Technical Summary: Introduction of Over/Under/Off Masses

1. Problem Statement

Traditional information fusion theories, such as Dempster-Shafer Theory (DST) and Dezert-Smarandache Theory (DSmT), operate under the constraint that belief masses must lie within the interval $[0, 1]$ and sum to exactly 1. While effective for standard uncertainty, these frameworks struggle to model specific real-world scenarios involving:

• Over-accumulation of evidence: Situations where multiple high-confidence sources reinforce a single hypothesis, creating a "consensus intensity" that exceeds the standard unit limit (analogous to overtime work).
• Under-accumulation/Indeterminacy: Scenarios with significant information gaps or missing data where the total belief is insufficient to reach a unit sum.
• Contradictory or Falsifying Evidence: Cases where evidence actively disproves a hypothesis or represents damage/counter-evidence, requiring negative values to mathematically "cancel out" false alarms.

Existing extensions (fuzzy, neutrosophic) have addressed these concepts in set theory and logic (OverSets, UnderSets, OffSets), but a corresponding formalism for Mass Functions in information fusion was previously undefined.

2. Methodology

The paper proposes a mathematical extension of classical mass functions to three new categories: OverMass, UnderMass, and OffMass. The methodology involves redefining the domain and range of the mass function $m: 2^\Theta \to \mathbb{R}$, where $\Theta$ is the frame of discernment.

Core Definitions

• OverMass ($m_{over}$): A mass function where the sum of masses exceeds 1 ($\sum m(x) = \psi > 1$). The range is $[0, \psi]$.
• UnderMass ($m_{under}$): A mass function where the sum of masses is less than 1 ($\sum m(x) = \varphi + 1$, where $\varphi < 0$). The range is $[\varphi, 1]$.
• OffMass ($m_{off}$): A mass function allowing values both below 0 and above 1 ($\sum m(x) = \varphi + \psi$). The range is $[\varphi, \psi]$ where $\varphi < 0 < 1 < \psi$.

Fusion Rules

The paper introduces specific fusion rules to handle these new mass types, moving beyond standard Dempster's rule:

1. OverConjunctive Rule: For fusing two OverMasses, the standard conjunctive rule is applied, followed by OverNormalization.

   • The raw product of masses is calculated.
   • The total sum is normalized by dividing by the product of the original sums (e.g., if both sources sum to 1.1, the result is divided by 1.1).
   • PCR5 (Proportional Conflict Redistribution Rule #5) is applied to redistribute the conflicting mass proportionally to the non-conflicting elements.

2. Average of Masses Rule: For fusing masses involving negative values (UnderMasses or OffMasses), the paper argues that multiplication is inappropriate (as negative $\times$ negative = positive). Instead, it proposes using the arithmetic average of the masses from different sources to preserve the sign and magnitude of the evidence.

3. Interval Unification: When fusing masses from different intervals (e.g., one source in $[0, 1.1]$ and another in $[0, 1.2]$), the resulting domain is defined as the union of the intervals (e.g., $[0, 1.2]$) to accommodate all possible values.

3. Key Contributions

• Formal Extension: First-time definition of OverMass, UnderMass, and OffMass, extending the concepts of Over/Under/Off Sets and Logic into the domain of evidence fusion.
• Novel Fusion Algorithms: Development of specific fusion rules (OverConjunctive with OverNormalization and Average Rule) tailored to handle values outside the $[0, 1]$ interval.
• Handling Negative Belief: Introduction of a mechanism to mathematically represent "falsity" or "damage" via negative mass values, allowing for the cancellation of erroneous data.
• Dynamic Normalization: A shift from fixed normalization (sum = 1) to dynamic normalization (sum = $\psi$ or $\varphi + 1$), preserving the "intensity" of the evidence rather than dampening it.

4. Results and Examples

The paper validates the methodology through three distinct scenarios:

A. Manager Selection (OverMass Fusion)

• Scenario: Two directors evaluate two candidates (A and B). Both directors assign high confidence, resulting in mass sums of 1.1.
• Process: The OverConjunctive rule was applied. The raw conflict was redistributed using PCR5. The final masses were OverNormalized by dividing by 1.1.
• Outcome: The system successfully identified the superior candidate (A) while maintaining the "over-confidence" signal (total mass = 1.1) rather than forcing it back to 1.0, which would have diluted the strength of the consensus.

B. Wildfire Evacuation (Application Scenarios)

• OverMass (Redundant Evidence): Satellite, IoT, and social media all confirm a fire. The sum of masses > 1 triggers a "Critical Priority" alert, justifying immediate deployment of heavy resources (aerial tankers) based on the "over-performance" of evidence.
• UnderMass (Coverage Gaps): In a remote area with missing sensors, the sum of masses is low (e.g., 0.3). The system interprets the "missing mass" (0.7) as high uncertainty, triggering a "Search and Reconnaissance" protocol rather than a blind evacuation.
• OffMass (Falsification): A sensor falsely reports a fire (positive mass), but thermal cameras confirm no fire (negative mass). The OffMass framework allows these to cancel out mathematically, preventing the dispatch of resources to a safe zone.

C. Police Investigation (OverMass with PCR5)

• Scenario: Two sources provide evidence on suspects A and B with mass sums of 1.1.
• Result: After applying the conjunctive rule, PCR5 redistribution, and OverNormalization, the system calculated Belief (Bel) and Plausibility (Pl) values. The final normalized masses (summing to 1.1) clearly indicated suspect B as the primary target, demonstrating the robustness of the method in handling conflicting high-confidence data.

5. Significance

The introduction of Over/Under/Off Masses represents a paradigm shift in information fusion for uncertain environments:

1. Enhanced Expressiveness: It allows systems to distinguish between "strong consensus" (OverMass) and "weak consensus" (UnderMass), which classical normalization obscures.
2. Robustness to Error: By incorporating negative values (OffMass), the framework provides a rigorous mathematical way to handle and discount false positives or malicious data, a critical feature for security and safety-critical systems.
3. Real-World Applicability: The methodology directly addresses complex, multi-source fusion problems in emergency management (wildfires), security monitoring, and personnel evaluation, where data is often redundant, incomplete, or contradictory.
4. Theoretical Foundation: It bridges the gap between Neutrosophic Set theory and practical evidence fusion, offering a unified framework for handling indeterminacy, contradiction, and excess information.

This work lays the groundwork for future research into the fusion of mixed mass types (e.g., OverMass with UnderMass) and the development of belief function interpretations for these extended mass values.