Probabilistic models of genetic variation in structured populations applied to global human studies

Probabilistic models of genetic v ariation in str uctured populations applied to global human studies W ei Hao 1 ∗ , Minsun Song 1 ∗ + , and John D . Storey 1 , 2 † 1. Lewis-Sigler Institute f or Integrativ e Genomics, Princeton University , Princeton, NJ 08544 2. Depar tment of Molecular Biology , Princeton University , Pr inceton, NJ 08544 ∗ These authors contributed equally to this work + Present address: Division of Cancer Epidemiology and Genetics, National Cancer Institute , National Institutes of Health, Rockville , MD 20850 † T o whom correspondence should be addressed: jstorey@princeton.edu Contents A B S T R A C T 2 1 I N T R O D U C T I O N 2 2 M E T H O D S 4 3 R E S U L T S 10 4 D I S C U S S I O N 13 5 F I G U R E S A N D T A B L E S 15 6 S U P P L E M E N T A RY M A T E R I A L 21 7 S U P P L E M E N T A RY F I G U R E S A N D T A B L E S 27 R E F E R E N C E S 33 1 A B S T R A C T Moder n population genetics studies typically inv olv e genome-wide genotyping of individuals from a di- v erse network of ancestr ies. An impor tant problem is how to f or m ulate and estimate probabilistic mod- els of obser v ed genotypes that account f or complex population str ucture . The most prominent work on this problem has focused on estimating a model of admixture propor tions of ancestral populations f or each individual. Here, we instead f ocus on modeling variation of the genotypes without requir ing a higher-le vel admixture inter pretation. We formulate two general probabilistic models, and we propose computationally efﬁcient algorithms to estimate them. First, w e sho w ho w principal component analysis (PCA) can be utilized to estimate a general model that includes the well-known Pritchard-Stephens- Donnelly admixture model as a special case. Noting some dra wbacks of this approach, we introduce a new “logistic f actor analysis” (LF A) frame work that seeks to directly model the logit transformation of probabilities underlying obser v ed genotypes in ter ms of latent variables that capture population str uc- ture. W e demonstrate these advances on data from the Human Genome Diversity P anel and 1000 Genomes Project, where we are able to identify SNPs that are highly differentiated with respect to structure while making minimal modeling assumptions. 1 I N T R O D U C T I O N Understanding genome-wide genetic variation among individuals is one of the pr imary goals of mod- er n human genetics. Genome-wide association studies aim to identify genetic v ariants throughout the entire genome that are associated with a complex trait [1 – 3]. One of the major challenges in analyzing these studies is the problem of spur ious associations due to population structure [4], and methods to deal with this are still in dev elopment [5 – 7]. A related effor t is underwa y to provide a comprehensive , genome-wide understanding of how genetic v ariation among humans is dr iv en b y ev olutionar y and de- mographic f orces [8]. A rigorous characterization of this variation will lead to a better understanding of the history of migration, e xpand our ability to identify signatures of natural selection, and provide impor- tant insights into the mechanisms of human disease [9, 10]. For e xample, the Human Genome Div ersity Project (HGDP) is an inter national project that has genotyped a large collection of DNA samples from individuals distributed around the world, aiming to assess wor ldwide genetic diversity at the genomic le vel [10 – 12]. The 1000 Genomes Project (TGP) is comprehensively cataloging human genetic varia- tion b y producing complete genome sequences of w ell ov er 1000 individuals of diverse ancestr ies [13]. Systematically characterizing genome-wide patter ns of genetic variation is difﬁcult due to the nu- merous and complex forces driving variation. There is a fundamental need to provide probabilistic models of obser v ed genotypes in the presence of comple x population str ucture . A series of inﬂuen- 2 tial publications hav e proposed methods to estimate a model of admixture, where the primar y focus is on the admixture propor tions themselves [14 – 16], which in tur n may produce estimates of the al- lele frequencies of ev er y genetic mar k er for each individual. Here, we instead focus directly on these individual-speciﬁc allele frequencies, which gives us potential advantages in ter ms of accuracy and computational efﬁciency . W e propose two ﬂexib le genome-wide models of individual-speciﬁc allele frequencies as well as methods to estimate them. First, we dev elop a model that includes as special cases the af orementioned models; speciﬁcally , the Balding-Nichols (BN) model [17] and its e xtension to the Pr itchard-Stephens- Donnelly (PSD) model [14]. How ev er , we identify some limitations of our method to estimate this model. W e therefore propose an alternative model based on the log-likelihood of the data that allows for rapid estimation of allele frequencies while maintaining a v alid probabilistic model of genotypes. The estimate of the ﬁrst model is based on principal component analysis (PCA), which is a tool often applied to genome-wide data of genetic v ar iation in order to unco ver structure. One of the earliest applications of PCA to population genetic data was carr ied out by Menozzi et al. [18]. Explor ator y analysis of complex population str ucture with PCA has been thoroughly studied [18 – 22]. W e show that a par ticular application of PCA can also be used to estimate allele frequencies in highly structured populations, although we hav e to deal with the fact that PCA is a real-valued operation and is not guaranteed to produce allele frequency estimates that lie in the unit interval [0,1]. The estimate of the second model is based on a generalized factor analysis approaches that di- rectly model latent str ucture in obser v ed data, including categor ical data [23] in which genotypes are included. W e utilize a f actor model of population str ucture [24] in ter ms of nonparametr ic latent v ar i- ables , and w e propose a method called “logistic f actor analysis” (LF A) that extends the PCA perspectiv e tow ards likelihood-based probabilistic models and statistical inf erence. LF A is shown to provide accu- rate and inter pretable estimates of individual-speciﬁc allele frequencies f or a wide range of population structures. At the same time, this proposed approach pro vides visualizations and numerical summaries of str ucture similar to that of PCA, building a conv enient br idge from e xplorator y data analysis to prob- abilistic modeling. W e compare our proposed methods to e xisting algorithms (ADMIXTURE [16] and f astStructure [25]) and show that when the goal is to estimate all individual-speciﬁc allele frequencies, our proposed ap- proaches are conclusiv ely superior in both accur acy and computational speed. We apply the proposed methods to the HGDP and TGP data sets , which allows us to estimate allele frequencies of e v er y SNP in an individual-speciﬁc manner . Using LF A, we are also ab le to r ank SNPs f or diff erentiation according to population structure based on the likelihoods of the ﬁtted models. In both data sets, the most diff er- entiated SNP is proximal to SLC24A5 , and the second most differentiated SNP is proximal to ED AR . V ariation in both of these genes has been hypothesiz ed to be under positive selection in humans. In 3 the TGP data set, the second most different SNP is rs3827760, which confers a missense mutation in ED AR and has been recently experimentally v alidated as ha ving a functional role in deter mining a phe- notype [26]. We also identify sev eral SNPs that are highly differentiated in these global human studies that hav e recently been associated with diseases such as cancer , obesity , and asthma. 