# rbfST: gaussian, exponential, trigonometric, thin plate spline,... In geosptdb: Spatio-Temporal; Inverse Distance Weighting and Radial Basis Functions with Distance-Based Regression

## Description

Function for spatio-temporal interpolation from radial basis function (rbfST), where rbfST is in a local neighbourhood.

exponential (EXPON)

φ(δ)=e^{-η δ}, η>0

gaussiano (GAU)

φ(δ)=e^{-η δ^{2}}, η\neq0

φ(δ)=√{η^2+δ^2}, η\neq0

φ(δ)=1/√{η^2+δ^2}, η\neq0

thin plate spline (TPS)

φ(δ)=(η\cdotδ)^{2}log(η\cdotδ), if: δ>0, η>0

φ(δ) = 0, otherwise

completely regularized spline (CRS)

φ(δ) = \ln(η\cdot δ/2)^{2}+E_{1}(η\cdot δ/2)^{2}+C_{E}, if: δ>0, η>0

φ(δ) = 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)

φ(δ)=\ln(η\cdot δ/2)+K_{0}(η\cdot δ)+C_{E}, if: δ>0

φ(δ) = 0, otherwise

where K_{0}(x) is the modified Bessel function and C_{E} is the Euler constant.

## Usage

 1 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

## Value

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]

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ## 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) 

geosptdb documentation built on May 1, 2019, 8:02 p.m.