Existing 20 ancestry deconvolution tools: ✓ indicates the ability of the software to perform a specified task, ✗ indicates the inapplicability of the task by a particular tool. Unless explicitly specified, LD refers to background LD.
\r\n\t
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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"39346",title:"Convolution Kernel for Fast CPU/GPU Computation of 2D/3D Isotropic Gradients on a Square/Cubic Lattice",doi:"10.5772/46455",slug:"convolution-kernel-for-fast-cpu-gpu-computation-of-2d-3d-isotropic-gradients-on-a-square-cubic-latti",body:'The design of discrete operators or filters for the calculation of gradients is a classical topic in scientific computing. Typical applications are gradient reconstruction in computational fluid dynamics, edge detection in computer graphics and biomedical imaging, and phase boundary definition in the modeling of multiphase flows.
Edge detection, which is widely performed in image analysis, is an operation that requires gradient calculation. Commonly used edge detection methods are Canny, Prewitt, Roberts and Sobel, which can be found in MATLAB’s platform. In this field, edge detection techniques rely on the application of convolution masks to provide a filter or kernel to calculate gradients in two perpendicular directions. A threshold is then applied to obtain an edge shape.
For multiphase flows, an edge or contour corresponds to the interface between the fluids. In this respect, traditional gradient calculation methods based on 1D edge detection are not necessarily suited for the underlying physics, because there is no direction in which the gradients of the phase contours tend to evolve over time. As a result, definition of the geometric progress of the interface requires many gradient estimation computations, as is the case in moving and deforming bubbles or droplets, for example. Although it can still be a viable tool, it is clear that the 1D-based method is becoming less useful for simulating these phenomena, which are not, in general, biased toward any particular direction.
To address this issue, we present an efficient computational method for obtaining discrete isotropic gradients that was previously applied to simulate two-phase flows within the lattice Boltzman framework [1, 2]. This "omnidirectional" approach makes it possible to improve the limitations inherent in handling high density ratios between the phases and to significantly reduce spurious currents at the interface. The method is based on a filter which is generally not split along any direction, and there is no need to make the assumption of a continuous filter to reach isotropy, as done by [3]. We also believe that optimal or maximal isotropy can only be reached with a discrete filter when the error terms of Taylor’s series expansion are isotropic, as explained in detail by [1, 2].
Below, we describe isotropic and anisotropic discretizations that will and will not conserve the isotropic property of the differentiated function respectively. This is followed by a description of how convolution can be used to reduce computer time in the gradient calculation. We then present details of the MATLAB implementation of these ideas, along with speedup comparisons of convolution performed on a single core of an Intel® Core i7-970 processor and on an Nvidia® GeForce GTX 580 GPU using the Jacket plugin for MATLAB developed by AccelerEyes®. The GPU with the Jacket plugin for MATLAB speeds up gradient computation by a factor of up to
Let us define the real scalar function
with
This finite difference discretization is very similar to Prewitt’s operator/kernel [4], which is used in image processing. Note that
As in Ref. [2] and without loss of generality, the function
with
To calculate the gradient in the
A similar expression is found for the
In the gradients of Eqs. (5) and (6), the leading
Using the following transformation from a Cartesian to a polar partial derivative operator:
And, by supposing that
Taking into consideration the previous anisotropy problem, it is possible to change the weights of the grid points when computing the gradients to make them isotropic, up to the second order in space, by defining gradients in the
With this new discretization, the dominant differential operator of the second order error term takes the form:
If a gradient has a small dependence on direction, this would imply that the dominant error term has only an axial dependence when the function being derived also only depends on the radius. That is, the operator in Eq. (12) applied on
with
which can be rewritten in the same form as Eq. (13). Similarly, the first differential operator of the fourth order error term in Eqs. (10) and (11) takes the form:
and the associated components in polar coordinates, when applied to a rotationally invariant function
which again meets the rotational invariance requirement, and can be rewritten in the same form as given in Eq. (13). The last differential error operator of the fourth order error term in Eqs. (10) and (11) is:
and can be shown to be anisotropic (i.e.
In this work, we only consider the gradient approximation of scalar functions over a square or cubic lattice of unit spacing, so we need to take
Here, we present the mathematical and computational aspects of convolution. As finite difference discretization and edge detection kernels are very similar, let us return to the mathematical foundations of these techniques. We often give examples involving 2D images, but we could give the same examples and talk about 2D discrete functions. We don’t know the exact coding in the MATLAB and Jacket libraries, as they are under license, but we do have a general idea about function algorithms, which is given in the next section.
The convolution product of two functions
If Eq. (20) defines the convolution of two functions in the space domain, it is also equivalent to the point wise product of these functions in the frequency domain. In mathematics, the convolution theorem for Fourier transforms [5, chap. 4] is formalized by the following equation, where
The Fourier transform of a function generates the frequency spectrum of a function. This representation can easily be extended to two dimensions, which is more suitable to image or spatial analysis if the image is a function of two spatial variables. In our case, the output of the Fourier transform will generate a 2D function in the frequency domain from the 2D spatial domain. High frequencies will correspond to information varying rapidly in the original function, while low frequencies correspond to slow variations.
By applying the inverse Fourier Transform to both sides of Eq. (21), we obtain a method for calculating the convolution of two functions:
Therefore, there are two possible ways of convoluting two functions: the Fourier transform pipeline in three steps (Fourier transform, frequency spectrum filtering, and Fourier inverse transform), or the application of a stencil in the spatial domain. Indeed, edge detection kernels acts as high pass filters, accentuating high frequency contributions to the image or to the 2D function in the form of edges and details.
