructed to sort similar photos into the same pile. A pair of photos is similar if both photos are sorted into the same pile. A pair of photos is dissimilar if the photos are sorted into different piles. An analysis of rating results shows that there is a large variance amongst raters in terms of the number of piles and the sizes of those piles (see Figure 1).