# gaussian, exponential, trigonometric, thin plate spline, inverse multiquadratic, and multiquadratic radial basis function for spatio-temporal prediction Function for spatio-temporal interpolation from radial basis function (**rbfST**), where **rbfST** is in a local neighbourhood. **exponential (EXPON)** [REMOVE_ME]$$ \phi(\delta)=e^{-\eta \delta}, \eta>0 [REMOVE_ME_2]$$ **gaussiano (GAU)** [REMOVE_ME]$$ \phi(\delta)=e^{-\eta \delta^{2}}, \eta\neq0 [REMOVE_ME_2]$$ **multiquadratic (M)** [REMOVE_ME]$$ \phi(\delta)=\sqrt{\eta^2+\delta^2}, \eta\neq0 [REMOVE_ME_2]$$ **inverse multiquadratic (IM)** [REMOVE_ME]$$ \phi(\delta)=1/\sqrt{\eta^2+\delta^2}, \eta\neq0 [REMOVE_ME_2]$$ **thin plate spline (TPS)** [REMOVE_ME]$$ \phi(\delta)=(\eta\cdot\delta)^{2}log(\eta\cdot\delta), if: \delta>0, \eta>0 [REMOVE_ME_2]$$ [REMOVE_ME]$$ \phi(\delta) = 0, otherwise [REMOVE_ME_2]$$ **completely regularized spline (CRS)** [REMOVE_ME]$$ \phi(\delta) = \ln(\eta\cdot \delta/2)^{2}+E_{1}(\eta\cdot \delta/2)^{2}+C_{E}, if: \delta>0, \eta>0 [REMOVE_ME_2]$$ [REMOVE_ME]$$ \phi(\delta) = 0, otherwise [REMOVE_ME_2]$$ where $\ln$ is natural logarithm, $E_{1}(x)$ is the exponential integral function, and $C_{E}$ is the Euler constant. **spline with tension (ST)** [REMOVE_ME]$$ \phi(\delta)=\ln(\eta\cdot \delta/2)+K_{0}(\eta\cdot \delta)+C_{E}, if: \delta>0 [REMOVE_ME_2]$$ [REMOVE_ME]$$ \phi(\delta) = 0, otherwise [REMOVE_ME_2]$$ where $K_{0}(x)$ is the modified Bessel function and $C_{E}$ is the Euler constant. ## Description Function for spatio-temporal interpolation from radial basis function (**rbfST**), where **rbfST** is in a local neighbourhood. **exponential (EXPON)** $$ \phi(\delta)=e^{-\eta \delta}, \eta>0 $$ **gaussiano (GAU)** $$ \phi(\delta)=e^{-\eta \delta^{2}}, \eta\neq0 $$ **multiquadratic (M)** $$ \phi(\delta)=\sqrt{\eta^2+\delta^2}, \eta\neq0 $$ **inverse multiquadratic (IM)** $$ \phi(\delta)=1/\sqrt{\eta^2+\delta^2}, \eta\neq0 $$ **thin plate spline (TPS)** $$ \phi(\delta)=(\eta\cdot\delta)^{2}log(\eta\cdot\delta), if: \delta>0, \eta>0 $$ $$ \phi(\delta) = 0, otherwise $$ **completely regularized spline (CRS)** $$ \phi(\delta) = \ln(\eta\cdot \delta/2)^{2}+E_{1}(\eta\cdot \delta/2)^{2}+C_{E}, if: \delta>0, \eta>0 $$ $$ \phi(\delta) = 0, otherwise $$ where $\ln$ is natural logarithm, $E_{1}(x)$ is the exponential integral function, and $C_{E}$ is the Euler constant. **spline with tension (ST)** $$ \phi(\delta)=\ln(\eta\cdot \delta/2)+K_{0}(\eta\cdot \delta)+C_{E}, if: \delta>0 $$ $$ \phi(\delta) = 0, otherwise $$ where $K_{0}(x)$ is the modified Bessel function and $C_{E}$ is the Euler constant. ```r rbfST(formula, data, eta, rho, newdata, n.neigh, func, progress) ``` ## Arguments - `formula`: formula that defines the dependent variable as a linear model of independent variables (covariates or principal coordinates); suppose the dependent variable has name $z_{st}$ for a **rbfST** detrended use $z_{st}$`~1`; for a **rbfST** with trend suppose $z_{st}$ is linearly dependent on `x` and `y`, use the formula $z_{st}$`~x+y` (linear trend). - `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. - `eta`: the optimal smoothing parameter, we recommend using the parameter found by minimizing the root-mean-square prediction errors using cross-validation - `rho`: optimal robustness parameter, we recommend using the value obtained by minimizing the root-mean-square prediction errors with cross-validation. **eta** and **rho** parameters can be optimized simultaneously, through the `bobyqa` function from `nloptr` or `minqa` packages - `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 **rbfST** prediction, where nearest is defined in terms of the spatio-temporal locations - `func`: spatio-temporal radial basis function; model type: "GAU", "EXPON", "TRI", "TPS", "CRS", "ST", "IM" and "M", are currently available - `progress`: whether a progress bar shall be printed for spatio-temporal radial basis functions; default=TRUE ## Details rbf.st function generates individual spatio-temporal predictions from gaussian (GAU), exponential (EXPON), trigonometric (TRI) thin plate spline (TPS), completely regularized spline (CRS), spline with tension (ST), inverse multiquadratic (IM), and multiquadratic (M) functions ## Returns Attributes columns contain coordinates, time, predictions, and the variance column contains NA's ## References Melo, C. E. (2012). **Analisis geoestadistico espacio tiempo basado en distancias y splines con aplicaciones**. PhD. Thesis. Universitat de Barcelona. 276 p. [[link]](http://www.tdx.cat/bitstream/handle/10803/101202/CEMM_TESIS.pdf) ## Examples ```r ## Not run: # considering 10 principal coordinates (constructed from a distance-based regression model) data(croatia.temp) data(croatiadb) # prediction case: one point point <- data.frame(670863,5043464,5,170,200,15.7,3) names(point) <- c("x","y","t","dem","dsea","twi","est") croatia.temp[,7] <- as.factor(croatia.temp[,7]) dblm1 <- dblm(data=croatia.temp,y=croatiadb$MTEMP) newdata1 <- t(cp.xnews(newdata=point,eigenvalues=dblm1$ev, data=croatia.temp,trend=dblm1$cp)) colnames(newdata1) <- c("X1","X2","X3","X4","X5","X6","X7","X8","X9","X10") newdata1 <- data.frame(point[,1:3],newdata1) data(croatiadb) coordinates(croatiadb) <- ~x+y coordinates(newdata1) <- ~x+y rbfST(MTEMP~X1+X2+X3+X4+X5+X6+X7+X8+X9+X10, data=croatiadb, eta=0.010076, rho=0.00004, newdata=newdata1, n.neigh=60, func="TPS") # prediction case: a grid of points Croatia (month july) data(croatia.grid7cp) coordinates(croatia.grid7cp) <- ~x+y rbf.t <- rbfST(MTEMP~X1+X2+X3+X4+X5+X6+X7+X8+X9+X10, croatiadb, eta=0.01076, rho=0.00004, newdata=croatia.grid7cp, n.neigh=30, func="TPS") coordinates(rbf.t) <- c("x", "y") gridded(rbf.t) <- TRUE # show prediction map spplot(rbf.t["var1.pred"], cuts=30, col.regions=bpy.colors(40), main = "Earth's average temperature TPS map\n (july month)", key.space=list(space="right", cex=0.8)) ## End(Not run) ```

rbfST function