Knowing that the properties of the Fourier transform also work for a sampled signal, we present the definition of the 2D discrete Fourier transform (2D DFT), where M and N represent the number of samples in each dimension, x and y are the discrete spatial variables, and u and v are the transform or frequency variables:
In addition to having the same properties as the continuous Fourier transform, which are outside the scope of this presentation and can be found in [6], there are two more in 2D DFT that are important: periodicity and separability. Since the discrete signal is sampled from the finite length sequences, its frequency spectrum will be periodic. In the spatial domain, this property allows us to slide the filter from one frontier to the opposite one, and then start again at the frontier where we began. Moreover, its Fourier transform is separable. The 2D DFT can be processed in two steps: applying 1D DFT on the lines, and then applying 1D DFT on the resulting columns. In the spatial domain, a 2D filter represented by
As an example, let us take kernelX, the 2D separable kernel in section (4.2):
We can easily deduce the main functionality of the 2D kernel, the gradient component in x that approximates the derivative in x, by calculating the difference between the first and third rows in the neighborhood of a point, instead of the first and third columns, because of the rotation of our axis system. The separated 1D filters give us further information about the kernel function: a smoothing mask is applied on the differentiating mask, in order to reduce noise that could be exaggerated by the first filter.
Now, to explain how to apply a circular convolution mask in the spatial domain, we go back to the definition of convolution for discrete functions, presented in 2D and 3D in Eqs. (25) and (26) respectively, where A is the image and B is the mask. Processing a convolution filter consists of computing the scalar product of the filter weights with the input values within a window of the filter dimensions surrounding each of the output values, after flipping the mask in each dimension, as described in [7, chap. 4].
In circular convolution, the equation is slightly modified to model periodic shifting. Here we consider a square image N by N:
The values on the image we see in the window of the K-by-K filter are distributed on a periodic surface. The 2D filter is rolled up in the opposite direction and will turn in a clockwise direction. For each output value, the stencil is lined up on the input values, the scalar product is applied, and then the stencil rotates to shift to the next position to compute. Therefore, in the spatial domain, an N-by-N image convoluted with a K-by-K filter requires
In the frequency domain, the 2D discrete Fourier transform and its inverse are computed using the divide-and-conquer algorithm of the Fast Fourier Transform (FFT and IFFT), which reduces the complexity from
In this section, we present a systematic method that can be applied in MATLAB to calculate the gradient of scalar functions on a square lattice, or in 2D, which is simply the gradient of the images. The 3D case is a straightforward extension of the 2D case. First, let us define a function with which it is possible to test the gradient discretization previously defined in section (2.2):
For the purpose of this test and for simplicity, we consider only a square image
Characteristics of the functionF(x,y).
Figures 1(a) and 1(b) show the surface shape and contour of this function.
When the gradient of a function needs to be computed, it may be tempting to apply Eqs. (10) and (11) directly. This is quite a simple approach, and in MATLAB can be applied as follows:
However, this is rather a naive and slow implementation for MATLAB. It is far more efficient to evaluate the gradient by using a convolution product, as presented in section (4.2).
Instead of calculating the gradient as previously shown, the use of a convolution product between the functions to differentiate them, and a kernel representing the weights and stencils of the finite difference approximation is more beneficial. For the isotropic discretization of Eqs. (10) and (11), this computation can be performed in MATLAB as follows:
A useful list of 2D and 3D kernels for calculating isotropic gradients is available in sections (6.1) and (6.2).
Sometimes the kernel is separable, which means that, instead of applying an nD convolution, we can apply an 1D convolution
Note that in the work of [1], one of the most important 3D isotropic kernels, which is the one that uses the nearest neighbors only, was not presented. In this work, two variants of this kernel were obtained, one using only 10 nearest neighbors and the other using only 18 nearest neighbors, and their MATLAB form are given in section (6.2). Moreover, the 3D kernel using 18 nearest neighbors has the advantage of being separable, which means that we should expect to be able to rapidly compute a 3D gradient of low isotropic order using the knowledge presented in this chapter. Also note that the higher isotropic order (>2nd) kernels given in sections (6.1) and (6.2) are not separable.
In this work, the order accuracy is defined by
The main point about the kernel that we make in this study is that, as its size increases, it becomes possible to set isotropic the higher order error term in the Taylor series expansion at fixed space second order accuracy. It is therefore important to be careful not to confuse space order accuracy and isotropic order accuracy. We believe that, based on this result, future research could provide other kernels (perhaps of similar size) for which the leading space order accuracy would be the same as the leading isotropic order accuracy. For some applications, finding such kernels could be a significant step forward. In fact, achieving leading higher space order accuracy with equal leading isotropic order accuracy might have greater impact than achieving leading low space order with very high isotropic order accuracy, as is currently the case. However, higher space order gradient discretizations may suffer from another non physical numerical artifacts, known as spurious oscillations.
As previously indicated, computation of the gradient by applying convolution is faster than using a simpler, more straightforward method, but a naive one. We present some numerical results in this section that will show that this is indeed true. We also show that using a GPU instead of a CPU significantly reduces computation time.
First, all performance testing consists in evaluating gradients of random 2D and 3D double precision images. Note that we suppose a periodic padding for these images. All these image gradients are computed using MATLAB with the -singleCompThread startup option. This is done for benchmarking purposes, because the reference case should be computed using a sequential algorithm, that is, with a single core only. The CPU used in this work is an Intel® Core i7-970 processor, while the GPU is an Nvidia® GeForce GTX 580. All computations on the GPU are performed in MATLAB via the Jacket plugin developed by AccelerEyes®. The version of MATLAB is R2010b, and the Jacket version is 2.0 (build a15607c).
The timing method on the CPU is the usual MATLAB tic; m-code; toc; procedure. However, this method is not suited for timing m-code on the GPU. For this, we refer the reader to the method proposed by [8]. In Figures 2-5, the time taken for one simulation "dot" or result "dot" is the average of a hundred simulations.