2 M E T H O D S 2.1 Models of Allele Frequencies It is often the case that human and other outbred populations are “structured” in the sense that the geno- type frequencies at a par ticular locus are not homogeneous throughout the population [5]. Geographic characterizations of ancestr y often explain differing genotype frequencies among subpopulations. For e xample, an individual of European ancestr y ma y receive a par ticular genotype according to a probabil- ity diff erent than an individual of Asian ancestry . This phenomenon has been obser v ed not only across continents, but on very ﬁne scales of geographic character izations of ancestr y . Recent studies ha ve shown that population structure in human populations is quite comple x, occurring more on a continuous rather than a discrete basis [10]. We can illustrate the spectr um of structural complexity with Figure 1, which shows dendrograms of hierarchically clustered individuals from the HapMap (phase II), HGDP , and TGP data sets. The HapMap samples strongly indicate explicit membership of each individual to one of three discrete subpopulations (due to the intended sampling scheme). On the other hand, the clusterings of the HGDP and TGP individuals show a ver y comple x conﬁguration, more representative of random sampling of global human populations . Let us introduce Z as an unobser v ed v ar iab le capturing an individual’ s structure. Let x ij be the obser v ed genotype for SNP i and individual j ( i = 1 , . . . , m , j = 1 , . . . , n ), and assume that x ij is coded to take the values 0 , 1 , 2 . W e will call the obser v ed m × n genotype matr ix X . F or SNP i , the allele frequency can viewed as a function of Z , i.e. π i ( Z ) . F or a sampled individual j from an ov erall population, we hav e “individual-speciﬁc allele frequencies” [27] deﬁned as π ij ≡ π i ( z j ) at SNP i . Each value of π ij inf or ms us as to the e xpectation of that par ticular SNP/individual pair , supposing we obser v ed a new individual at that locus with the same str ucture; i.e. E[ x ij ] / 2 = π ij . If an observed SNP genotype x ij is treated as a random variable , then under Hardy-Weinberg Equilibr ium π ij ser v es to model x ij as a Binomial parameter: x ij ∼ Binomial (2 , π ij ) . The focus of this paper is on the simultaneously estimation of all m × n π ij v alues. The ﬂexib le, accurate, and computationally efﬁcient estimation of individual-speciﬁc allele frequen- cies is impor tant for population genetic analyses. F or e xample, Corona et al. (2013) [28] recently show ed that considering the worldwide distr ib ution of allele frequencies of SNPs known to be asso- 4 ciated with human diseases ma y be a fundamental component to understanding the relationship be- tween ancestr y and disease. T esting f or Hardy-W einberg equilibrium reduces to testing whether the genotype frequencies for SNP i follo w probabilities π 2 ij , 2 π ij (1 − π ij ) , and (1 − π ij ) 2 f or all individuals j = 1 , . . . , n . It can be shown that the well-known F ST measure can be characterized for SNP i using v alues of π ij , j = 1 , 2 , . . . , n (Section 6.5). Finally , we hav e recently dev eloped a test of association that corrects f or population structure and in volv es the estimation of log  π ij 1 − π ij  [29]. Theref ore, ﬂe xib le and well-beha v ed estimates of the individual-speciﬁc allele frequencies π ij are needed for do wnstream population genetic analyses. It is straightf orward to wr ite other models of population structure in ter ms of Z . F or the Balding- Nichols model, each individual is assigned to a population, thus z j indicates individual j ’ s population assignment. F or the Pritchard-Stephens-Donnelly (PSD) model, each individual is considered to be an admixture of a ﬁnite set of ancestral populations. Follo wing the notation of [14], we can wr ite z j as a v ector with elements q kj , where k index es the ancestral populations, and we constrain q kj to be between 0 and 1 subject to P k q kj = 1 . Assuming the PSD model allows us to write each π ij = P k p ik q kj and leads to a matrix f or m: F = PQ , where F is the m × n matrix of allele frequencies with ( i, j ) entr y π ij , P is the m × d matr ix of ancestral population allele frequencies p ik , and Q is the d × n matr ix of admixture propor tions. The elements of P and Q are e xplicitly restricted to the range [0 , 1] . The PSD model is f ocused on the matr ices P and Q , which ha v e standalone inter pretations , b ut w e aim instead to estimate all π ij with a high lev el of accuracy and computational efﬁciency . Writing the structure of the allele frequency matr ix F as a linear basis, we ha ve: Model 1: F = ΓS (1) where Γ is m × d and S is d × n with d ≤ n . The d × n matrix S encapsulates the genetic population structure for these individuals since S is not SNP-speciﬁc. The m × d matrix Γ maps how the structure S is manif ested in the allele frequencies. Operationally , each SNP’ s allele frequency are a linear com- bination of the rows of S , where the linear weights f or SNP i are contained in row i of Γ . We deﬁne the dimension d so that d = 1 corresponds to the case of no str ucture: when d = 1 , S = (1 , 1 , . . . , 1) and Γ is the column vector of marginal allele frequencies. This model is not necessarily the most eff ective wa y to estimate π ij when working in the conte xt of a probabilistic model or with the likelihood function given the data. Model 1 resembles linear regression, where the allele frequencies are treated as a real-valued response variable that is linear ly dependent on the structure. A version of regression f or the case of categor ical response v ar iab les (e.g., genotypes) with under lying probability parameters is logistic regression. W e de veloped an approach we call “logistic f actor analysis”, which is essentially an e xtension of nonparametr ic factor analysis to { 0 , 1 , 2 } v alued genotype data. Much of the justiﬁcation f or LF A is similar to that of generalized linear models [30]. 5 The log-likelihood is the pref erred mathematical framework for representing the information the data contain about unknown parameters [31]. Suppose that Hardy-W einberg equilibrium holds such that x ij ∼ Binomial (2 , π ij ) . W e can wr ite the log-lik elihood of the data f or SNP i and individual j as: ` ( π ij | x ij ) = log (Pr( x ij | π ij )) ∝ log  π x ij ij (1 − π ij ) 2 − x ij  = x ij log  π ij 1 − π ij  + 2 log(1 − π ij ) . The log-likelihood of SNP i for all unrelated individuals is the sum: P n j =1 ` ( π ij | x ij ) . The ter m log  π ij 1 − π ij  is the logit function and is written as logit( π ij ) . logit( π ij ) is called the “natur al parameter” or “canonical parameter” of the Binomial distribution and is the key component of logistic regression. An immediate beneﬁt of w or king with logit( π ij ) is that it is real valued, which allows us to directly model logit( π ij ) with a linear basis. Let L be the m × n matrix with ( i, j ) entr y equal to logit( π ij ) . W e formed the f ollowing parameteri- zation of L : Model 2: L = AH (2) where A is m × d and H is d × n with d ≤ n . In this case we can wr ite logit( π ij ) = d X k =1 a ik h kj , where all parameters are free to span the real numbers R . W e call the rows of H “logistic latent f actors” or just “logistic factors” as they represent unobser v ed v ar iab les that explain the inter-individual diff erences in allele frequencies. In other words, the logit of the vector of individual-speciﬁc allele frequencies for SNP i can be wr itten as a linear combination of the rows of H : [logit( π i 1 ) , . . . , logit( π in )] = logit( π i ) = d X k =1 a ik h k , where h k is the k th ro w of H . Like wise, we can write: ( π i 1 , . . . , π in ) = π i = exp h P d k =1 a ik h k i 1 + exp h P d k =1 a ik h k i . The relationship between our proposed LF A approach and existing approaches of estimating latent v ar iab les in categorical data is detailed in Section 6.6. Speciﬁcally , it should be noted that ev en though we propose calling the approach “logistic f actor analysis”, we do not make any assumptions about the distribution of the f actors (which are often assumed to be Nor mal). A technically more detailed name of the method is a “logistic nonparametric linear latent variab le model for Binomial data. ” 6 2.2 Estimation and Algorithms The two models presented earlier make minimal assumptions as to the nature of the str ucture . F or e xample, in Model 1, both Γ nor S are real valued. This allo ws us to apply an efﬁcient PCA-based algo- rithm directly to the genotype matrix X , obtaining estimates of e F , e Γ , and e S . In essence, e F is estimated by forming the projection of X / 2 onto the top d principal components of X with an explicit intercept f or the d = 1 case. One dra wback of this approach is that because PCA is designed f or continuous data, we hav e to ar tiﬁcially constrain e F to be in the range [0 , 1] . Howe v er , we show below that e F is still an extremely accurate estimate of the allele frequencies F f or all formulations of F considered here, including the PSD model. Algorithm 1 : Estimating F from PCA 1. Let e µ i be the sample mean of row i of X . Set x ∗ ij = x ij − e µ i and let X ∗ be the m × n matrix with ( i, j ) entr y x ∗ ij . 