In this document we present details of the benchmark of the
tensorEVD() routine against the eigen()
function of the ‘base’ R-package (R Core Team 2021) in performing the
eigenvalue decomposition (EVD) of a Hadamard matrix product involving
two covariance structure matrices \(\textbf{K}_1\) and \(\textbf{K}_2\),
\[
\textbf{K} =
(\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2).
\] We assessed the computational time used to derive
eigenvectors, the accuracy of the approximation provided by
tensorEVD(), and the dimension of the resulting basis.
Likewise, we evaluated the performance of the approximation of the
Hadamard \(\textbf{K}\) provided by the
tensorEVD() method in Gaussian linear models in terms of
variance components estimates and cross-validation prediction
accuracies.
The data used in these benchmarks was generated by the Genomes-To-Fields (G2F) Initiative (Lima et al. 2023) which was curated and expanded by adding environmental covariates (EC) by Lopez-Cruz et al. (2023). We used the subset of the data corresponding to the northern testing locations that includes \(n=59,069\) records for 4 traits (grain yield, anthesis, silking, and anthesis-silking interval) from \(n_G = 4,344\) hybrids and \(n_E = 97\) environments.
We used a data analysis pipeline as shown below with folders
code, data, output,
parms, and source.
pipeline
├── code
│ ├── 1_simulation.R
│ ├── 2_model_components.R
│ ├── 3_ANOVA_GxE_model.R
│ ├── 4_get_variance_GxE_model.R
│ └── 5_10F_CV_GxE_model.R
├── data
├── output
├── parms
└── source
├── ECOV.csv
├── GENO.csv
└── PHENO.csvFolder source contains the phenotypic (file
PHENO.csv), SNPs (file GENO.csv), and ECs
(file ECOV.csv) data from G2F. These files can be
downloaded from the Figshare repository (https://doi.org/10.6084/m9.figshare.22776806).
Folder code contains the R-scripts to implement the
sequence of analyses detailed in the next sections and can be downloaded
from this link.
The R-scripts were run on the MSU high-performance computing center
(HPCC) (https://icer.msu.edu/hpcc/hardware) as a batch job
script. The header of the scripts contains the shebang line
#!/usr/bin/env Rscript in the first line followed by job
requirements (e.g., memory, number of CPUs, run time) that are specified
using the SLURM scheduler by adding the prefix
#SBATCH at the beginning of each request instruction line,
for example
#!/usr/bin/env Rscript
#SBATCH --time=03:59:00
#SBATCH --cpus-per-task=1
#SBATCH --mem-per-cpu=84G
#SBATCH --constraint=intel18Each R-script is submitted to the HPCC using the sbatch
command, for instance
cd /mnt/scratch/quantgen/TENSOR_EVD/pipeline/code
sbatch 1_simulation.RThe R-code below show how to obtain the subset of the data corresponding to the northern locations. Next, this data subset is used to derive a genetic (GRM, VanRaden 2008) for the \(n_G = 4,344\) hybrids as \(\textbf{K}_G = \textbf{X}\textbf{X}'/trace(\textbf{X}\textbf{X}')\), where \(\textbf{X}\) is the matrix of centered SNPs (hybrids in rows, SNPs in columns). Likewise, an environmental relationship matrix (ERM) is derived for the \(n_E = 97\) environments as \(\textbf{K}_E = \textbf{W}\textbf{W}'/trace(\textbf{W}\textbf{W}')\) where \(\textbf{W}\) is the matrix of centered and scaled ECs (environments in rows, ECs in columns).
