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arxiv: 1907.04679 · v1 · pith:W5EPE5F3new · submitted 2019-07-09 · 💻 cs.AI

Measuring Inter-group Agreement on zSlice Based General Type-2 Fuzzy Sets

Javier Navarro , Christian Wagner This is my paper

Pith reviewed 2026-05-25 00:21 UTC · model grok-4.3

classification 💻 cs.AI
keywords fuzzy setsagreement ratiointerval agreement approachgeneral type-2 fuzzy setszSlicesinter-group agreementuncertainty modelingsimilarity measures
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The pith

The Agreement Ratio is extended to General Type-2 Fuzzy Sets by applying similarity measures across their zSlice levels to quantify inter-group agreement from interval data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a way to measure how much different groups agree when their responses are modeled as General Type-2 Fuzzy Sets using the Interval Agreement Approach. It builds on an earlier Agreement Ratio that worked only for simpler type-1 sets by introducing a similarity calculation that compares agreement strengths at successive vertical slices of the more complex set. This matters because many real perception tasks involve ambiguous concepts where participants from varied backgrounds produce overlapping but distinct intervals of uncertainty. If the extension works, analysts gain a single numeric indicator of consensus strength that accounts for both the range of opinions and the secondary uncertainties within them. Synthetic cases and one real dataset illustrate how the calculation proceeds and what values it produces for differing group alignments.

Core claim

The central claim is that the Agreement Ratio can be extended to General Type-2 Fuzzy Sets by using a similarity measure to relate the distinct levels of agreement encoded at each zSlice, thereby producing a quantitative expression of the inter-group agreement captured by the fuzzy set model.

What carries the argument

The zSlice Agreement Ratio extension, which applies a similarity measure to the vertical slices of a General Type-2 Fuzzy Set to relate agreement strengths across participant groups.

If this is right

  • The measure produces a numeric value for agreement strength that incorporates both primary and secondary membership information in the fuzzy set.
  • It can be calculated for any General Type-2 Fuzzy Set generated by the Interval Agreement Approach without requiring outlier removal or preset membership shapes.
  • Application to real data shows it can highlight divergence between heterogeneous participant groups on the same concept.
  • The extension preserves the original ratio's focus on position and strength of agreement while adding the inter-group dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same similarity-based slicing approach might be tested on other constructions of General Type-2 Fuzzy Sets to see whether agreement readings remain stable.
  • If the measure is applied repeatedly to evolving data sets it could track changes in group consensus over time.
  • The technique opens a route to compare agreement across different ambiguity levels within a single modeling framework.

Load-bearing premise

The method assumes that a similarity measure applied directly to the zSlice levels of the General Type-2 Fuzzy Set accurately reflects inter-group agreement without further conditions on how the underlying interval data are distributed.

What would settle it

Collect interval responses from two groups known to hold sharply opposing views on an ambiguous concept, construct the corresponding General Type-2 Fuzzy Set via the Interval Agreement Approach, compute the extended ratio, and check whether it returns a low value consistent with the known opposition.

Figures

Figures reproduced from arXiv: 1907.04679 by Christian Wagner, Javier Navarro.

Figure 1
Figure 1. Figure 1: Fuzzy set generated from 4 intervals: D1 = [2, 5] and D2 = [3, 5], D¯ 3 = [6, 8] and D4 = [3, 7] Finally, the overall summation is divided by the sum of ‘weights’ yα so the final ratio is normalised to a number in the range [0,1]. As can be seen in (4), these yα terms act as indicators of the level of agreement weighting overlapping α -cuts in proportion to the number of intervals considered in the interse… view at source ↗
Figure 2
Figure 2. Figure 2: T1 FSs generated from Patients, Physiotherapists [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The GT2 FSs T˜, N˜ and M˜ illustrating three synthetic cases of Total, Moderate and Null inter-group agreement from the intervals shown in Table I. B. Inter-group agreement in the IV TESS case In this last section, we consider the case of three groups of participants (i.e., patients, surgeons and physiotherapists) who provided IV responses in response to their perception about 5 linguistic descriptors used… view at source ↗
Figure 4
Figure 4. Figure 4: T1 FSs modelling the perception of different groups [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Recently, there has been much research into modelling of uncertainty in human perception through Fuzzy Sets (FSs). Most of this research has focused on allowing respondents to express their (intra) uncertainty using intervals. Here, depending on the technique used and types of uncertainties being modelled different types of FSs can be obtained (e.g., Type-1, Interval Type-2, General Type-2). Arguably, one of the most flexible techniques is the Interval Agreement Approach (IAA) as it allows to model the perception of all respondents without making assumptions such as outlier removal or predefined membership function types (e.g. Gaussian). A key aspect in the analysis of interval-valued data and indeed, IAA based agreement models of said data, is to determine the position and strengths of agreement across all the sources/participants. While previously, the Agreement Ratio was proposed to measure the strength of agreement in fuzzy set based models of interval data, said measure has only been applicable to type-1 fuzzy sets. In this paper, we extend the Agreement Ratio to capture the degree of inter-group agreement modelled by a General Type-2 Fuzzy Set when using the IAA. This measure relies on using a similarity measure to quantitatively express the relation between the different levels of agreement in a given FS. Synthetic examples are provided in order to demonstrate both behaviour and calculation of the measure. Finally, an application to real-world data is provided in order to show the potential of this measure to assess the divergence of opinions for ambiguous concepts when heterogeneous groups of participants are involved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the Agreement Ratio (previously limited to Type-1 fuzzy sets) to General Type-2 Fuzzy Sets constructed via the Interval Agreement Approach (IAA) using zSlices. The extension applies a similarity measure across the agreement levels at successive zSlices to quantify inter-group agreement for ambiguous concepts when participant groups are heterogeneous. The method is illustrated with synthetic examples showing calculation and behavior, followed by one real-world application demonstrating its use in assessing opinion divergence.