2. P erform singular value decomposition (SVD) on X ∗ which decomposes X ∗ = U∆V T . Note that the rows of ∆V T are the n row-wise principal components of X ∗ and U are the pr incipal component loadings. 3. Let e X ∗ d − 1 be the projection of X ∗ on the top d − 1 eigen-v ectors of this SVD , e X ∗ d − 1 = U 1:( d − 1) ∆ 1:( d − 1) V T 1:( d − 1) . 4. Constr uct e F ∗ by adding e µ i to row i of e X ∗ d − 1 (f or i = 1 , . . . , n ) and multiplying the resulting matr ix by 1 / 2 . In mathematical ter ms , e F ∗ = e Γ e S where 7 e Γ =     1 2 e µ 1 1 2 U 1:( d − 1) ∆ 1:( d − 1) . . . 1 2 e µ m     =        1 2 u 11 δ 1 · · · 1 2 u 1 ,d − 1 δ d − 1 1 2 e µ 1 1 2 u 21 δ 1 · · · 1 2 u 2 ,d − 1 δ d − 1 1 2 e µ 2 . . . . . . . . . 1 2 u m 1 δ 1 · · · 1 2 u m,d − 1 δ d − 1 1 2 e µ m        , e S = V T 1:( d − 1) 1 1 . . . 1 ! =           v 11 v 21 · · · v n 1 v 12 v 22 · · · v n 2 . . . . . . . . . v 1 ,d − 1 v 2 ,d − 1 · · · v n,d − 1 1 1 · · · 1           , and δ i is the i th diagonal entr y of ∆ . Let e π ∗ ij to be the ( i, j ) entr y of e F ∗ . 5. Since it may be the case that some e π ∗ ij are such that e π ∗ ij < 0 or e π ∗ ij > 1 , we tr uncate these. The ﬁnal PCA based estimate of F is formed as e F where the ( i, j ) entr y e π ij is deﬁned to be e π ij =          C if e π ∗ ij ≤ C e π ∗ ij if C < e π ∗ ij < 1 − C 1 − C if e π ∗ ij ≥ 1 − C f or some C & 0 . An estimate of L can be formed as e L = logit( e F ) . Here we used C = 1 2 n . In summar y , e F is a projection of X into its top pr incipal components , scaled by 1 / 2 , and truncated so that all values lie in the interval (0 , 1) . F or Model 2, we propose a method for estimating the latent variables H . Star ting from the e F f ound by Algor ithm 1, we apply the logit transf or mation to the subset of rows where we did not hav e to adjust the values that were < 0 or > 1 , and then extr act the r ight singular v ectors of this transf or med subset. As long as the subset is large enough to span the same space as the row space of L , this approach accurately estimates the basis of H . Ne xt, we calculate the maximum likelihood estimation of A parametrized by b H to yield b A and then b L = b A b H . This inv olv es performing a logistic regression of each SNP’ s data on b H . In order to estimate the individual-speciﬁc allele frequency matr ix F , we 8 calculate b F = logit − 1 ( b L ) . An impor tant proper ty to note is that all b π ij ∈ [0 , 1] due to the fact that we are modeling the natural parameter . Algorithm 2 : Estimating Logistic F actors 1. Apply Algor ithm 1 to obtain the estimate e F ∗ from Step 4. 2. Recalling that e π ∗ ij is the ( i, j ) entr y of e F ∗ , we choose some C & 0 and form S = { i : C < e π ∗ ij < 1 − C , ∀ j = 1 , ..., n } . S identiﬁes the rows of e F ∗ where the logit function can be applied stably . Here we use C = 1 2 n . 3. Deﬁne e F S to be the corresponding subset of rows of e F ∗ , and calculate e L S = logit  e F S  . Let e L 0 S be the row-wise mean centered and standard de viation scaled matrix e L S . 4. P erform SVD on e L 0 S resulting in e L 0 S = TΛW T . Set b H to be the d × n matr ix composed of the top d − 1 right singular vectors of the SVD of b L 0 S stack ed on the row n -v ector (1 , 1 , · · · , 1) : b H = W T 1:( d − 1) 1 1 · · · 1 1 ! =           w 11 w 21 · · · w n 1 w 12 w 22 · · · w n 2 . . . . . . . . . w 1 ,d − 1 w 2 ,d − 1 · · · w n,d − 1 1 1 · · · 1           . Algorithm 3 : Estimating F and L from LF A 1. Apply Algor ithm 2 to X to obtain b H . 2. For each SNP i , perform a logistic regression of the SNP genotypes x i = ( x i 1 , x i 2 , . . . , x in ) on the rows of b H , speciﬁcally by maximizing the log-lik elihood ` ( π i | x i , b H ) = n X j =1 x ij log  π ij 1 − π ij  + 2 log(1 − π ij ) under the constraint that logit( π ij ) = P d k =1 a ik b h kj . It should be noted that an intercept is included because b h d j = 1 ∀ j by construction. 9 3. Set b a ij ( j = 1 , . . . , n ) to be equal to the maximum likelihood estimates from the abov e model ﬁt, f or each of i = 1 , . . . , m . Let b L = b A b H , b F = logit − 1 ( b L ) , and b π ij be the ( i, j ) entr y of b F : b π ij = exp n P d k =1 b a ik b h kj o 1 + exp n P d k =1 b a ik b h kj o . PCA-based estimation of Model 1 requires one application of singular value decomposition (SVD) and LF A requires two applications of SVD . We lev erage the f act that n  d to utilize Lanczos bidiago- nalization which is an iterative method f or computing the singular value decomposition of a matrix [32]. Lanczos bidiagonalization excels at computing a fe w of the largest singular values and corresponding singular vectors of a sparse matrix. While the sparsity of genotype matrices is fairly low , we ﬁnd that in practice using this method to perf or m the abov e estimation algorithms is more effectiv e than using methods that require the calculation of all the singular values and vectors. This results in a dramatic reduction of the computational time needed f or the implementation of our methods. 3 R E S U L T S W e applied our methods to a comprehensive set of simulation studies and to the HGDP and TGP data sets. 3.1 Simulation Studies T o directly ev aluate the perf or mance of the estimation methods (Section 2.2), we devised a simulation study where we generated synthetic genotype data with var ying lev els of comple xity in population struc- ture. Genotypes were simulated based on allele frequencies subject to str ucture from the BN model, the PSD model, spatially structure populations, and real data sets. For the ﬁrst three types of simu- lations, the allele frequencies were parameterized by Model 1, while for the real data simulations, the allele frequencies were tak en from model ﬁts on the data themselves . A ke y proper ty to assess is how well the estimation methods capture the ov erall str ucture . One wa y to e valuate this is to deter mine how well e S from the PCA based method (Algor ithm 1) estimates the tr ue under lying S , and like wise how well b H from LF A estimates the true H . Note that ev en though the genotype data was generated from the F of Model 1, we can e valuate b H by con ver ting with L = logit( F ) . T o ev aluate PCA, we regressed each row of F on e S and calculated the av erage R 2 ; similarly , f or LF A we regressed each row of L on b H and calculated the av erage R 2 v alue. The results are presented in T able 1. Both methods estimate the tr ue latent structure well. 10 W e speciﬁcally note that when the PSD model was utilized to simulate structure, we were able to recov er the structure S very well (Supplementar y Figure 6) without needing to emplo y the computa- tionally intensiv e and assumption-heavy Bay esian model ﬁtting techniques from ref. [14]. Additionally , it seems that the e S largely captures the geometr y of S where it ma y be the case that S can be reco vered with a high degree of accuracy by transforming e S back into the simplex. By comparing the results on the real data (Figures 2-3) with the simulated data (Supplementar y Figure 6), one is ab le to visually assess how closely the assumptions of the PSD model resemble real data sets. When structure w as simulated that differed substantially from the assumptions of the PSD model, our estimation methods were able to capture that structure just as well (Supplementar y Figure 7). This demonstrates the ﬂe xibility of the proposed approaches. W e also compared PCA and LF A to two methods of ﬁtting the PSD model, ADMIXTURE [16] and f astStr ucture [25], by seeing how well the methods estimated the individual speciﬁc allele frequencies π ij (T able 3). F or the real data scenarios, we generated synthetic genotypes based on estimates of F from the four different methods, thus giving each method an oppor tunity to ﬁt its own simulation. The methods were compared by computing three different error metr ics with respect to the oracle F : K ullback-Leib ler divergence , absolute error , and root mean squared error . PCA and LF A signiﬁcantly outperf or med ADMIXTURE and f astStr ucture , which conﬁr ms the intuitive understanding of the differ- ences between the models: the goal of Model 1 and 2 is to estimate the allele frequencies π ij , while the PSD model provides a probabilistic inter pretation of the structure by modeling them as admixture propor tions. The computational time required to perf orm the proposed methods w as also signiﬁcantly better than ADMIXTURE and f astStructure. Both proposed methods completed calculations on av erage over 10 times f aster than ADMIXTURE and fastStructure, with some scenarios as high as 150 times faster . This is notable in that both ADMIXTURE and fastStructure are described as computationally efﬁcient implementations of methods to estimate the PSD model [16, 25]. 