library(data.table)
setwd("/mnt/scratch/quantgen/TENSOR_EVD/pipeline")
PHENO <- read.csv("source/PHENO.csv")
GENO <- fread("source/GENO.csv", data.table=FALSE)
ECOV <- read.csv("source/ECOV.csv", row.names=1)
# Select North region
PHENO <- PHENO[PHENO$region %in% 'North',]
PHENO$year_loc <- factor(as.character(PHENO$year_loc))
PHENO$genotype <- factor(as.character(PHENO$genotype))
save(PHENO, file="data/pheno.RData")
# Calculate the GRM
ID <- GENO[,1]
GENO <- as.matrix(GENO[,-1])
rownames(GENO) <- ID
X <- scale(GENO, center=TRUE, scale=FALSE)
KG <- tcrossprod(X)
KG <- KG[levels(PHENO$genotype),levels(PHENO$genotype)]
KG <- KG/mean(diag(KG))
save(KG, file="data/GRM.RData")
# Calculate the ERM
ECOV <- ECOV[,-grep("HI30_",colnames(ECOV))]
KE <- tcrossprod(scale(ECOV))
KE <- KE[levels(PHENO$year_loc),levels(PHENO$year_loc)]
KE <- KE/mean(diag(KE))
save(KE, file="data/ERM.RData")After running this code, R-files pheno.RData,
GRM.RData, and ERM.RData with the phenotypic
northern subset, GRM, and ERM, respectively, are to be saved in the
folder data.
pipeline
:
├── data
: ├── ERM.RData
├── GRM.RData
└── pheno.RDataWe formed Hadamard products \(\textbf{K} =
(\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2)\)
between the GRM (as \(\textbf{K}_1\))
and the ERM (as \(\textbf{K}_2\)) of
various sizes by sampling hybrids (\(n_G=100,500,1000\)), environments (\(n_E=10,30,50\)), and the level of
replication needed to complete a total sample size of \(n=10000,20000,30000\). Then, we factorized
the resulting Hadamard product matrix using the R-base function
eigen() as well as using tensorEVD(), deriving
as many eigenvectors as needed to explain a proportion \(\alpha=0.90,0.95,0.98\) of the total
variance. We implemented 10 replicates of each experiment.
The R-script 1_simulation.R
was used to perform the analysis for a given combination of parameters
(nG, nE, n, alpha,
and replicate). To do this, first, we created an array with
all combinations of the parameters. A data.frame object
called JOBS containing \(3\times3\times3\times3\times10=810\) rows
and the \(5\) parameters in columns,
was created using the expand.grid R-function and saved in
folder parms
JOBS <- expand.grid(nG = c(100,500,1000),
nE = c(10,30,50),
n = c(10000,20000,30000),
alpha = c(0.90,0.95,0.98),
replicate = 1:10)
dim(JOBS); head(JOBS)
#[1] 810 5
# nG nE n alpha replicate
#1 100 10 10000 0.9 1
#2 500 10 10000 0.9 1
#3 1000 10 10000 0.9 1
#4 100 30 10000 0.9 1
#5 500 30 10000 0.9 1
#6 1000 30 10000 0.9 1
save(JOBS, file="/mnt/scratch/quantgen/TENSOR_EVD/pipeline/parms/JOBS1.RData")To perform all the \(3\times3\times3\times3\times10=810\) cases
(each row of the object JOBS) we submitted the R-script for
multi-job implementation by specifying in the script header a job array
through the SLURM option
#SBATCH --array=1-810, each value of the array is read in
the R-code with the instruction
Sys.getenv("SLURM_ARRAY_TASK_ID").
The output file simulation_results.txt generated after
running the previous R-script contains at each row the results from each
experiment case. The information of the experiment are given in the
first columns followed by the results on computation time, approximation
accuracy (measured with the Frobenius and CMD metrics), and the number
of eigenvectors (associated to the \(\alpha\)-value) for the
eigen() and tensorEVD() methods.
pipeline
:
├── output
: └── simulation
└── simulation_results.txtThe first rows of this file are shown below for the results presented in the manuscript (the file can be found in this link).
