Author: Kai Yang, Peihua Qiu
Date: 2023-09-29

# SpTe2M: Nonparametric Modeling and Monitoring of Spatio-Temporal Data

#### Kai Yang, Peihua Qiu

#### September 29, 2023

## Introduction

Spatio-temporal data are common in practice. Such data often have complex structures that are difficult to describe and estimate. To overcome this difficulty, the `R` package SpTe2M has been developed to estimate the underlying spatio-temporal mean and covariance structures. This package also includes tools of online process monitoring to detect change-points in a spatio-temporal process over time. More specifically, it implements the nonparametric spatio-temporal data modeling methods described in Yang and Qiu (2018, 2019, and 2022), as well as the online spatio-temporal process monitoring methods discussed in Qiu and Yang (2021 and 2023) and Yang and Qiu (2020). In this vignette, we will demonstrate the main functions in SpTe2M by using the Florida influenza-like illness (ILI) data, which is a built-in dataset of the package.

## Florida ILI Data

The Florida ILI dataset, named as `ili_dat` in SpTe2M, contains daily ILI incidence rates at 67 Florida counties during years 2012-2014. The observed IIL incidence rates was collected by the Electronic Surveillance System for the Early Notification of Community-based Epidemics of the Florida Department of Health. In addition to observations of the variable `Rate` (i.e., ILI incidence rate), the dataset also includes observations of the following 7 variables: `County`, `Date`, `Lat` (i.e., latitude of the geometric center of a county), `Long` (i.e., longitude of the geometric center of a county), `Time`, `Temp` (i.e., air temperature), and `RH` (i.e., relative humidity). First of all, let us install and load the package SpTe2M and take a quick look at the ILI data.

### Load ILI data

First, we run the following `R` code to install the package from CRAN and load the package in `R`:

    install.packages("SpTe2M")
    library(SpTe2M)

Then, we read the built-in dataset `ili_dat` and use the `R` function `head()` to display its first 6 rows.

    data(ili_dat)
    head(ili_dat)
    #>     County     Date      Lat      Long       Time         Rate Temp   RH
    #> 1  alachua 1/1/2012 29.65320 -82.32440 0.00273224 8.020372e-06 60.5 78.6
    #> 2    baker 1/1/2012 30.27236 -82.21351 0.00273224 0.000000e+00 62.0 78.6
    #> 3      bay 1/1/2012 30.16190 -85.65297 0.00273224 3.532404e-05 61.5 78.6
    #> 4 bradford 1/1/2012 29.96893 -82.12255 0.00273224 0.000000e+00 61.0 78.6
    #> 5  brevard 1/1/2012 28.70765 -80.89049 0.00273224 0.000000e+00 68.0 78.6
    #> 6  broward 1/1/2012 26.05192 -80.14526 0.00273224 2.527846e-05 71.0 78.6

### Create ILI maps

Next, we make some geographic maps to investigate the spatio-temporal patterns of the ILI incidence. To do this, we first combine `ili_dat` with the map shape files included in the packages maps and mapproj and then create maps by using the function `ggplot()` in the package ggplot2.

    library(maps)
    library(ggplot2)
    library(mapproj)
    # turn shape files in the maps packages into a data frame
    FL <- map_data('county','florida')
    names(FL)[6] <- 'County'
    # Only plot maps on Jun 15 and Dec 15
    subdat <- subset(ili_dat,Date%in%c('6/15/2012','6/15/2013','6/15/2014',
                              '12/15/2012','12/15/2013','12/15/2014'))
    subdat$Date <- factor(subdat$Date,levels=c('6/15/2012','6/15/2013','6/15/2014',
                                           '12/15/2012','12/15/2013','12/15/2014'))
    # merge ILI data with map data
    mydat <- merge(FL,subdat)
    # make maps using ggplot
    maps <- ggplot(data=mydat,aes(long,lat,group=group,fill=Rate))+geom_polygon()+
      facet_wrap(~Date,ncol=3)+geom_path(colour='grey10',lwd=0.5)+
      scale_fill_gradient2('',low='cyan',mid='white',high='navy',
                           guide='colorbar',limits=c(0,0.0003),na.value='orange',
                           breaks=c(0,0.0001,0.0002,0.0003),
                           labels=c('0','1e-4','2e-4','3e-4'))+
      guides(fill=guide_colorbar(barwidth=25,barheight=1,direction='horizontal'),
             cex=1)+theme_bw(base_size=15)+xlab('longitude')+ylab('latitude')+
      theme(legend.position='bottom',
            axis.ticks=element_blank(),
            line=element_blank(),
            axis.text=element_blank(),
            panel.border=element_rect(color='black',linewidth=1.2),
            axis.line=element_line(colour='black'),
            legend.margin=margin(-10,0,0,0),
            legend.box.margin=margin(0,0,0,0))
    # display the maps
    maps

