Inferring Population Structure and Admixture Proportions ......Many NGS studies are based on medium (

GENETICS | INVESTIGATION

Inferring Population Structure and AdmixtureProportions in Low Depth NGS Data

Jonas Meisner∗,1 and Anders Albrechtsen∗∗The Bioinformatics Centre, Department of Biology, University of Copenhagen, DK-2200 Copenhagen N, Denmark

ABSTRACT We here present two methods for inferring population structure and admixture proportions in low depth next-generation sequencing data. Inference of population structure is essential in both population genetics and association studiesand is often performed using principal component analysis or clustering-based approaches. Next-generation sequencingmethods provide large amounts of genetic data but are associated with statistical uncertainty for especially low depth sequencingdata. Models can account for this uncertainty by working directly on genotype likelihoods of the unobserved genotypes. Wepropose a method for inferring population structure through principal component analysis in an iterative heuristic approach ofestimating individual allele frequencies, where we demonstrate improved accuracy in samples with low and variable sequencingdepth for both simulated and real datasets. We also use the estimated individual allele frequencies in a fast non-negative matrixfactorization method to estimate admixture proportions. Both methods have been implemented in the PCAngsd frameworkavailable at http://www.popgen.dk/software/.

KEYWORDS Population structure; PCA; Admixture; Ancestry; Next-generation sequencing; Genotype likelihoods; Low depth

P OPULATION genetic studies often consist of individualsof diverse ancestries, and inference of population structure

therefore plays an important role in population genetics andassociation studies. Population stratification can act as a con-founding factor in association studies as it can lead to spuriousassociations (Marchini et al. 2004). Principal component analy-sis (PCA) has been used in genetics for a long time, such as inMenozzi et al. (1978) where synthetic maps were produced in anexploratory analysis of genetic variation. PCA is now a commontool in population genetic studies, where its dimension reduc-tion properties can be used to visualize population structureby summarizing the genetic variation through principal com-ponents (Novembre and Stephens 2008), correct for populationstratification in association studies, investigate demographic his-tory (Patterson et al. 2006; Fumagalli et al. 2013; Price et al. 2006)as well as perform genome selection scans (Galinsky et al. 2016;Hao et al. 2015; Luu et al. 2017). PCA is an appealing approachto infer population structure as the aim is not to classify theindividuals into discrete populations, however instead describe

doi: 10.1534/genetics.XXX.XXXXXXManuscript compiled: Thursday 9th August, 20181Corresponding author: The Bioinformatics Centre, Department of Biology, Universityof Copenhagen, Ole Maaloes Vej 5, DK-2200 Copenhagen N, Denmark.E-mail: [email protected]

continuous axes of genetic variation such that heterogeneouspopulations and admixed individuals can be better represented(Patterson et al. 2006). Another successful approach in modelingcomplex population structure is to estimate admixture propor-tions based on clustering-based methods (Pritchard et al. 2000;Tang et al. 2005; Alexander et al. 2009; Skotte et al. 2013), such asthe popular software ADMIXTURE, which have also been usedfor correction of population stratification in association studies(Price et al. 2010).

Next-generation sequencing (NGS) methods (Metzker 2010)produce a large amount of DNA sequencing data at low costand are commonly used in population genetic studies (Nielsenet al. 2012). But NGS methods are associated with high errorrates usually caused by several factors such as sampling, align-ment and sequencing errors. Many NGS studies are based onmedium (<15X) and low (<5X) depth data due to the demandfor large sample sizes as seen in large-scale sequencing studies,e.g. 1000 Genomes Project Consortium (Consortium et al. 2010,2012). However, the use of medium and especially low depthsequencing data introduces challenges rooted in the statisticaluncertainty induced when calling genotypes and variants inthese scenarios (Nielsen et al. 2012). The statistical uncertaintyincreases for low depth samples due to the increased difficultyof distinguishing a variable site from a sequencing error with theinformation provided. Problems can arise due to chromosomes

Genetics 1

Genetics: Early Online, published on August 21, 2018 as 10.1534/genetics.118.301336

Copyright 2018.

http://www.popgen.dk/software/

being sampled with replacement in the sequencing process, andboth alleles may not have been sampled for a heterozygous indi-vidual in low depth scenarios. Homozygous genotypes may alsobe wrongly inferred as heterozygous due to sequencing errors.Thus, genotype calling will associate individuals with a statis-tical uncertainty which should be taken into account (Nielsenet al. 2011, 2012).

To overcome these problems related to NGS data and geno-type calling, probabilistic methods have been developed to takeuse of genotype likelihoods in combination with external in-formation for various population genetic parameters (Kim et al.2011; Nielsen et al. 2012; Korneliussen et al. 2014; Skotte et al. 2013;Fumagalli et al. 2013; Vieira et al. 2013; Kousathanas et al. 2017),such that posterior genotype probabilities can be used to modelthe related uncertainty. Genotype likelihoods can be estimatedto incorporate errors of the sequencing process such as the basequality scores as well as the allele sampling (McKenna et al. 2010).These posterior genotype probabilities have also been used tocall genotypes with a higher accuracy than previous methodsfor low depth NGS data (Nielsen et al. 2011, 2012).

We present two new methods for low depth NGS data usinggenotype likelihoods to model complex population structurethat connect the results of PCA with the admixture proportionsof clustering-based approaches. One method performs a variantof PCA using an iterative heuristic approach of estimating indi-vidual allele frequencies to compute a covariance matrix, whilethe other method uses the estimated individual allele frequen-cies in an accelerated non-negative matrix factorization (NMF)algorithm to estimate admixture proportions. The performancesof the two methods are assessed on both simulated and realdatasets in regards to existing methods for both low depth NGSand genotype data. The methods have been implemented in aframework called PCAngsd (Principal Component Analysis ofNext-Generation Sequencing Data).

