Robust Localization with Crowd Sensors: A Data Cleansing Approach

Abstract

In this paper, the source localization problem with crowd of anchor nodes is investigated, under the circumstances that abnormal data could be sporadically and randomly produced for the reason of either accidental equipment failures or random malicious behaviors. To cope with the problem that abnormal data brings, we formulate a generalized modeling of abnormal data in localization problem, which involves the impacts of both unexpected equipment failures and malicious data falsifications. The corresponding Cramer-Rao lower bound (CRLB) of the specific localization problem is derived. For the localization enhancement, we propose a data cleansing-based robust localization algorithm which exploits the low occupancy of channel band by sources and the sparsity of abnormal data. The data cleansing approach achieves both the new sensing data matrix that cleansed out abnormal data component and the estimated abnormal data matrix, which are respectively used for the correct source detection process and the position estimation process of the final source localization. The root mean squared error (RMSE) is derived to assess the performance of the proposed robust localization algorithm. Computer simulations show that the proposed data cleansing-based robust localization algorithm can effectively eliminate the impairment of the abnormal data and hence improve the localization performance evidently.

Appendix

1.1 Cramer-rao lower bound analysis of robust localization

In this section, the effect of the abnormal data on Cramer-Rao lower bound is analyzed. The CRLB sets a lower limit on the convariance matrix of an unbiased estimator and is applied as a benchmark for evaluating the performance of the estimator [33, 34]. Since the abnormal data position is random in data matrix and the signal strength is un-constant for different unexpected conditions, the abnormal data model in Eq. (9) is difficult to be analyzed for CRLB. Hence, we consider a simplified abnormal data model from Eq. (9) and perform the CRLB analysis. Here we consider the unbiased condition that the mean of abnormal data sets zeroμ _A  = 0 and all sensing data suffers a possibility to be the abnormal one. Thus, the abnormal data A _ij follows the distribution of Gaussian distribution which is${A}_{ij}\sim N\left(0,{\sigma}_{ij}^2\right)$. The Fisher Information Matrix (FIM) of the abnormal data model is written as:

$$ \operatorname{cov}\left(\boldsymbol{\upvarphi} \right)\ge {\mathrm{J}}^{-1}, $$

(18)

where J is the FIM, φis the unknown vector which contain the coordinates$\mathbf{s}=\left[{\mathbf{s}}_1^T,\cdots {\mathbf{s}}_i^T,\cdots {\mathbf{s}}_M^T\right]$ and the transmit powerP ₀  = [P ₁₀, ⋯P _i0, ⋯P _M0] of the Nsources. Thus the unknown vector is written as $\boldsymbol{\upvarphi} =\left[{\mathbf{s}}^T,{\mathbf{P}}_{\mathbf{0}}^T\right]$, which is a 3Mvector.

From the above signal model, the joint probability density function of the sensing measurement can be written as

$$ g\left(\mathbf{P}\left|\boldsymbol{\upvarphi} \right.\right)={\left(2\pi \right)}^{-\frac{NM}{2}}{\left|\mathbf{C}\right|}^{-\frac{1}{2}} \exp \left(-\frac{1}{2}{\left(\mathbf{P}-\boldsymbol{\upmu} \right)}^T{C}^{-1}\left(\mathbf{P}-\boldsymbol{\upmu} \right)\right), $$

(19)

where P is the measured signal strength, Cis the covariance matrix, μ is the mean of the measurement vector P.

The matrix element C _mn of C is

$$ {C}_{mn}={\sigma}_W^2+{\sigma}_{ij}^2.\kern1.5em m= n $$

(20)

The element in the matrix μ is

$$ {\mu}_{i j}={P}_{i0}^{\frac{1}{2}}{d}_{i j}^{-\frac{1}{2}\gamma}.\kern0.6em i=1,2,\cdots M\kern0.3em ; j=1,2,\cdots N $$

(21)

Furthermore, the logarithm of the joint pdf is obtained:

$$ \ln \left( g\left(\mathbf{P}\left|\boldsymbol{\upvarphi} \right.\right)\right)= K-\frac{1}{2}{\sigma}_{ov}^{-2}{\left(\mathbf{P}-\boldsymbol{\upmu} \right)}^T\left(\mathbf{P}-\boldsymbol{\upmu} \right), $$

(22)

where $K=-\frac{1}{2} \ln \left[{\left(2\pi \right)}^{NM}\left|\mathbf{C}\right|\right]$ is a constant that does not depend on the unknown value φ.

As can be seen in Eq. (18), CRLB is actually the inverse of the FIM, and the FIM is calculated as follows

$$ \mathrm{J}\left(\boldsymbol{\upvarphi} \right)=- E\left[\frac{\partial^2 \ln g\left(\mathbf{P}\left|\boldsymbol{\upvarphi} \right.\right)}{\partial \boldsymbol{\upvarphi} \partial {\boldsymbol{\upvarphi}}^T}\right]={\sigma}_{ov}^{-2}{\left(\frac{\partial \boldsymbol{\upmu}}{\partial \boldsymbol{\upvarphi}}\right)}^T\left(\frac{\partial \boldsymbol{\upmu}}{\partial \boldsymbol{\upvarphi}}\right) $$

(23)

Denote $\mathbf{F}=\frac{\partial \boldsymbol{\upmu}}{\partial \boldsymbol{\upvarphi}}$ for simply, thus

$$ \begin{array}{l}\frac{\partial \boldsymbol{\upmu}}{\partial \boldsymbol{\upvarphi}}={\left[{\mathbf{F}}_1,\cdots {\mathbf{F}}_i,\cdots {\mathbf{F}}_M\right]}^T,\hfill \\ {}{\mathbf{F}}_i={\left[{\mathbf{F}}_{i-1},\cdots {\mathbf{F}}_{i- j},\cdots {\mathbf{F}}_{i- N}\right]}^T. i=1,2,\cdots M\kern0.3em ; j=1,2,\cdots N\hfill \end{array} $$

(24)

For detailed,

$$ \begin{array}{l}{\mathbf{F}}_{i- j}=\left[{0}_{1\times 3\left( i-1\right)},\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {s}_{i x}},\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {s}_{i y}},\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {P}_i},{0}_{1\times 3\left( N- i\right)}\right]\kern0.3em .\hfill \\ {}\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {s}_{i x}}=-\frac{1}{2}\gamma {P}_{i0}^{\frac{1}{2}}\frac{s_{i x}-{t}_{jx}}{d_{i j}^{\frac{1}{2}\left(\gamma +2\right)}},\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {s}_{i y}}=-\frac{1}{2}\gamma {P}_{i0}^{\frac{1}{2}}\frac{s_{i y}-{t}_{jy}}{d_{i j}^{\frac{1}{2}\left(\gamma +2\right)}},\frac{\partial {\boldsymbol{\upmu}}_{i- j}}{\partial {P}_{i0}}=\frac{1}{2}{P}_{i0}^{-\frac{1}{2}}{d}_{i j}^{-\frac{1}{2}\gamma}\hfill \end{array} $$

(25)

Once the FIM J(φ) is figured out, the CRLB of the unknown vectorφcan be calculated as J⁻¹(φ) ∈ ℜ ^3N × 3N.