Dempster-Shafer Fusion

The exhaustive set of mutually exclusive hypotheses, denoted Θ. In spectrum sensing, Θ = {H₀, H₁} where H₀ represents 'channel vacant' and H₁ represents 'channel occupied'. The power set 2^Θ includes all subsets: {∅}, {H₀}, {H₁}, {H₀, H₁}. The subset {H₀, H₁} explicitly represents total ignorance or uncertainty about the true state.

A function m: 2^Θ → [0,1] that distributes a unit of belief across the power set. Key properties:

• m(∅) = 0 (no belief in the empty set)
• Σ m(A) = 1 for all A ⊆ Θ

A sensing node might assign:

• m({H₁}) = 0.6 (evidence for occupied)
• m({H₀}) = 0.2 (evidence for vacant)
• m({H₀, H₁}) = 0.2 (uncommitted belief, representing uncertainty)

The orthogonal sum m₁ ⊕ m₂ fuses two independent BBAs into a joint belief. For two sources, the combined mass for hypothesis A is:

m₁₂(A) = (1 / (1 - K)) * Σ m₁(B) × m₂(C) for B ∩ C = A

Where K is the conflict coefficient, measuring the total mass assigned to empty intersections. The denominator (1 - K) normalizes the result, redistributing conflicting belief proportionally across agreeing hypotheses.

The conflict coefficient K = Σ m₁(B) × m₂(C) for all B ∩ C = ∅ directly measures the degree of contradiction between evidence sources. A high K value (near 1) indicates severe disagreement, which may signal:

• A Spectrum Sensing Data Falsification (SSDF) attack
• Deep correlated shadowing affecting multiple nodes
• A malfunctioning sensor

This explicit conflict metric is a key advantage over Bayesian fusion, which can mask disagreement.

Two derived measures bound the true probability of a hypothesis:

• Belief Bel(A) = Σ m(B) for all B ⊆ A. The total evidence that definitively supports A.
• Plausibility Pl(A) = Σ m(B) for all B ∩ A ≠ ∅. The total evidence that does not refute A.

The interval [Bel(A), Pl(A)] represents the evidential uncertainty range. For spectrum sensing, a narrow interval indicates high confidence; a wide interval signals the need for additional sensing.

Dempster-Shafer fusion is inherently robust against Spectrum Sensing Data Falsification (SSDF) attacks. A malicious node reporting falsified data will generate high conflict with honest nodes. The fusion center can:

• Detect the attack by monitoring the conflict coefficient K
• Apply discounting factors to reduce the weight of consistently conflicting sources
• Integrate with reputation management systems that dynamically adjust node trust scores based on historical conflict patterns

The foundational evidence structure in Dempster-Shafer theory, also called a mass function. A BPA assigns probability mass to subsets of the frame of discernment, not just singletons.

• m({H₁}) = 0.6: 60% belief that channel is occupied
• m({H₀, H₁}) = 0.4: 40% belief assigned to uncertainty (either state possible)
• m(∅) = 0: No belief in impossible outcomes

This explicit representation of ignorance is the key advantage over Bayesian priors, which force probability distribution across all hypotheses even when evidence is weak.

The orthogonal sum operator (⊕) that fuses two independent BPAs into a single combined mass function. It normalizes the conjunctive product by the conflict factor.

Formula: m₁₂(A) = (1 / (1-K)) × Σ m₁(B)·m₂(C) for B∩C=A

• K: Total conflict mass between evidence sources
• 1-K: Normalization factor redistributing conflicting mass
• Handles high-conflict scenarios where Bayesian updates would produce counterintuitive posteriors

The rule is commutative and associative, allowing incremental fusion as new sensing reports arrive at the fusion center.

Dual uncertainty measures derived from the BPA that bound the true probability of a hypothesis, forming an evidence interval.

• Belief Bel(A): Sum of all mass committed to subsets of A — the lower bound of support
• Plausibility Pl(A): Sum of all mass that does not contradict A — the upper bound
• Evidence Interval: [Bel(A), Pl(A)] represents the range of possible probability

For spectrum sensing, a wide interval signals high uncertainty from conflicting or sparse reports, enabling the fusion center to defer decisions rather than forcing a false classification.

High conflict (K → 1) between sensing nodes can produce counterintuitive results under standard Dempster's rule. Alternative combination methods address this:

• Yager's Rule: Assigns conflicting mass to the universal set Θ rather than normalizing, preserving uncertainty
• Dempster-Shafer with Discounting: Pre-processes BPAs by applying a discount factor α based on node reliability before combination
• Transferable Belief Model: Separates the credal level (belief representation) from the pignistic level (decision-making)

These variants make DST robust against Spectrum Sensing Data Falsification (SSDF) attacks where malicious nodes inject conflicting evidence.

The exhaustive set of mutually exclusive hypotheses Θ that defines the decision space. In cooperative spectrum sensing, this is typically binary but can be extended.

Standard CSS Frame: Θ = {H₀, H₁}

• H₀: Primary user absent (channel vacant)
• H₁: Primary user present (channel occupied)

Extended Frames may include:

• H₂: Jamming signal detected
• H₃: Primary User Emulation attack suspected

The power set 2^Θ contains all subsets, enabling nodes to express compound hypotheses when unable to discriminate between specific signal types.

Key architectural differences between evidence-theoretic and probabilistic fusion for cooperative sensing:

• Prior Requirements: DST uses vacuous belief (m(Θ)=1) when no prior exists; Bayesian requires explicit priors
• Uncertainty Modeling: DST separates aleatory uncertainty (randomness) from epistemic uncertainty (ignorance) via mass on non-singleton sets
• Computational Cost: DST combination is O(2^|Θ|) per fusion round vs. O(|Θ|) for Bayesian updates
• Conflict Handling: DST explicitly quantifies conflict via K; Bayesian methods conflate conflict with low probability

DST is preferred when sensing nodes have heterogeneous reliability or operate in environments with unknown interference characteristics.