Materials and Methods

We will analyze NGS data of n diploid individuals across mvariable sites. These sites will either be known or called single-nucleotide polymorphisms (SNPs), which are assumed to bediallelic such that the major and minor allele of each SNP havebeen inferred. This can either be done from sequencing reads(Kim et al. 2011) or from genotype likelihoods (Korneliussen et al.2014) and only three different genotypes will be possible. Thus,we assume that a genotype G can be seen as a Binomial randomvariable with realizations 0, 1 and 2 that represent the numberof copies of the minor allele in a site for a given individual in theabsence of population structure. The expectation and variance ofG can therefore be defined as E[G] = 2p and Var[G] = 2p(1− p)with p representing the allele frequency of a population, whichwe also refer to as population allele frequency.

However, genotypes are not observed in NGS data and wewill instead work on genotype likelihoods that also includeinformation of the sequencing process. The genotype likelihoodsare the probability of the observed sequencing data X given thethree different possible genotypes, P(X |G = g), for g = 0, 1, 2.One method to compute genotype likelihoods from sequencingreads is described in the supplementary material based on thesimple GATK model (McKenna et al. 2010).

External information can be incorporated to define posteriorgenotype probabilities using Bayes’ theorem in combinationwith genotype likelihoods (Nielsen et al. 2011). The populationallele frequency is often used as information in the estimation of

prior genotype probability P(Gis | ps), for an individual i in sites (Kim et al. 2011; Nielsen et al. 2012; Fumagalli et al. 2013; Vieiraet al. 2013). Assuming the population is in Hardy-WeinbergEquilibrium (HWE) for a site s, the prior genotype probabilityis then given as P(Gis = 0 | ps) = (1− ps)2, P(Gis = 1 | ps) =2ps(1 − ps) and P(Gis = 2 | ps) = p2

s for the three differentpossible genotypes. As defined in Kim et al. (2011), using theestimated population allele frequency ps, the posterior genotypeprobability is computed as follows for individual i in site s:

P(Gis = g |Xis, ps) =P(Xis |Gis = g)P(Gis = g | ps)

∑2g′=0 P(Xis |Gis = g′)P(Gis = g′ | ps)

.

(1)

PCAThe standard way of performing PCA in population geneticsand using it to infer population structure is based on the methoddefined in Patterson et al. (2006). For a genotype matrix G ofn individuals and m variable sites, the n × n covariance ma-trix C, also known as the genetic relationship matrix (GRM), iscomputed as follows for two individuals i and j:

cij =1m

m

∑s=1

(gis − 2ps)(gjs − 2ps)

2ps(1− ps). (2)

Here gis is the observed genotype for individual i in site s todistinguish it from G defined above for unobserved genotypes,and p is the estimated population allele frequency. The principalcomponents are then inferred by performing an eigendecom-position of the covariance matrix, such that C = VΣVT withV being the matrix of eigenvectors and Σ the diagonal matrixof the corresponding eigenvalues. Principal components andeigenvectors will be used interchangeably throughout this study.The top principal components capture most of the populationstructure as they represent the projection of the individuals onaxes of genetic variation in the dataset (Patterson et al. 2006;Engelhardt and Stephens 2010).

This method has been extended to NGS data in Fumagalliet al. (2013), as well as in Skotte et al. (2012), using the proba-bilistic framework described in equation 1, by summing overthe genotypes of each individual weighted by the joint poste-rior genotype probabilities under the assumption of HWE inthe whole sample. The method has been implemented in thengsTools framework (Fumagalli et al. 2014). The covariancematrix is estimated as follows for NGS data using only knownvariable sites for two individuals i and j:

cij =1m

m

∑s=1

∑2gi=0 ∑2

gj=0(gi − 2ps)(gj − 2ps)P(Gis = gi , Gjs = gj |Xis , Xjs , ps)

2ps(1− ps).

(3)

ngsTools splits up the joint posterior probability,P(Gis, Gjs |Xis, Xjs, ps), into P(Gis |Xis, ps)P(Gjs |Xjs, ps)for i 6= j by assuming conditional independence betweenindividuals given the estimated population allele frequencies.The non-diagonal entries in the covariance matrix are nowdirectly estimated from the posterior expectations of thegenotype instead of the observed genotypes as describedin equation 2. The original method weighs each site by itsprobability of being a variable site such that SNP calling isnot needed prior to the covariance matrix estimation. Thisis not taken into account in this study as we are using calledvariable sites to infer population structure. The population

2 Meisner and Albrechtsen

allele frequencies are estimated from the genotype likelihoodsusing an expectation maximization (EM) algorithm (Kim et al.2011) as described in the supplementary material.

The problem with this approach is that the assumption ofconditional independence between individuals given the popu-lation allele frequency is only valid when there is no populationstructure. Here we propose a novel approach of estimating thecovariance matrix using iteratively estimated individual allelefrequencies to update the prior information of the posteriorgenotype probability. Thereby we condition on the individualallele frequencies as in the clustering-based approaches such asPritchard et al. (2000); Tang et al. (2005); Alexander et al. (2009);Skotte et al. (2013).