To test performance, five different algorithms are considered for computing the gradient:
MATLAB singlethread naive (section 4.1)
MATLAB singlethread convolution (section 4.2) [REFERENCE CASE]
MATLAB Jacket naive
MATLAB Jacket convolution
MATLAB Jacket GFOR + convolution
Note that all results differ with respect to machine accuracy in double precision, and that the padding of the images is computed on the fly to save computer memory. This is because padding is very cheap in terms of computing cost, when compared to the cost of evaluating the gradient.
Case (4), MATLAB Jacket convolution, is the GPU equivalent of reference case (2) with a CPU. Case (3) is the GPU equivalent of case (1) with a CPU. The last case (5), MATLAB Jacket GFOR + convolution, is a special case that is not available on the CPU. To explain this, let us suppose that the user wishes to evaluate the gradient of
In all the figures showing performance behavior, the y-axis is in log scale. Figures 2(a) and 2(b) show the performance speedup with the 2D 2nd and 14th order isotropic gradients as a function of image size. For large images, speedups of
Figures 3(a) and 3(b) show the same situation, but with the 3D 2nd and 8th order isotropic gradients. For large images, a speedup of
Figures 4(a) and 4(b) show the speedup with the 2D and 3D isotropic gradient as function of the isotropy order at a fixed image size. In 2D, images are 992x992, and in 3D they are 110x110x110. For high isotropy, speedups of
Figure 5 shows the speedup with the 3D 8th order isotropic gradient as a function of
For both 2D and 3D cases, as the isotropy order or the number of images to evaluate simultaneously increases, the speedup that can be achieved using the GPU also increases. This is to be expected, since the computational complexity increases. Nevertheless, the chances of obtaining a speedup of
We must remember that the speedups were computed with the reference case, which uses convolution. It is important to note that a speedup of
Speedup as a function of image size (2D isotropic gradient).
Speedup as a function of image size (3D isotropic gradient).
Speedup as a function of the gradient isotropy order.
Speedup as a function of the number of cubic images (110x110x110) to be evaluated simultaneously using the 3D 8th order isotropic gradient.
The lattice Boltzmann method is a computational approach that is mainly used for fluid flow simulation with its roots in the field of cellular automata [9]. This method is particularly useful for solving complex flow systems, such as multiphase flows, in porous media where classical approaches based on the Navier-Stokes equations, like finite volumes or finite elements, encounter some difficulties. The method we use is based on the original Ph.D. thesis of [10]. Since then, there have been several improvements. However, these enhancements are outside the scope of this chapter, and will not be described.
The isotropic gradients we present here are useful for simulating immiscible multiphase flows, where the orientation of the various fluid interfaces has to be computed very frequently. The gradients of the density of each fluid color (phase) define the normal orientation of the interface, and special operators are used to keep the different interfaces between the fluids defined. Moreover, the norm of these gradients serves as a means to introduce a certain amount of surface tension between the fluids.
Suppose we wish to simulate the behavior of three fluids. At a certain point in the algorithm, three image gradients need to be computed, corresponding to the interface normal of each fluid density. Here is where the methods presented in this chapter become useful, a situation described in section (4.5) with
In multiphase flow simulation, calculation of a high order isotropic gradient is the most expensive part of the method, and the use of Jacket, has enabled us to reduce the computational cost by an astonishing amount. This type of calculation would not have been possible, in a reasonable time, by applying plain MATLAB.
We end this section with a simulation example that shows the spinodal decomposition of a three phase flow. This flow consists of an initial random mixture, where each phase self-agglomerates until a steady state is achieved. Figure 6 shows the spinodal decomposition at various times (in lattice Boltzmann units). Note that this simulation is given for illustration purposes only, and that spinodal decomposition has been quantitatively studied by [11].
Random spinodal decomposition of a three phase flow.
In this section, we give several convolution kernels in the form of MATLAB m-code, which is very useful for calculating 2D/3D isotropic gradients on a square or cubic lattice. Most of the weights were taken from Ref. [1].
In this work, a detailed description of isotropic gradient discretizations and convolution products has been presented. These isotropic gradients are useful, and superior to anisotropic discretizations. This is especially true in the field of flow simulation, when the lattice Boltzmann method is used. However, high order isotropic gradients are computationally expensive. To address this issue, we combined the convolution product with the Jacket plugin in MATLAB and GPU hardware, which enabled us to achieve high computational speedups (up to
We extend our special thanks to Pavan Yalamanchili from AccelerEyes for his quick response to our queries and his generous support. We applied the sequence-determines-credit (SDC) approach to our listing of authors [12]. This work was supported by a grant from the NSERC (Natural Sciences and Engineering Research Council of Canada).
Today, advances in high-throughput technologies have generated huge amounts of human genomics data in public domains. These data are useful for medical and population genetics to understand the population history, human evolution and demographics, susceptibility to disease, and response to drug. Over time, humanity has experienced the exchange of genetic materials across populations, mainly due to population migrations [1], which have led to wide human genetic variations as results of interbreeding or mating between different populations previously isolated. These genetic variations observed in the human deoxyribonucleic acid (DNA) sequences are caused by inheritance processes, such as mutation and recombination. Generally, the mating process yields the genetic recombination break points, introduces some variations, and creates mixed DNA segments. As a consequence, current human populations are admixed [2, 3] with specific genomes displaying a mosaic of segments originating from different ancestral populations [1, 2, 4], wide phenotypic variations, divergent genetic ancestry, and different traits observed among individuals in worldwide population groups. Thus, it is critical to understand the dynamics related to the origin of these variations, the evolution process, and its consequences in human heredity and health.