From the maps, it can be seen that ILI incidence rates have obvious seasonal patterns with more ILI cases in the winters and fewer ILI cases in the summers. Moreover, there seems to be an unusual pattern of ILI incidence rates in the winter of 2014 because the incidence rates on 12/15/2014 are much higher than those on 12/15/2012 and 12/15/2013. Next, we appply the package SpTe2M to this dataset to estimate the spatio-temporal mean and covariance structures and monitor the ILI incidence sequentially over time.

## ILI Data Modeling

In this part, we estimate the spatio-temporal mean and covariance structures in year 2013 by using the functions `spte_meanest()` and `spte_covest()` in SpTe2M. To this end, we first use the following code to extract the observed ILI data in year 2013.

    n <- 365; m <- 67; N1 <- (n+1)*m; N2 <- n*m
    # extract the observed ILI data in year 2013
    ili13 <- ili_dat[(N1+1):(N1+N2),]
    y13 <- ili13$Rate; st13 <- ili13[,c('Lat','Long','Time')]

### Estimate spatio-temporal mean

After obtaining the ILI data in year 2013, the spatio-temporal mean structure can be estimated by applying the function `spte_meanest()` to the extracted data.

    # estimate the mean
    mu.est <- spte_meanest(y=y13,st=st13)

Note that we don’t specify the arguments `ht` and `hs` when using the function `spte_meanest()`. In such a case, the two bandwidths `ht` and `hs` would be selected automatically by the modified cross-validation procedure (cf., Yang and Qiu 2018).

To visually check whether the estimated means describe the observed ILI data well, we use the code below to plot the estimated means for the 4 Florida counties Broward, Lake, Pinellas, and Seminole, along with the observed ILI incidence rates at the 4 counties.

    mu <- mu.est$muhat; mu <- t(matrix(mu,m,n))
    obs <- t(matrix(y13,m,n))
    ids <- c(6,34,52,57) # IDs for Broward, Lake, Pinellas, Seminole
    par(mfrow=c(2,2),mgp=c(2.4,1,0))
    par(mar=c(3.5,3.5,1.5,0.5))
    plot(1:365,mu[,ids[1]],type='l',lty=1,lwd=1.5,xaxt='n',
         ylim=c(0,8e-5),main='Broward',xlab='Time',ylab='Incidence rate',
         cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
    points(1:365,obs[,ids[1]],cex=1)
    axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
         label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
    par(mar=c(3.5,3,1.5,1))
    plot(1:365,mu[,ids[2]],type='l',lty=1,lwd=1.5,xaxt='n',
         ylim=c(0,8e-5),main='Lake',xlab='Time',ylab='',
         cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
    points(1:365,obs[,ids[2]],cex=1)
    axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
         label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
    par(mar=c(3.5,3.5,1.5,0.5))
    plot(1:365,mu[,ids[3]],type='l',lty=1,lwd=1.5,xaxt='n',
         ylim=c(0,8e-5),main='Pinellas',xlab='Time',ylab='Incidence rate',
         cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
    points(1:365,obs[,ids[3]],cex=1)
    axis(1,cex.axis=1.2,at=c(1+c(1,62,123,184,245,306, 367)),
         label=c('Jan','Mar','May','July','Sep','Nov','Jan'))
    par(mar=c(3.5,3,1.5,1))
    plot(1:365,mu[,ids[4]],type='l',lty=1,lwd=1.5,xaxt='n',
         ylim=c(0,8e-5),main='Seminole',xlab='Time',ylab=' ',
         cex=1.2,cex.lab=1.3,cex.axis=1.2,cex.main=1.3)
    points(1:365,obs[,ids[4]],cex=1)
    axis(1.2,at=c(1+c(1,62,123,184,245,306, 367)),
         label=c('Jan','Mar','May','July','Sep','Nov','Jan'))

It can be seen from the figure that the estimated spatio-temporal mean structure can well describe the longitudinal pattern of the observed ILI incidence rates. Thus, the function `spte_meanest()` provides a reliable tool for estimating the spatio-temporal mean function of the ILI data.

