## 2 The Detection and Isolation Method of the Parity Space Method
### The Relation Between Detection Function and Anomaly Vector  
In Section 3, the anomaly recognition method is proposed, and the method is based on the analysis of the detection function. To verify the feasibility of the proposed method, the relation between the detection function and the anomaly vector is deduced here. Take (3), (6), and (7) into (8), and the detection function is given as follows:  
\[\mathbf{F}_{\text{D}}=\frac{[\mathbf{V}(\mathbf{H}\mathbf{x}+\mathbf{b}+\mathbf{\varepsilon})]^{T}[\bm {V}(\mathbf{H}\mathbf{x}+\mathbf{b}+\mathbf{\varepsilon})]}{\sigma^{2}} \tag{14}\]  
The singular value decomposition of \(\mathbf{H}^{T}\) is given by  
\[\mathbf{H}^{T}=\mathbf{U}_{HT}[\mathbf{S}_{HT}\vdots\mathbf{O}]\mathbf{L}_{HT}^{T} \tag{15}\]  
where \(\mathbf{L}_{HT}=[\mathbf{L}_{1}\vdots\mathbf{L}_{2}]\).  
Compared with Equation (4), the relations can be expressed as follows:  
\[\begin{array}{l}\mathbf{L}_{H}=\mathbf{U}_{HT}\\ \mathbf{L}_{HT}=\mathbf{U}_{H}\\ \mathbf{S}_{H}=\mathbf{S}_{HT}^{T}\end{array} \tag{16}\]  
Further, we can deduce out that \(\mathbf{L}_{1}=\mathbf{U}_{1}\), \(\mathbf{L}_{2}=\mathbf{U}_{2}\). Equations (4) and (15) can be simplified as  
\[\begin{array}{l}\mathbf{H}^{T}=\mathbf{U}_{HT}\mathbf{S}_{HT}\mathbf{L}_{1}^{T}\\ \mathbf{H}=\mathbf{U}_{1}\mathbf{S}_{H}\mathbf{L}_{H}^{T}\end{array} \tag{17}\]  
The null space of the matrix \(\mathbf{H}^{T}\) is \(N(\mathbf{H}^{T})=\left\{\mathbf{x}\big{|}\mathbf{H}^{T}\mathbf{x}=0\right\}\), and it can be expressed as  
\[N(\mathbf{H}^{T})=\left\{\mathbf{x}\big{|}\mathbf{U}_{HT}\mathbf{S}_{HT}\mathbf{L}_{1}^{T}\mathbf{x}=0 \right\} \tag{18}\]  
\(\mathbf{S}_{HT}\) is a non-singular matrix. From the conclusion of SVD, there is \(\textit{rank}(\mathbf{S}_{HT})=\textit{rank}(\mathbf{L}_{1}^{T})\), so \(\mathbf{L}_{1}^{T}\) is equivalent to \(\mathbf{U}_{HT}\mathbf{S}_{HT}\mathbf{L}_{1}^{T}\). Equation (18) can be simplified as  
\[N(\mathbf{H}^{T})=\left\{\mathbf{x}\big{|}\mathbf{L}_{1}^{T}\mathbf{x}=0\right\} \tag{19}\]Because \(L_{HT}\) is an orthogonal matrix, the solution space of \(L_{1}^{T}x\) is the range of the column vectors of \(L_{2}\), which is \(R(L_{2})\). Consequently, the equations are established as follows:  
\[N(\mathbf{H}^{T})=R(\mathbf{L}_{2})=R(\mathbf{U}_{2}) \tag{20}\]  
\[\mathbf{H}^{T}\mathbf{U}_{2}=0 \tag{21}\]  
From Equation (21), it is concluded that \(\mathbf{V}\mathbf{H}=0\). The detection function can be simplified as  
\[\mathbf{F}_{\rm D}=\frac{[\mathbf{V}(\mathbf{b}+\mathbf{\varepsilon})]^{T}[\mathbf{V}(\mathbf{b}+\mathbf{ \varepsilon})]}{\sigma^{2}}=\frac{(\mathbf{b}+\mathbf{\varepsilon})^{T}\mathbf{V}^{T}\mathbf{ V}(\mathbf{b}+\mathbf{\varepsilon})}{\sigma^{2}} \tag{22}\]  
Because \(\mathbf{L}_{HT}\) is an orthogonal matrix, and each column vector is a unit vector, the column vectors of \(L_{2}\) are orthogonal to each other. Then another equation can be given by  
\[\mathbf{L}_{2}^{T}L_{2}=\mathbf{I} \tag{23}\]  
Taking Equations (11) and (16) into (23), the relation above can be expressed as  
\[\mathbf{V}\mathbf{V}^{T}=\mathbf{I} \tag{24}\]  
The error vector and noise vector can be replaced by  
\[\mathbf{b}_{\rm e}=\mathbf{b}+\mathbf{\varepsilon} \tag{25}\]  
So the detection function is expressed as follows:  
\[\mathbf{F}_{\rm D}=\frac{\mathbf{b}_{\rm e}{}^{T}\mathbf{b}_{\rm e}}{\sigma^{2}} \tag{26}\]  
Since the \(\mathbf{F}_{\rm D}\) obeys the Chi-square distribution, the variation of \(\mathbf{F}_{\rm D}\) that exceeds the threshold can be thought to be caused by the anomaly vector \(\mathbf{b}\). The anomaly vectors \(\mathbf{b}\) for various types of data anomalies are definitely different, so the distributions and variation trends of \(\mathbf{F}_{\rm D}\) also differ. The patterns of data anomalies can be identified by analyzing the detection function.