3.2 Analysis of the HGDP and TGP Data W e analyzed the HGDP and TGP data using the proposed methods . The HGDP data consisted of n = 940 individuals and m = 431 , 345 SNPs, and the TGP data consisted of n = 1500 and m = 339 , 100 (see Supplementar y Section 6.1 for details). We ﬁrst applied PCA and LF A to these data sets and made bi-plots of the top three PCs and top three LFs (Figures 2 and 3). It can be seen that PCA and LF A provide similar visualizations of the structure present in these data. We next chose a dimension d for the LF A model (Model 2) for each data set. This was done by identifying the value of d that provides the best ov erall goodness of ﬁt with Hardy-Weinberg equilibr ium (Supplementar y Section 6.2). W e identiﬁed d = 15 for HGDP and d = 7 for TGP based on this criterion. 11 One dr awbac k of utilizing a PCA based approach (Algor ithm 1) f or estimating the individual-speciﬁc allele frequencies F is that we are not guaranteed that all v alues of the estimates lie in [0 , 1] , so some f or m of tr uncation is necessar y . We found that 65.4% of the SNPs in the HGDP data set and 26.5% in the TGP data set resulted in at least one estimated individual-speciﬁc allele frequency < 0 or > 1 bef ore the tr uncation was applied. Theref ore, the tr uncation in forming the estimate e F is necessar y when employing Algorithm 1 to estimate F from Model 1. On the other hand, due to the f or mulation of Model 2, all estimated allele frequencies f all in the valid range when applying LF A (Algorithms 2 and 3). The LF A framew or k provides a natural computational method f or ranking SNPs according to how diff erentiated they are with respect to structure. Note that e xisting methods typically require one to ﬁrst assign each individual to one of K discrete subpopulations [33] which ma y make unnecessar y assumptions on moder n data sets such as HGDP and TGP . In order to rank SNPs for diff erentiation, we calculate the de viance statistic when perf or ming a logistic regression of the SNPs genotypes on the logistic f actors. Speciﬁcally we calculated the deviance b y comparing the models logit( π i ) = a id h d vs. logit( π i ) = P d k =1 a ik h k , where the f or mer model is intercept only (i.e ., d = 1 , no structure). Our application of LF A to identify SNPs with allele frequencies differentiated according to str ucture can be dev eloped fur ther . First, the recently proposed “jac kstraw” approach [34] provides a manner in which statistical signiﬁcance can be assigned to these SNPs. Assigning statistical signiﬁcance to the population differentiation of SNPs has traditionally been a difﬁcult problem [35]. Second, we found the de viance measure tends to ha ve more extreme values for SNPs with larger minor allele frequencies (MAFs). Therefore , the ranking of SNPs may be made more informative if MAF is taken into account. Third, although this ranking is identifying diff erentiation and not speciﬁcally selection, it ma y provide a useful star ting point in understanding methods that attempt to detect selection. The most diff erentiated SNPs (Supplementar y T ables 4 and 5) rev eal some notewor th y results, especially considering the ﬂe xible approach to forming the ranking. SNPs located within or very close to SLC24A5 were the top ranked in both HGDP and TGP . This gene is well known to be inv olv ed in deter mining skin pigmentation in humans [36] and is hypothesized to hav e been subject to positive selection [37]. The ne xt most highly rank ed SNPs in both studies are located in ED AR , which pla ys a major role in distinguishing phenotypes (e.g., hair f ollicles) among Asians. SNP rs3827760 is the second most differentiated SNP in the TGP data, which has also been hypothesiz ed to be under positive selection in humans and whose causal role in the hair f ollicle phenotype has been v er iﬁed in a mouse model [26]. SNPs corresponding to these two genes for both studies are plotted in increasing order of b π ij v alues, rev ealing subtle variation within each major ancestral group in addition to coarser diff erences in allele frequency (Figure 4). Other note wor th y genes with highly diff erentiated proximal SNPs include: • FO XP1 , which is a candidate gene f or inv olv ement in tumor progression and plays an impor tant regulator y role with FO XP2 [38, 39]; 12 • TBC1D1 in which genetic variation has been shown to confer risk f or se vere obesity in females [40]; • KIF3C , a nov el kinesin-like protein, which has been hypothesiz ed to be inv olved in microtubule- based transpor t in neuronal cells [41]; • KCNMA1 , a recently identiﬁed susceptibility locus for obesity [42]; • CTNNA3 in which genetic v ariation has been shown to be associated with diisocy anate-induced occupational asthma [43]; • PTK6 , breast tumor kinase (Br k), which is known to function in cell-type and context-dependent processes gov er ning normal diff erentiation [44]. W e hav e provided information on the 5000 most diff erentiated SNPs f or both TGP and HGDP in sup- plementar y ﬁles . 3.3 Software An R package called lfa is av ailable at https://github.com/StoreyLab/lfa . 4 D I S C U S S I O N W e hav e investigated two latent variable models of population structure to simultaneously estimate all individual-speciﬁc allele frequencies from genome-wide genotyping data. Model 1, a direct model of allele frequencies, can be estimated by using a modiﬁed PCA and Model 2, a model of the logit trans- f or mation of allele frequencies, is estimated through a new approach we called “logistic factor analysis” (LF A). For both models, the latent variables are estimated in a nonparametr ic fashion, meaning we do not mak e any assumptions about the underlying structure captured by the latent variables. These mod- els are general in that the y allow f or each individual’ s genotype to be generated from an allele frequency speciﬁc to that individual, which includes discretely str uctured populations, admixed populations, and spatially structured populations. In LF A, we construct a model of the logit of these allele frequencies in ter ms of underlying factors that capture the population structure. W e hav e proposed a computationally efﬁcient method to estimate this model that requires only two applications of SVD . This approach builds on the success of PCA in that we are ab le to capture population str ucture in terms of a low-dimensional basis. It improv es on PCA in that the latent v ar iab les we estimate can be straightf orwardly incor po- rated into downstream statistical inference procedures that require well-behav ed estimates of allele 13 frequencies. In par ticular , statistical inf erences of Hardy-W einberg equilibr ium, F ST , and mar k er-trait associations are amenable to comple x population structures within our frame work. W e demonstrated our proposed approach on the HGDP and TGP data sets and sev eral simulated data sets motiv ated b y the HapMap , HGDP , and TGP data sets as w ell as the PSD model and spatially distributed structures. It was shown that our method estimates the under lying logistic f actors with a high degree of accuracy . W e also showed that applying PCA to genotype data estimates a row basis of population structure on the or iginal allele frequency scale to a high degree of accuracy . Howe v er , problems occur when trying to recov er estimates of individual-speciﬁc allele frequencies because PCA is a real-v alued model that does not alwa ys result in allele frequency estimates lying between 0 and 1. Although PCA has become v ery popular f or genome-wide genotype data, it should be stressed that PCA is fundamentally a method for characterizing variance and special care should be taken when applying it to estimate latent variables . The authoritative treatment of PCA [45] eloquently makes this point throughout the te xt and considers cases where factor analysis is more appropriate than PCA through e xamples reminiscent of the population structure problem. Here, we ha v e sho wn that modeling and estimating population structure can be understood from the factor analysis perspective, leading to estimates of individual-speciﬁc allele frequencies through their natural parameter on the logit scale. At the same time, we hav e av oided some of the difﬁculties of traditional parameter ic factor analysis by maintaining the rele vant nonparametric proper ties of PCA, speciﬁcally in making no assumptions about the underlying probability distr ib utions of the logistic f actors that capture population str ucture . 