Individual allele frequencies

A model for estimating individual allele frequencies based onpopulation structure was introduced in STRUCTURE (Pritchardet al. 2000) as later described in equation 13. Hao et al. proposeda different model for estimating individual allele frequencies Π

by using the information in the principal components insteadof having an assumption of K ancestral populations (Hao et al.2015). The model is defined as the matrix product,

Π = SA, (4)

where S represents the population structure such that A rep-resents the mapping of the population structure S to the allelefrequencies. Hao et al. estimated the individual allele frequen-cies through a singular value decomposition (SVD) method,where genotypes are reconstructed using only the top D princi-pal components such that they will be modeled by populationstructure. A similar approach has been proposed by Conomoset al. (Conomos et al. 2016) where the inferred principal com-ponents are used to estimate individual allele frequencies in asimple linear regression model. However, due to working onNGS data and not knowing the genotypes, we are extending themethod of Hao et al. to NGS data by using posterior expecta-tions of the genotypes, referred to as genotype dosages, insteadof genotypes. Thus we will be using,

E[Gis |Xis, ps] =2

∑g=0

g P(Gis = g |Xis, ps), (5)

for individual i in site s.The individual allele frequencies are then estimated by per-

forming a SVD on the centered genotype dosages and recon-structing them using only the top D principal components. 2pis then added to the reconstruction and scaled by 1

2 based ona Binomial distribution assumption of Gis, for i = 1, . . . , n ands = 1, . . . , m, to produce the individual allele frequencies. SinceSVD is a method that takes real-valued input, we will have totruncate the estimated individual allele frequencies in order toconstrain them in the range [0, 1]. However, Hao et al. showedthat the resulting estimates were still very accurate for commonvariants considering this limitation.

For ease of notation, let E be the n×m matrix of genotypedosages, eis = E[Gis |Xis, ps], for i = 1, . . . , n and s = 1, . . . , m.The following steps for estimating the individual allele frequen-cies are adopted from the SVD method (Hao et al. 2015) to workon NGS data:

Algorithm 1: SVD method for estimating individual allele frequencies.

1. The centered genotype dosages are constructed as E(C)i = Ei − 2p

for i = 1, . . . , n.

2. Perform SVD on the centered genotype dosages, E(C) = W∆UT ,where W will represent population structure similarly to V.

3. Define E(C)D to be the prediction of the centered genotype dosages

using only the top D principal components, E(C)D = W1:D∆1:DUT

1:D .

4. Estimate Π by adding 2p to E(C)D row-wise and scaling by 1

2 , basedon πis ≈ 1

2 E[Gis].

For matrix notations define S = [1, W1, . . . , WD] and AT =12 [2p, U1δ1, . . . , UDδD], all representing column vectors, suchthat equation 4 can be approximated as Π = SA. Finally, Π

is truncated to constrain allele frequency estimates in a rangebased on a small value γ (1.0× 10−4), such that πis ∈ [γ, 1− γ]for i = 1, . . . , n and s = 1, . . . , m.

We now incorporate the individual allele frequencies into theestimation of posterior genotype probabilities. The estimatedindividual allele frequencies are used as updated prior informa-tion instead of the population allele frequencies, and will be ableto model missing data with the inferred population structure ofthe individuals. Thus, the posterior genotype probabilities areestimated as follows for individual i in site s:

P(Gis = g |Xis, πis) =P(Xis |Gis = g)P(Gis = g | πis)

∑2g′=0 P(Xis |Gis = g′)P(Gis = g′ | πis)

.

(6)Each individual are now seen as a single population with

allele frequency πis, where as the prior genotype probability areestimated assuming HWE, such that P(G = 0 | πis) = (1− πis)

2,P(G = 1 | πis) = 2(1− πis)πis and P(G = 2 | πis) = π2

is. Anupdated definition of the posterior expectations of the genotypesare then given as:

E[G |Xis, πis] =2

∑g=0

g P(G = g |Xis, πis). (7)

This procedure of updating the prior information can beiterated to estimate new individual allele frequencies on thebasis of updated population structure. Therefore, we proposethe following algorithm for an iterative procedure of estimatingthe individual allele frequencies.

Algorithm 2: Iterative estimation of individual allele frequencies.1. Estimate population allele frequencies p from genotype likelihoods

(See supplementary materials).

2. Estimate posterior genotype probabilities and genotype dosages Ebased on genotype likelihoods and p.

3. Estimate Π using the SVD based method on E as described inAlgorithm 1.

4. Estimate posterior genotype probabilities and genotype dosages Eusing updated prior information, Π.

5. Repeat step 3 and 4 until individual allele frequencies have con-verged.

Convergence of our iterative method is defined as when theroot-mean-square deviation (RMSD) of the inferred population

GENETICS Journal Template on Overleaf 3

structure in the SVD W is smaller than a value µ (1.0× 10−5)between two successive iterations. The RMSD of iteration t + 1for D principal components is given as,

RMSD =

√√√√ 1nD

n

∑i=1

D

∑d=1

(w(t+1)

id − w(t)id

)2. (8)

Covariance matrixWe now use the final set of individual allele frequencies to es-timate an updated covariance matrix in a similar model as inequation 3, but incorporating the individual allele frequenciesinto the joint posterior probability. The entries of the covariancematrix C are now defined as follow for individuals i and j:

cij =1m

m

∑s=1

∑2gi=0 ∑2

gj=0(gi − 2ps)(gj − 2ps)P(Gi = gi , Gj = gj |Xis , Xjs , πis , πjs)

2ps(1− ps).

(9)

For i 6= j, the joint posterior probability can be computedas P(Gi |Xis, πis)P(Gj |Xjs, πjs), since the individuals are condi-tionally independent given the individual allele frequencies incontrary to the assumption made in the model of Fumagalli et al.(2013) using population allele frequencies. The above equationcan be expressed in terms of the genotype dosages for ease ofnotation and computation for i 6= j:

cij =1m

m

∑s=1

(E[Gi |Xis, πis]− 2ps)(E[Gj |Xjs, πjs]− 2ps)

2ps(1− ps).

