Recall the eigenvalue decomposition (EVD) of an \(N\times N\) positive semi-definite matrix \(\textbf{K}\) which has the form
\[\begin{equation*} \textbf{K} = \textbf{V}\textbf{D}\textbf{V}' = \sum_{k=1}^N{d_k}\boldsymbol{v}_k\boldsymbol{v}'_k \end{equation*}\]
where \(\textbf{V}=[\boldsymbol{v}_1,...,\boldsymbol{v}_N]\) is an orthonormal matrix (i.e., \(\textbf{V}'\textbf{V}=\textbf{I}\)) whose columns \(\boldsymbol{v}_k\) (\(k=1,...,N\)) are the eigenvectors and \(\textbf{D}=diag(d_1,...,d_N)\) is a diagonal matrix with the eigenvalues \(d_1\geq\cdots \geq d_N\geq0\).
Assume that \(\textbf{K}\) can be expressed as the Kronecker product (‘\(\otimes\)’) of two symmetric positive semi-definite matrices, \(\textbf{K}_1\) and \(\textbf{K}_2\) of dimensions \(n_1\) and \(n_2\) such that \(n_1\times n_2=N\), respectively. This is
\[\begin{equation*} \textbf{K} = \textbf{K}_1\otimes\textbf{K}_2 \end{equation*}\]
Let the EVD of the two matrices in the right-hand side be \(\textbf{K}_1 = \textbf{V}_1\textbf{D}_1\textbf{V}'_1\) and \(\textbf{K}_2 = \textbf{V}_2\textbf{D}_2\textbf{V}'_2\), respectively. Using properties of Kronecker products (e.g., Searle 1982, p. 265), it can be shown that the eigenvectors and eigenvalues of \(\textbf{K}\) are Kronecker products of the eigenvectors and eigenvalues of \(\textbf{K}_1\) and \(\textbf{K}_2\). Specifically, we have that:
\[\begin{equation*} \textbf{K} = (\textbf{V}_1\otimes\textbf{V}_2)(\textbf{D}_1\otimes\textbf{D}_2)(\textbf{V}_1\otimes\textbf{V}_2)' \end{equation*}\]
where \(\textbf{V} = \textbf{V}_1\otimes\textbf{V}_2\) are the eigenvectors and \(\textbf{D} = \textbf{D}_1\otimes\textbf{D}_2\) are the eigenvalues.
Consider an \(n\times n\) matrix formed as a Hadamard product (‘\(\odot\)’) of two matrices involving \(\textbf{K}_1\) and \(\textbf{K}_2\),
\[\begin{equation*} (\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2) \end{equation*}\]
where \(\textbf{Z}_1\) and \(\textbf{Z}_2\) are (\(n\times n_1\) and \(n\times n_2\), respectively) incidence matrices connecting rows/columns in the Hadamard with the rows/columns of \(\textbf{K}_1\) and \(\textbf{K}_2\), respectively. This Hadamard \((\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2)\) is a sub-matrix of the Kronecker \(\textbf{K}_1\otimes\textbf{K}_2\).
Therefore, it can be shown that
\[\begin{equation*} (\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2) = \tilde{\textbf{V}}\tilde{\textbf{D}}\tilde{\textbf{V}}' \end{equation*}\]
where \(\tilde{\textbf{V}} = (\textbf{Z}_1\star\textbf{Z}_2)(\textbf{V}_1\otimes\textbf{V}_2) = [\tilde{\boldsymbol{v}}_1,...,\tilde{\boldsymbol{v}}_N]\) are the eigenvectors (each of length \(n\) and not necessary orthogonal/orthonormal) and \(\tilde{\textbf{D}} = \textbf{D}_1\otimes\textbf{D}_2 = diag(\tilde{d}_1,...,\tilde{d}_N)\) are the eigenvalues.
Here, the term \(\textbf{Z}_1\star\textbf{Z}_2\) is an \(n\times N\) matrix obtained as the “face-splitting product” (aka “transposed Khatri–Rao product”) of matrices \(\textbf{Z}1\) and \(\textbf{Z}_2\), which is defined as a row-by-row Kronecker product
\[\begin{equation} \textbf{Z}_1\star\textbf{Z}_2 = \begin{pmatrix} \boldsymbol{z}_{11}\otimes\boldsymbol{z}_{12} \\ \boldsymbol{z}_{21}\otimes\boldsymbol{z}_{22}\\ \vdots \\ \boldsymbol{z}_{n1}\otimes\boldsymbol{z}_{n2} \end{pmatrix} \end{equation}\]
with \(\boldsymbol{z}_{i1}\) and \(\boldsymbol{z}_{i2}\) being the \(i^{th}\) row of \(\textbf{Z}1\) and \(\textbf{Z}_2\), respectively.