### Estimate spatio-temporal covariance

To estimate the spatio-temporal covariance, we can use the following code:

    # estimate the covariance
    cov.est <- spte_covest(y=y13,st=st13)

In the above command, we don’t specify the arguments `gt` and `gs`. Then, the two bandwidths `gt` and `gs` would be determined automatically by minimizing the mean squared prediction error, as discussed in Yang and Qiu (2019).

## ILI Data Monitoring

In this part, we discuss how to monitor the observed ILI data by using the functions `sptemnt_cusum()`, `sptemnt_ewmac()` and `sptemnt_ewsl()`, which correspond to the CUSUM chart (Yang and Qiu 2020), the EWMAC chart (Qiu and Yang 2021) and the EWSL chart (Qiu and Yang 2023), respectively.

### Use the function sptemnt_cusum()

Notice that the construction of the CUSUM chart (cf., Yang and Qiu 2020) consists of 2 main steps. The first step is to model an in-control (IC) spatio-temporal dataset and estimate the regular spatio-temporal pattern of the IC process. Then, in the second step, the control limit of the chart is first determined by the block bootstrap procedure (cf., Yang and Qiu 2020) and the chart can then be used for online process monitoring. In this example, we use the data in years 2012 and 2013 as the IC data for setting up the chart. These IC observations are divided into two parts: the IC data in year 2013 are used for estimating the mean and covariance functions of the IC spatio-temporal data, and the data in year 2012 are used for determining the control limit of the chart. After that, the chart starts to monitor the ILI incidence rates from the beginning of year 2014.

    y <- ili_dat$Rate; st <- ili_dat[,c('Lat','Long','Time')]
    # specify the argument type
    # data with type=IC1 are used to determine the control limit
    # data with type=IC2 are used to estimate the regular pattern
    # data with type =Mnt are used for sequential process monitoring
    type <- rep(c('IC1','IC2','Mnt'),c(N1,N2,N2))

By using the code below, we can monitor the ILI incidence rates in year 2014 by the CUSUM chart:

    ili.cusum <- sptemnt_cusum(y,st,type,ht=0.05,hs=6.5,gt=0.25,gs=1.5)

In this example, the bandwidths `ht` and `hs` for estimating the mean function are chosen based on the modified cross-validation procedure (cf., Yang and Qiu 2018), which can be implemented by using the function `mod_cv()` in SpTe2M, and the selected bandwidths for `ht` and `hs` are 0.05 and 6.50, respectively. Regarding the bandwidths `gt` and `gs` for estimating the covariance function, they are determined by the mean square prediction error criterion (cf., Yang and Qiu 2019). After running the function `cv_mspe()`, they are chosen to be 0.25 and 1.50, respectively, in this example.

After obtaining the charting statistics from the above code, we can plot the charting statistic values versus the observation times by using the following code:

    cstat <- ili.cusum$cstat; cl <- ili.cusum$cl
    par(mfrow=c(1,2),mgp=c(2.1,1,0))
    # plot the CUSUM chart
    par(mar=c(3.5,3.5,1.5,0.5))
    plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
         main='CUSUM',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.7,at=c(1,59,121,182,243,304,365),
         labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
    # plot the zoom-in part
    par(mar=c(3.5,3,1.5,1))
    plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
         main='First signal time: 10/16/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.7,at=c(259,274,289,304),
         label=c('Sep 15','Sep 30','Oct 15','Oct 31'))

The resulting plot is shown in the left panel of the above figure. To better perceive charting statistic values around its first signal time 10/16/2014, the charting statistic values during the time period from 09/15/2014 to 10/31/2014 are presented in the right panel of the figure.

