---
title: "Filtering by Kriging"
author: "RGeostats Team"
date: "September 2019"
output:
  beamer_presentation:
    colortheme: beaver
    fig_height: 6
    theme: Boadilla
fontsize: 8pt
theme: Boadilla
colortheme: beaver
---

\frametitle{Loading Data}

```{r Loading_Library, include=FALSE}
library(knitr)
library(RGeostats)
constant.define("asp",1)
rm(list=ls())
```

The Grid containing the information is downloaded from the distribution


```{r Loading_Data}
rg.load("Exdemo_Filter","grid")
```

The loaded file (called **grid **) contains 3 variables:

- **P** (phosphorus) which is the *variable of interest*
- **Cr** (chromium) is an auxiliary variable
- **Ni** (nickel) another auxiliary variable

---

\frametitle{Statistics on main variable}

```{r Statistics_P}
print(grid,flag.resume=FALSE,flag.stats=TRUE,names="P")
```

Note that some pixels are not informed for variable **P**.

---

\frametitle{Statistics on auxiliary variables}


```{r Statistics_Cr}
print(grid,flag.resume=FALSE,flag.stats=TRUE,names="Cr")
```

---

\frametitle{Statistics on auxiliary variables}

```{r Statistics_Ni}
print(grid,flag.resume=FALSE,flag.stats=TRUE,names="Ni")
```

---

\frametitle{Correlation between Cr and P}

```{r Correlation_Cr_P,results="hide"}
correlation(grid,"Cr","P",title="")
```

---

\frametitle{Correlation between Ni and P}

```{r Correlation_Ni_P,results="hide"}
correlation(grid,"Ni","P",title="")
```

---

\frametitle{Correlation between Ni and CR}

```{r Correlation_Ni_Cr,results="hide"}
correlation(grid,"Ni","Cr",title="")
```

---

\frametitle{Completing the variable **P**}

Using inverse square distance

```{r Completing_P}
grid = db.locate(grid,"P","z")
grid = db.grid.fill(grid,radix="Fill")
```

We concentrate on the variable of interest **P** (completed) in the next operations

```{r Locate_P}
grid = db.locate(grid,"Fill.P","z")
```

---

\frametitle{Display P}

```{r Visualize_P}
plot(grid,pos.legend=1,title="P after completion")
```

---

\frametitle{Variogram Calculation}

```{r Variogram_calculation}
varioP = vario.grid(grid,nlag=50)
modelP = model.auto(varioP,struct=c(1,2,12),auth.aniso = FALSE,flag.noreduce=TRUE)
```

---

\frametitle{Variogram Model}

```{r Variogram_Model}
modelP
```

---

\frametitle{Monovariate Factorial Kriging Analysis}

We must define the **Neighborhood**

```{r Neighborhood_Definition}
neigh = neigh.create(ndim=2,type=3,radimg=c(10,10))
```

The **image** neighborhood is based on $(2*10+1)^2=441$ pixels (centered on the target pixel).

During the estimation, only the contribution of second and third basic structures are kept (Nugget Effect is filtered out): ** Factorial Kriging Analysis**.

```{r Performing_monovariate_FKA}
means = db.stat(grid,flag.mono=TRUE,names="Fill.P")
mono  = krimage(grid,modelP,neigh,cov.extract = c(2,3),
                uc="",mean=means)
```

---

\frametitle{Monovariate Factorial Kriging Analysis}

```{r Visualize_P_after_FKA_monovariate}
plot(mono,pos.legend=1,title="P denoised (monovariate)",zlim=c(0,150))
```

---

\frametitle{Correlation between Initial and Filtered images}

Correlation for **P** variable between Initial image and its Filtered version (monovariate FKA)

```{r Correlation_P_denoised_monovariate}
correlation(mono,"Fill.P","Kimage.Fill.P.estim",
            title="",xlab="P Filled",ylab="P Filtered")
```

---

\frametitle{Multivariate Approach}


```{r Locate_Multivariate, fig.align='center', fig.height=5}
grid = db.locate(grid,c("Fill.P","Cr","Ni"),"z")
varioM = vario.grid(grid,nlag=50)
modelM = model.auto(varioM,struct=c(1,2,12),auth.aniso = FALSE,flag.noreduce=TRUE)
```

---

\frametitle{Multivariate Model}

```{r Multivariate_Model}
modelM
```

--- 

\frametitle{Multivariable Factorial Kriging Analysis}

```{r Performing_Multivariate_FKA}
means = db.stat(grid,flag.mono=TRUE,names=c("Fill.P","Cr","Ni"))
multi = krimage(grid,modelM,neigh,cov.extract = c(2,3),
                uc="",mean=means)
```

Note that, using the same **neigh** as in monovariate, the dimension of the Kriging System is now $3 * 441 = 1323$

---

\frametitle{Multivariable Factorial Kriging Analysis}

```{r Visualize_P_after_FKA_multivariate}
plot(multi,pos.legend=1,title="P denoised (multivariate)",zlim=c(0,150))
```

---

\frametitle{Correlation between Initial and Filtered images}

Correlation for **P** variable between Initial image and its Filtered version (multivariate FKA)

```{r Correlation_P_denoised_multivariate}
correlation(multi,"Fill.P","Kimage.Fill.P.estim")
```