14 5 F I G U R E S A N D T A B L E S HapMap ● ● ● YRI CEU JPT+CHB Human Genome Diversity Project (HGDP) ● ● ● ● ● ● ● AFRICA AMERICA CENTRAL_SOUTH_ASIA EAST_ASIA EUROPE MIDDLE_EAST OCEANIA 1000 Genomes Project (TGP) ● ● ● ● ● ● ● AFRICA_IN_AFRICA AFRICA_IN_AM EAST_ASIA_IN_AM/ASIA EUROPE_IN_AM/EU MEXICO_IN_AM SOUTH_AMERICA SOUTH_ASIA_IN_AM/ASIA Figure 1: A hierarchical clustering of individuals from the HapMap, HGDP , and TGP data sets. A den- drogram was dr awn from a hier archical clustering using W ard distance based on SNP genotypes (MAF > 5% ). Whereas the HapMap project shows a deﬁnitive discrete population str ucture (by sampling design), the HGDP and TGP data show the comple x structure of human populations. 15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC1 PC2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC1 PC3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC2 PC3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF1 LF2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF1 LF3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF2 LF3 ● ● ● ● ● ● ● AFRICA AMERICA CENTRAL_SOUTH_ASIA EAST_ASIA EUROPE MIDDLE_EAST OCEANIA Figure 2: Principal components versus logistic f actors for the HGDP data set. The top three pr incipal components from the HGDP data are plotted in a pairwise f ashion in the top panel. The top three logistic f actors are plotted analogously in the bottom panel. It can be seen that both approaches yield similar visualizations of structure. 16 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC1 PC2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC1 PC3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC2 PC3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF1 LF2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF1 LF3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● LF2 LF3 ● ● ● ● ● ● ● AFRICA_IN_AFRICA AFRICA_IN_AM EAST_ASIA_IN_AM/ASIA EUROPE_IN_AM/EU MEXICO_IN_AM SOUTH_AMERICA SOUTH_ASIA_IN_AM/ASIA Figure 3: Pr incipal components versus logistic f actors for the TGP data set. The top three principal components from the TGP data are plotted in a pairwise f ashion in the top panel. The top three logistic f actors are plotted analogously in the bottom panel. It can be seen that both approaches yield similar visualizations of structure. 17 rs1834640 − SLC24A5 rs260690 − EDAR 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 250 500 750 Individuals Ordered by Allele Frequency Estimated Individual−Specific Allele Frequencies geo AFRICA AMERICA CENTRAL_SOUTH_ASIA EAST_ASIA EUROPE MIDDLE_EAST OCEANIA HGDP rs1426654 − SLC24A5 rs3827760 − EDAR 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 500 1000 1500 Individuals Ordered by Allele Frequency Estimated Individual−Specific Allele Frequencies geo AFRICA_IN_AFRICA AFRICA_IN_AM EAST_ASIA_IN_AM/ASIA EUROPE_IN_EU MEXICO_IN_AM SOUTH_AMERICA SOUTH_ASIA_IN_AM/ASIA TGP Figure 4: SNPs with highly differentiated allele frequencies with respect to structure. T wo of the most highly different SNPs according to LF A are shown for the HGDP and TGP data sets. F or each SNP , the b π ij v alues are ordered and they are colored according repor ted ancestr y . The hor iz ontal bars on the sides of the plots denote the usual allele frequency estimates formed within each ancestral group . 18 T able 1: Accuracy in estimating linear bases for S . Column 1 shows the scenar io from which the data were simulated. Columns 2 and 3 display the estimation accuracy of the PCA based method (Column 2) and LF A (Column 3). Column 2 shows the mean R 2 v alue when regressing the true ( π i 1 , π i 2 , . . . , π in ) on b S from PCA, av eraging across all SNPs. Column 3 shows the mean R 2 v alue when regressing the true (logit( π i 1 ) , logit( π i 2 ) , . . . , logit( π in )) on b H from LF A, av eraging across all SNPs. All estimated standard errors f ell betw een 10 − 6 and 10 − 8 so these are not shown. Note f or each scenario , R 2 v alues are higher for the method from which the tr ue F matrix was generated. All but the two scenar ios marked with an aster isk (*) are from Model 1, while the two mar k ed scenarios are from Model 2, where we took F = logit − 1 L . Mean R 2 Scenario F ∼ e S logit( F ) ∼ b H TGP ﬁt by PCA 0.9998 0.9722 TGP ﬁt by LF A * 0.9912 0.9990 HGDP ﬁt by PCA 0.9996 0.9614 HGDP ﬁt by LF A * 0.9835 0.9983 BN 0.9999 0.9999 PSD α = 0 . 01 0.9998 0.9974 PSD α = 0 . 1 0.9998 0.9879 PSD α = 0 . 5 0.9996 0.9827 PSD α = 1 0.9993 0.9844 Spatial a = 0 . 1 0.9999 0.9964 Spatial a = 0 . 25 0.9999 0.9962 Spatial a = 0 . 5 0.9999 0.9964 Spatial a = 1 0.9998 0.9970 19 T able 2: Accuracy in estimating π ij parameters by the PCA based method and LF A. Each row is a different simulation scenar io . Each column is the accuracy of a method’ s ﬁts with the given metric. Scenario Median KL Mean Abs. Err . RMSE PCA LF A ADX FS PCA LF A ADX FS PCA LF A ADX FS BN 6.9 E -5 6.8 E -5 2.6 E -3 2.6 E -3 5.8 E -3 5.8 E -3 3.7 E -2 3.7 E -2 7.5 E -3 7.5 E -3 5.8 E -2 5.8 E -2 PSD α = 0 . 01 7.0 E -5 7.3 E -5 1.6 E -2 1.6 E -2 5.6 E -3 5.8 E -3 9.7 E -2 9.7 E -2 7.2 E -3 7.6 E -3 1.7 E -1 1.7 E -1 α = 0 . 1 6.7 E -5 9.2 E -5 3.6 E -2 3.6 E -2 5.6 E -3 6.9 E -3 1.6 E -1 1.6 E -1 7.2 E -3 9.3 E -3 2.4 E -1 2.4 E -1 α = 0 . 5 6.3 E -5 8.5 E -5 5.4 E -2 5.4 E -2 5.6 E -3 6.8 E -3 1.4 E -1 1.4 E -1 7.3 E -3 9.0 E -3 1.8 E -1 1.8 E -1 α = 1 . 0 6.1 E -5 7.4 E -5 3.3 E -2 3.3 E -2 5.6 E -3 6.3 E -3 1.4 E -1 1.4 E -1 7.4 E -3 8.4 E -3 2.2 E -1 2.2 E -1 Spatial a = 0 . 1 7.3 E -5 1.2 E -4 8.2 E -3 8.1 E -3 5.5 E -3 7.6 E -3 7.4 E -2 7.4 E -2 7.0 E -3 1.0 E -2 1.2 E -1 1.2 E -1 a = 0 . 25 6.9 E -5 1.1 E -4 8.6 E -3 8.6 E -3 5.6 E -3 7.4 E -3 9.3 E -2 9.3 E -2 7.2 E -3 9.8 E -3 1.6 E -1 1.6 E -1 a = 0 . 5 6.6 E -5 9.5 E -5 1.0 E -2 1.0 E -2 5.6 E -3 6.9 E -3 6.7 E -2 6.7 E -2 7.2 E -3 9.2 E -3 1.0 E -1 1.0 E -1 a = 1 . 0 6.3 E -5 7.8 E -5 1.2 E -2 1.2 E -2 5.7 E -3 6.4 E -3 1.1 E -1 1.1 E -1 7.4 E -3 8.5 E -3 1.7 E -1 1.7 E -1 TGP ﬁt PCA 4.1 E -4 5.2 E -4 2.8 E -3 3.4 E -3 1.3 E -2 1.5 E -2 8.1 E -2 8.3 E -2 1.8 E -2 2.1 E -2 1.5 E -1 1.5 E -1 LF A 4.3 E -4 4.8 E -4 2.4 E -3 2.7 E -3 1.3 E -2 1.4 E -2 7.9 E -2 8.1 E -2 1.8 E -2 2.0 E -2 1.4 E -1 1.5 E -1 ADX 5.4 E -4 4.4 E -4 5.0 E -3 5.5 E -3 1.5 E -2 1.3 E -2 1.1 E -1 1.1 E -1 2.0 E -2 1.9 E -2 2.0 E -1 2.0 E -1 FS 4.1 E -4 5.5 E -4 7.8 E -4 9.2 E -4 1.3 E -2 1.5 E -2 5.6 E -2 5.8 E -2 1.8 E -2 2.1 E -2 1.3 E -1 1.3 E -1 HGDP ﬁt PCA 1.0 E -3 1.2 E -3 1.3 E -2 1.4 E -2 2.3 E -2 2.5 E -2 1.2 E -1 1.2 E -1 3.4 E -2 3.6 E -2 2.2 E -1 2.2 E -1 LF A 9.9 E -4 1.1 E -3 1.3 E -2 1.2 E -2 2.2 E -2 2.4 E -2 1.2 E -1 1.2 E -1 3.5 E -2 3.7 E -2 2.2 E -1 2.2 E -1 ADX 1.6 E -3 1.4 E -3 2.3 E -3 2.3 E -3 2.6 E -2 2.6 E -2 5.6 E -2 5.6 E -2 3.6 E -2 3.7 E -2 1.0 E -1 1.0 E -1 FS 1.4 E -3 1.6 E -3 3.1 E -2 2.9 E -2 2.6 E -2 2.7 E -2 1.4 E -1 1.3 E -1 3.6 E -2 3.8 E -2 2.2 E -1 2.1 E -1 20 6 S U P P L E M E N T A R Y M A T E R I A L 6.1 Data sets The HGDP data set was constr ucted by intersecting the data av ailable from the HGDP web site, http://www.hagsc.org/hgdp/files.html , with the set of individuals “H952” identiﬁed b y Rosenberg (2006) [46] with a high conﬁdence as containing no ﬁrst and second-degree relativ e pairs . This yielded complete SNP genotype data on 431,345 SNPs f or 940 individuals. In order to obtain data from the TGP we ﬁrst obtained the genotype data that had been measured through the Omni Platform, 2011-11-17, ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/technical/ working . We remov ed related individuals based on the TGP sample information. W e then sor ted in- dividuals according to least percentage of SNPs with missing data, and we selected the top 1500 individuals. This yielded complete SNP genotype data on 339,100 SNPs for 1500 individuals . W e utilized the HapMap data set in the simulated data described below . W e obtained the HapMap data release 23a, NCBI b uild 36 from www.hapmap.org consisting of unrelated individuals: 60 from Eu- ropean ancestr y group (CEU), 60 from Y oruba, Africa (YRI) , and 90 from Japan and China (JPT+CHB). W e identiﬁed all SNPs with obser v ed minor allele frequency ≥ 5% and with no missing data. The total number of SNPs used after ﬁlter ing in each population were CEU: 1,416,940, YRI: 1,539,314, JPT+CHB: 759,452. W e then identiﬁed all SNPs common to all three populations resulting in a total of 363,955. 