out <- read.csv(url("https://raw.githubusercontent.com/MarcooLopez/tensorEVD/main/inst/extdata/results_simulation.txt"))
head(out[,1:7])## alpha nG nE n replicate nComb nPosEigen
## 1 0.9 100 10 10000 1 1000 1000
## 2 0.9 500 10 10000 1 4329 4329
## 3 0.9 1000 10 10000 1 6376 6376
## 4 0.9 100 30 10000 1 2899 2899
## 5 0.9 500 30 10000 1 7253 7253
## 6 0.9 1000 30 10000 1 8483 8483head(out[,8:12]) # results from the eigen function## time_eigen Frobenius_eigen CMD_eigen nPC_eigen pPC_eigen
## 1 242.120 144.4331 0.004126274 406 0.4060000
## 2 243.954 139.0350 0.001844430 968 0.2236082
## 3 248.545 124.0434 0.002342244 1566 0.2456085
## 4 240.986 105.3499 0.002186993 765 0.2638841
## 5 251.789 111.8236 0.001962019 1737 0.2394871
## 6 254.031 104.0453 0.001778629 2298 0.2708947head(out[,13:17]) # results from the tensorEVD function## time_tensorEVD Frobenius_tensorEVD CMD_tensorEVD nPC_tensorEVD pPC_tensorEVD
## 1 0.122 139.11850 0.003451564 423 0.4230000
## 2 0.283 129.80934 0.001340386 1128 0.2605683
## 3 0.856 114.21707 0.001431199 2079 0.3260665
## 4 0.241 98.92667 0.001561782 837 0.2887202
## 5 0.836 98.57860 0.001082447 2397 0.3304839
## 6 1.512 88.48203 0.000876179 3828 0.4512555The following R-code chunks can be used to the reproduce Figures 1-3 that are presented in the manuscript. The code for the plots can be found in this link and can be loaded into R using
source("https://raw.githubusercontent.com/MarcooLopez/tensorEVD/main/misc/functions.R")Computation time
R-code below creates a plot for the EVD computation time ratio
(eigen()/tensorEVD()) (Figure 1).
# Some data edits
out$alpha <- factor(100*out$alpha)
out$nG <- paste0("n[G]*' = '*",out$nG,"L")
out$nE <- paste0("n[E]*' = '*",out$nE,"L")
out$nG <- factor(out$nG, levels = unique(out$nG))
out$nE <- factor(out$nE, levels = unique(out$nE))
# Reshaping the data
measure <- c("time","Frobenius","CMD","nPC","pPC")
dat <- melt_data(out, id=c("nG","nE","n","alpha"),
measure=paste0(measure,"_"),
value.name=measure, variable.name="method")
color1 <- c('90%'="navajowhite2", '95%'="chocolate1", '98%'="red4")
color2 <- c(eigen="#E69F00", tensorEVD="#009E73", eigs="#56B4E9",
trlan="#CC79A7", chol="#D55E00")
# Figure 1: Computation time ratio (eigen/tensorEVD)
dat0 <- out[out$alpha != "100",]
dat0$alpha <- factor(paste0(dat0$alpha,"%"))
dat0$ratio <- log10(dat0$time_eigen/dat0$time_tensorEVD)
dat0$n <- dat0$n/1000
breaks0 <- seq(1,4,by=1)
figure1 <- make_plot(dat0, type="line", x='n', y='ratio', group="alpha",
group.label=NULL, facet="nG", facet2="nE", facet.type="grid",
xlab="Sample size (x1000)",
ylab="Computation time ratio (eigen/tensorEVD)",
group.color=color1, nSD=0, errorbar.size=0,
breaks.y=breaks0, labels.y=sprintf("%.f",10^breaks0),
scales="fixed")
#print(figure1)Approximation accuracy
R-code below produces the approximation accuracy plots using the
Frobenius norm (Figure 2) of the difference between the Hadamard matrix
and the approximation provided by eigen() and
tensorEVD().
dat0 <- dat[dat$method %in% c("eigen","tensorEVD") & dat$alpha!="100",]
dat0$method <- factor(as.character(dat0$method))
dat0$alpha <- factor(as.character(dat0$alpha))
# Figure 2: Approximation accuracy using Frobenious norm
figure2 <- make_plot(dat0, x='alpha', y='Frobenius',
group="method", by="n", facet="nG", facet2="nE", facet.type="grid",
xlab=bquote(alpha~"x100% of variance of K"),
ylab=expression("Frobenius norm ("~abs(abs(K-hat(K)))[F]~")"),
by.label="Sample size", breaks.y=seq(0,500,by=100),
group.color=color2, rect.by.height=-0.05, ylim=c(0,NA))
#print(figure2)Dimension reduction
The following R-code creates the plot (Figure 3) showing the number
of eigenvectors produced by the eigen() and
tensorEVD() methods, relative to the rank of the Hadamard
matrix (number of eigenvectors with positive eigenvalue).