(10)However for i = j (diagonal of the covariance matrix), the

joint posterior probability is simplified to P(Gi |Xis, πis) suchthat the estimation of the diagonal covariance entries is given as:

cii =1m

m

∑s=1

∑2gi=0(gi − 2ps)2P(Gi = gi |Xis, πis)

2ps(1− ps). (11)

An eigendecomposition of the updated estimated covariancematrix is then performed to obtain the principal componentsas described earlier, C = VΣVT . Note that V and W fromalgorithm 1 are not the same even though they both representpopulation structure through axes of genetic variation in thedataset. This is due to a different scaling and the joint posteriorprobability of equation 11 is not taken into account in W fori = j.

Number of principal componentsIt can be hard to determine the optimal number of principalcomponents that represent population structure. In our method,we are using Velicier’s minimum average partial (MAP) testas proposed by Shriner (Shriner 2011) to automatically detectthe number of top principal components D used for estimatingthe individual allele frequencies. Shriner showed that the testbased on a Tracy-Widom distribution (Patterson et al. 2006) sys-tematically overestimates the number of significant principalcomponents and even performs worse for datasets includingadmixed individuals. However, in order to be able to performthe MAP test and detect the optimal D, an initial covariancematrix is estimated based on the model in equation 3.

The MAP test is performed on the estimated initial covari-ance matrix C for NGS data as an approximation of the Pearsoncorrelation matrix used by Shriner. Using the notation of Shriner,

C∗d is defined as the matrix of partial correlations after havingpartialed out the first d principal components. Velicer (1976) pro-

posed the summary statistic ld = ∑ni=1,i 6=j ∑n

j=1(C∗d,ij)

2

n(n−1) , whereC∗d,ij represents the entry in C∗d for individuals i and j. Thus, thetest statistic ld represents the average squared correlation afterpartialing out the top d principal components. The number oftop principal components that represent population structure isthen chosen as argmind ld, for d = 0, . . . , m− 1. We have usedthe same implementation of the MAP test as Shriner.

The MAP test and the preceding estimation of the initialcovariance matrix can be avoided by having prior knowledgeof an optimal D for the dataset being analyzed and manuallyselecting D.

Genotype calling

As previously shown in Nielsen et al. (2012); Fumagalli et al.(2013), genotypes can be called from posterior genotype proba-bilities to achieve higher accuracy in low depth NGS scenarios.We can adapt this concept to our posterior genotype probabili-ties based on individual allele frequencies, such that genotypescan be called at a higher accuracy in structured populations fromlow depth NGS data. The genotype for individual i in site s iscalled as follows:

gis = argmaxg∈{0,1,2}

P(Gis = g |Xis, πis). (12)

Admixture proportions

Based on the likelihood model defined in STRUCTURE(Pritchard et al. 2000), individual allele frequencies Π can be esti-mated using admixture proportions Q and population-specificallele frequencies F (Alexander et al. 2009), such that:

πis =K

∑k=1

qik fsk, (13)

for an individual i in a variable site s. This is based on anassumption of K ancestral populations where ∑K

k=1 qik = 1 and0 ≤ q, f ≤ 1 ∀ q, f ∈ (Q, F). Here Q and F must be inferredin order to estimate the individual allele frequencies, whereas K is assumed to be known. One probabilistic approach forinferring population structure through admixture proportionsfor low depth NGS data has been implemented in the NGSadmixsoftware (Skotte et al. 2013). Here both parameters, Q and F, arejointly estimated in an EM algorithm using genotype likelihoods.

In our case, we have already estimated the individual al-lele frequencies based on our iterative procedure using PCAdescribed above. K can be chosen as the number of princi-pal components D + 1, since it would explain the number ofdistinct ancestral population from which the individual allelefrequencies have been estimated from. There is however not al-ways a direct interpretation between principal components andadmixture proportions (Alexander et al. 2009; Engelhardt andStephens 2010). Therefore, we propose an approach based onnon-negative matrix factorization (NMF) to infer Q and F usingonly our estimated individual allele frequencies as informationfor low depth NGS data. NMF has previously been applieddirectly on genotype data to infer population structure and ad-mixture proportions by Frichot et al. (Frichot et al. 2014), wheretheir method showed comparable accuracy and faster run-timein comparison to ADMIXTURE.


NMF is a dimension reduction and factor analysis methodfor finding a low-rank approximation of a matrix, which is sim-ilar to PCA, but NMF is constrained to find non-negative lowdimensional matrices. For an non-negative matrix Π ∈ Rn×m

+ ,the goal of NMF is to find an approximation of Π based on twonon-negative factor matrices Q ∈ Rn×K

+ and F ∈ Rm×K+ , such

that:

Π ≈ QFT . (14)

Q will consist of columns of non-negative basis vectors suchthat linear combinations of these approximates Π through F.Thus based on the non-negative nature of our parameters, wecan apply the ideas of NMF to infer admixture proportions Qand population-specific allele frequencies F from our individualallele frequencies. We use a combination of recent research inNMF to minimize the following least squares problem with asparseness constraint on Q:

minQ,F

∥∥∥Π−QFT∥∥∥2

F+ α

m

∑i=1

K

∑k=1|qik|, (15)

for Q ≥ 0, F ≥ 0 and α ≥ 0. Here ‖ . ‖F is the Frobenius normof a matrix and α is the regularization parameter controlling thesparseness enforced as also introduced in Frichot et al. (2014).