6.2 Choosing the model dimension The model dimension d was deter mined for the HGDP and TGP data sets under the rationale that when d is large enough, then the great major ity of SNPs should appear to be in HWE. When d is too small, then the structure which has not been accounted f or will lead to spurious deviations from HWE. V alues d = 1 , 2 , . . . , 20 were considered for each data set, and we ended up identifying d = 15 f or HGDP and d = 7 for TGP . W e note that these choices could also be inter preted as reasonable according to a scree plot when PCA was applied to the genotype data. F or a giv en d value , we formed b F using the LF A method. We calculated a HWE goodness of ﬁt statistic f or each SNP i as f ollows: 2 X k =0 h P n j =1 1( x ij = k ) − P n j =1  2 k  b π k ij (1 − b π ij ) 2 − k i 2 P n j =1  2 k  b π k ij (1 − b π ij ) 2 − k , where P n j =1 1( x ij = k ) is the observed number of genotypes equal to k and P n j =1  2 k  b π k ij (1 − b π ij ) 2 − k is the e xpected n umber of genotypes equal to k under HWE. W e then utilized b F to simulate ﬁve instances 21 of a genotype matrix X 0 under HWE where we simulated x 0 ij ∼ Binomial (2 , b π ij ) . On each simulated genotype matrix X 0 , we again applied LF A to obtain b F 0 and calculate HWE goodness of ﬁt statistics. These goodness of ﬁt statistics were then pooled across all ﬁve simulated data sets and across all SNPs to form the null distr ib ution, which then allow ed us to calculate a HWE p-value f or each obser ved SNP . (It should be noted that we also f or med a separate null distribution according to minor allele frequency bins of length 0.05, and we arr iv ed at the same conclusion.) We then compared these p-values to the Unif or m(0,1) distribution and also against the p-values from the d + 1 case. This allowed us to identify a value of d where the HWE p-values were both close to the Uniform(0,1) distribution and to the HWE p-v alues from the d + 1 case. 6.3 Simulated data F or each simulation scenario , genotypes X were simulated such that x ij ∼ Binomial (2 , π ij ) , where π ij were elements of the allele frequency matr ix F . The results from the simulated data are summarized in T ables 1 and 2. Balding-Nichols (BN). For each SNP in the HapMap data set, w e estimated its marginal allele frequency according to the obser ved frequency and estimated its F ST v alue using the W eir & Cockerham estimate [47]. We set the simulated data to hav e m = 100 , 000 SNPs and n = 5000 individuals with d = 3 . Using Model 1, the S matr ix was generated by sampling its columns s j i.i.d. from (1 , 0 , 0) T , (0 , 1 , 0) T , and (0 , 0 , 1) T with respective probabilities 60 / 210 , 60 / 210 , and 90 / 210 to reﬂect the or iginal data’ s subpopulation propor tions. F or each row i of Γ , we simulated i.i.d. dra ws from the Balding-Nichols model: γ i 1 , γ i 2 , γ i 3 i.i.d. ∼ BN ( p i , F i ) , where the pair ( p i , F i ) was randomly selected from among the marginal allele frequency and F ST pairs calculated on the HapMap data set. PSD . W e analyzed each SNP in the HGDP data set to estimate its marginal allele frequency according to the obser ved marginal frequency and F ST using the Weir & Cock erham estimate [47]. T o estimate F ST , each individual in the HGDP data set was assigned to one of K = 5 subpopulations according to the analysis in Rosenberg et al. (2002) [10]. W e set m = 100 , 000 SNPs and n = 5000 individuals with d = 3 . Again utilizing Model 1, each row i of Γ was simulated according to γ i 1 , γ i 2 , γ i 3 i.i.d. ∼ BN ( p i , F i ) , where the pair ( p i , F i ) was randomly selected from among the marginal allele frequency and F ST pairs calculated on the HGDP data set. T o generate S , we simulated ( s 1 j , s 2 j , s 3 j ) i.i.d. ∼ Dirichlet ( α ) f or j = 1 , . . . , 5000 . We considered α = (0 . 01 , 0 . 01 , 0 . 01) , α = (0 . 1 , 0 . 1 , 0 . 1) , α = (0 . 5 , 0 . 5 , 0 . 5) , and α = (1 , 1 , 1) . It should be noted that as α → 0 , the draws from the Dir ichlet distr ib ution become increasingly closer to assigning each individual to one of three discrete subpopulations with equal probability . When α = (1 , 1 , 1) , the admixture propor tions are distributed unif ormly ov er the simplex. Spatial. This scenar io is meant to create population structure that is dr iv en by spatial position of the 22 individual. W e set the sim ulated data to ha v e m = 100 , 000 SNPs and n = 5000 individuals with d = 3 . Rows i = 1 , 2 of S were simulated as s ij i.i.d. ∼ Beta ( a, a ) for j = 1 , . . . , 5000 , and row 3 of S contained the intercept ter m, s 3 j = 1 . W e considered four values of a : 0.1, 0.25, 0.5, and 1. The ﬁrst two rows of S place each individual in a two-dimensional space (Figure 7), where the ancestr y of individual j is located at ( s 1 j , s 2 j ) in the unit square. When a = 1 , the Beta ( a, a ) distribution is Uniform (0 , 1) , so this scenario represents a unif orm distribution of individuals in unit square. As a → 0 , the Beta ( a, a ) places each individual with equal probabilities in one of the four cor ners of the unit square. The matr ix Γ was created by sampling γ ij i.i.d. ∼ 0 . 9 × Unif or m (0 , 1 / 2) f or j = 1 , 2 and γ i 3 = 0 . 05 . It should be noted that all π ij ∈ [0 . 05 , 0 . 95] by construction. Real Data. For the HGDP and TGP scenarios, we estimated an allele frequency matr ix F from the real data via four diff erent methods. For HGDP we had m = 431 , 345 SNPs by n = 940 individuals with d = 15 , and for TGP we had m = 339 , 100 and n = 1 , 500 with d = 7 . The four methods are: • PCA : F was taken to be the matrix e F estimated via Algor ithm 1. • LF A : F = logit − 1 ( b L ) , where b L was estimated via Algorithm 3. • ADX : F was taken to be the matr ix formed by computing the marginal allele frequencies in the Pritchard-Stephens-Donnelly model, i.e . F = PQ , and P and Q were estimated via the software ADMIXTURE [16]. • FS : Same as above e xcept P and Q are estimated via the software f astStructure [25]. 6.4 Error Measures Used to Ev aluate Estimates of F and L Estimates of π ij were ev aluated with three diff erent metrics. Let b π ij be the estimate for any given method. The K ullback-Leib ler divergence f or the binomial distribution allows us to measure the difference between the distribution from the estimated allele frequencies to the distr ib ution from the oracle allele frequencies: KL = π ij ln  π ij b π ij  + (1 − π ij ) ln  1 − π ij 1 − b π ij  . Mean absolute error compares the allele frequencies directly: MAE = 1 m × n m X i =1 n X j =1 | π ij − b π ij | . Root mean squared error : 23 RMSE = v u u t 1 m × n m X i =1 n X j =1 (logit( π ij ) − logit( b π ij )) 2 . 6.5 F ST for individual-speciﬁc allele frequencies By consider ing the der iv ation of F ST f or K discrete populations as described in W eir (1984, 1996) [47, 48], it can be seen that a potential gener alization of F ST to arbitrary population str ucture is F ST = 1 − E Z [V ar( x | Z )] V ar( x ) , where, as descr ibed in Section , Z is a latent variable capturing an individual’ s population structure position or membership . The allele frequency of a SNP conditional on Z can be viewed as being a function of Z , which we hav e denoted by π ( Z ) . If n individuals are sampled independently and homogeneously from the population 1 such that z 1 , . . . , z n are i.i.d. from the distribution on Z , then f or SNP i in HWE, it f ollows that V ar( x ij | z j ) = 2 π ij (1 − π ij ) and F ST a.s. = lim n →∞ 1 − 1 n P n j =1 π ij (1 − π ij ) π i (1 − π i ) , where π i = P n j =1 π ij /n is the marginal allele frequency among the n individuals. Thus, good estimates of the π ij v alues ma y be useful for estimating F ST in this general setting. One example would be to f or m a plug-in estimate of F ST by replacing π ij with b π ij from the proposed LF A method. 