figure3 <- make_plot(dat0, x='alpha', y='pPC',
group="method", by="n", facet="nG", facet2="nE", facet.type="grid",
xlab=bquote(alpha~"x100% of variance of K"),
ylab="Number of eigenvectors/rank", by.label="Sample size",
group.color=color2, rect.by.height=-0.05,
hline=1, hline.color="red2", ylim=c(0,NA))
#print(figure3)We analyzed each trait (grain yield, anthesis, silking, and anthesis-silking interval) with a Gaussian reaction norm \(G\times E\) model (Jarquín et al. 2014) in which the trait phenotype (\(y_{ijk}\)) is modeled as the sum of the main effect of hybrid (\(G_i\)), main effect of environment (\(E_j\)), and the hybrid\(\times\)environment interaction (\(GE_{ij}\)) term, this is
\[ y_{ijk} = \mu + G_i + E_j + GE_{ij} + \varepsilon_{ijk}. \]
Above, \(\mu\) is an intercept and \(i\), \(j\), and \(k\) are indices for the hybrids, environment, and replicate, respectively. The term \(\varepsilon_{ijk}\) is an error term assumed to be independently and identically Gaussian distributed as \(\varepsilon_{ijk} \sim N(0,\sigma_{\varepsilon}^2)\), with \(\sigma_{\varepsilon}^2\) variance parameter associated to the error. Hybrid, environment, and interaction effects were assumed to be multivariate normally distributed with zero mean and effect-specific covariance matrices, specifically \(\textbf{G}\sim MVN(\textbf{0},\sigma_G^2 \textbf{K}_G)\), \(\textbf{E}\sim MVN(\textbf{0},\sigma_E^2 \textbf{K}_E)\), and \(\textbf{GE}\sim MVN(\textbf{0},\sigma_{GE}^2 \textbf{K})\), where
\[ \textbf{K} = (\textbf{Z}_1\textbf{K}_G\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_E\textbf{Z}'_2) \]
is a Hadamard product between the GRM \(\textbf{K}_G\) and the ERM \(\textbf{K}_E\), and \(\sigma_G^2\), \(\sigma_E^2\), and \(\sigma_{GE}^2\) are variance parameters associated to \(\textbf{G}\), \(\textbf{E}\) and \(\textbf{GE}\), respectively.
We used a ‘BRR’ equivalence of the above model by fitting the model
\[ \boldsymbol{y} = \boldsymbol{\mu}+\textbf{X}_G\boldsymbol{\beta}_1+\textbf{X}_E\boldsymbol{\beta}_2 + \textbf{X}_{GE}\boldsymbol{\beta}_3 + \boldsymbol{\varepsilon} \]
where the predictors \(\textbf{X}_G\), \(\textbf{X}_E\), and \(\textbf{X}_{GE}\) are the scaled
eigenvectors of the covariance matrices \(\textbf{K}_G\), \(\textbf{K}_E\), and \(\textbf{K}\), respectively, and the
regression coefficients are assumed to be distributed \(\boldsymbol{\beta}_1\sim
MVN(\textbf{0},\sigma_{G}^2 \textbf{I})\), \(\boldsymbol{\beta}_2\sim
MVN(\textbf{0},\sigma_{E}^2 \textbf{I})\), and \(\boldsymbol{\beta}_3\sim
MVN(\textbf{0},\sigma_{GE}^2 \textbf{I})\). First, we obtained
the decomposition \(\textbf{K}_G=\textbf{V}_1\textbf{D}_1\textbf{V}'_1\)
and \(\textbf{K}_E=\textbf{V}_2\textbf{D}_2\textbf{D}'_2\)
using the eigen R-function. Next, for a given proportion \(\alpha\) of variance, we obtained the
decomposition of the Hadamard \(\tilde{\textbf{K}}_{\alpha} =
\tilde{\textbf{V}}_{\alpha}\tilde{\textbf{D}}_{\alpha}\tilde{\textbf{V}}_{\alpha}\)
using the eigen() and tensorEVD() approaches.