Lee and Seung (Lee and Seung 1999, 2001) proposed an mul-tiplicative update (MU) algorithm to solve the standard NMFproblem without the sparseness constraint included above. Theirupdate rules can be seen as conservative steps in a gradient de-scent optimization problem for updating F and Q, which ensurethat the non-negative constraint holds for each update. Hoyer(Hoyer 2002) extended the MU to incorporate the sparsenessconstraint described in equation 15 for Q. For α > 0, the regu-larization parameter is used to reduce noise, especially inducedby the uncertainty of low depth NGS data, in the estimated ad-mixture proportions by enforcing sparseness in the solution. Aniteration of using the MU rules are then described as follows:

F(t+1) = F(t) ⊗ ΠTQ(t)

F(t)Q(t) TQ(t), (16)

Q(t+1) = Q(t) ⊗ ΠF(t+1)

Q(t)F(t+1) T F(t+1) + α. (17)

where⊗ represents element-wise multiplication and the divisionoperator is element-wise as well.

However, MU has been shown to have a slow convergencerate, especially for dense matrices, and our approach is thereforeto accelerate MU by combining two different techniques. Wepropose an algorithm of combining the acceleration scheme de-scribed by Gillis and Glineur (Gillis and Glineur 2012) with theasymmetric stochastic gradient descent algorithm (ASG-MU) ofSerizel et al. (Serizel et al. 2016) for updating F and Q in a fastapproach. The acceleration scheme of Gillis and Glineur (2011)updates each matrix F and Q a fixed number of times at a lowercomputational cost without losing the convergence propertiesof MU. We simply incorporate this acceleration scheme insideASG-MU that works by randomly assigning the columns of Π

into a set of B mini-batches, which are then updated sequentiallyin a permuted order to improve the convergence rate and perfor-mance of MU (Serizel et al. 2016). After each update, we truncatethe entries of both F and Q to be in range [0, 1] and normalizethe rows of Q to sum to one. The concept of combining an accel-eration scheme with a stochastic gradient descent approach forMU has also been explored in Kasai (2017).

The algorithm is iterated until the admixture proportionshas converged. Convergence is defined as when the RMSD ofestimated admixture proportions of two successive iterationsare smaller than a value φ (1.0× 10−4). The RMSD of iterationt + 1 is given as,

RMSD =

√√√√ 1nK

n

∑i=1

K

∑k=1

(q(t+1)ik − q(t)ik )2. (18)

The α parameter enforcing sparseness in the estimated so-lution of Q is arbitrarily specified. However the use of thelikelihood measure in the NGSdamix (Skotte et al. 2013) modelcan be used to determine the α parameter fitting the dataset. Thelikelihood measure is defined as:

L(Q, F) =n

∏i=1

m

∏s=1

2

∑g=0

P(Xis |Gis = g)P(Gis = g | πis), (19)

where πis = ∑Kk=1 qik fsk. Based on the fast estimation of admix-

ture proportions using our NMF algorithm, an appropriate α caneasily be found by scanning a specified interval in an automatedfashion based on the likelihood measure. This can be performedwithout sacrificing significant run-time compared to NGSadmixdue to already having estimated the individual allele frequenciesfor a particular K.

ImplementationBoth presented methods have been implemented in a Pythonframework named PCAngsd (Principal Component Analysisof Next Generation Sequencing Data). The framework is freelyavailable at http://www.popgen.dk/software/.

The memory requirements of PCAngsd isO(mn) as the entirematrix of genotype likelihoods needs to be stored in memoryfor both methods. The most computational expensive step is theestimation of individual allele frequencies and covariance matrix(O(m2n)). However, a fast SVD method for only computing thetop D eigenvectors, implemented in the Scipy library (Jones et al.2014) using ARPACK (Lehoucq et al. 1998) as an eigensolver, hasbeen used to speed up the iterative estimations of the individualallele frequencies. PCAngsd is multithreaded as well to takeadvantage of several cores and the backbone of the frameworkis based on Numpy data structures (Walt et al. 2011) using theNumba library (Lam et al. 2015) to speed up bottlenecks withjust-in-time (JIT) compilation.

Simple simulation of genotypes and sequencing dataTo test the capabilities of our two presented methods, we simu-lated low depth NGS data and generated genotype likelihoods.Allele frequencies of the reference panel of the Human GenomeDiversity Project (HGDP) (Cann et al. 2002) have been used togenerate a total of 380 individuals from three distinct popula-tions (French, Han Chinese, Yoruba) including admixed indi-viduals in approximately 0.4 million SNPs across all autosomes.As the allele frequencies are known for each population, thegenotypes of each individual can be sampled from a Binomialdistribution for each diallelic SNP, using the population-specificallele frequency or an admixed allele frequency as parameter.No LD has been simulated. The genotypes are therefore knownand are used in the evaluation of our methods in our low depthscenarios. The number of reads in each SNP are sampled froma Poisson distribution with a mean parameter resembling the


http://www.popgen.dk/software/

Figure 1 PCA plots of the top 2 principal components in the simulated dataset consisting of 380 individuals and 0.4 million variablesites. The left-hand plot shows the PCA performed on the known genotypes using equation 2. The middle plot shows the PCAperformed by PCAngsd, and the right-hand plot displays the PCA performed by the ngsTools model (equation 3).

Figure 2 Admixture plots for K = 3 of the simulated dataset where each bar represents a single individual and the different colorsreflect each of the K components. The first plot is the admixture proportions estimated in ADMIXTURE using the known geno-types, which we use as the ground-truth in our simulation studies. The second plot shows admixture proportions estimated usingPCAngsd with parameter α = 0 and the bottom plot using NGSadmix.


average sequencing depth of the individual, and the genotype isused to sample the number of derived alleles from a Binomialdistribution using the sampled depth as parameter. The averagesequencing depth of each individual is sampled uniformly ran-dom from a range of [0.5, 5]. Sequencing errors are incorporatedby sampling each read with a probability ε = 0.01 of being anerror. The genotype likelihoods are then finally generated fromthe probability mass function of a Binomial distribution usingthe sampled parameters and ε. This approach of genotype likeli-hood simulation has previously been used in Kim et al. (2011);Vieira et al. (2013); Skotte et al. (2013).

