---
title: "Evolutionary Inference using Supervised Learning"
subtitle: "Part 1. Population structure inference using discriminant analysis"
author: "Xinghu Qin  --School of Biology, University of St Andrews"
#date: "`r Sys.Date()`"
date: "30-03-2020"
URL: https://xinghuq.github.io/DA/index.html
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Instruction for Package DA }
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
# Instruction

This vignette demonstrates the implementation of supervised learning in ecological and evolutionary inference. In this vignette, we take the microsatellite genotypes of 15 cattle breeds (Laloë et al. 2007) as an example. We aim to use different supervised leaning techniques to identify the population structure of 15 cattle breeds.


# Practical analyses

We use the microsatellite genotypes of 15 cattle breeds (Laloë et al. 2007) as an example to show population structure inference and visualization. We compare six approaches that are feasible and suitable for population structure inference here. We use the commonly used unsupervised learning technique, PCA, as the benchmark. We demonstrate how to use these five supervised learning approaches, including DAPC, LFDAPC, LFDA, LFDAKPC, and KLFDA, to identify population structure. These five supervised learning techniques are all from the same discriminant family.  

First, we need to install and load the package.

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

##Install DA

```{r, library}
#Install from CRAN
#install.packages("DA")
## or you can get the latest version of HierDpart from github
#library(devtools)
#install_github("xinghuq/DA")

library("DA")
library("kernlab")

```

```{r}
# example genepop file
f <- system.file('extdata',package='DA')
infile <- file.path(f, "Cattle_breeds_allele_frequency.csv")
Cattle_pop=file.path(f, "Cattle_pop.csv")
cattle_geno=read.csv(infile,h=T)
cattle_pop=read.csv(Cattle_pop,h=T)

```
## Principal Component Analysis (PCA)
PCA is still one of the most commonly used approaches to study population structure. However, PCs represent the global structure of the data without consideration of variation within classes.

```{r fig1, fig.height = 5, fig.width = 8.5, fig.align = "center"} 
cattle_pop$x=factor(cattle_pop$x,levels = unique(cattle_pop$x))
### PCA
cattle_pc=princomp(cattle_geno[,-1])

#plot the data projection on the components
library(plotly)
   cols=rainbow(length(unique(cattle_pop$x)))
   p0 <- plot_ly(as.data.frame(cattle_pc$scores), x =cattle_pc$scores[,1], y =cattle_pc$scores[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'PC1'),
                         yaxis = list(title = 'PC2')))

  
p0

```
Fig.1 PCA plot of 15 cattle breeds.

## Discriminant Analysis of Principal Components (DAPC)
Using DAPC to display the pop structure is a common means in population genetics. This can be achieved through "adegenet" package.
```{r fig2, fig.height = 5, fig.width = 8.5, fig.align = "center"}
library(adegenet)
cattle_pop$x=factor(cattle_pop$x,levels = unique(cattle_pop$x))
###DAPC
cattle_dapc=dapc(cattle_geno[,-1],grp=cattle_pop$x,n.pca=10, n.da=3)

#plot the data projection on the components
library(plotly)
   cols=rainbow(length(unique(cattle_pop$x)))
   p1 <- plot_ly(as.data.frame(cattle_dapc$ind.coord), x =cattle_dapc$ind.coord[,1], y =cattle_dapc$ind.coord[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'LDA1'),
                         yaxis = list(title = 'LDA2')))

   p1

```

Fig.2 DAPC plot of 15 cattle breeds.
This is an interactive plot that allows you to point the data values and display the value as you wish.

## Discriminant Analysis of Kernel Principal Components (DAKPC)

Discriminant analysis of kernel principal components (DAKPC) is a variant of DAPC. However, people try to incorporate the non-linear relationship between loci and samples, so that the kernel principal component analysis is emolyed to achieve this goal.
Below is the implementation of DAKPC.

```{r fig3, fig.height = 5, fig.width = 8.5, fig.align = "center"} 

cattle_ldakpc=LDAKPC(cattle_geno[,-1],cattle_pop$x,n.pc=3)

 cols=rainbow(length(unique(cattle_pop$x)))
   p2 <- plot_ly(as.data.frame(cattle_ldakpc$LDs), x =cattle_ldakpc$LDs[,1], y =cattle_ldakpc$LDs[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'LDA1'),
                         yaxis = list(title = 'LDA2')))
p2
```
Fig.3 LDAKPC plot of 15 cattle breeds.

LDAKPC has the similar result with DAPC.

##  Local Fisher Discriminant Analysis  (LFDA)
In comparison to LDA, LFDA not only considers the variation between classes, but also the variation within classes. Thus, LFDA can discriminate the multimodal data while LDA can not. LFDA is an upgraded version of LDA.

```{r fig4, fig.height = 5, fig.width = 8.5, fig.align = "center"} 

cattle_lfda=LFDA(cattle_geno[,-1],cattle_pop$x,r=3,tol=1E-3)

cols=rainbow(length(unique(cattle_pop$x)))
p3 <- plot_ly(as.data.frame(cattle_lfda$Z), x =cattle_lfda$Z[,1], y =cattle_lfda$Z[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'LDA1'),
                         yaxis = list(title = 'LDA2')))
p3
```
Fig.4 LFDA plot of 15 cattle breeds.