6.6 Relationship of LF A to existing models and methods The problem of modeling a genotype matrix X in order to uncov er latent variables that explain cryptic structure is a special case of a much more general prob lem that has been considered f or se ver al years in the statistics literature [49, 50]. Under a latent v ar iab le model, it is assumed that the “manif est” (obser v ed) variables are the result of the “latent” (unobser v ed) variab les. Diff erent types of the latent v ar iab le models can be g rouped according to whether the manif est and latent v ariables are categorical or continuous. For e xample, factor analysis is a latent v ariable method f or the case where both manifest v ar iab le and latent variable are continuous. A proposed naming conv ention [23] is summarized as f ollows: 1 When the individuals are not sampled homogeneously throughout the population (e.g., in the HapMap data with 60, 60, and 90 observations from three discretely deﬁned subpopulations), then it ma y be the case that the abo ve quantity should be modiﬁed to reﬂect the stratiﬁed or non-homogeneous sampling. 24 Manif est variables Latent v ar iab les Continuous Categorical Continuous F actor analysis Latent trait analysis Categorical Latent proﬁle analysis Latent class analysis The prob lem we consider is that the manifest v ariables (observed gento ypes) are categorical, and the y are dr iv en b y latent variables (population structure) that may either be categor ical (discrete population structure) or continuous (complex population structure). Therefore , the LF A method may be descr ibed as a nonparametric latent v ariable estimation method that jointly captures latent trait analysis and latent class analysis. Another naming convention that we could apply to LF A would be to call it a nonpara- metric latent variab le model f or Binomial data. The naming conv entions of latent variable models are inconsistent and often confusing [23]. Bar tholome w (1980) [51] proposed a model related to equation (2) to identify latent variables that inﬂuence the probabilities of a collection of Binomial random variables. See also Bar tholomew et al. 2011 f or a comprehensiv e treatment of this area, which the y call “general linear latent variable models” (GLL VM). In par ticular , when the manifest variables x ij ∼ Bernoulli( π ij ) and the latent variables h kj are continuous variables , the GLL VM in this case is Model 2 , logit( π ij ) = P d k =1 a ik h kj . While we begin with this model, there are some ke y differences . The number of manifest v ar iab les in the data considered in Bar tholomew (1980) and related wor k is notably smaller than genome-wide genotype data, so the assumptions and estimation approach differ substantially . Model assumptions are typically made about the probability distributions of the latent v ar iab les; we consider these model assumptions too strong and also unnecessar y for the genome-wide genotype data considered here, although they ma y be quite reasonable f or the problems considered in other contexts . Existing methods typically estimate Model 2 by calculating the joint poster ior distr ib ution of the h kj based on an assumed pr ior distribution of the latent v ar iab les. Our LF A approach for estimating the row basis of L is nonparametric since it does not require a pr ior assumption on the distr ib ution of latent v ariables, H . The model ﬁtting methods of ref . [23] are too com- putationally intensive for high-dimensional data, requir ing many iterations and potential conv ergence issues. Our proposed algor ithm requires perf or ming SVD twice, which leads to a dramatic reduction in computational burden and difﬁculties. Engelhardt and Stephens (2010) [24] make an interesting con- nection betw een classical f actor analysis models of F and other models of population structure, b ut the f actor analysis model runs into the difﬁculty that the latent f actors are assumed to be Normal distributed, and the constraint that alleles frequencies are in [0 , 1] is not easily accommodated by this continuous, real-v alued model. Se ver al extensions of PCA to categor ical data hav e been proposed [52 – 54]. We found that the algorithms perf or m v er y slowly on genome-wide genotyping data, and the estimation can be quite poor 25 when d > 1 . Also , PCA is essentially a method for characterizing v ar iance in data [45], and the latent v ariable approach is more directly aimed at uncov er ing latent population str ucture . Non-negative matrix f actorization (NMF) [55] is another matr ix factorization for count data (e.g., P oisson random v ar iab les). This identiﬁes two non-negative matrices whose product approximates the original matr ix. How ev er , similarly to PCA, we do not ﬁnd that this approach easily translates into inter pretable models of population and it is computationally intensive . NMF has pro ven to be quite useful as a n umerical tool f or decomposing images into par ts humans recognize as distinct [56]. 26 7 S U P P L E M E N T A R Y F I G U R E S A N D T A B L E S L""""""""""" = """"""A""""H"""""""""""""""" ≈ """""A"""""H # F""""""""""" = """"""" Γ """""S"""""""""""""""" ≈ """" Γ """"""" S"""" X" € logit π ij ( ) € π ij 0#2#2#1#1# 0#1#0#2#1# 0#1#2#.#.#.# Logis&c(Factor(Analysis(( (LFA)# Principal(Components(Analysis( (PCA)# SNP#Data# ⇒ #values#in#(−∞,∞)# ⇒ #values#in#(0,1)# ⌃ " ⌃ " ~ " ~ " Figure 5: A comparison of LF A model (2) and its estimate to model (1) and its PCA estimate. The proposed LF A approach ﬁrst models the logit of the individual-speciﬁc allele frequencies in ter ms of the product of two matr ices , the left matr ix establishing how population structure is present in allele frequencies, and the right matrix giving the str ucture . Whereas the LF A approach preser ves the scale of the model through the estimate (all real-valued numbers), the same is not tr ue to PCA. This leads to issues in the estimation of individual-speciﬁc allele frequencies when utilizing PCA. We ha v e shown, how ev er , that PCA estimates v er y well a row basis f or S from Model 1. This connects PCA to an e xplicit model of population structure. 27 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● alpha=0.01 alpha=0.1 alpha=0.5 alpha=1 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 S 1 S 2 S ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● alpha=0.01 alpha=0.1 alpha=0.5 alpha=1 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 S 1 S 2 \tilde{S} 0.00 0.25 0.50 0.75 1.00 S3 Figure 6: A mapping from S to e S f or four simulated S matrices under the PSD model. The left col- umn shows the simulated str ucture S f or each of four scenar ios (a–d) and the right column shows the resulting estimated row basis of S produced from PCA. It can be seen that the scale on which S was generated, all v alues in (0,1), is lost in the principal components, values in R . 28 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● a=0.1 a=0.25 a=0.5 a=1 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 S 1 S 2 S ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● a=0.1 a=0.25 a=0.5 a=1 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 −0.02 0.00 0.02 S 1 S 2 \tilde{S} Figure 7: A mapping from S to e S f or four simulated S matrices under the Spatial model. The left column shows the simulated structure S for each of four scenarios (a–d) and the r ight column shows the resulting estimated row basis of S produced from PCA. It can be seen that the scale on which S was gener ated, all values in (0,1), is lost in the principal components, v alues in R . 29 T able 3: Accuracy in estimating π ij parameters by the PCA based method and LF A. Each row is a different simulation scenar io . Each column is the accuracy of a method’ s ﬁts with the given metric. Scenario Median KL Mean Abs. Err . RMSE PCA LF A ADX FS PCA LF A ADX FS PCA LF A ADX FS BN 6.9 E -5 6.8 E -5 2.6 E -3 2.6 E -3 5.8 E -3 5.8 E -3 3.7 E -2 3.7 E -2 7.5 E -3 7.5 E -3 5.8 E -2 5.8 E -2 PSD α = 0 . 01 7.0 E -5 7.3 E -5 1.6 E -2 1.6 E -2 5.6 E -3 5.8 E -3 9.7 E -2 9.7 E -2 7.2 E -3 7.6 E -3 1.7 E -1 1.7 E -1 α = 0 . 1 6.7 E -5 9.2 E -5 3.6 E -2 3.6 E -2 5.6 E -3 6.9 E -3 1.6 E -1 1.6 E -1 7.2 E -3 9.3 E -3 2.4 E -1 2.4 E -1 α = 0 . 5 6.3 E -5 8.5 E -5 5.4 E -2 5.4 E -2 5.6 E -3 6.8 E -3 1.4 E -1 1.4 E -1 7.3 E -3 9.0 E -3 1.8 E -1 1.8 E -1 α = 1 . 