A complex admixture scenario has been constructed to testthe capabilities of our methods. 100 individuals have been sam-pled directly from each of the population-specific allele frequen-cies (non-admixed), while 50 individuals have been sampledto have equal ancestry from each of the three distinct popula-tions (three-way admixture). At last, 30 individuals have beensampled from a gradient of ancestry between all pairs of theancestral populations (two-way admixture).

1000 Genomes low depth sequencing dataWe also analyze human low coverage NGS data of 193 individ-uals from the 1000 Genomes Project Consortium (Consortiumet al. 2010, 2012). The individuals are from four different popula-tions consisting of 41 from CEU (Utah residents with Northernand Western European ancestry), 40 from CHB (Han Chinesein Beijing), 48 from YRI (Yoruba in Ibadan) and 64 individualsfrom MXL (Mexican ancestry in Los Angeles) representing anadmixed scenario of European and Native American ancestry.The individuals from the low coverage datasets have a varyingsequencing depth from 1.5− 12.5X after site filtering. An advan-tage of using the low coverage data of the 1000 Genomes Projectdata is that reliable genotypes are available which can be usedfor validation purposes.

SNP calling and estimation of genotype likelihoods of the1000 Genomes dataset has been performed in ANGSD (Kor-neliussen et al. 2014) using simple read quality filters. A signif-icance threshold of 1.0× 10−6 has been used for SNP callingalongside a MAF threshold of 0.05 to remove rare variants. Atotal number of 8 million variable sites across all autosomeshave been used in the analyses. The full ANGSD commandused to generate the genotype likelihoods is provided in thesupplementary material.

Waterbuck low depth sequencing dataLastly, an animal dataset (non-model organism) has also beenincluded in our study. A reduced low depth NGS dataset of thewaterbuck (Kobus ellipsiprymnus) originating from Pedersen etal. (unpublished) has been analyzed. The dataset consists of73 samples that have been sampled at 5 different sites in Africawith a varying sequencing depth from 2.2 − 4.7X aligned to88935 scaffolds. The dataset has been reduced to only includesampling sites with more than 10 samples such that the inferredaxes of genetic variation will reflect true population structure. Asperformed for the 1000 Genomes dataset, genotype likelihoodshas been estimated in ANGSD with the same SNP and MAFfilters. A total number of 9.4 million SNPs across the autosomesof the waterbuck is analyzed in this study.

Data availabilityThe authors affirm that all data necessary for confirming theconclusions of the article are present within the article, figures,

and tables. The waterbuck dataset analyzed in our study ispublicly available in the European Nucleotide Archive (ENA)repository (PRJEB28089).

Results

For the simulated and 1000 Genomes datasets, results estimatedin PCAngsd on low depth NGS data are evaluated against theresults estimated from genotype data as well as naively calledgenotypes from genotype likelihoods. The model in equation 2is used to perform PCA, while ADMIXTURE is used to estimateadmixture proportions on the "true" genotype datasets. Theperformance of PCAngsd is also compared to existing genotypelikelihood methods with the ngsTools model (equation 3) forperforming PCA, and NGSadmix (equation 19) for estimatingadmixture proportions. In all the following cases of admixtureplots estimated by PCAngsd, we have used B = 5 and α has beenchosen as the one maximizing the likelihood measure describedabove (equation 19), also shown in Figure S5.

RMSD is used to evaluate the performances of both NGSmethods for estimating admixture proportions in terms of accu-racy:

RMSD =

√√√√ 1nK

n

∑i=1

K

∑k=1

(q(geno)

ik − q(NGS)ik

)2, (20)

where q(geno)ik and q(NGS)

ik represents the estimated admixtureproportion for individual i in ancestral population k from knowngenotypes and NGS data, respectively. The accuracy of the in-ferred PCA plots of both NGS methods are also compared to thePCA plots of the known genotypes for the simulated and 1000Genomes datasets using RMSD. However, a Procrustes analysis(Wang et al. 2010; Fumagalli et al. 2013) must be performed priorto the comparison as the direction of the principal componentscan differ based on the eigendecomposition of the covariancematrices.

All tests in this study have been performed server-side using32 threads (Intel® Xeon® CPU E5-2690) for both PCAngsd andNGSadmix.

SimulationThe results of performing PCA on the simulated dataset based onfrequencies from 3 human populations are displayed in Figure1, where we simulated unadmixed, two-way admixed and three-way admixed individuals. The MAP test reported 2 significantprincipal components which was also expected for individualssimulated from three distinct populations. The inferred principalcomponents clearly shows the importance of taking individualallele frequencies into account in the probabilistic framework.Here PCAngsd is able to infer the population structure of in-dividuals from distinct populations and admixed individualsnicely as also verified by a Procrustes analysis obtaining a RMSDof 0.00121, when compared to the PCA inferred from the truegenotypes. There is clear bias in the results of the ngsTools modelwhere the patterns are representing sequencing depth instead ofpopulation structure as made apparent in Figure S1. The indi-viduals are acting as a gradient towards the origin due to theirvarying sequencing depth. The biased performance of ngsToolsis also reflected in the corresponding Procrustes analysis with aRMSD of 0.0174.

To ensure that the individual allele frequencies estimated us-ing PCAngsd are representative estimates, we compare them to


Figure 3 PCA plots of the top 2 principal components for the 1000 Genomes dataset with 193 individuals and 8 million variablesites. The left-hand plot is based on the reliable genotypes of the overlapping variable sites in the low depth NGS data, the middleplot is performed by PCAngsd and the right-hand plot is performed by the ngsTools model.