Finally, we derived the scaled eigenvectors \(\textbf{X}_G =
\textbf{V}_1\textbf{D}_1^{1/2}\), \(\textbf{X}_E =
\textbf{V}_2\textbf{D}_2^{1/2}\), and \(\tilde{\textbf{X}}_{GE,\alpha} =
\tilde{\textbf{V}}_{\alpha}\tilde{\textbf{D}}_{\alpha}^{1/2}\).
The later was done for values \(\alpha=0.90,0.95,0.98,1.00\).
The calculation of all these matrices was carried out using the
R-script 2_model_components.R
for a given \(\alpha\)-value. We
replicated this task 5 times to obtain an average (and SD) computing
time of the decomposition, to this end, we created a job array for each
combination of parameters alpha and replicate
as follows
JOBS <- expand.grid(alpha = c(0.90,0.95,0.98,1.00),
replicate = 1:5)
dim(JOBS); head(JOBS)
#[1] 20 2
# alpha replicate
#1 0.90 1
#2 0.95 1
#3 0.98 1
#4 1.00 1
#5 0.90 2
#6 0.95 2
save(JOBS, file="/mnt/scratch/quantgen/TENSOR_EVD/pipeline/parms/JOBS2.RData")The script was submitted using a job array for the \(4\times5=20\) combination of parameters
(each row of the object JOBS) using the SLURM
option #SBATCH --array=1-20. After running the R-script,
the following files are created in the folder
output/genomic_prediction/model_comps.
pipeline
:
├── output
: └── genomic_prediction
└── model_comps
├── XE.RData
├── XG.RData
├── XGE_90_eigen.RData
├── XGE_90_tensorEVD.RData
├── XGE_95_eigen.RData
├── XGE_95_tensorEVD.RData
├── XGE_98_eigen.RData
├── XGE_98_tensorEVD.RData
├── XGE_100_eigen.RData
└── timing_EVD.txtThe \(G\times E\) model was
implemented as a BRR using the BLRXy() function from the ‘BGLR’
R-package (Pérez-Rodríguez and de los Campos 2022) for each combination
of trait, method, and \(\alpha\)-value,
with 5 replicates each to present an average (and SD) of the results.
The model was run for \(\alpha=1.0\)
for the eigen method only. The BLRXy() function fits
the model and generates samples from the posterior distribution of the
regression coefficients \(\boldsymbol{\beta}_1\), \(\boldsymbol{\beta}_2\), and \(\boldsymbol{\beta}_3\) using the Gibbs
sampler. We run the BGLR with nIter=50000 and
burnIn=5000 parameters.
We implemented the model using the R-script 3a_fit_GxE_model.R
for a given combination of parameters (trait,
method, alpha, and replicate). We
created an array with all combinations of parameters as follows
JOBS <- expand.grid(trait = c("yield","anthesis","silking","ASI"),
method = c("eigen","tensorEVD"),
alpha = c(0.90,0.95,0.98,1.00),
replicate = 1:5)
JOBS <- JOBS[-which(JOBS$alpha==1.00 & JOBS$method=="tensorEVD"),]
dim(JOBS); head(JOBS)
#[1] 140 4
# trait method alpha replicate
#1 yield eigen 0.9 1
#2 anthesis eigen 0.9 1
#3 silking eigen 0.9 1
#4 ASI eigen 0.9 1
#5 yield tensorEVD 0.9 1
#6 anthesis tensorEVD 0.9 1
save(JOBS, file="/mnt/scratch/quantgen/TENSOR_EVD/pipeline/parms/JOBS3.RData")The script was submitted using a job array for all the \((4\times2\times4\times5) -
(4\times1\times1\times5)=140\) combination of parameters (each
row of object JOBS) using the SLURM option
#SBATCH --array=1-140. The outputs generated by the
R-script are saved in folder
output/genomic_prediction/ANOVA in a sub-folder
corresponding to each trait, method,
alpha, and replicate combination as shown
below. The posterior samples for coefficients \(\boldsymbol{\beta}_1\), \(\boldsymbol{\beta}_2\), and \(\boldsymbol{\beta}_3\) are stored in files
ETA_G_b.bin, ETA_E_b.bin, and
ETA_GE_b.bin, respectively. For instance, the file
ETA_G_b.bin is a \(q\times
p\) matrix containing at each row the samples \(\hat{\boldsymbol{\beta}}_1^{(1)},\hat{\boldsymbol{\beta}}_1^{(2)},...,\hat{\boldsymbol{\beta}}_1^{(q)}\).