### Use the function sptemnt_ewmac()

In the ILI dataset, there are observations of the two covariates `Temp` and `RH` available. It is our belief that the performance of a control chart can be improved by using the information in the covariates. To take advantage of the covariate information, we can use the EWMAC chart (Qiu and Yang 2021) for process monitoring, which can be implemented by using the function `sptemnt_ewmac()`. The following code is an example about how to use the function `sptemnt_ewmac()` to monitor the ILI incidence rates in 2014:

    x <- as.matrix(ili_dat[,c('Temp','RH')]) # the covariates
    ili.ewmac <- sptemnt_ewmac(y,x,st,type,ht=0.05,hs=6.5,gt=0.25,gs=1.5)

After running the above code, we use the code below to plot the EWMAC chart, as well as its zoom-in part during the time period from 09/15/2014 to 10/31/2014.

    cstat <- ili.ewmac$cstat; cl <- ili.ewmac$cl
    par(mfrow=c(1,2),mgp=c(2.1,1,0))
    # plot the EWMAC chart
    par(mar=c(3.5,3.5,1.5,0.5))
    plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
         main='EWMAC',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.7,at=c(1,59,121,182,243,304,365),
         labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
    # plot its zoom-in part
    par(mar=c(3.5,3,1.5,1))
    plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
         main='First signal time: 9/23/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.7,at=c(259,274,289,304),
         label=c('Sep 15','Sep 30','Oct 15','Oct 31'))

From the figure, it is clear that the EWMAC chart gives its first signal on 09/23/2014, which is 23 days earlier than the first signal of the CUSUM chart. So, it is beneficial to use the covariate information when monitoring the ILI incidence rates.

### Use the function sptemnt_ewsl()

In many applications, anomalies in spatio-temporal processes start in some small regions that are spatially clustered. To take into account this spatial feature, the EWSL chart was developed for spatio-temporal process monitoring, which is effective in cases when anomalies occur in spatially clustered regions.

Next, we demonstrate the application of the EWSL chart (cf., Qiu and Yang 2023) by using the function sptemnt_ewsl(). To this end, let us use the code as follows:

    ili.ewsl <- sptemnt_ewsl(y,st,type,ht=0.05,hs=6.5,gt=0.25,gs=1.5)

The plot of the EWSL chart and its zoom-in part during the period from 09/15/2014 to 10/31/2014 can be generated by the code given below.

    cstat <- ili.ewsl$cstat; cl <- ili.ewsl$cl
    par(mfrow=c(1,2),mgp=c(2.1,1,0))
    # plot the EWSL chart
    par(mar=c(3.5,3.5,1.5,0.5))
    plot(1:n,cstat,type="l",xlab="Time",xaxt="n",ylab="Charting statistic",
         main='EWSL',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.8,at=c(1,59,121,182,243,304,365),
         labe= c('Jan','Mar','May','July','Sep','Nov','Jan'))
    # plot its zoom-in part
    par(mar=c(3.5,3,1.5,1))
    plot(259:305,cstat[259:305],type="b",xlab="Time",xaxt="n",ylab=" ",
         main='First signal time: 10/6/2014',cex=0.6,lwd=1.5,cex.lab=0.7,
         cex.axis=0.7,cex.main=0.8)
    abline(h=cl,lty=2,lwd=1.5)
    axis(1,cex.axis=0.7,at=c(259,274,289,304),
         label=c('Sep 15','Sep 30','Oct 15','Oct 31'))

It can be seen from the figure that the EWSL chart gives its first signal on 10/6/2014, which is 10 days earlier than that of the CUSUM chart.

## References

- Qiu, P. and Yang, K. (2021). Effective Disease Surveillance by Using Covariate Information. Statistics in Medicine, 40, 5725-5745.

- Qiu, P. and Yang, K. (2023). Spatio-Temporal Process Monitoring Using Exponentially Weighted Spatial LASSO. Journal of Quality Technology, 55, 163-180.

- Yang, K. and Qiu, P. (2018). Spatio-Temporal Incidence Rate Data Analysis by Nonparametric Regression. Statistics in Medicine, 37, 2094-2107.

- Yang, K. and Qiu, P. (2019). Nonparametric Estimation of the Spatio-Temporal Covariance Structure. Statistics in Medicine, 38, 4555-4565.

- Yang, K. and Qiu, P. (2020). Online Sequential Monitoring of Spatio-Temporal Disease Incidence Rates. IISE Transactions, 52, 1218-1233.

- Yang, K. and Qiu, P. (2022). A Three-Step Local Smoothing Approach for Estimating the Mean and Covariance Functions of Spatio-Temporal Data. Annals of the Institute of Statistical Mathematics, 74, 49-68.