It holds that the sum of the \(N\) eigenvalues of the matrix \(\textbf{K}\) equals to the sum of the diagonal values of \(\textbf{K}\), this is \(\sum_{k=1}^N{d_k} = trace(\textbf{K})\).
Therefore, for \(\textbf{K}\) being a variance structure matrix, we say that the term \(d_k/\sum_{k=1}^N{d_j}\) is the proportion of the total variability of \(\textbf{K}\) associated to the \(k^{th}\) eigenvector. A low-rank approximation (\(\tilde{\textbf{K}}\)) of \(\textbf{K}\) can be formed by retaining the top \(m\) eigenvectors (with \(m<N\)), those that jointly explain a proportion \(\alpha\) of the total variance (\(0<\alpha\leq1\), e.g., \(\alpha=0.95\)). This approximation is obtained by summing over the first \(m\) terms in the EVD of \(\textbf{K}\) above described, this is \(\tilde{\textbf{K}}_{\alpha}=\sum_{k=1}^m{d_k}\boldsymbol{v}_k\boldsymbol{v}'_k\). This is also equivalent to
\[\begin{equation*} \tilde{\textbf{K}}_{\alpha} = \tilde{\textbf{V}}_{\alpha}\tilde{\textbf{D}}_{\alpha}\tilde{\textbf{V}}_{\alpha} \end{equation*}\]
where \(\tilde{\textbf{V}}_{\alpha}=[\tilde{\boldsymbol{v}}_1,...,\tilde{\boldsymbol{v}}_m]\) and \(\tilde{\textbf{D}}_{\alpha}=diag(d_1,...,d_m)\) are matrices formed with the top \(m\) eigenvectors and eigenvalues in \(\textbf{V}\) and \(\textbf{D}\), respectively, explaining a proportion α of the variance.
The tensorEVD algorithm derives an approximate decomposition of a Hadamard product \((\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_2\textbf{Z}'_2) = \tilde{\textbf{V}}\tilde{\textbf{D}}\tilde{\textbf{V}}'\) using as inputs:
K1
and
K2
of dimensions \(n_1\)
and \(n_2\), respectively.ID1
and ID2
are
\(n\)-vectors (\(n\) here is the sample size) mapping from
observations to the rows and columns of K1
and
K2
, respectively. (The row- and column-numbers of
K1
and K2
are the unique entries of
ID1
and ID2
, respectively.) These IDs are used
to form the incidence matrices \(\textbf{Z}_1\) and \(\textbf{Z}_1\). For instance, the matrix
\(\textbf{Z}_1\textbf{K}_1\textbf{Z}'_1\)
can be obtained by indexing rows and columns of K1
by
ID1
, this is K1[ID1,ID1]
.alpha
is a
proportion (\(0<\alpha\leq1\), e.g.,
\(\alpha=0.95\)) to build only the
eigenvectors needed to achieve such proportion of variance. A value
alpha = 1
will build all the \(N\) eigenvectors.The following script shows how to perform EVD using the tensorEVD() function
EVD = tensorEVD(K1, K2, ID1, ID2, alpha = 0.95)
ncol(EVD$vectors) # Number of eigenvectors
sum(EVD$values)/EVD$totalVar # Variance explained
We measure the approximation error by calculating two metrics:
The Frobenius norm and CMD were calculated after transforming the covariance matrices \(\textbf{K}\) and \(\tilde{\textbf{K}}_{\alpha}\) into correlation matrices.
