Age-at-death estimation of an individual skeleton is important to forensic and biological anthropologists for identification and demographic analysis, but it has been shown that the current aging methods are often unreliable because of skeletal variation and taphonomic factors. Multifactorial methods have been shown to produce better results when determining age-at-death than single indicator methods. However, multifactorial methods are difficult to apply to single or poorly preserved skeletons, and they rarely provide the investigator with information about the reliability of the estimate. The goal of this research is to examine the validity of the Sugeno fuzzy integral as a multifactorial method for modeling age-at-death of an individual skeleton. This approach is novel because it produces an informed decision of age-at-death utilizing multiple age indicators while also taking into consideration the accuracies of the methods and the condition of the bone being examined. Additionally, the Sugeno fuzzy integral does not require the use of a population and it qualitatively produces easily interpreted graphical results. Examples are presented applying three commonly used aging methods on a known-age skeletal sample from the Terry Anatomical Collection. This method produces results that are more accurate and with smaller intervals than single indicator methods.

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Abstract-The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multi-criteria decision making, skeletal ageat-death estimation and multi-source (e.g., feature, algorithm, sensor, confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher-order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number-(FN), i.e., convex and normal fuzzy set-(FS), valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non EP-based FI extension called the non-direct FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multi-source fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications.

In prior work, Grabisch put forth a direct (i.e., result of the Extension Principle) generalization of the Sugeno fuzzy integral (FI) for fuzzy set (FS)-valued normal (height equal to one) integrands and number-based fuzzy measures (FMs). Grabisch's proof is based in large on Dubois and Prade's analysis of functions on intervals, fuzzy numbers (thus normal FSs) and fuzzy arithmetic. However, a case not studied is the extension of the FI for sub-normal FS integrands. In prior work, we described a real-world forensic application in anthropology that requires fusion and has sub-normal FS inputs. We put forth an alternative non-direct approach for calculating FS results from sub-normal FS inputs based on the use of the number-valued integrand and number-valued FM Sugeno FI. In this article, we discuss a direct generalization of the Sugeno FI for sub-normal FS integrands and numeric FMs, called the Sub-normal Fuzzy Integral (SuFI). To no great surprise, it turns out that the SuFI algorithm is a special case of Grabisch's generalization. An algorithm for calculating SuFI and its mathematical properties are compared to our prior method, the Non-Direct Fuzzy Integral (NDFI). It turns out that SuFI and NDFI fuse in very different ways. We assert that in some settings, e.g., skeletal age-at-death estimation, NDFI is preferred to SuFI. Numeric examples are provided to stress important inner workings and differences between the FI generalizations.

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