\nStudying admixture patterns in human populations consists of characterization of admixture features in human populations, including admixture mapping and date to admixture events. Admixture mapping combines both the identification of genetic variants underlying the ethnic difference in disease risk and inference of ancestry estimates associated with these genetic variants. Estimation of ancestry is commonly known as genetic ancestry inference, which is either global or local ancestry inference. Global ancestry inference estimates the overall proportion contributed by each ancestral population to the admixed genome; while, local ancestry deconvolution (local ancestry inference) estimates the number of copies from a particular population at a given site [5]. Together, admixture mapping and date to admixture events provide a better understanding of the genetic variation features throughout modern human evolution, the demographics, and adaptive processes of human populations. Currently, analyzing admixture patterns has become central to genomics research, contributing to a wide range of biomedical applications. Current advance in technologies is facilitating the movement of people worldwide, thus influencing the complexity of population admixture dynamics and leading to multi-faceted admixture events. On the other hand, the determination of local ancestry through genotyping and microarray datasets has empowered the approaches for dating mutation, selection, and admixture events [6, 7].
\nThe significance of the local ancestry inference topic is viewed through the research interests it has raised over the last two decades. Several models exist for local ancestry deconvolution, including ANCESTRYMAP [8], ADMIXMAP [9], SABER [10], LAMP [11], LAMPLD/LAMPHAP [12], SUPPORTMIX [13], EILA [14], LOTER [15], etc. Figure 1 displays the implementation dynamics of different local ancestry deconvolution models graphically, indicating the time each model was introduced. Local ancestry inference is relevant in personalizing medicines, understanding complex diseases, localizing missing sequences in reference genomes and understanding the population history and demographics. Subsequently, several studies have particularly been focusing on dating past admixture events, relevant to population migrations, heritable genes associated to some diseases, and responses to treatment [16]. The date of admixture in a given population can be predicted by analyzing the ancestral track, break-points, and linkage disequilibrium (LD) [17]. Also, distinction between date of admixture events is made with the use of LD and ancestral tracts in the admixed genomes [17]. Nowadays, there are several models for predicting the age of an admixture event, which are classified into two main groups: LD-based approaches and haplotype-based approaches [17, 18]. These models use information from genomes of several population groups around the world as representative or equivalent ancient populations known to influence the migration and/or admixture processes, yielding observed admixed population patterns worldwide (Figure 2).
\nThe evolution of local ancestry deconvolution since 2003 to 2017.
A partial worldwide admixture painting map. The figure shows several worldwide admixed populations with patterns identified through published paper on population structure from 2008 to 2018. The population migrations within and between continents have resulted in different admixed populations ranging from one- to five-way admixtures.
In this chapter, we survey current models for deconvoluting local ancestry and dating admixture events and explore computational techniques used in these models. We highlight advances made so far in this genomic era and opportunities behind these models and challenges or gaps that still need to be addressed. This informs users and researchers on the current state of research, and orient future trends in designing more effective models, which account for current challenges and produce more accurate and biological relevant estimates. In the subsequent sections, we provide an overview of existing methods used for inferring local ancestry estimates and dating admixture events.
\nIn this section, we survey current models used to elucidate admixture patterns, including local ancestry estimates (deconvolution) and dating admixture events. These models assume that the T genotyped sites are biallelic and the genotype information of the K reference candidate ancestral and admixed populations are considered known. Ancestry at different sites or windows follows a Markov chain. Recombination is assumed to occur at every generation resulting in Poison recombination points with a rate which depends on both the recombination rate, \n
As pointed out previously, existing local ancestry inference models can be categorized into two main groups based on whether the model makes use of admixture/background linkage disequilibrium (LD) or not.
\nLD-based models account for LD in local ancestry deconvolution, and due to the importance of LD in disease mapping, the first local ancestry methods fall into this category. They assume that ancestry along an admixed individual genome follows a first order Markov chain. This means that the immediate past state captures all the information on past states [19]. As a result, LD-based models assume that, at every site, the observed admixed genotypes are generated by the unobserved ancestry, and hence, Hidden Markov Model (HMM) and its extensions are used to infer the unobserved (hidden) states. Thus, to deconvolute ancestry along the admixed genome, these models have three model parameters, namely the initial, transition and observation, or emission probability models. Due to uncertainty and the number of parameters involved, LD-based methods use Markov Chain Monte Carlo (MCMC), forward-backward, or Viterbi algorithms to determine the hidden ancestry sequence for a given individual. Falush et al. and Patterson et al. modeled ancestry switch between ancestry populations at a given site, \n
representing the first marker, and the transition probability between consecutive markers with \n
Admixture LD-based methods are models that account for LD that resulted from the admixture process. They do not model background LD. Admixture LD-based methods include the early methods, for example, STRUCTURE V2 [20], ANCESTRYMAP [8], and ADMIXMAP [9], which are based on the Bayesian framework. Early methods rely on markers that show significant difference in frequency between ancestral populations (AIMs). Admixture LD-based models assume that markers are independent and the global and ancestral allele frequencies are known. They integrate HMM with MCMC, and their switch model and initial and transition models are as in Eqs. (1) and (2), respectively. Since LD-based methods do not model background LD, their observation model depends on only the allele frequency of the ancestry at that site. For instance, assuming K = 2, Patterson et al. defined the emission probability by
where y and \n
SUPPORTMIX [11] models only admixture LD by combining support vector machines (SVMs) and HMM. It was proposed in 2012 to improve on the computational time and address the challenge of a few typed or nonexistent reference panels, which overall improve multi-way local ancestry deconvolution. SUPPORTMIX is the first model to allow the learning of ancestral surrogates given a pool of reference panels. As a result, it is capable to train ancestral populations that are bigger in size than those that are mixed. Since SVMs can handle huge datasets, SUPPORTMIX is faster than early methods. It uses the rich haplotype information. Also proposed in 2012, PCADMIX [24] divides the genome into contiguous windows of SNPs as in SUPPORTMIX. It leverages principal component analysis from proxy ancestral haplotypes to model admixture LD under a standard HMM. Similar to SUPPORTMIX, PCADMIX is fast and requires phased data. Nevertheless, SUPPORTMIX and PCADMIX do not model phase switch errors, and as a result, in 2013, SEQMIX [23] was proposed. Unlike all other admixture LD-based methods, SEQMIX is based on exome sequence, reads data, and uses HMM. SEQMIX models only admixture LD and prunes SNPs in background LD. As a result, to reduce noise and systematic biases from using all SNPs [10] whilst not fully modeling LD (background), admixture and background LD methods emerged [22].