Figure 4 Admixture plots for K = 4 of the 1000 Genomes dataset where each bar represents a single individual and the differentcolors reflect each of the K components. The first plot is the admixture proportions estimated in ADMIXTURE using the reliablegenotypes, the second plot shows admixture proportions estimated in PCAngsd with parameter α = 1500 and the last plot is theadmixture proportions estimated in NGSadmix.


Figure 5 PCA plots of the top 4 principal components for the waterbuck dataset with 73 individuals and 9.4 million variable sites.The first row displays the plots of the first and second principal components for PCAngsd and the ngsTools model, respectively,while the second row displays the plots of the third and fourth principal components.

Figure 6 Admixture plots for K = 5 of the waterbuck dataset where each bar represents a single individual and the different colorsreflect each of the K components. The first plot is the admixture proportions estimated in PCAngsd with parameter α = 5000 andthe second plot shows the admixture proportions estimated in NGSadmix.


the allele frequencies of the HGDP reference panel from whichthe genotypes of each individual has been sampled from. Sam-pling errors are therefore not taken into account in the compari-son. The estimates obtained from NGSadmix are compared aswell. The estimates of PCAngsd obtain a RMSD value of 0.0330and the estimates of NGSadmix a value of 0.0327 based on thelow depth NGS data. The results of PCAngsd are displayed inFigure S9.

The estimated admixture proportions of the simulateddataset are displayed in Figure 2. PCAngsd estimates the admix-ture proportions well with a RMSD of 0.00476 compared to theADMIXTURE estimates of the known genotypes, but is howeveroutperformed by NGSadmix with a RMSD of 0.00184. For the380 individuals and 0.4 million SNPs using K = 3, PCAngsdhad an average run-time of only 2.9 minutes while NGSadmixhad an average run-time of 7.9 minutes.

1000 GenomesWe have also applied the methods of PCAngsd to the CEU (Euro-pean ancestry), CHB (Chinese ancestry), YRI (Nigerian ancestry)and MXL (Mexican ancestry) populations of the low coverage1000 Genomes dataset. The MAP test indicated evidence of 3significant principal components meaning that the Native Amer-ican ancestry explains enough genetic variance in the datasetto represent an axis of its own. The results of the PCA are dis-played in Figure 3. As was also seen for the simulated dataset,PCAngsd is able to cluster all individuals almost perfectly, whilethe ngsTools model is only able to capture some of the same pop-ulation structure patterns with some of the populations lookingadmixed. Its results are still biased by the variable sequencingdepth as seen as well in Figure S2. The RMSD values of theProcrustes analyses verify the observations, where PCAngsdhas a RMSD of 0.00182 compared to ngsTools with a RMSD of0.0075.

The admixture plots are displayed in Figure 4. PCAngsd isnot able to outperform NGSadmix in terms of accuracy, howeverit is still able to estimate a very similar result. PCAngsd hassome issues with noise in its estimation but is however able toreduce it with the use of the sparseness parameter α = 1500. Thelikelihood measure in equation 19 has been used to easily find anoptimal α as seen in Figure S10. PCAngsd estimates the admix-ture proportions with a RMSD of 0.0108 compared to NGSadmixwith a RMSD of 0.007148. The average run-time for 193 indi-viduals and 8 million SNPs using K = 4 was 27.3 minutes forPCAngsd and 7.1 hours for NGSadmix, making PCAngsd morethan 15x faster than NGSadmix while both performing PCA andestimating admixture proportions.

WaterbuckLastly, we have analyzed the low depth whole genome sequenc-ing waterbuck dataset consisting of 73 individuals from 5 locali-ties. The MAP test reported 4 significant principal componentsfor explaining the genetic variation in the dataset which also fitswith having 5 distinct waterbuck sampling sites. The PCA plotsare visualized in Figure 5, where the top 4 principal componentshave been plotted for each method. Once again, PCAngsd is ableto cluster the populations much better than the ngsTools model,however the effect is not as apparent as for the other datasets.Interestingly, populations can switch positions between the twomethods as seen with Samole on the second principal componentand Samburu and Matetsi on the third principal component.

As some few clusters are not so well defined, they will affect

the admixture plots seen in Figure 6, where the increased level ofnoise is hard to remove without also affecting the true ancestrysignals. Still, PCAngsd is capturing the same ancestry signalsas NGSadmix with the use of the sparseness parameter. It isworth noting that an admixed individual of Ugalla and QENP iscaptured in both PCA and admixture estimation of PCAngsd asalso verified by the NGSadmix method. The run-times for thewaterbuck dataset consisting of 73 samples and 9.4 million SNPsusing K = 5 was an average of 14.5 minutes for PCAngsd whileNGSadmix had an average run-time of 3.2 hours, thus makingPCAngsd more than 13x faster.

Naively called genotypes

We have also inferred population structure from naively calledgenotypes of the simulated and 1000 Genomes datasets, andthe results are visualized in Figure S7 and S8. Genotypes arecalled by choosing the genotypes with the highest genotypelikelihoods. No filters have been applied in the genotype calling,since Skotte et al. (2013) showed that the naively called geno-types had higher accuracy of the inferred admixture proportionwhen no filters were used. The Procrustes analyses report RMSDvalues of 0.0123 and 0.00310 for performing PCA on the simu-lated and the 1000 Genomes dataset, respectively (compared toRMSD values of 0.00121 and 0.00182 using PCAngsd). Here thenaively called genotypes perform slightly better than ngsToolsin both cases but the results are still biased by sequencing depth.ADMIXTURE estimates admixture proportions from the calledgenotypes with RMSD values of 0.00995 and 0.00865 for thetwo datasets, respectively, thus performing slightly better thanPCAngsd for the 1000 Genomes dataset.