We benchmarked the tensorEVD() routine against the eigen() function of the ‘base’ R-package (R Core Team 2021) in terms of the computational time used to derive eigenvectors of the Hadamard \(\textbf{K}\), 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 \(n=59069\) records for 4 traits (grain yield, anthesis, silking, and anthesis-silking interval) from \(n_G = 4344\) 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.csv
Folder 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 scripts can be downloaded using the following instruction in the
command line
cd /mnt/scratch/quantgen/TENSOR_EVD/pipeline
curl https://codeload.github.com/MarcooLopez/tensorEVD/tar.gz/main | tar -xz --strip=2 tensorEVD-main/misc/code
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
Each R-script is submitted to the HPCC using the sbatch
command, for instance
cd /mnt/scratch/quantgen/TENSOR_EVD/pipeline/code
sbatch 1_simulation.R
The 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 \(n_G = 4344\) 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.RData
We 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 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 full header added to the R-script containing all the job requirements is the following
#!/usr/bin/env Rscript
#SBATCH --job-name=simulation
#SBATCH --output=../log/%x_%A_%a
#SBATCH --time=03:59:00
#SBATCH --cpus-per-task=1
#SBATCH --mem-per-cpu=84G
#SBATCH --constraint=intel18
#SBATCH --array=1-810
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.txt
The 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[,1:8])
## job alpha nG nE n replicate nComb nPosEigen
## 1 1 0.9 100 10 10000 1 1000 1000
## 2 2 0.9 500 10 10000 1 4314 4314
## 3 3 0.9 1000 10 10000 1 6290 6290
## 4 4 0.9 100 30 10000 1 2904 2904
## 5 5 0.9 500 30 10000 1 7295 7295
## 6 6 0.9 1000 30 10000 1 8509 8509
head(out[,9:13]) # results from the eigen function
## time_eigen Frobenius_eigen CMD_eigen nPC_eigen pPC_eigen
## 1 255.960 148.8676 0.003611943 404 0.4040000
## 2 170.438 193.1440 0.004002020 936 0.2169680
## 3 171.677 148.1347 0.002807314 1417 0.2252782
## 4 166.713 122.9745 0.003348740 780 0.2685950
## 5 178.486 118.3712 0.003481097 1895 0.2597670
## 6 185.491 128.3336 0.003024160 2160 0.2538489
head(out[,14:18]) # results from the tensorEVD function
## time_tensorEVD Frobenius_tensorEVD CMD_tensorEVD nPC_tensorEVD pPC_tensorEVD
## 1 0.165 146.7451 0.003268131 419 0.4190000
## 2 0.283 184.9391 0.003106224 1096 0.2540566
## 3 0.772 136.8840 0.001843963 1881 0.2990461
## 4 0.248 119.0399 0.002760687 853 0.2937328
## 5 0.911 100.7295 0.001879908 2749 0.3768334
## 6 2.056 102.6804 0.001429371 3779 0.4441180
The following R-code chunks can be used to the reproduce the plots 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")
R-code below creates a plot for the EVD computation times of the eigen and tensorEVD methods (Supplementary Figure 1) and a plot for the computation time ratio (eigen/tensorEVD) (Figure 1).
# Some data edits
out$alpha <- factor(100*out$alpha)
out$nGE <- paste0("n[G]*' = '*",out$nG,"L*', '*n[E]*' = '*",out$nE,"L")
out$nGE <- factor(out$nGE, levels = unique(out$nGE))
# Reshaping the data
measure <- c("time","Frobenius","CMD","nPC","pPC")
dat <- melt_data(out, id=c("nGE","n","alpha"),
measure=paste0(measure,"_"),
value.name=measure, variable.name="method")
## Loading required namespace: reshape2
color1 <- c('90%'="skyblue1",'95%'="royalblue1",'98%'="navy")
color2 <- c(eigen="#E69F00", tensorEVD="#009E73")
# Supplementary Figure 1: Computation times of the eigen and tensorEVD
figureS1 <- make_plot(dat, x='alpha', y='time',
group="method", by="n", facet="nGE",
xlab=bquote(alpha~"x100% of variance of K"),
by.label="Sample size", ylab="Time (seconds) eigen",
ylab2="Time (seconds) tensorEVD",
sec.axis=TRUE, group.color=color2,
scales="fixed", ylim=c(0,NA))
## Loading required namespace: RColorBrewer
## Loading required namespace: ggplot2
# Figure 1: Computation time ratio (eigen/tensorEVD)
out$alpha2 <- factor(paste0(out$alpha,"%"))
out$time_ratio <- out$time_eigen/out$time_tensorEVD
figure1 <- make_plot(out, type="line", x='n', y='time_ratio',
group="alpha2", group.label=NULL, facet="nGE",
xlab="Sample size",
ylab="Computation time ratio (eigen/tensorEVD)",
group.color=color1, nSD=0, errorbar.size=0,
hline=1, scales="free_y", ylim=c(0,NA))
print(figure1)
R-code below produces the approximation accuracy plots using the Frobenius norm (Figure 2) of the difference between the Hadamard matrix and the approximation, and using the CMD metric (Supplementary Figure 2), provided by the eigen and tensorEVD.