Significance. If the similarity-to-agreement mapping is valid and independent of data distribution assumptions, the measure would offer a practical tool for quantifying inter-group divergence in IAA-based GT2FS models without requiring outlier removal or predefined membership functions. The synthetic examples and real-world case provide concrete demonstrations, and the parameter-free nature of the underlying IAA is a strength. However, the absence of a derivation linking similarity on zSlice levels directly to group heterogeneity reduces the result's immediate applicability.

major comments (2)
  1. [Proposed measure definition] The central extension (described after the review of the T1 Agreement Ratio) defines the new inter-group measure via a similarity function applied to zSlice agreement levels, yet provides no derivation or axiomatic justification showing why this similarity equals inter-group agreement strength rather than an artifact of the chosen similarity function or zSlice discretization. This assumption is load-bearing for the claim that the measure captures 'the degree of inter-group agreement'.
  2. [Real-world application] In the real-world application section, the measure is applied to heterogeneous participant groups but no comparison is made to alternative inter-group metrics (e.g., statistical divergence tests or direct IAA agreement ratios per group) nor any validation against known ground-truth opinion splits, leaving the mapping untested.
minor comments (2)
  1. [Method] Notation for the zSlice levels and the similarity function should be introduced with explicit equations rather than inline description to improve readability.
  2. [Abstract and examples] The abstract states the measure 'relies on using a similarity measure' but does not name the specific similarity function employed in the examples; this should be stated at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment point-by-point below.

read point-by-point responses

  1. Referee: [Proposed measure definition] The central extension (described after the review of the T1 Agreement Ratio) defines the new inter-group measure via a similarity function applied to zSlice agreement levels, yet provides no derivation or axiomatic justification showing why this similarity equals inter-group agreement strength rather than an artifact of the chosen similarity function or zSlice discretization. This assumption is load-bearing for the claim that the measure captures 'the degree of inter-group agreement'.

    Authors: We acknowledge that the manuscript introduces the measure as a direct structural extension without a formal axiomatic derivation. The rationale is that zSlices in the IAA explicitly encode successive levels of agreement strength, so applying a similarity measure across these levels quantifies congruence in agreement profiles between groups; the synthetic examples illustrate that the resulting values align with intuitive notions of inter-group agreement. To strengthen the presentation, we will add a short explanatory subsection deriving this mapping from the properties of zSlices and the original T1 Agreement Ratio. revision: yes

  2. Referee: [Real-world application] In the real-world application section, the measure is applied to heterogeneous participant groups but no comparison is made to alternative inter-group metrics (e.g., statistical divergence tests or direct IAA agreement ratios per group) nor any validation against known ground-truth opinion splits, leaving the mapping untested.

    Authors: The real-world section is intended as an illustrative demonstration rather than a validation study. We agree that comparisons would improve the paper and will add, in revision, a brief quantitative comparison between the proposed inter-group measure and the per-group Agreement Ratios computed separately on each participant group. A controlled validation against known ground-truth opinion splits would require new data collection and is noted as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain.

full rationale

The paper defines a new inter-group agreement measure by extending the prior Agreement Ratio (for T1 FS) to GT2FS via IAA and zSlices, using an (unspecified) similarity measure across z-levels. This is a definitional construction of a new quantity rather than any reduction of a claimed prediction or result back to its inputs by construction. No equations or steps are shown that equate the output to a fitted parameter or prior result via self-citation load-bearing; the abstract presents the extension as a novel application with synthetic and real-world demonstrations. The derivation remains self-contained as an independent modeling choice.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper extends existing concepts without introducing new free parameters or invented entities; it relies on standard assumptions in fuzzy set theory and similarity measures.

axioms (2)
  • domain assumption Fuzzy sets can model uncertainty in human perception using interval data without outlier removal or predefined membership functions
    This underpins the use of IAA to obtain General Type-2 Fuzzy Sets.
  • domain assumption Similarity measures can quantitatively relate agreement levels across zSlice levels in a General Type-2 Fuzzy Set
    This is the core mechanism for the extended Agreement Ratio.

pith-pipeline@v0.9.0 · 5803 in / 1277 out tokens · 30157 ms · 2026-05-25T00:21:10.934261+00:00 · methodology

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Reference graph

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