Multivariate Gaussian Subspatial Regression

This repository contains the required functions to apply the MGSR.

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Table of Contents

1. Summary
2. Algorithm
3. Installation
4. Reference
5. Example
6. How to cite

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Summary

Our proposal is a mixture between Factorial Techniques and Gaussian Processes.

Factorial techniques (Biplot, PCA, CA ...) are useful to represent our observations in terms of unobserved variables called factors. All these techniques provide a set of coordinates linked to the observations, which display information of our analyzed variables. These kind of procedures are merely descriptive and have a low predictive power.

On the other hand, Gaussian Processes are statistical methods where observations occur in a continuous domain (mainly time or space). Furthermore, variables have a multivariate normal distribution. Gaussian Processes use similarity between points to predict the value in an unobserved point.

The MGSR applies a Gaussian Process Regression to the created subspatial of the Factorial Technique. We make use of this subspace to simulate a continuous domain that permit the application of Gaussian Processes, such as Cokriging.

Algorithm

1. Factorial Technique
2. Cross-variogram
   • CV plot (cross variogram and LMC)
3. Linear Model of Coregionalization
4. Subspatial Grid
5. Cokriging

Note:

• Cross-variograms usually follows a "Power Distribution" due to the small scale of Factorial Techniques.
• Unlike geostatistical analyses, we don't have a real field where boundaries restrict our study. On the other hand, this aspect is more positive than negative because we can portray a more simple grid without losing information.

Installation

install.packages('devtools')
library(devtools)
install_github("victorvicpal/MGSR")
library(MGSR)

Reference

Example

Iris Data

data(iris)

Data Exploration

summary(iris)
apply(iris[which(iris$Species=='versicolor'),1:4],2,function(x,y) plot(density(x))) #density function

"Versicolor" Train/subset

Versicolor <- iris[which(iris$Species=='versicolor'),-5]

ind_subTrain <- sample(50,10)

subTrain <- Versicolor[ind_subTrain,]
Train <- Versicolor[-ind_subTrain,]

Data Standardization

means_train <- apply(Train,2,mean)
sd_train <- apply(Train,2,sd)
Train_st <- Train

for (i in 1:length(Train[1,]))
{Train_st[,i] <- (Train[,i]-means_train[i])/sd_train[i]}

Principal Component Analysis

PC_train <- princomp(Train_st)
biplot(PC_train)

CrossVariogram

CV_train <- crossvariogram(as.data.frame(PC_train$scores[,1:2]),as.data.frame(Train_st),10)
plot.crossvariogram(CV_train)

lmc fitting

Tip: Range value may vary. Check different values within the "Power" function.

RES_train <- lmc(CV_train,'Pow',1.7)
plot.crossvariogram(CV_train,RES_train)

Grid

xygrid <- GRID_MGSR(as.data.frame(PC_train$scores[,1:2]),0.05)

Cokriging

Z_train_st <- cokrig(RES_train,xygrid)
Z_train <- Z_train_st

for (i in 1:length(Train[1,]))
{Z_train[,i+2] <- Z_train_st[,i+2]*sd_train[i]+means_train[i]}

Predicting Subset values

ind_pred <- apply(dist2(subTrain[,1:4],Z_train[,3:6]),1,which.min)
residuales <- subTrain[,1:4]-Z_train[ind_pred,3:6]

par(mfrow=c(2,2))
apply(residuales,2,function(x) plot(density(x)))

for (i in 1:4)
{
  qqnorm(residuales[,i])
  qqline(residuales[,i])
}

Try to fit a model with virginica and setosa species.

Citation

@book{mgsr,
      author    = "Vicente-Palacios, V",
      publisher = "victorvicpal/MGSR",
      year      = "2016",
      doi       = "(http://doi.org/10.5281/zenodo.264102)",
      title     = "Multivariate Gaussian Subspatial Regression"}