# A Task-Centric Cooperative Sensing Scheme for Mobile Crowdsourcing Systems
## 3 Problem Statements
### Estimating Sensing Data Using Spatial Correlation  
As mentioned earlier, for ecological and environmental monitoring applications, the MCS data will need to be collected and archived for future analysis. When some active participants are turned to idle mode, the data samples that would be measured by these active participants will be lost if the sensing data of these participants are statistically independent on the sensing data of remaining active participants. However, by exploiting spatial correlation among sensing data, it is possible to obtain an estimate of these missing data by using the observed sensing data. Since the correlation may not be perfect, approximation error is likely to occur. We propose to use the expected approximation error per active participant as a criterion to measure the loss of data integrity resulted by putting a portion of active participant s into idle state. Denote: {_grid_}; 1 \(\leqslant\) _k \(\leqslant\) M_} is the location of these \(M\) grids {_yk(t)egrid_; 1 \(\leqslant\) _k \(\leqslant\) M_, 0 \(\leqslant\) _t \(\leqslant\) T_ \(-\) 1} is the set of participants measurements of _grid_\({}_{k}\) during the training period; The selection of grids set in a task time period _I_\({}_{R}\) = {1,2... _R_}; and the all grids set \(I\) = {1,2,... _M_}. The set of training sensing data is organized in a _T \(\times\) M_ matrix \(\mathbf{Data_{M}}\)_in which the _kth_ column consists of the _kth_ grid's observations over time period [0, \(T\) \(-\) 1]. _I_\({}_{R}\) is used to denote a set of \(R\) (<_M_) indices in {1 \(\leqslant\) _k \(\leqslant\) M_}. _T \(\times\) R_ matrix \(\mathbf{Data_{M}}\) is obtained from \(\mathbf{Data_{M}}\) by keeping columns whose indices are in _I_\({}_{R}\). The remaining columns of \(\mathbf{Data_{M}}\) then will be denoted as a _T \(\times\)_(M \(-\) _R_) matrix \(\mathbf{Data_{M}}\)\({}_{-\)_R_. The relationship between \(\mathbf{Data_{M}}\), \(\mathbf{Data_{R}}\), and \(\mathbf{Data_{M}}\)\({}_{-\)_R_ is expressed as:  
\[\left[ \begin{array}{cc}Data_{R}&\shortmid&Data_{M-R}\end{array} \right]=Data _{M}*P \tag{2}\]  
where \(P\) is a permutation matrix determined by _I_\({}_{R}\). _I_\({}_{R}\) corresponds to the \(R\) active participants. By exploiting the spatial correlation among sensing data, given \(\mathbf{Data_{R}}\), one may estimate the missing data \(\mathbf{Data_{M}}\)\({}_{-\)_R_ by projecting it onto the subspace spanned by \(\mathbf{Data_{R}}\):  
\[Data_{M-R}=Data_{R}*\left(Data_{R}^{T}*Data_{R}\right)^{-1}Data_{R}^{T}*Data_{ M-R}=Data_{R}*\textbf{B} \tag{3}\]  
where the _R_\(\times\) (_M \(-\)_R_) matrix:  
\[\textbf{B}=\left(Data_{R}^{T}*Data_{R}\right)^{-1}Data_{R}^{T}*Data_{M-R} \tag{4}\]  
which is the spatial correlation matrix between data measured at the active participants (_I_\({}_{R}\)) and data measured at remaining active participants. Thus, for any _R_\(\times\) (_M \(-\)_R_) matrix **D**:  
\[\left\lvert\,\right\rvert Data_{M-R}-Data_{M-R}\left\lvert\,\right\rvert_{F}= \left\lvert\,\right\rvert Data_{M-R}-Data_{R}*\textbf{B}\left\lvert\,\right\rvert _{F}\leqslant\left\lvert\,\right\rvert Data_{M-R}-Data_{R}*\textbf{D}\left\lvert \,\right\rvert_{F} \tag{5}\]  
where \(\left\lvert\,\right\rvert Data\left\lvert\,\right\rvert_{F}\) indicates that the Frobenius norm be the same with the square root of the sum of squares of all elements of \(\mathbf{Data}\). However, from Equation (4), it is clear that to compute the **B** matrix, \(\mathbf{Data_{M}}\)\({}_{-\)_R_ must be known. During the training period where all sensing data are known, the **B** matrix may be estimated. An important assumption of the data centric approach is that this estimated spatial correlation matrix **B** shall remain unchanged in later normal operations when the corresponding \(\mathbf{Data_{M}}\)\({}_{-\)_R_ matrix is unavailable.