# Online spatio-temporal process monitoring by a CUSUM chart The function `sptemnt_cusum` implements the sequential online monitoring procedure described in Yang and Qiu (2020). ```r sptemnt_cusum( y, st, type, ARL0 = 200, gamma = 0.1, B = 1000, bs = 5, T = 1, ht = NULL, hs = NULL, gt = NULL, gs = NULL ) ``` ## Arguments - `y`: A vector of $N$ spatio-temporal observations. - `st`: An $N\times 3$ matrix specifying the spatial locations and times for all the spatio-temporal observations in `y`. - `type`: A vector of $N$ characters specifying the types of the observations. Here, `type` could be `IC1`, `IC2` or `Mnt`, where `type='IC1'` denotes the in-control (IC) observations used to perform the block bootstrap procedure to determine the control limit of the CUSUM chart, `type='IC2'` denotes the IC observations used to estimate the spatio-temporal mean and covariance functions by `spte_meanest` and `spte_covest`, and `type='Mnt'` denotes the observations used for online process monitoring (cf., Yang and Qiu 2020). If there are only data points with either `type='IC1'` or `type='IC2'`, then these data points will be used to estimate the model and conduct the bootstrap procedure as well. This function will return an error if there are no observations with `type='IC1'` or `type='IC2'`. - `ARL0`: The pre-specified IC average run length. Default is 200. - `gamma`: The pre-specified allowance constant in the CUSUM chart. Default is 0.1. - `B`: The bootstrap sizes used in the block bootstrap procedure for determining the control limit. Default value is 1,000. - `bs`: The block size of the block bootstrap procedure. Default value is 5. - `T`: The period of the spatio-temporal mean and covariance. Default value is 1. - `ht`: The temporal kernel bandwidth `ht`; default is `NULL` and it will be chosen by the modified cross-validation via `mod_cv` if `ht=NULL`. - `hs`: The spatial kernel bandwidth `hs`; default is `NULL`, and it will be chosen by the function `mod_cv` if `hs=NULL`. - `gt`: The temporal kernel bandwidth `gt`; default is `NULL` and it will be chosen by minimizing the mean squared prediction error via `cv_mspe` if `gt=NULL`. - `gs`: The spatial kernel bandwidth `gs`; default is `NULL`, and it will be chosen by the function `cv_mspe` if `gs=NULL`. ## Returns - **ARL0**: Same as the one in the arguments. - **gamma**: Same as the one in the arguments. - **cstat**: The charting statistics which can be used to make a plot for the control chart. - **cl**: The control limit that is determined by the block bootstrap. - **signal_time**: The signal time (i.e., the first time point when the charting statistic `cstat` exceeds the control limit `cl`). ## Examples ```r library(SpTe2M) data(ili_dat) n <- 365; m <- 67 y <- ili_dat$Rate; st <- ili_dat[,3:5] type <- rep(c('IC1','IC2','Mnt'),c(m*(n+1),(m*n),(m*n))) ids <- c(1:(5*m),((n+1)*m+1):(m*(n+6)),((2*n+1)*m+1):(m*(2*n+6))) y.sub <- y[ids]; st.sub <- st[ids,]; type.sub <- type[ids] ili.cusum <- sptemnt_cusum(y.sub,st.sub,type.sub,ht=0.05,hs=6.5,gt=0.25,gs=1.5) ``` ## References Yang, K. and Qiu, P. (2020). Online Sequential Monitoring of Spatio-Temporal Disease Incidence Rates. **IISE Transactions**, 52 , 1218-1233. ## Author(s) Kai Yang kayang@mcw.edu and Peihua Qiu

sptemnt_cusum function