# Supplementary Figure 2: Approximation accuracy using CMD
dat$CMD <- 1000*dat$CMD
figureS2 <- make_plot(dat, x='alpha', y='CMD',
group="method", by="n", facet="nGE",
xlab=bquote(alpha~"x100% of variance of K"),
ylab=expression("Correlation Matrix Distance (x"~10^{-3}~")"),
by.label="Sample size", group.color=color2,
scales="free_y", ylim=c(0,NA))
# Figure 2: Approximation accuracy using Frobenious norm
figure2 <- make_plot(dat, x='alpha', y='Frobenius',
group="method", by="n", facet="nGE",
xlab=bquote(alpha~"x100% of variance of K"),
ylab=expression("Frobenius norm ("~abs(abs(K-hat(K)))[F]~")"),
by.label="Sample size", group.color=color2,
scales="free_y", ylim=c(0,NA))
print(figure2)
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(dat, x='alpha', y='pPC',
group="method", by="n", facet="nGE",
xlab=bquote(alpha~"x100% of variance of K"),
ylab="Number of eigenvectors/rank",
by.label="Sample size", group.color=color2,
hline=1, scales="fixed")
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
\[\begin{equation*} y_{ijk} = \mu + G_i + E_j + GE_{ij} + \varepsilon_{ijk}. \end{equation*}\]
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
\[\begin{equation*} \textbf{K} = (\textbf{Z}_1\textbf{K}_G\textbf{Z}'_1)\odot(\textbf{Z}_2\textbf{K}_E\textbf{Z}'_2) \end{equation*}\]
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
\[\begin{equation*} \boldsymbol{y} = \boldsymbol{\mu}+\textbf{X}_G\boldsymbol{\beta}_1+\textbf{X}_E\boldsymbol{\beta}_2 + \textbf{X}_{GE}\boldsymbol{\beta}_3 + \boldsymbol{\varepsilon} \end{equation*}\]
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 following job
requirements
#!/usr/bin/env Rscript
#SBATCH --job-name=comps
#SBATCH --output=../log/%x_%A_%a
#SBATCH --time=06:59:00
#SBATCH --cpus-per-task=1
#SBATCH --mem-per-cpu=160G
#SBATCH --constraint=intel18
#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.txt
The \(G\times E\) 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
). The full job requirements used are the
following
#!/usr/bin/env Rscript
#SBATCH --job-name=ANOVA
#SBATCH --output=../log/%x_%A_%a
#SBATCH --time=20:59:00
#SBATCH --cpus-per-task=3
#SBATCH --mem-per-cpu=96G
#SBATCH --constraint=intel16
#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.dat
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
.
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 0.008838661
R-code below can be used to create a plot showing the average proportion of phenotypic variance of grain yield explained by each model (Figure 4). The same code can be used for traits anthesis, silking, and anthesis-silking interval (Supplementary Figures 4, 5, and 6, 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)
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 5975
We 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 following
job requirements
#!/usr/bin/env Rscript
#SBATCH --job-name=10F_CV
#SBATCH --output=../log/%x_%A_%a
#SBATCH --time=20:59:00
#SBATCH --cpus-per-task=3
#SBATCH --mem-per-cpu=96G
#SBATCH --constraint=intel16
#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.RData
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 0.04627769
R-code below can be used to create a plot showing the within environment prediction correlation using the tensorEVD and eigen methods for each combination of trait and \(\alpha\)-value (Figure 5).
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")){
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)
Golub G. H., and C. F. Van Loan, 1996 Matrix computations. Johns Hopkings University, Baltimore, MD.
Henderson C. R., 1985 Best linear unbiased prediction of nonadditive genetic merits in noninbred populations. J. Anim. Sci. 60: 111–117.
Herdin M., N. Czink, H. Özcelik, and E. Bonek, 2005 Correlation Matrix Distance, a Meaningful Measure for Evaluation of Non-Stationary MIMO Channels, pp. 136–140 in IEEE 61st Vehicular Technology Conference, Stockholm, Sweden.
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.
Searle S. R., 1982 Matrix Algebra Useful for Statistics. John Wiley & Sons, Inc, New Jersey.