---
title: "Simple Kriging"
author: "Pedro Guarderas"
date: "January 2, 2018"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Simple Kriging}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

An example presenting an application of Simple Kriging.

Loading packages
```{r,eval=TRUE,message=FALSE, warning=FALSE}
library( KRIG )
```

Defining objective function to approach.
```{r,eval=TRUE,message=FALSE, warning=FALSE}
m<-100
XLim<-c( -3, 6 )
x<-seq( XLim[1], XLim[2], length.out = m )
f<-function(x){
  return( 1 + exp(-x^2) + 0.4 * exp(-(x-2)^2 ) - 0.5 * exp( -(x-4)^2 ) ) 
}
z<-sapply( x, f )
```

Sampling function values.
```{r,eval=TRUE,message=FALSE, warning=FALSE}
n<-10
X<-matrix( runif( n, XLim[1], XLim[2] ), n, 1 )

Z<-matrix( sapply( X, f ), n, 1 )

m<-100
YLim<-c(-6,8)
Y<-matrix( seq( YLim[1], YLim[2], length.out = m ), m, 1 )
```

Kernel definition.
```{r,eval=TRUE,message=FALSE, warning=FALSE}
s<-0.1
t<-1

Kern<-function( x, y ) {
  h<-sqrt( sum( ( x - y )^2 ) )
  return( gaussian_kernel( h, s, t ) )
}
```

Computing Kriging.
```{r,eval=TRUE,message=FALSE, warning=FALSE}
K = Kov( X, X, Kern, TRUE )
k = Kov( Y, X, Kern );

KRIG<-Krig( Z = Z,
            K = K, 
            k = k,
            G = matrix( 0, 1, 1),
            g = matrix( 0, 1, 1),
            type = "simple", 
            cinv = 'syminv' )
```

Plotting results.
```{r,eval=TRUE,fig.show='hold', fig.show='hold', fig.align="center", fig.width=7, fig.height=4}
ymin<-min( z, KRIG$Z[,1], Z[,1] )
ymax<-max( z, KRIG$Z[,1], Z[,1] )
plot( x, z, type = 'l', lwd = 2, col='gold', ylim = c( ymin, ymax ), xlim = YLim )
points( Y, KRIG$Z[,1], col='darkgreen', type = 'l', lwd = 2 )
points( X, Z[,1], col = 'dodgerblue3', pch = 16, cex = 1.2 )
```