\nSince the biological data often have some dependences that violate the independence assumption in standard HMM, admixture LD-based methods are often not realistic. To relax the independence assumption, the HMM is extended to either Markov HMM, factorial HMM, hierarchical HMM, or two-layer HMM or other multivariate statistical models such as multivariate normal distribution (MVN) and a rich ancestral haplotype data are used unlike early methods. This is the case for SABER [10], SWITCH [25], HAPAA [26], HAPMIX [4], MULTIMIX [27], ALLOY [28], and ELAI [29]. MHMMs were the first HMM extension in local ancestry. They were first implemented in SABER and later in SWITCH. SABER was the first method to model background LD in the genetic ancestry inference. MHMM assumes that the current observed haplotype depends on both the current ancestry and the immediate past observation. The difference in the MHMM and admixture LD HMM-based is that when ancestry switches between sites t − 1 and t, then the MHMM observation model depends on the joint distribution of allele frequencies at the two sites [6, 30], defined as follows [10]:
where \n
where \n
However, SABER has a large parameter set, and does not explicitly model background LD as it models background LD using first order Markov chain [22]; other methods such as SWITCH were proposed. SWITCH takes into recombination even if it does not result in an ancestry switch, emerged. In contrast to SABER, SWITCH conditions the MHMM on recombination. Similar to early methods, probability of recombination depends on the admixture generations, genetic distance between consecutive SNPs, and the recombination rate. Thus, if the transition probability model in SWITCH is marginalized over recombination, then it is similar to Eq. (2) for two-way and Eq. (5) for multi-way. Although SWITCH models background LD and estimates recombination rates, the authors recommended richer MHMM or other different models that would outperform the SWITCH and SABER pairwise models [25]. As a result, methods that use both large- and small-scale HMM, referred to as the HHMM, were introduced.
\nNon-LD methods neither model background nor admixture LD. They either remove SNPs in LD which is the case for LAMP [11] and WINPOP [31], or use all SNPs (linked and unlinked SNPs) without modeling LD; this is the case for EILA [14], RFMIX [32], and LOTER [15]. Since MHMMs have a large number of parameters and do not model LD explicitly, an algorithmic approach that divides genome into windows of SNPs, LAMP [11], emerged in 2008. LAMP is fast and robust, and can infer local ancestry even without proxy ancestral genotypes. This is the case for two-way admixtures. It uses the naive Bayes classifier and a clustering algorithm known as the iterative conditional modes. LAMP estimates the most probable ancestry at a site by applying the majority vote for each SNP [11]. Although accuracy is comprised, LAMP does not suffer from challenges of HMM and extension. As a result, LAMP underperforms in closely related populations, and hence it was extended to WINPOP [31], a dynamic programming algorithm. Unlike LAMP, WINPOP assumes at least one recombination event within each window and varies the window length depending on the genetic distance between populations. Hence, WINPOP and LAMP outperform other methods in closely and distantly related populations, respectively. Both LAMP and WINPOP assume unlinked markers and discards SNPs in LD.
\nAs the admixed sequence data availability increases, Maples et al. proposed a discriminative approach to estimate local ancestry, RFMIX [32]. A discriminative approach estimates the posterior probability directly and not via the joint probability distribution. In contrast to generative ancestry inference models, RFMIX uses the information contained in admixed individuals. This is advantageous in cases of genotyped few reference panels. This is the case for Native Americans [32]. RFMIX uses conditional random fields (CRFs) parametrized on random forests. It outperforms in multi-way admixtures maybe due to modeling phase switch errors. In 2013, EILA [14], a multivariate statistic based method, was proposed particularly to increase inference power through addressing three common challenges in local ancestry. Addressed challenges are the independence of SNP assumption, difficulties in identifying break points, and the use of three genotype values. Instead of raw genotypes, EILA uses a numerical value between 0 and 1. The score determines how close SNPs are to the ancestral populations. Breakpoints are a challenge to identify, but EILA identifies them by fused quantile regression facilitating the use of estimates in admixture dating. Finally, k-means classifiers are used to infer ancestry using all genotyped SNPs [14].
\nRecently, a software package that deconvolves local ancestry in multi-way admixtures for a wide range of species, LOTER [15], was proposed. LOTER can account for phase errors in two-way admixture only. It facilitates the local ancestry inference process and its application in non-model species [15]. Unlike other methods, LOTER needs no biological such as admixture time and recombination rate or statistical parameters such as, number of hidden states and misfit probabilities to deconvolve ancestry [15]. Although it uses the Li and Stephen’s copying model [33] as in LAMPLD/LAMPHAP, LOTER is a nonprobabilistic approach formulated from an optimization problem. Its solution is obtained through dynamic programming.
\nFinally, different existing LD and non-LD-based local ancestry inference models are summarized in Table 1 extracted from Geza et al. [34].