pipeline
:
├── output
: └── genomic_prediction
:
└── ANOVA
├── yield
: ├── eigen
: ├── alpha_90
: ├── rep_1
: ├── ETA_E_b.bin
├── ETA_E_varB.dat
├── ETA_G_b.bin
├── ETA_G_varB.dat
├── ETA_GE_b.bin
├── ETA_GE_varB.dat
├── fm.RData
├── mu.dat
└── varE.datObtaining total variance
We used the sample files to obtain the total variance of hybrid
(\(\textbf{X}_G\boldsymbol{\beta}_1^{(k)}\)),
environment (\(\textbf{X}_E\boldsymbol{\beta}_2^{(k)}\)),
hybrid\(\times\)environment interaction
(\(\textbf{X}_{GE}\boldsymbol{\beta}_3^{(k)}\)),
and error (\(\boldsymbol{\varepsilon}\)) terms in the
BRR model, and reported the average across all samples. As we
standardized the phenotype to have unit variance, these variances can be
seen as the proportion of the phenotypic variance explained by each
model component. We performed this task using the R-script 3b_get_variance_GxE_model.R
using a job array for all the \(140\)
jobs (each row in the object JOBS). The code will create a
data.frame stored in the file VC.RData in the
corresponding sub-folder in folder
output/genomic_prediction/ANOVA.
Collecting results
Once the previous R-script has been run for all the jobs, the following code can be used to collect into a single table all the individual variance components results from all jobs.
setwd("/mnt/scratch/quantgen/TENSOR_EVD/pipeline")
load("parms/JOBS3.RData")
prefix <- "output/genomic_prediction/ANOVA"
out <- c()
for(k in 1:nrow(JOBS))
{
trait <- as.vector(JOBS[k,"trait"])
method <- as.vector(JOBS[k,"method"])
alpha <- as.vector(JOBS[k,"alpha"])
replicate <- as.vector(JOBS[k,"replicate"])
suffix <- paste0(trait,"/",method,"/alpha_",100*alpha,"/rep_",replicate,"/VC.RData")
filename <- paste0(prefix,"/",suffix)
if(file.exists(filename)){
load(filename)
out <- rbind(out, VC)
}else{
message("File not found: '",suffix,"'")
}
}The first rows of this data.frame are displayed below
for the results presented in the manuscript (the file can be found in
this link).
out <- read.csv(url("https://raw.githubusercontent.com/MarcooLopez/tensorEVD/main/inst/extdata/results_ANOVA.txt"))
head(out)## trait method alpha replicate source mean SD
## 1 yield eigen 0.9 1 G 0.06956522 0.004202744
## 2 yield eigen 0.9 1 E 0.48419062 0.011957294
## 3 yield eigen 0.9 1 GE 0.07582496 0.004068338
## 4 yield eigen 0.9 1 Error 0.39192312 0.002409890
## 5 anthesis eigen 0.9 1 G 0.12780639 0.004926835
## 6 anthesis eigen 0.9 1 E 0.85284787 0.008838661Visualizing results
R-code below can be used to create Figure 4 in the manuscript showing the average proportion of phenotypic variance of grain yield explained by each model. The same code can be used for traits anthesis, silking, and anthesis-silking interval (Supplementary Figures 8, 9, and 10, respectively).