0 6.1 E -5 7.4 E -5 3.3 E -2 3.3 E -2 5.6 E -3 6.3 E -3 1.4 E -1 1.4 E -1 7.4 E -3 8.4 E -3 2.2 E -1 2.2 E -1 Spatial a = 0 . 1 7.3 E -5 1.2 E -4 8.2 E -3 8.1 E -3 5.5 E -3 7.6 E -3 7.4 E -2 7.4 E -2 7.0 E -3 1.0 E -2 1.2 E -1 1.2 E -1 a = 0 . 25 6.9 E -5 1.1 E -4 8.6 E -3 8.6 E -3 5.6 E -3 7.4 E -3 9.3 E -2 9.3 E -2 7.2 E -3 9.8 E -3 1.6 E -1 1.6 E -1 a = 0 . 5 6.6 E -5 9.5 E -5 1.0 E -2 1.0 E -2 5.6 E -3 6.9 E -3 6.7 E -2 6.7 E -2 7.2 E -3 9.2 E -3 1.0 E -1 1.0 E -1 a = 1 . 0 6.3 E -5 7.8 E -5 1.2 E -2 1.2 E -2 5.7 E -3 6.4 E -3 1.1 E -1 1.1 E -1 7.4 E -3 8.5 E -3 1.7 E -1 1.7 E -1 TGP ﬁt PCA 4.1 E -4 5.2 E -4 2.8 E -3 3.4 E -3 1.3 E -2 1.5 E -2 8.1 E -2 8.3 E -2 1.8 E -2 2.1 E -2 1.5 E -1 1.5 E -1 LF A 4.3 E -4 4.8 E -4 2.4 E -3 2.7 E -3 1.3 E -2 1.4 E -2 7.9 E -2 8.1 E -2 1.8 E -2 2.0 E -2 1.4 E -1 1.5 E -1 ADX 5.4 E -4 4.4 E -4 5.0 E -3 5.5 E -3 1.5 E -2 1.3 E -2 1.1 E -1 1.1 E -1 2.0 E -2 1.9 E -2 2.0 E -1 2.0 E -1 FS 4.1 E -4 5.5 E -4 7.8 E -4 9.2 E -4 1.3 E -2 1.5 E -2 5.6 E -2 5.8 E -2 1.8 E -2 2.1 E -2 1.3 E -1 1.3 E -1 HGDP ﬁt PCA 1.0 E -3 1.2 E -3 1.3 E -2 1.4 E -2 2.3 E -2 2.5 E -2 1.2 E -1 1.2 E -1 3.4 E -2 3.6 E -2 2.2 E -1 2.2 E -1 LF A 9.9 E -4 1.1 E -3 1.3 E -2 1.2 E -2 2.2 E -2 2.4 E -2 1.2 E -1 1.2 E -1 3.5 E -2 3.7 E -2 2.2 E -1 2.2 E -1 ADX 1.6 E -3 1.4 E -3 2.3 E -3 2.3 E -3 2.6 E -2 2.6 E -2 5.6 E -2 5.6 E -2 3.6 E -2 3.7 E -2 1.0 E -1 1.0 E -1 FS 1.4 E -3 1.6 E -3 3.1 E -2 2.9 E -2 2.6 E -2 2.7 E -2 1.4 E -1 1.3 E -1 3.6 E -2 3.8 E -2 2.2 E -1 2.1 E -1 30 T able 4: The top 50 SNPs most associated with structure in the HGDP data, identiﬁed by perf or ming a logistic regression of SNP genotypes on the logistic f actors. Shown are the SNP ID and location, de viance measure of diff erentiation, gene closest to the SNP , distance to gene (rounded to nearest 10bp), and the variant type (if none shown, then intergenic). rsid chr position de viance genesymbol locusID distance variant type 1 rs1834640 15 48392165 1605.28 SLC24A5 283652 21000 2 rs2250072 15 48384907 1313.82 SLC24A5 283652 28260 3 rs12440301 15 48389924 1263.83 SLC24A5 283652 23240 4 rs260690 2 109579738 1262.72 ED AR 10913 0 intron-variant 5 rs9837708 3 71487582 1189.48 FO XP1 27086 0 intron-variant 6 rs260714 2 109562495 1184.50 ED AR 10913 0 intron-variant 7 rs4918664 10 94921065 1178.40 XRCC6P1 387703 45340 8 rs10882168 10 94929434 1160.99 XRCC6P1 387703 36970 9 rs300153 2 17986417 1143.48 MSGN1 343930 11360 10 rs9809818 3 71480566 1135.58 FO XP1 27086 0 intron-variant 11 rs6583859 10 94893473 1119.25 NIP7P1 389997 26290 12 rs11187300 10 94920291 1114.22 XRCC6P1 387703 46120 13 rs260698 2 109566759 1111.64 ED AR 10913 0 intron-variant 14 rs1834619 2 17901485 1111.40 SMC6 79677 0 intron-variant 15 rs11637235 15 48633153 1104.45 DUT 1854 0 intron-variant 16 rs4497887 2 125859777 1097.13 RNA5SP102 100873373 169180 17 rs7091054 10 95018444 1085.45 RPL17P34 643863 25280 18 rs7090105 10 75131545 1075.50 ANXA7 310 3640 19 rs973787 4 38263893 1074.57 TBC1D1 23216 123090 20 rs4279220 4 38254182 1070.43 TBC1D1 23216 113380 21 rs7556886 2 17908130 1062.58 SMC6 79677 0 intron-variant 22 rs12473565 2 175163335 1056.31 LOC644158 644158 1390 23 rs6500380 16 48375777 1051.10 LONP2 83752 0 intron-variant 24 rs2384319 2 26206255 1033.88 KIF3C 3797 810 upstream-variant-2KB 25 rs12220128 10 94975011 1023.79 XRCC6P1 387703 6090 26 rs17034770 2 109616376 1019.03 EDAR 10913 10540 27 rs3792006 2 26498222 998.96 HADHB 3032 0 intron-variant 28 rs4918924 10 94976956 994.79 XRCC6P1 387703 8030 29 rs1984996 10 95008745 990.92 RPL17P34 643863 34980 30 rs3751631 15 52534344 987.33 MY O5C 55930 0 reference,synon ymous-codon 31 rs4578856 2 17853388 987.29 SMC6 79677 0 intron-variant 32 rs13397666 2 109544052 986.80 EDAR 10913 0 intron-variant 33 rs12619554 2 17352372 986.20 ZFYVE9P2 100420972 113180 34 rs3736508 11 45975130 981.05 PHF21A 51317 0 missense,ref erence 35 rs12472075 2 177691130 973.02 RPL29P8 100131991 16650 36 rs9522149 13 111827167 965.50 ARHGEF7 8874 0 intron-variant 37 rs2917454 10 78892415 964.40 KCNMA1 3778 0 intron-variant 38 rs10882183 10 94974083 961.04 XRCC6P1 387703 5160 39 rs10079352 5 117494640 960.33 LOC100505811 100505811 123620 40 rs10935320 3 139056584 958.33 MRPS22 56945 6270 41 rs9571407 13 34886039 957.04 LINC00457 100874179 123540 42 rs6542787 2 109556365 955.56 ED AR 10913 0 intron-variant 43 rs953035 1 36079508 954.67 PSMB2 5690 0 intron-variant 44 rs4657449 1 165465281 951.72 LOC400794 400794 0 intron-variant 45 rs9960403 18 13437993 949.43 LDLRAD4 753 0 intron-variant 46 rs203150 18 38037221 944.32 RPL17P45 100271414 312750 47 rs2823882 21 17934419 942.05 LINC00478 388815 0 intron-v ariant 48 rs10886189 10 119753963 937.81 RAB11FIP2 22841 10460 49 rs2441727 10 68224886 937.08 CTNNA3 29119 0 intron-variant 50 rs310644 20 62159504 931.90 PTK6 5753 260 downstream-v ariant-500B T able 5: The top 50 SNPs most associated with str ucture in the TGP data, identiﬁed by performing a logistic regression of SNP genotypes on the logistic f actors. Shown are the SNP ID and location, de viance measure of diff erentiation, gene closest to the SNP , distance to gene (rounded to nearest 10bp), and the variant type (if none shown, then intergenic). rsid chr position deviance genesymbol locusID distance v ariant type 1 rs1426654 15 48426484 3129.76 SLC24A5 283652 0 missense,ref erence 2 rs3827760 2 109513601 2395.27 ED AR 10913 0 missense,reference 3 rs922452 2 109543883 2338.38 ED AR 10913 0 intron-variant 4 rs372985703 17 19172196 1975.16 EPN2 22905 0 intron-variant 5 rs4924987 17 19247075 1949.03 B9D1 27077 0 intron-variant,missense,ref erence 6 rs260687 2 109578855 1925.18 ED AR 10913 0 intron-variant 7 rs7209202 17 58532239 1890.67 APPBP2 10513 0 8 rs7211872 17 58550725 1890.67 APPBP2 10513 0 9 rs67929453 3 139109825 1890.57 LOC100507291 100507291 0 intron-variant,upstream-variant-2KB 10 rs260643 2 109539653 1850.71 ED AR 10913 0 intron-variant 11 rs260707 2 109574150 1838.37 ED AR 10913 0 intron-variant 12 rs1545071 18 67695505 1821.35 RTTN 25914 0 intron-variant 13 rs12729599 1 1323078 1812.91 CCNL2 81669 0 intron-variant 14 rs12347078 9 344508 1811.16 DOCK8 81704 0 intron-variant 15 rs12142199 1 1249187 1779.28 CPSF3L 54973 0 reference,synon ymous-codon 16 rs12953952 18 67737927 1750.15 RTTN 25914 0 intron-variant 17 rs9467091 6 10651772 1746.75 GCNT6 644378 4270 18 rs7165971 15 55921013 1736.83 PRTG 283659 0 intron-variant 19 rs6132532 20 2315543 1730.64 TGM3 7053 0 intron-variant 20 rs959071 17 19142226 1729.18 EPN2 22905 0 intron-variant 21 rs10962599 9 16795286 1726.24 BNC2 54796 0 intron-variant 22 rs967377 20 53222217 1724.93 DOK5 55816 0 intron-variant 23 rs4891381 18 67595449 1723.79 CD226 10666 0 intron-variant 24 rs377561427 15 63988357 1713.98 HERC1 8925 0 frameshift-variant,reference 25 rs73889254 22 46762214 1711.40 CELSR1 9620 0 intron-v ar iant 26 rs4918664 10 94921065 1700.64 XRCC6P1 387703 45340 27 rs2759281 1 204866365 1691.03 NF ASC 23114 0 intron-variant 28 rs12065033 1 173579034 1682.54 ANKRD45 339416 0 utr-variant-3-prime 29 rs9796793 16 30495652 1681.28 ITGAL 3683 0 intron-variant 30 rs1240708 1 1335790 1675.48 LOC148413 148413 0 intron-variant,upstream-variant-2KB 31 rs2615876 10 117665860 1670.53 A TRNL1 26033 0 intron-variant 32 rs2823882 21 17934419 1669.32 LINC00478 388815 0 intron-v ariant 33 rs8097206 18 38024931 1663.29 RPL17P45 100271414 300460 34 rs8071181 17 58508582 1662.44 C17orf64 124773 0 reference,synonymous-codon 35 rs1075389 15 64174177 1661.21 MIR422A 494334 10950 36 rs6875659 5 175158653 1657.54 HRH2 3274 22410 37 rs7171940 15 64170986 1654.01 MIR422A 494334 7760 38 rs2148359 9 7385508 1652.16 RPL4P5 158345 91440 39 rs7531501 1 234338303 1648.15 SLC35F3 148641 0 intron-variant 40 rs57742857 15 93567352 1645.21 CHD2 1106 0 intron-v ar iant 41 rs931564 17 58631702 1636.86 LOC388406 388406 10200 42 rs4738296 8 73857539 1632.70 LOC100288310 100288310 0 intron-variant 43 rs4402785 2 104766351 1631.33 LOC100287010 100287010 228950 44 rs12988506 2 33162854 1630.14 LOC100271832 100271832 0 intron-variant 45 rs9410664 9 91196828 1625.48 NXNL2 158046 6120 46 rs2041564 2 72453847 1623.91 EXOC6B 23233 0 intron-variant 47 rs6024103 20 54034601 1623.41 LOC101927796 101927796 2270 48 rs6583859 10 94893473 1619.79 NIP7P1 389997 26290 49 rs12913832 15 28365618 1611.23 HERC2 8924 0 intron-v ar iant 50 rs632876 2 216572452 1610.26 LINC00607 646324 0 intron-v ariant References [1] McCar thy , M. 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