\nSoftware | \nMulti-way | \nAccount LD | \nLD model | \nBiological/statistical parameters | \nReference populations | \nAdmixed populations | \nYear of publication | \n
---|---|---|---|---|---|---|---|
STRUCTURE V2* | \n✓ | \n✓ | \nHMM | \nMarkers, and ancestry proportions | \nUnphased | \nUnphased | \nAugust 2003 | \n
ANCESTRYMAP* | \n✗ | \n✓ | \nHMM | \nPhysical map, recombination and ancestry proportions | \nUnphased | \nUnphased | \nMay 2004 | \n
ADMIXMAP* | \n✓ | \n✓ | \nHMM | \nPhysical map and ancestry proportions | \nUnphased | \nUnphased | \nMay 2004 | \n
SABER | \n✓ | \n✓ | \nMHMM | \nPhysical map or recombination distance | \nPhased/unphased | \nPhased/unphased | \nJuly 2006 | \n
“LAMP” | \n✓ | \n✗ | \n✗ | \nAdmixture generations, LD threshold, and physical map | \nUnphased | \nUnphased | \nFebruary 2008 | \n
HAPAA | \n✓ | \n✓ | \nHHMM | \nAdmixture generations and genetic divergence | \nPhased | \nPhased | \nFebruary 2008 | \n
SWITCH | \n✓ | \n✓ | \nMHMM | \nRecombination rate | \nPhased | \nPhased | \nFebruary 2008 | \n
GEDI-ADMX | \n✓ | \n✓ | \nFixed size FHMM | \nAdmixed and ancestral SNPs (physical map) | \nPhased | \nUnphased | \nMay 2009 | \n
WINPOP | \n✓ | \n✗ | \n✗ | \nRecombination, admixture generations, LD threshold, and physical map | \nUnphased | \nUnphased | \nJune 2009 | \n
HAPMIX | \n✗ | \n✓ | \nHHMM | \nGenetic map mutation rate and admixed and ancestral SNPs | \nPhased | \nUnphased | \nJune 2009 | \n
CHROMOPAINTER | \n✓ | \n✓ | \nCo-ancestry matrix | \nRecombination rate | \nPhased | \nPhased | \nJanuary 2012 | \n
LAMPLD | \n✓ | \n✓ | \nHHMM | \nNumber of hidden states, window size and physical map | \nPhased | \nUnphased | \nMay 2012 | \n
SUPPORTMIX* | \n✓ | \n✓ | \nHMM | \nAdmixture generations and genetic map | \nPhased | \nPhased | \nJune 2012 | \n
PCADMIX* | \n✓ | \n✓ | \nWindows of blocks of SNPs | \nGenetic map and window size | \nPhased | \nPhased | \nAugust 2012 | \n
mSPECTRUM | \n✓ | \n✓ | \n\n | SNPs, mutation and recombination rate | \nPhased | \nPhased | \nAugust 2012 | \n
MULTIMIX | \n✓ | \n✓ | \nMVN | \nGenetic map, legend file and misfitting probabilities | \nPhased/unphased | \nPhased/unphased | \nNovember 2012 | \n
ALLOY | \n✓ | \n✓ | \nNon-homogeneous VLMC | \nMarkers, ancestral proportions, admixture generations, and genetic map | \nPhased | \nPhased | \nFebruary 2013 | \n
RFMIX | \n✓ | \n✗ | \n✗ | \nGenetic map, window size, and admixture generations | \nPhased | \nPhased | \nAugust 2013 | \n
EILA | \n✓ | \n✗ | \n✗ | \nPhysical map | \nUnphased (no missing values) | \nUnphased (no missing values) | \nNovember 2013 | \n
SEQMIX | \n✓ | \n✗ | \n✗ | \nGenetic map | \nUnphased | \nUnphased | \nNovember 2013 | \n
ELAI | \n✓ | \n✓ | \nTwo layer HMM | \nAdmixture generations, lower and upper cluster | \nPhased/unphased | \nPhased/unphased | \nMay 2014 | \n
LOTER | \n✓ | \n✗ | \n✗ | \n— | \nPhased | \nPhased | \nNovember 2017 | \n
Existing 20 ancestry deconvolution tools: ✓ indicates the ability of the software to perform a specified task, ✗ indicates the inapplicability of the task by a particular tool. Unless explicitly specified, LD refers to background LD.
Several models are now available to determine the date of admixture events in a given admixed genome. Breakpoints of haplotypes are used by some models while others focus on the ancestry blocks. Models based on ancestry blocks for dating admixture are formulated using either an empirical criteria or variants associated with a specific population. In order to determine the average length of the admixture block, these methods then assign ancestry on predefined windows using either wavelet transformation or conditional random fields [35]. On the other hand, there are models requiring rapid decrease in haplotype block sizes to estimate the date of the admixture event [36]. This suggests that, in general, models used for dating admixture events can be subdivided in two main classes [17, 18], namely those based on LD and those based on the haplotype distribution, as mentioned earlier.
\nAn admixture event is mainly characterized by the transfer of genes from the ancestral populations to the admixed ones. This leads to the appearance of linkage disequilibrium with regard to the ancestral populations. However, this LD formed often decreases with time. Also, the rate of decay of LD is a function of recombination and the proportion of the admixture [35]. Inversely, many methods employ this rate to calculate the time since the admixture event occurs.
\nIn 2011, Moorjani et al. introduced a method to determine the weighted correlation for a pair of SNPs [36]. This correlation coefficient is further used to measure the LD with ancestral populations [37]. The time of admixture is then determined by analyzing the correlation with respect to the genetic distance, and also fitting using a least squares method the decay of the correlation [35]. This method got improved in 2011 by Loh et al. [18]. The major improvements are in terms of computation. Loh et al. employed instead a fast Fourier transform and other faster techniques to determine the optimal distance to the fitting curve. This method has another advantage that it reduces considerable biases in the estimation of the time of admixture [18, 36]. Later, Loh et al.’s method was improved by Pickrell et al. [38] by introducing the notion of mixture exponential decay in order to take into account the admixture events in the given admixed population history. It mainly focuses on the decay of the LD.
