## 6 Conclusions
\begin{tabular}{l l} \(N\) & The ideal number of sensors. \\ \(n_{t}\) & Number of effective observed sensors at time \(t\). \\ \(x\) & Space position. \\ \(X\) & Set of interested sites in quantity \(N\). \\ \(Z_{t}(x)\) & State variables at location x and time \(t\). \\ \(Y_{t}(x)\) & Observations at location x and time \(t\). \\ \(\mathbf{Z}_{t}\) & The state vector at time \(t\) for \(N\) sensors. \\ \(\mathbf{Y}_{t}\) & Effective observed vector at time \(t\) for \(n_{t}\) sensors. \\ \(\tilde{Z}_{t}(x)\), \(\tilde{Y}_{t}(x)\), \(\tilde{\mathbf{Z}}_{t}(x)\), \(\tilde{\mathbf{Y}}_{t}(x)\) & Estimate of \(Z_{t}(x)\), \(Y_{t}(x)\), \(\mathbf{Z}_{t}(x)\), \(\mathbf{Y}_{t}(x)\), respectively. \\ \(\mathbf{W}_{t}\) & The selection matrix is used to represent the conversion relationship between ideal sensors and effective observation sensors, and its matrix size is \(n_{t}*N\) and its rank is \(n_{t}\). \\ \(f(x)=[f_{1}(x)\dots f_{r}(x)]^{T}\) & Known drift functions at location x with size \(r*1\). \\ \(F=[f(x_{1}),\dots,f(x_{n})]\) & Known drift functions at \(N\) ideal sensors location with size \(r*N\). \\ \(\tilde{\beta}_{t}=[\beta_{t1}\dots\beta_{tr}]^{T}\) & Unknown drift coefficients vector at time \(t\) with size \(r*1\). \\ \(\varepsilon_{t}(x)\) & The spatially correlated errors with zero mean and known \\  & covariance function \(E[\varepsilon_{t}(x_{a})\varepsilon_{t}(x_{b})]=k(x_{a},x_{b})\). \\ \(\varepsilon_{t}\) & Vector of \(\varepsilon_{t}(x)\) for ideal sensors at time \(t\) with length \(N\). \\ \(k(x_{a},x_{b})\) & Kernel function in terms of locations \(x_{a}\) and \(x_{b}\). \\ \(K\) & Covariance matrix of \(\varepsilon_{t}(x)\) with size \(N*N\). \\ \(\varepsilon_{t}(x)\) & a latent variable at location x and time t for simple calculation. \\ \(\varepsilon_{t}\) & The latent state vector at time \(t\) for \(N\) sensors. \\ \(\tilde{c}_{t}(x)\), \(\tilde{\varepsilon}_{t}\) & Estimate of \(c_{t}(x)\) and \(c_{t}\), respectively. \\ \(\tilde{\mathbf{Y}}_{t}\) & A simple representation of the computational process, and its \\  & expression is \(\tilde{\mathbf{Y}}_{t}=\mathbf{Y}_{t}-\mathbf{W}_{t}\tilde{c}_{t-1}\). \\ \(v_{t}(x)\) & System random error at location \(x\) and time \(t\), generally regarded as white noise, is independent of time and position. \\ \(\mathbf{v}_{t}\) & Vector of \(v_{t}(x)\) for effective observed vector at time \(t\) with length \(n_{t}\). \\ \(\mathbf{R}_{t}\) & Covariance matrix of \(v_{t}\) with size \(n_{t}*n_{t}\), which is diagonal matrix. \\ \(\mathbf{P}_{t}\) & Covariance matrix of \(\varepsilon_{t}-\tilde{c}_{t}\) with size \(N*N\). \\ \(\mathbf{p}_{t}(x)\) & Covariance vector between \(\varepsilon_{t}-\tilde{\alpha}_{t}\) and \(c_{t}(x)-\tilde{c}_{t}(x)\) with length \(N\). \\ \(\mathbf{k}(x)\) & Covariance vector between \(\varepsilon_{t}\) and \(\varepsilon_{t}(x)\) with length \(N\). \\ \(\mathbf{q}_{t}(x)\) & The weight value when using BLUE estimation for location x and \\  & time \(t\). \\ \(\zeta_{t}\), \(\tau\), \(\gamma_{t}\),\(L\) & Parameters of the Holt-Winters model. \\ \(S_{t}\), \(B_{t}\), \(C_{t-L+1}\) & Intermediate variable vector of the Holt-Winters model. \\ \(L_{t}\), \(M_{t}\), \(m_{t}\), \(G_{t}\), \(g_{t}\) & Simplified variables in calculation process. \\ \end{tabular}   
**Appendix A**  
For \(\mathbf{P}_{t}\), we can write it as:  
\[ \begin{array}{c}\mathbf{P}_{t}=E[(\varepsilon_{t}-\widehat{\varepsilon}_{t}) (\mathbf{c}_{t}-\widehat{\varepsilon}_{t})T]\\ =E[(\varepsilon_{t}-\widehat{\varepsilon}_{t})(\mathbf{c}_{t}-\widehat{ \varepsilon}_{t-1}-G_{t}\widetilde{\mathbf{Y}}_{t})^{T}]\\ =E[(\varepsilon_{t}-\widehat{\varepsilon}_{t})(\mathbf{c}_{t}-\widehat{ \varepsilon}_{t-1})^{T}]-E[(\varepsilon_{t}-\widehat{\varepsilon}_{t})( \mathbf{G}_{t}\widetilde{\mathbf{Y}}_{t})^{T}]\end{array} .\] (\[