SNP effects from Single Step GBLUP with APY

 APY is a technique that allows representing the inverse of a genomic relationship matrix in a sparse format by choosing a "core" of animals (here and here). The authoritative guide to APY is Bermann et al. 2022: this paper . 

One of the key aspects of genomic models is the ability of estimating SNP effects and then apply them to newly genotyped animals, what is know as Indirect Predictions. Matias and I found out that it's easier than we thought as shown here.

Among many other things Bermann et al. show that one can write indirect predictions of "non-core" animals as (I use the original equation numbering)

eq. (10) : $\mathbf{{u}_n}= \mathbf{Z}_n \mathbf{Z}'_c (\mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c  \mathbf{{a}} + \boldsymbol{\xi}$

where $\mathbf{{a}}$ are SNP effects and $\boldsymbol{\xi}$ is an error term that does not depend on $\mathbf{{u}_c}$. 

we obtain SNP effects from eq. 21 in Bermann et al:

$\mathbf{\hat{a}}=k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}$

we plug that into (10) and expand:

$\mathbf{\hat{u}_n}= \mathbf{Z}_n \mathbf{Z}'_c (\mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c \mathbf{\hat{a}} =k \mathbf{Z}_n \mathbf{Z}'_c (k \mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c \mathbf{\hat{a}}$

we substitute for $\mathbf{\hat{a}}$:

$\mathbf{\hat{u}_n} =k \mathbf{Z}_n \mathbf{Z}'_c (k \mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}=k \mathbf{Z}_n \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c} =  \mathbf{Z}_n \mathbf{\hat{a}}$

which is the original eq. 21. This is because $\hat{a}$  is part of the  column space of  $\mathbf{Z}'_c$, then $\mathcal{P}\mathbf{\hat{a}} = \mathbf{\hat{a}}$.

Finally, obtaining Indirect Predictions from APY is deadly simple and intuitive:

• $\mathbf{\hat{a}}=k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}$
• $\mathbf{\hat{u}_n} =  \mathbf{Z}_n \mathbf{\hat{a}}$

compiling macs in MacBook with Ventura

 I wanted to compile the coalescent simulator macs in the Mac running Ventura with M2 chips. This was of course the beginning of a great adventure. I end up doing the following:

- install library boost using homebrew

- dig out the path for the boost library e.g. as here, which turned out to be /opt/homebrew/Cellar/boost/1.81.0_1/include

- finally, modify the makefile as follows

# compile options

CFLAGS = -Wall -g

#CFLAGS = -Wall -O3

# Add location of any library locations below with -L

LINKFLAGS =

#LINKFLAGS = -static

# compiler

CC = g++

# libraries. For a local Boost installation

# Example:

#LIB = -I /Users/garychen/software/boost_1_36_0

# Default:

#LIB = -I .

LIB = -I /opt/homebrew/Cellar/boost/1.81.0_1/include

# simulator name

SIM=macs

OBJS = simulator.o algorithm.o datastructures.o

$(SIM) : $(OBJS)

$(CC) -o $(SIM) $(OBJS) $(LINKFLAGS)

simulator.o: simulator.cpp simulator.h

$(CC) $(CFLAGS) $(LIB) -c $<

algorithm.o: algorithm.cpp simulator.h

$(CC) $(CFLAGS) $(LIB) -c $<

datastructures.o: datastructures.cpp simulator.h

$(CC) $(CFLAGS) $(LIB) -c $<

draw many elements julia

 # a vector of frequencies

p=rand(Beta(2,2),10)

# a vector of "distributions"

a=Binomial.(2,p)

# a vector of vectors

 b=rand.(a,1)

# collapsed (row vector)

reduce(hcat,b)

# draw 5 animals at once

b=rand.(a,5)

# collapse

bb=reduce(hcat,b)

#compute frequencies

pest = mean(bb,dims=1)/2