# Inverse Distance Weighting (IDW) function for spatio-temporal prediction. This function performs spatio-temporal interpolation. Here **idwST** is in a local neighborhood. This interpolation method considers the value of a point can be obtained from the weighted sum of values of the regionalized variable of closest neighbors. The general formula for the IDW is given by: [REMOVE_ME]$$ \hat{z}_0(st)=\sum_{i=1}^n \lambda_i z_i(st) [REMOVE_ME_2]$$ The expression for determining the weights is: [REMOVE_ME]$$ \lambda_i = \frac{d_{i0}^{-p}}{\sum_{i=1}^n d_{i0}^{-p}} [REMOVE_ME_2]$$ The weight is controlled by a factor **p** with each increment of the distance, $d_{i0}$ is the distance between the prediction position and each of the measured positions. The expression $d_{i0}$ can be obtained by: [REMOVE_ME]$$ d_{i0}=\sqrt{(x_{i}-x_{0})^2+(y_{i}-y_{0})^2+C\cdot (t_{i}-t_{0})^2} [REMOVE_ME_2]$$ $x$, $y$ and $t$ correspond to the spatio-temporal coordinates, **p** (factor.p) and **C** factors defined below. ## Description This function performs spatio-temporal interpolation. Here **idwST** is in a local neighborhood. This interpolation method considers the value of a point can be obtained from the weighted sum of values of the regionalized variable of closest neighbors. The general formula for the IDW is given by: $$ \hat{z}_0(st)=\sum_{i=1}^n \lambda_i z_i(st) $$ The expression for determining the weights is: $$ \lambda_i = \frac{d_{i0}^{-p}}{\sum_{i=1}^n d_{i0}^{-p}} $$ The weight is controlled by a factor **p** with each increment of the distance, $d_{i0}$ is the distance between the prediction position and each of the measured positions. The expression $d_{i0}$ can be obtained by: $$ d_{i0}=\sqrt{(x_{i}-x_{0})^2+(y_{i}-y_{0})^2+C\cdot (t_{i}-t_{0})^2} $$ $x$, $y$ and $t$ correspond to the spatio-temporal coordinates, **p** (factor.p) and **C** factors defined below. ```r idwST(formula, data, newdata, n.neigh, C, factor.p, progress) ``` ## Arguments - `formula`: formula that defines a detrended linear model, use $z_{st}$`~1`. - `data`: SpatialPointsDataFrame: should contain the spatio-temporal dependent variable, independent variables (statics and/or dynamics), spatial coordinates and the time as an integer or numerical variable. - `newdata`: data frame or spatial object with prediction/simulation spatio-temporal locations; should contain attribute columns with the independent variables (if present) and (if locations is a formula) the coordinates and time with names, as defined in locations where you want to generate new predictions - `n.neigh`: number of nearest observations that should be used for a **idwST** prediction, where nearest is defined in terms of the spatio-temporal locations - `C`: numeric; associated to time factor, we recommend using the parameter found by minimizing the root-mean-square prediction errors using cross-validation. Using **idwST.cv** and `optimize` - `factor.p`: numeric; specify the inverse distance weighting power (**p** is the exponent that influences the weighting or optimal smoothing parameter) - `progress`: whether a progress bar shall be printed for spatio-temporal inverse-distance weighted function; default=TRUE ## Details idwST function generates individual spatio-temporal predictions from IDW spatio-temporal interpolation. IDW is a type of deterministic method for interpolation, the assigned values to unknown points are calculated with a weighted average of the values available at the known points. ## Returns Attributes columns contain coordinates, time, predictions, and the variance column contains NA's ## References Li L, Losser T, Yorke C, Piltner R. (2014). **Fast inverse distance weighting-based spatiotemporal interpolation: a web-based application of interpolating daily fine particulate matter PM2:5 in the contiguous U.S. using parallel programming and k-d tree.** Int. J. Environ. Res. Public Health, 11: 9101-9141. [[link]](https://www.mdpi.com/journal/ijerph) ## Examples ```r # Loading Croatia data data(croatia2008) coordinates(croatia2008) <- ~x+y # prediction case: one point point <- data.frame(670863,5043464,5) names(point) <- c("x","y","t") coordinates(point) <- ~x+y idwST(MTEMP~1, data=croatia2008, newdata=point, n.neigh=60, C=1, factor.p=2) ## Not run: # prediction case: a grid of points Croatia (year 2008) data(croatia) points <- spsample(croatia, n=5000, type="regular") data(croatia2008) coordinates(croatia2008)<-~x+y GridsT <- vector(mode = "list", length = 12) for(i in 1:12){ GridsT[[i]] <- data.frame(points@coords,i) names(GridsT[[i]]) <- c("x","y","t") } idw.croatia <- data.frame(matrix(NA, ncol = 14, nrow=nrow(GridsT[[1]]))) pb <- txtProgressBar(min = 0, max = 12, char = "=", style = 3) for(i in 1:12){ coordinates(GridsT[[i]]) <- c("x", "y") idw.croatia[,i+2] <- idwST(MTEMP~1, croatia2008, newdata=GridsT[[i]], n.neigh=10, C=1, factor.p=2, progress=FALSE)[,4] setTxtProgressBar(pb, i) } close(pb) idw.croatia[,1:2] <- GridsT[[1]]@coords nam <- paste(c("ENE","FEB","MAR","ABR","MAY","JUN","JUL","AGO","SEP","OCT","NOV","DIC"), 2008,sep="") names(idw.croatia) <- c("x","y",nam) coordinates(idw.croatia) <- c("x", "y") gridded(idw.croatia) <- TRUE # show prediction map pal2 <- colorRampPalette(c("blue3", "wheat1", "red3")) p1 <- spplot(idw.croatia[,1:12], cuts=30, col.regions=pal2(35), colorkey=F, scales = list(draw =T,cex=0.6, abbreviate=TRUE,minlength=1), pch=0.3, cex.lab=0.3, cex.title=0.3, auto.key = F, main = "Earth's average temperature IDW map 2008", key.space=list(space="right", cex=0.8)) split.screen( rbind(c(0, 1,0,1), c(1,1,0,1))) split.screen(c(1,2), screen=1)-> ind screen( ind[1]) p1 screen( ind[2]) image.plot(legend.only=TRUE, legend.width=0.5, col=pal2(100), smallplot=c(0.7,0.75, 0.3,0.7), zlim=c(min(idw.croatia@data), max(idw.croatia@data)), axis.args = list(cex.axis = 0.7)) close.screen( all=TRUE) ## End(Not run) ```