## Local Fisher Discriminant Analysis of Kernel Principal Components (LFDAKPC)

As LFDA is more advanced than LDA, I adopt LFDA for discriminant analysis on the basis of LDAKPC. Now we get LFDAKPC, Local (Fisher) Discriminant Analysis of Kernel Principal Components (LFDAKPC). Below is the implementation of LFDAKPC.

```{r fig5, fig.height = 5, fig.width = 8.5, fig.align = "center"} 


cattle_lfdakpc=LFDAKPC(cattle_geno[,-1],cattle_pop$x,n.pc=3,tol=1E-3)

cols=rainbow(length(unique(cattle_pop$x)))
p4 <- plot_ly(as.data.frame(cattle_lfdakpc$LDs), x =cattle_lfdakpc$LDs[,1], y =cattle_lfdakpc$LDs[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'LDA1'),
                         yaxis = list(title = 'LDA2')))
p4
```


Fig.5 LFDAKPC plot of 15 cattle breeds.

The LFDAKPC also produces the similar results as LDAKPC and DAPC.


##  Kernel Local Fisher Discriminant Analysis (KLFDA)

Kernel local (Fisher) discriminant analysis (KLFDA) is a kernelized version of local Fisher discriminant analysis (LFDA). KLFAD can capature the non-linear relationships between samples. It was reported that the discrimintory power of KLFDA was significantly improved compared to LDA.


```{r fig6, fig.height = 5, fig.width = 8.5, fig.align = "center"} 
cattle_klfda=KLFDA(as.matrix(cattle_geno[,-1]),as.factor(cattle_pop$x),r=3,tol=1E-10,prior = NULL)

cols=rainbow(length(unique(cattle_pop$x)))
p5 <- plot_ly(as.data.frame(cattle_klfda$Z), x =cattle_klfda$Z[,1], y =cattle_klfda$Z[,2], color = cattle_pop$x,colors=cols[cattle_pop$x],symbol = cattle_pop$x,symbols = 1:15L) %>% 
     add_markers() %>%
     layout(scene = list(xaxis = list(title = 'LDA1'),
                         yaxis = list(title = 'LDA2')))
p5
```
Fig.6 KLFDA plot of 15 cattle breeds.

KLFDA seems present the aggregates that are more convergent than the above methods.

All the above methods show the same global structure for 15 cattle breeds. 

##  Individual assignment using Kernel Local Fisher Discriminant Analysis (KLFDA)

Kernel local (Fisher) discriminant analysis (KLFDA) is the optimal approach for population structurte inference when tested using this cattle data. Now, we plot the cattle individual membership representing the posterior possibilities of individuals as the population structure. 
This gives the similar plot produced from STRUCTURE software.


```{r fig7, fig.height = 5, fig.width = 8.5, fig.align = "center"} 


library(adegenet)
## asignment plot
compoplot(as.matrix(cattle_klfda$bayes_assigment$posterior),show.lab = TRUE, posi=list(x=5,y=-0.01),txt.leg = unique(cattle_pop$x))

```

Fig. 7 The population structure of Cattle breeds (individual assignment)

More tutorials can be found at URL: https://xinghuq.github.io/DA/articles/index.html.

# References

Laloë, D., Jombart, T., Dufour, A.-B. & Moazami-Goudarzi, K. (2007). Consensus genetic structuring and typological value of markers using multiple co-inertia analysis. Genetics Selection Evolution, 39, 545.

Jombart, T. (2008). adegenet: a R package for the multivariate analysis of genetic markers. Bioinformatics, 24, 1403-1405.
Sugiyama, M (2007).Dimensionality reduction of multimodal labeled data by local Fisher discriminant analysis. Journal of Machine Learning Research, vol.8, 1027-1061.

Sugiyama, M (2006). Local Fisher discriminant analysis for supervised dimensionality reduction. In W. W. Cohen and A. Moore (Eds.), Proceedings of 23rd International Conference on Machine Learning (ICML2006), 905-912.


Tang, Y., & Li, W. (2019). lfda: Local Fisher Discriminant Analysis inR. Journal of Open Source Software, 4(39), 1572.

Moore, A. W. (2004). Naive Bayes Classifiers. In School of Computer Science. Carnegie Mellon University.

Pierre Enel (2020). Kernel Fisher Discriminant Analysis (https://github.com/p-enel/MatlabKFDA), GitHub. Retrieved March 30, 2020.

Karatzoglou, A., Smola, A., Hornik, K., & Zeileis, A. (2004). kernlab-an S4 package for kernel methods in R. Journal of statistical software, 11(9), 1-20.

Bingpei Wu, 2012, WMDB 1.0: Discriminant Analysis Methods by Weight Mahalanobis Distance and bayes.

Ito, Y., Srinivasan, C., & Izumi, H. (2006, September). Discriminant analysis by a neural network with Mahalanobis distance. In International Conference on Artificial Neural Networks (pp. 350-360). Springer, Berlin, Heidelberg.

Wölfel, M., & Ekenel, H. K. (2005, September). Feature weighted Mahalanobis distance: improved robustness for Gaussian classifiers. In 2005 13th European signal processing conference (pp. 1-4). IEEE.

Qin, X., Wu, M., Lock, R., Kallenbach, R. (2020). DA: Ecological and Evolutionary Inference Using Supervised Discriminant Analysis. Authorea.DOI: 10.22541/au.159256808.83862168


