Theoretical Analysis of Brooks–Iyengar Algorithm: Accuracy and Precision Bound

Abstract

Sensors have limited precision and accuracy. They extract data from the physical environment, which contains noise. The goal of sensor fusion is to make the final decision robust, minimizing the influence of noise and system errors. One problem that has not been adequately addressed is establishing the bounds of fusion result precision. Precision is the maximum range of disagreement that can be introduced by one or more faulty inputs. This definition of precision is consistent both with Lamport’s Byzantine Generals problem and the mini-max criteria commonly found in game theory. This article considers the precision bounds of several fault tolerant information fusion approaches, including Byzantine agreement, Marzullo’s interval-based approach, and the Brooks–Iyengar fusion algorithm. We derive precision bounds for these fusion algorithms. The analysis provides insight into the limits imposed by fault tolerance and guidance for applying fusion approaches to applications.

The following article with permission has been reproduced from the original copy: Ao, Buke, et al. “On precision bound of distributed fault tolerant sensor fusion algorithms.” ACM Computing Surveys (CSUR) 49.1 (2016): 5.

Appendix: Proof of Theorem 4.3—Precision Bound of BVC

The following proof is based on the proof sketch in [27]. The proof of precision bound is based on that Z _i and Z _j both contain one identical point. Suppose that m = N − τ PEs are non-faulty and v _i[t], v _j[t] are estimate vectors of two non-faulty PEs at round t. In [27], Observations 1 and 3 in Part III of the proof of Theorem 4.5 imply that:

$$\displaystyle \begin{aligned} {\mathbf{v}}_i[t] = \sum_{k=1}^{m} \alpha_k {\mathbf{v}}_k[t-1] \end{aligned} $$

(4.34)

where $\sum _{k=1}^{m} \alpha _k = 1, \alpha _k \geq 0$, and

$$\displaystyle \begin{aligned} {\mathbf{v}}_j[t] = \sum_{k=1}^{m} \beta_k v_k[t-1] \end{aligned} $$

(4.35)

where $\sum _{k=1}^{m} \beta _k = 1, \beta _k \geq 0$. Let g denote the index that satisfies α _g ≥ γ and α _g ≥ γ. The existence proof of g is in [27], where $\gamma = 1/N\mathrm {C}_{N-\tau }^{N}$.

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} v_{il}[t] &\displaystyle =&\displaystyle \sum_{k=1}^{m} \alpha_k v_{kl}[t-1] \\ &\displaystyle \leq&\displaystyle \alpha_g v_{gl}[t-1] + (1-\alpha_g)\varOmega_l[t-1] \\ \end{array} \end{aligned} $$

(4.36)

Since v _kl[t − 1] ≤ Ω _l[t − 1], ∀k, (4.36) can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_{il}[t] &\displaystyle \leq &\displaystyle \! \gamma v_{gl}[t-1]\! +\! (\alpha_g\! -\! \gamma)v_{gl}[t-1]\! +\! (1-\alpha_g)\varOmega_l[t-1] \\ &\displaystyle \leq&\displaystyle \! \gamma v_{gl}[t-1]\! +\! (\alpha_g\! -\! \gamma)\varOmega_{l}[t-1]\! +\! (1-\alpha_g)\varOmega_l[t-1] \\ &\displaystyle \leq&\displaystyle \gamma v_{gl}[t-1] + (1-\gamma)\varOmega_l[t-1] \\ {} \end{array} \end{aligned} $$

(4.37)

Similarly, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_{jl}[t]&\displaystyle =&\displaystyle \sum_{k=1}^{m} \beta_k v_{kl}[t-1] \\ &\displaystyle \geq&\displaystyle \beta_g v_{gl}[t-1] + (1-\beta_g)\mu_l[t-1] \\ &\displaystyle \geq&\displaystyle \! \gamma v_{gl}[t-1]\! +\! (\beta_g\! -\! \gamma)v_{gl}[t-1]\! +\! (1-\beta_g)\mu_l[t-1] \\ &\displaystyle \geq&\displaystyle \! \gamma v_{gl}[t-1]\! +\! (\beta_g\! -\! \gamma)\mu_{l}[t-1]\! +\! (1-\beta_g)\mu_l[t-1] \\ &\displaystyle \geq&\displaystyle \gamma v_{gl}[t-1] + (1-\gamma)\mu_l[t-1] \\ {} \end{array} \end{aligned} $$

(4.38)

Subtracting (4.37) from (4.38) yields

$$\displaystyle \begin{aligned} \left| v_{il}[t] - v_{jl}[t] \right| \leq (1-\gamma)(\varOmega_l[t-1] - \mu_l[t-1])\end{aligned} $$

(4.39)