out$alpha <- factor(paste0(100*out$alpha,"%"), levels=c("100%","98%","95%","90%"))
out$source <- factor(out$source, levels=c("G","E","GE","Error"))
trait <- c("yield", "anthesis", "silking", "ASI")[1]
myfun <- function(x) sprintf('%.3f', x)
# Figure 4: Phenotypic variance of yield
dat <- out[out$trait==trait,]
figure4 <- make_plot(dat, x='alpha', y='mean', SD="SD",
group="method", facet="source",
xlab=bquote(alpha~"x100% of variance of K"),
ylab=paste0("Proportion of variance of ",trait),
group.color=color2, scales="free_y",
ylabels=myfun, text=myfun, ylim=c(0,NA))
#print(figure4)We evaluated the performance of the approximation \(\tilde{\textbf{K}}_{\alpha} =
\tilde{\textbf{V}}_{\alpha}\tilde{\textbf{D}}_{\alpha}\tilde{\textbf{V}}_{\alpha}\)
of the kernel \(\textbf{K}\) provided
by the eigen() and tensorEVD() methods in the
\(G\times E\) model in terms of
prediction accuracy. We conducted a 10-fold cross-validation (CV) with
hybrids assigned to folds. We predicted all the records of hybrids in
the \(k^{th}\) fold using a model
trained with all records from hybrids in the remaining 9 folds. The
model was implemented for each combination of trait and method for \(\alpha=0.90,0.95,0.98\).
The folds were previously created by Lopez-Cruz et al.
(2023) and are provided in the column CV_10fold of the
phenotypic data file. The number records in each fold ranges between
5436 and 6277.
setwd("/mnt/scratch/quantgen/TENSOR_EVD/pipeline")
load("data/pheno.RData")
table(PHENO$CV_10fold)
# 1 2 3 4 5 6 7 8 9 10
#6180 6277 6246 5785 6160 5858 5492 5660 5436 5975We implemented this CV using the R-script 4_10F_CV_GxE_model.R
which fits the model for a given fold for a given combination of trait,
method, and \(\alpha\)-value. To this
end, we created an array with all combinations of parameters
(trait, method, alpha, and
fold) as follows
JOBS <- expand.grid(trait = c("yield","anthesis","silking","ASI"),
method = c("eigen","tensorEVD"),
alpha = c(0.90,0.95,0.98),
fold = 1:10)
dim(JOBS); head(JOBS)
#[1] 240 4
# trait method alpha fold
#1 yield eigen 0.9 1
#2 anthesis eigen 0.9 1
#3 silking eigen 0.9 1
#4 ASI eigen 0.9 1
#5 yield tensorEVD 0.9 1
#6 anthesis tensorEVD 0.9 1
save(JOBS, file="/mnt/scratch/quantgen/TENSOR_EVD/pipeline/parms/JOBS4.RData")The script was submitted using a job array for all the \(4\times2\times3\times10=240\) combination
of parameters (each row of object JOBS) using the
SLURM option #SBATCH --array=1-240. The
outputs generated by the R-script are saved in folder
output/genomic_prediction/10F_CV in a sub-folder
corresponding to each trait, method, and
alpha combination as shown below. Each file
results_fold_*.RData contains a table with the predicted
and observed values within each fold.
pipeline
:
├── output
: └── genomic_prediction
:
└── 10F_CV
├── yield
: ├── eigen
: ├── alpha_90
├── results_fold_1.RData
├── results_fold_2.RData
├── results_fold_3.RData
├── results_fold_4.RData
├── results_fold_5.RData
├── results_fold_6.RData
├── results_fold_7.RData
├── results_fold_8.RData
├── results_fold_9.RData
└── results_fold_10.RDataWithin-environment prediction accuracy
The following R-code can be used to calculate the within environment
(column year_loc) correlation between observed and
predicted phenotypes. The code will collect first the results in files
results_fold_*.RData for all folds from each job (a
combination of trait, method, and
alpha). The results from all jobs are to be saved in a
single table.
