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.
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.
##Install DA
#Install from CRAN
#install.packages("DA")
## or you can get the latest version of HierDpart from github
library(devtools)
#> Warning: package 'devtools' was built under R version 3.6.2
#> Loading required package: usethis
#> Warning: package 'usethis' was built under R version 3.6.2
#install_github("xinghuq/DA")
library("DA")# 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)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.
cattle_pop$x=factor(cattle_pop$x,levels = unique(cattle_pop$x))
### PCA
cattle_pc=princomp(cattle_geno[,-1])
#plot the data projection on the components
library(plotly)
#> Warning: package 'plotly' was built under R version 3.6.2
#> Loading required package: ggplot2
#>
#> Attaching package: 'plotly'
#> The following object is masked from 'package:ggplot2':
#>
#> last_plot
#> The following object is masked from 'package:stats':
#>
#> filter
#> The following object is masked from 'package:graphics':
#>
#> layout
cols=rainbow(length(unique(cattle_pop$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
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.Using DAPC to display the pop structure is a common means in population genetics. This can be achieved through “adegenet” package.
library(adegenet)
#> Loading required package: ade4
#> Registered S3 method overwritten by 'spdep':
#> method from
#> plot.mst ape
#> Registered S3 methods overwritten by 'vegan':
#> method from
#> plot.rda klaR
#> predict.rda klaR
#> print.rda klaR
#>
#> /// adegenet 2.1.1 is loaded ////////////
#>
#> > overview: '?adegenet'
#> > tutorials/doc/questions: 'adegenetWeb()'
#> > bug reports/feature requests: adegenetIssues()
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$x,symbols = 1:15L) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'LDA1'),
yaxis = list(title = 'LDA2')))
p1
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.Fig.2 DAPC plot of 15 cattle breeds. This is an interactive plot that allows you to point the data values and display the value as you wish.
Discriminant analysis of kernel principal components (DAKPC) is a variant of DAPC. However, people try to incorporate the non-linear relationship between loci and samples, so that the kernel principal component analysis is emolyed to achieve this goal. Below is the implementation of DAKPC.
cattle_ldakpc=LDAKPC(cattle_geno[,-1],cattle_pop$x,n.pc=3)
#> Loading required package: kernlab
#>
#> Attaching package: 'kernlab'
#> The following object is masked from 'package:ggplot2':
#>
#> alpha
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$x,symbols = 1:15L) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'LDA1'),
yaxis = list(title = 'LDA2')))
p2
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.LDAKPC has the similar result with DAPC.
In comparison to LDA, LFDA not only considers the variation between classes, but also the variation within classes. Thus, LFDA can discriminate the multimodal data while LDA can not. LFDA is an upgraded version of LDA.
cattle_lfda=LFDA(cattle_geno[,-1],cattle_pop$x,r=3,tol=1E-3)
#> Loading required package: lfda
#> Loading required package: klaR
#> Warning: package 'klaR' was built under R version 3.6.3
#> Loading required package: MASS
#>
#> Attaching package: 'MASS'
#> The following object is masked from 'package:plotly':
#>
#> select
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$x,symbols = 1:15L) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'LDA1'),
yaxis = list(title = 'LDA2')))
p3
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.As LFDA is more advanced than LDA, I adopt LFDA for discriminant analysis on the basis of LDAKPC. Now we get LFDAKPC, Local (Fisher) Discriminant Analysis of Kernel Principal Components (LFDAKPC). Below is the implementation of LFDAKPC.
cattle_lfdakpc=LFDAKPC(cattle_geno[,-1],cattle_pop$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$x,symbols = 1:15L) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'LDA1'),
yaxis = list(title = 'LDA2')))
p4
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.Fig.5 LFDAKPC plot of 15 cattle breeds.
The LFDAKPC also produces the similar results as LDAKPC and DAPC.
Kernel local (Fisher) discriminant analysis (KLFDA) is a kernelized version of local Fisher discriminant analysis (LFDA). KLFAD can capature the non-linear relationships between samples. It was reported that the discrimintory power of KLFDA was significantly improved compared to LDA.
cattle_klfda=klfda_1(as.matrix(cattle_geno[,-1]),as.matrix(cattle_pop$x),r=3,tol=1E-10,prior = NULL)
#> Loading required package: WMDB
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
#> Warning: The shape palette can deal with a maximum of 6 discrete values because
#> more than 6 becomes difficult to discriminate; you have 15. Consider
#> specifying shapes manually if you must have them.All the above methods show the same global structure for 15 cattle breeds.
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.
Original Matlab Implementation: http://www.ms.k.u-tokyo.ac.jp/software.html#LFDA
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://www.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.