## 3 Proposed Resilient Demand Forecasting Systems
\begin{table}
\begin{tabular}{c l} \hline \hline
**Term** & **Definition** \\ \hline \(N\) & Total number of DPVs \\ \hline \(t\) & Current time step \\ \hline \(DPV_{i}\) & Data provenance of the ith measurement \\ \hline \(DPV_{j}\) & Data provenance of the jth measurement \\ \hline \(nc\) & The total number of nodes common to \(DPV_{i}\) and \(DPV_{j}\) \\ \hline \(n_{1}\) & The lenght of the longer DPV between \(DPV_{i}\) and \(DPV_{j}\) \\ \hline \(\overrightarrow{mv_{i}(t)}\) & Time series of the \(i\)th DPV’s measurements containing both past and current measurements \\ \hline \(\overrightarrow{mv_{i}(t-1)}\) & Time series of the \(i\)th DPV’s past measurements \\ \hline \(\overrightarrow{mv_{i}(t)}\) & The received measurement of the \(i\)th DPV at time step \(t\) \\ \hline \(\overrightarrow{mv_{j}(t)}\) & The received measurement of the \(j\)th DPV at time step \(t\) \\ \hline \(DS_{ij}(t)\) & The data similarity between \(\overrightarrow{mv_{i}(t)}\) and \(\overrightarrow{mv_{j}(t)}\) estimated using MAAPE \\ \hline \(PS_{ij}(t)\) & Provenance similarity between the DPVs of \(\overrightarrow{mv_{j}(t)}\) and \(\overrightarrow{mv_{i}(t)}\): \(PS_{ij}(t)=nc/n_{1}\) \\ \hline \(\overrightarrow{mv_{i}(t)}\) & One step ahead forecast of the conditional mean of \(\overrightarrow{mv_{i}(t)}\) \\ \hline \(\varepsilon_{i}(t)\) & \(\varepsilon_{i}(t)=\overrightarrow{mv_{i}(t)}-\overrightarrow{mv_{i}(t)}\) \\ \hline \(\overrightarrow{\varepsilon_{i}(t)}\) & The residual series for the one step ahead forecasts of \(i\)th DPVs measurements containing both the past and current residuals \\ \hline \(\overrightarrow{\varepsilon_{i}(t-1)}\) & The residual series for the one step ahead forecasts of \(i\)th DPVs measurements containing the past residuals \\ \hline \(\sigma_{i}^{2}(t)\) & The observed conditional variance of \(\overrightarrow{mv_{i}(t)}\): \(\sigma_{i}^{2}(t)=\varepsilon_{i}^{2}(t)\) \\ \hline \(\widehat{\sigma_{i}^{2}(t)}\) & One step ahead forecast of \(\sigma_{i}^{2}(t)\) \\ \hline \(ENel_{i}^{M}\) & The ensemble of CNNs forecasting \(\overrightarrow{mv_{i}(t)}\) \\ \hline \(ENel_{i}^{V}\) & The ensemble of CNNs forecasting \(\widehat{\sigma_{i}^{2}(t)}\) based on \(\widehat{\varepsilon_{i}^{2}(t-1)}\) \\ \hline \(\varepsilon_{i}(t)\) & \(\varepsilon_{i}(t)=\sigma_{i}^{2}(t)-\widehat{\sigma_{i}^{2}(t)}\) \\ \hline \(f\Big{(}\overrightarrow{mv_{i}(t)},\sqrt{\widehat{\sigma_{i}^{2}(t)}}\Big{)}\) & A gaussian distribution with mean \(\widehat{mv_{i}(t)}\) and standard deviation \(\sqrt{\widehat{\sigma_{i}^{2}(t)}}\) \\ \hline \(SE\) & A matrix containing the residual series for the one step ahead forecasts of all DPVs measurements \\ \hline \(SK\) & A matrix containing the time series of skewness of the series in \(SE\) \\ \hline \(\overrightarrow{MV(t)}\) & The measurements of all DPVs at \(t\) : \(\overrightarrow{MV(t)}=[\overrightarrow{mv_{1}(t),\ldots,mv_{N}(t)}]\) \\ \hline \(D(t)\) & The most trustworthy aggregate demand measurement at time step \(t\) based on \(\overrightarrow{MV(t)}\) \\ \hline \(\overrightarrow{D(t)}\) & The time series of aggregate demand \\ \hline \(\overrightarrow{tr_{i}(t)}\) & Time series of the assessed trust scores of the \(i\)th DPV including its assessed trust score at time step \(t\) \\ \hline \(\overrightarrow{tr_{i}(t-1)}\) & Time series of the assessed trust scores of the \(i\)th DPV, not including its assessed trust score at time step \(t\) \\ \hline \(\overrightarrow{tr_{i}(t)}\) & The assessed trust score of the \(i\)th DPV at time step \(t\) \\ \hline \(\overrightarrow{tr_{i}(t)}\) & The forecasted trust score of the \(i\)th sensor at time step \(t\) \\ \hline \(m_{i}(t)\) & The forecasted conditional mean of \(\overrightarrow{tr_{i}(t)}\) based on \(\overrightarrow{tr_{i}(t-1)}\) \\ \hline \(\delta_{i}^{2}(t)\) & The forecasted conditional variance of \(m_{i}(t)\) \\ \hline \(GRNN_{i}^{M}\) & The \(i\)th member of GRNN of the ensemble of GRNNs forecasting \(m_{i}(t)\) \\ \hline \(GRNN_{i}^{V}\) & The \(i\)th member of GRNN of the ensemble of GRNNs forecasting \(\delta_{i}^{2}(t)\) \\ \hline \hline \end{tabular}
\end{table}