setwd("/mnt/scratch/quantgen/TENSOR_EVD/pipeline")
source("https://raw.githubusercontent.com/MarcooLopez/tensorEVD/main/misc/functions.R")
load("parms/JOBS4.RData")
prefix <- "output/genomic_prediction/10F_CV"
dat <- c()
for(trait in levels(JOBS$trait)){
for(method in levels(JOBS$method)){
for(alpha in unique(JOBS$alpha)){
out0 <- c()
for(fold in unique(JOBS$fold)){
suffix <- paste0(trait,"/",method,"/alpha_",100*alpha,"/results_fold_",fold,".RData")
filename <- paste0(prefix,"/",suffix)
if(file.exists(filename)){
load(filename)
out0 <- rbind(out0, out)
}else{
message("File not found: '",suffix,"'")
}
}
tmp <- get_corr(out0, by="year_loc")
dat <- rbind(dat, data.frame(trait,method,alpha,tmp))
}
}
}
# Reshaping the data
dat$trait <- factor(dat$trait, levels=levels(JOBS$trait))
out <- reshape2::dcast(dat, trait+alpha+year_loc+nRecords~method, value.var="correlation")
tmp <- reshape2::dcast(dat, trait+alpha+year_loc+nRecords~method, value.var="SE")[,levels(JOBS$method)]
colnames(tmp) <- paste0(colnames(tmp),".SE")
out <- data.frame(out, tmp)The first rows of this table are displayed below for the results presented in the manuscript (the file can be found in this link).
out <- read.csv(url("https://raw.githubusercontent.com/MarcooLopez/tensorEVD/main/inst/extdata/results_10F_CV.txt"))
head(out)## trait alpha year_loc nRecords eigen tensorEVD eigen.SE tensorEVD.SE
## 1 yield 0.9 2014-IAH1 1557 0.3847606 0.3837099 0.02340692 0.02341801
## 2 yield 0.9 2014-ILH1 462 0.3956203 0.3982038 0.04282128 0.04276919
## 3 yield 0.9 2014-INH1 477 0.5379250 0.5497122 0.03867916 0.03832868
## 4 yield 0.9 2014-MNH1 443 0.5814797 0.5814698 0.03874100 0.03874133
## 5 yield 0.9 2014-MOH2 495 0.5256614 0.5253639 0.03831333 0.03832160
## 6 yield 0.9 2014-NEH1 448 0.2133686 0.2117438 0.04626095 0.04627769Visualizing results
R-code below can be used to create Figure 5 in the manuscript showing
the within environment prediction correlation using the
tensorEVD() and eigen() methods for each
combination of trait and \(\alpha\)-value.
out$trait <- factor(out$trait, levels=unique(out$trait))
out$alpha <- factor(paste0(100*out$alpha,"%"), levels=c("98%","95%","90%"))
# Figure 5: Within environment prediction correlation
rg <- range(c(out$eigen,out$tensorEVD))
if(requireNamespace("ggplot2", quietly=TRUE)){
figure5 <- ggplot2::ggplot(out, ggplot2::aes(tensorEVD, eigen)) +
ggplot2::geom_abline(color="gray70", linetype="dashed") +
ggplot2::geom_point(fill="#56B4E9", shape=21, size=1.4) +
ggplot2::facet_grid(trait ~ alpha) +
ggplot2::theme_bw() + ggplot2::xlim(rg) + ggplot2::ylim(rg)
}
#print(figure5)Henderson C. R., 1985 Best linear unbiased prediction of nonadditive genetic merits in noninbred populations. J. Anim. Sci. 60: 111–117.
Jarquín D., J. Crossa, X. Lacaze, P. Du Cheyron, J. Daucourt, et al., 2014 A reaction norm model for genomic selection using high-dimensional genomic and environmental data. Theor. Appl. Genet. 127: 595–607.
Lima D. C., J. D. Washburn, J. I. Varela, Q. Chen, J. L. Gage, et al., 2023 Genomes to Fields 2022 Maize genotype by Environment Prediction Competition. BMC Res. Notes 16.
Lopez-Cruz M., F. Aguate, J. Washburn, S. K. Dayane, C. Lima, et al., 2023 Leveraging Data from the Genomes to Fields Initiative to Investigate G×E in Maize in North America. Nat. Comm. (in press)
Perez-Rodriguez P., and G. de los Campos, 2022 Additions to the BGLR R-package: a new function for biobank size data and Bayesian multivariate models, pp. 1486–1489 in Proceedings of 12th World Congress on Genetics Applied to Livestock Production (WCGALP), Rotterdam.
R Core Team, 2021 R: A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.