Discussion

We have presented two methods for inferring population struc-ture and admixture proportions in low depth NGS data andboth methods have been implemented in a framework namedPCAngsd. We have developed a method to iteratively estimateindividual allele frequencies based on PCA using genotype like-lihoods in a heuristic approach. We have connected principalcomponents to admixture proportions such that we are able toinfer and estimate both in a very fast approach making it feasibleto analyze large datasets.

Based on the results when inferring population structure us-ing PCA, it is clear that the increased uncertainty of low depthsequencing data biases the clustering of populations using thengsTools model which also takes genotype uncertainty into ac-count. Contrary to PCAngsd, population structure is not takeninto account when using the posterior genotype probabilitiesto estimate the covariance matrix. The ngsTools model usespopulation allele frequencies as prior information for all indi-viduals such that individuals are assumed to be sampled from ahomogeneous population. This assumption is of course violatedwhen individuals are sampled from structured populations withdiverse ancestries. Missing data is therefore modeled by popula-tion allele frequencies that resemble an average across the entiresample, which is similar to setting standardized genotypes to0 in the estimation of the covariance matrix for genotype data.As an effect of this, the low depth individuals are modeled bysequencing depth instead of population structure. These resultsmay lead to misinterpretations of population structure or admix-ture only due to low and variable sequencing depth. But thebias is not seen for individuals with equal sequencing depth as


Dataset n m K PCAngsd NGSadmix Depth

Simulated 380 0.4 million 3 2.9 min (2.1 min) 7.9 min 0.5− 5X

1000 Genomes 193 8 million 4 27.3 min (19.5 min) 424.9 min 1.5− 12.5X

Waterbuck 73 9.4 million 5 14.5 min (9.3 min) 192 min 2.2− 4.7X

Table 1 Average run-times of 10 initializations for both PCAngsd and NGSadmix. The run-times reported for PCAngsd includereading of data and estimation of covariance matrix and admixture proportions, while run-times listed in parentheses only in-clude estimation of admixture proportions, when parsing previously estimated individual allele frequencies. All tests have beenperformed server-side using 32 threads.

shown in Figure S4 for the ngsTools model. Here all individu-als have been simulated with an average sequencing depth of2.5X such that individuals will approximately inherit the sameamount of missing data. However, PCAngsd is able to overcomethe observed bias of low and variable sequencing depth by usingindividual allele frequencies as prior information, which leads tomore accurate results in all datasets of the study, as missing datais modeled accounting for inferred population structure. Theassumption of conditional independence between individuals inthe estimation of the covariance matrix (equation 10) also holdsfor structured populations by conditioning on individual allelefrequencies.

The number of significant eigenvectors used in the estimationof individual allele frequencies is determined by the MAP test.The MAP test is performed on the covariance matrix estimatedfrom the ngsTools model. Thus in cases of complex populationstructure and low and variable sequencing depth, it is possiblethat the MAP test will not find a suitable number of significanteigenvectors to represent the genetic variation of the dataset. Itcould therefore be more relevant to use prior information regard-ing the number of eigenvectors needed for the dataset instead.However for each of the cases analyzed in this study, the MAPtest inferred the expected number of significant eigenvectors todescribe the population structure.

PCAngsd is able to approximate the results of NGSadmix to ahigh degree when estimating admixture proportions using solelythe estimated individual allele frequencies. However, PCAngsdis not able to outperform NGSadmix in terms of accuracy, butit is however able to capture the exact same ancestry patternsas the clustering-based methods in a much faster approach, asshown by the run-times of each method. Another advantage ofPCAngsd is that the estimated individual allele frequencies areonly needed to be computed once for a specific K, thus multipledifferent random seeds can be tested in the same run for aneven greater speed advantage over NGSadmix, as the iterativeestimation of individual allele frequencies is the most computa-tional expensive step in PCAngsd. A proper α value, controllingthe sparseness enforced in the estimated admixture proportions,can also be found through an automated scan implemented inour framework based on the likelihood measure of NGSadmix.PCAngsd is therefore an appealing alternative for estimatingadmixture proportions for low depth NGS data as convergenceand run-time can be a problem for a large number of parametersin NGSadmix. PCAngsd was only seen to converge to a singlesolution for all our practical tests, where we used five batchesfor all analyses (B = 5).

Both methods of the PCAngsd framework rely on an repre-sentative set of individual allele frequencies which we modelusing the inferred principal components of the SVD on the geno-

type dosages. The number of individuals representing eachpopulation or subpopulation is essential for inferring principalcomponents that describe true population structure as each indi-vidual will contribute to the construction of these axes of geneticvariation. This particular effect can be seen in the PCA results ofthe waterbuck dataset where the populations are only describedby a low number of individuals such that some of the clustersare not so well defined as for the other datasets. The admixtureproportions estimated from the waterbuck dataset are thereforeaffected as well which can be seen by the additional noise in theadmixture plots.

The PCAngsd framework may be able to push the lowerboundaries of sequencing depth required to perform popula-tion genetic analyses on NGS data in large-scale genetic studies.This is also demonstrated by downsampling the 1000 Genomesdataset in Figure S5 and S6, which display the robustness ofPCAngsd in fairly low sequencing depth. However when down-sampling to only 1% of the reads, the PCA and admixture resultsbecome very noisy. PCAngsd demonstrates an effective ap-proach for dealing with merged datasets of various sequencingdepths as well, as missing data will be modeled by populationstructure. Further, the estimated individual allele frequenciesopen up for the development and extension of population ge-netic models based on a similar probabilistic framework, suchthat population structure can be taken into account in heteroge-neous populations.

Acknowledgments

This project was funded by the Lundbeck foundation (R215-2015-4174).

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