\nLet us consider three ancestral populations \n
The date of admixture between population \n
with \n
Among the haplotype-based approaches, there is the likelihood method introduced in 2009 by Price et al. [4]. It basically determines the number of breakpoints using Hidden Markov Model. It is also able to determine the number of alleles at a particular site inherited from a given ancestor in a population. This is done in two steps. First, the method consists in identifying haplotype from the proxy ancestry populations, and secondly, the origin of each haplotype bock is identified by comparing their likelihood for one ancestral population versus the others. Considering an admixed genome, the likelihood of an observed allele is given by
with \n
where ζ is the total Morgan length, γ the proportion of admixture, and C the observed number of breakpoints [4].
\nOn the other hand, Pugach et al. [17] employed the wavelet transform to design a haplotype block approach. The aim of this method is to derive the time of admixture of a given population using the simple hybrid isolation model. It proceeds in two main steps. First, it obtains a signal of admixture from the admixed data using the principal component technique. The second step consists in deriving the date of admixture using the signal obtained in the first step [17].
\nPool and Nielsen also built a haplotype-based approach. It used precautionary ancestral populations to infer the date of admixture from the genome of an admixed population [39]. It assumed that after a number of generation g, the distribution of the ancestral haplotypes follows exponential distribution given by
where \n
Further methods include that of Gravel developed in 2012 for the identification of multiple ancestral populations in a given admixture dataset [40]. Also, Jin et al. [41] came up with a similar method to explain admixture dynamics. The method incorporates several models including gradual admixture (GA), hybrid isolated (HI), and continuous gene flow (CGF) models [41], which can be extended to GA-Isolation (GA-I) and CGF-Isolation (CGF-I) by considering isolation after admixture [42]. Hellenthal et al. [43] on the other hand built up on the work of Lawson et al. [44] on dating admixture. This method particularly considers the genome of an admixed individual to be a set chunk DNA coming from other individuals. The scheme of this method is mainly made of two stages. The first stage consists in dividing the genome into chunks and matching each of them to the proper ancestral individual. This stage is achieved with the help of Hidden Markov Model. The second stage consists in identifying haplotypes and determining their respective ancestral population [43, 44]. Moreover, the admixture event and its date are derived by fitting the decay of the ancestral haplotype with an exponential distribution curve. Moreover, Ni et al. developed a method based on the observation that the date of admixture events is related to the model used. Their method consists in using the likelihood ratio test to identify the best model for the inference of the date of admixture. Furthermore, they are able to estimate several admixture events with the given optimal model [35].
\nFinally, different existing models and tools for dating admixture events are summarized in Table 2 extracted from Chimusa et al. [35].
\nTool | \nCategory | \nAdmixture model | \nPriori proxy ancestral raw data | \nMulti-way events | \nOnline link | \n
---|---|---|---|---|---|
ROLLOFF | \nLD-based model | \nHI | \nYes | \nNo | \n\n | \n
ALDER | \nHI | \nYes | \nNo | \n\n | \n|
MALDER | \nHI | \nYes | \nYes | \n\n | \n|
CAMer | \nHI, GA, CGF, GA-I, CGF-I | \nYes | \nYes | \n\n | \n|
IMAAPs | \nHI, GA, CGF, GA-I, CGF-I | \nYes | \nYes | \n\n | \n|
StepPCO | \nHaplotype/ancestry block size distribution-based model | \nHI | \nYes | \nYes | \n\n | \n
Adware | \n\n | \nHI, Dual-admixture | \nYes | \nYes | \n\n | \n
HAPMIX | \n\n | \nHI | \nYes | \nYes | \n\n | \n
MultiWaveIner | \n\n | \nHI | \nYes | \nYes | \n\n \n | \n
GLOBBERTROTTER | \n\n | \nHI, GA, CGF | \nNo | \nYes | \n\n | \n
Tracts | \n\n | \nHI, GA, CGF | \nNo | \nYes | \n\n | \n
Ancestry_HMM | \n\n | \nHI | \nNo | \nNo | \n\n | \n
Existing dating admixture genomic tools.
Although several models exist to deconvolve local ancestry, most studies that evaluate such models showed that deviations in local ancestry estimates still exist in multi-way admixtures. Deviations in local ancestry also result from genetic drift, miscalling true ancestry, and genotyping errors. However, the signals from these factors affect the whole genome while that of unmodelled natural selection affects particular regions. For example, Chen et al. using four local ancestry inference models to scan for disease-related loci through admixture mapping showed that although all of them are LD based and divide the genome into windows of continuous SNPs, MULTIMIX and LAMPLD estimates differed in almost 20% of the analyzed SNPs. This results from the differences in the biological and statistical parameters they require and the mathematical approaches they use. Another association study by Chimusa et al. [45] also pointed out that admixture mapping is still limited by inaccuracies in multi-way local ancestry deconvolution when they evaluated one LD-based and one non-LD-based local ancestry models, WINPOP and LAMPLD.
\nInaccuracies in local ancestry estimates may result from the use of statistical or biological parameters in the estimation process, which are not always accurate when provided. It could also be due to the dependence of models on reference panels which for some populations are few or even not sampled for others. This is the case for the Native Americans. More so for other admixed populations, their history is not well known. When applied to ancient admixtures, existing methods may yield spurious estimates as they were designed for recent admixtures. Existing methods do not account for natural selection; hence, some deviations exist in regions that are under selection [45]. Also, most of them are benchmarked for three-way admixtures.
\nSince each model was introduced to address a particular challenge that models before it faced, it is clearly expected that no model or tool can achieve the best performance in all admixture scenarios and not trading estimate accuracy with computational speed. Using existing studies by Geza et al. [34], more than 50% of studies that either introduced a model or evaluated methods for association mapping showed that LAMPLD/LAMPHAP outperforms most LD-based methods. And the only LD-based method than outperformed LAMPLD is ELAI; however, this is the only study that assessed ELAI with other models. In cases where LD-based models were compared to non-LD-based models, RFMIX outperformed LAMPLD in three cases highlighted in [34], while another separate study aiming to determine the place of admixture of an admixed population RFMIX also outperformed. This could be because RFMIX can deconvolve ancestry in closely related populations [46]. However, a recent assessment between RFMIX and LOTER resulted in LOTER outperforming in ancient admixtures [15].
\nGenerally, each model is implemented as a tool in local ancestry deconvolution, existing as individual scripts requiring unique inputs and producing unique outputs. This challenges researchers with a limited computational background; thus, there is lack of a unified framework which can require a standard easy to manipulate input files and output results in a way that is easy to process for further application. In conclusion, for informed decisions on models and algorithms, existing models or tools should be assessed within a unified framework. This will allow them to be tested on different admixture scenarios and also incorporating most state-of-the-art LD and non-LD based models.
\nThe evolution of human populations and the history of the mixture of these populations have been deciphered using statistical and computational methods. These methods have been found to perform well when dealing with single point admixture event in two-way admixed populations [35]. However, as any method, they not only have advantages but also pitfalls regarding the estimation of admixture dates in some cases. It is challenging to fit to real admixed populations (for more than 3-way admixture context) in the existing models dating admixture events due to several reasons, including reliance to optimal local ancestry estimates and accurate ancestry breakpoints. This suggests that there is still a need for designing an integrative or a new model to dating admixture events for current multi-way admixed populations to further advance our understanding of human demographics and movement, and facilitate admixture mapping and estimation of the age of a disease locus contributing to disease risk.
\nIn addition, it have been discovered that the mixture exponential decay model over-estimates the date of older admixture events [35] and was suggested to detect at most three admixture events. As mentioned earlier, Ni et al. [47] dealt with the optimization of the method used in dating admixture estimation. They took into account several models but the evaluation of their technique is not effective in the estimation of ancient and multi admixture events [35, 47]. On the other hand, several practical considerations can further limit these approaches including the use of proxy ancestry populations in the estimations which could bias the accuracy of the result. This is the case when dealing for instance with low sample size and inappropriate proxy ancestral populations [35]; the requirement of having accurate LD patterns, ancestry haplotypes distribution, and a big sample size of the admixed population. Thus, there is a need for an adequate model for inferring different dates of admixture events and matching real admixture history using proxy ancestry-based methods [35].
\nCurrently, more than 20 models exist and are implemented as software to deconvolve local ancestry and 12 tools for dating admixture events. In this chapter, we discussed in detail and summarized the most commonly used models, the model assumptions, statistical and biological parameters they require, and existing challenges. This discussion highlights the need for designing more effective models, which account for current challenges and produce more accurate and biologically relevant estimates. Furthermore, it provides useful information for the implementation of practical tools, which consider current medical and population genetic demands. More importantly, this may guide users in the choice of appropriate tools for specific applications and can assist software developers in designing more advanced tools for local ancestry deconvolution and dating admixture events.
\nSome of the authors are supported in part by the National Institutes of Health (NIH) Common Fund [grant numbers U24HG006941 (H3ABioNet) and 1U01HG007459?01 (SADaCC)]. One of the authors is fully funded by the Organization for Women in Science for the Developing World (OWSD) and Swedish International Development Cooperation Agency (Sida). The content of this publication is solely the responsibility of the authors and does not necessarily represent the official views of the funders.
\nThe authors declare that they have no competing interest.
IntechOpen publishes different types of publications
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\\n\\nCompacts provide a mid-length publishing format that bridges the gap between journal articles, book chapters, and monographs, and cover content across all scientific disciplines.
\\n\\nCompacts are the preferred publishing option for brief research reports on new topics, in-depth case studies, dissertations, or essays exploring new ideas, issues, or broader topics on the research subject. Compacts usually have between 50 and 130 pages.
\\n\\nCollection of papers presented at conferences, workshops, symposiums, or scientific courses, published in book format
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\n\nRESEARCH CHAPTER – A research chapter reports the results of original research thus contributing to the body of knowledge in a particular area of study.
\n\nREVIEW CHAPTER – A review chapter analyzes or examines research previously published by other scientists, rather than reporting new findings thus summarizing the current state of understanding on a topic.
\n\nCASE STUDY – A case study involves an in-depth, and detailed examination of a particular topic.
\n\nPERSPECTIVE CHAPTER – A perspective chapter offers a new point of view on existing problems, fundamental concepts, or common opinions on a specific topic. Perspective chapters can propose or support new hypotheses, or discuss the significance of newly achieved innovations. Perspective chapters can focus on current advances and future directions on a topic and include both original data and personal opinion.
\n\nINTRODUCTORY CHAPTER – An introductory chapter states the purpose and goals of the book. The introductory chapter is written by the Academic Editor.
\n\nMonographs is a self-contained work on a particular subject, or an aspect of it, written by one or more authors. Monographs usually have between 130 and 500 pages.
\n\nTYPES OF MONOGRAPHS:
\n\nSingle or multiple author manuscript
\n\nCompacts provide a mid-length publishing format that bridges the gap between journal articles, book chapters, and monographs, and cover content across all scientific disciplines.
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