Wavelet Screening: a novel approach to analyzing GWAS data

Background Traditional methods for single-variant genome-wide association study (GWAS) incur a substantial multiple-testing burden because of the need to test for associations with a vast number of single-nucleotide polymorphisms (SNPs) simultaneously. Further, by ignoring more complex joint effects of nearby SNPs within a given region, these methods fail to consider the genomic context of an association with the outcome. Results To address these shortcomings, we present a more powerful method for GWAS, coined ‘Wavelet Screening’ (WS), that greatly reduces the number of tests to be performed. This is achieved through the use of a sliding-window approach based on wavelets to sequentially screen the entire genome for associations. Wavelets are oscillatory functions that are useful for analyzing the local frequency and time behavior of signals. The signals can then be divided into different scale components and analyzed separately. In the current setting, we consider a sequence of SNPs as a genetic signal, and for each screened region, we transform the genetic signal into the wavelet space. The null and alternative hypotheses are modeled using the posterior distribution of the wavelet coefficients. WS is enhanced by using additional information from the regression coefficients and by taking advantage of the pyramidal structure of wavelets. When faced with more complex genetic signals than single-SNP associations, we show via simulations that WS provides a substantial gain in power compared to both the traditional GWAS modeling and another popular regional association test called SNP-set (Sequence) Kernel Association Test (SKAT). To demonstrate feasibility, we applied WS to a large Norwegian cohort (N=8006) with genotypes and information available on gestational duration. Conclusions WS is a powerful and versatile approach to analyzing whole-genome data and lends itself easily to investigating various omics data types. Given its broader focus on the genomic context of an association, WS may provide additional insight into trait etiology by revealing genes and loci that might have been missed by previous efforts.

CoverageMaster: comprehensive CNV detection and visualization from NGS short reads for genetic medicine applications

CoverageMaster (CoM) is a Copy Number Variation (CNV) calling algorithm based on depth-of-coverage maps designed to detect CNVs of any size in exome (WES) and genome (WGS) data. The core of the algorithm is the compression of sequencing coverage data in a multiscale Wavelet space and the analysis through an iterative Hidden Markov Model (HMM). CoM processes WES and WGS data at nucleotide scale resolution and accurately detect and visualize full size range CNVs, including single or partial exon deletions and duplications. The results obtained with this approach support the possibility for coverage-based CNV callers to replace probe-based methods such array CGH and MLPA in the near future.

On splitting for cubic spline wavelets with four zero moments on an interval

В пространстве кубических сплайнов построены вейвлеты, удовлетворяющие однородным граничным условиям Дирихле и обнулению первых четырех моментов. Получены неявные соотношения, связывающие сплайн-коэффициенты разложения на начальном уровне со сплайн-коэффициентами и вейвлет-коэффициентами на вложенном уровне ленточной системой линейных алгебраических уравнений с невырожденной матрицей. После расщепления на четные и нечетные уравнения матрица преобразования имеет пять (вместо трех в случае двух нулевых моментов) диагоналей. Доказано наличие строгого диагонального доминирования по столбцам. Для сравнения использованы вейвлеты с двумя нулевыми моментами и интерполяционные кубические сплайновые вейвлеты. Результаты численных экспериментов показывают, что схема с четырьмя нулевыми моментами точнее при аппроксимации функций, но грубее при аппроксимации второй производной. The article examines the problem of constructing a splitting algorithm for cubic spline wavelets. First, a cubic spline space is constructed for splines with homogeneous Dirichlet boundary conditions. Then, using the first four zero moments, the corresponding wavelet space is constructed. The resulting space consists of cubic spline wavelets that satisfy the orthogonality conditions for all thirddegree polynomials. The originality of the research lies in obtaining implicit relations connecting the coefficients of the spline expansion at the initial level with the spline coefficients and wavelet coefficients at the embedded level by a band system of linear algebraic equations with a nondegenerate matrix. Excluding the even rows of the system, the resulting transformation algorithm is obtained as a solution to a sequence of band systems of linear algebraic equations with five (instead of three in the case of two zero moments) diagonals. The presence of strict diagonal dominance over the columns is proved, which confirms the stability of the computational process. For comparison, we adopt the results of calculations using wavelets orthogonal to first-degree polynomials and interpolating cubic spline wavelets with the property of the best mean-square approximation of the second derivative of the function being approximated. The results of numerical experiments show that the scheme with four zero moments is more accurate in the approximation of functions, but becomes inferior in accuracy to the approximation of the second derivative.

Inversion of electromagnetic induction data using a novel wavelet-based and scale-dependent regularization term

<p>In frequency domain Electromagnetic Induction (EMI) surveys, an image of the electrical conductivity of the subsurface is obtained non-invasively. The electrical conductivity can be related to important subsurface properties such as the porosity, saturation or water conductivity via Archie’s law. The advantage of geophysical EMI surveys is its cost-effectiveness because it is a non-contacting method, one can easily walk with the device or mount in on a vehicle or a helicopter (AEM).</p><p>The process of finding the conductivity profile from the collected field data is an ill-posed inverse problem. Regularization improves the stability of the inversion and, based on Occam’s razor principle, a smoothing constraint is typically used with a very large number of thin layers. However, the conductivity profiles are not always expected to be smooth. Another alternative is to use a predefined number of layers and to invert for their conductivity and thickness. This can yield sharp contrasts in conductivity. In practice however, the real underground might be either blocky or smooth, or somewhere in between. Those standard constraints are thus not always appropriate.</p><p>We develop a new minimum-structure inversion scheme in which we transform the model into the wavelet space and impose a sparsity constraint. This sparsity constrained inversion scheme minimizes an objective function with a least-squares data misfit and a sparsity measure of the model in the wavelet domain. With a solid understanding of wavelet theory, a novel and intuitive model misfit term was developed, allowing for both smooth and blocky models, depending on the chosen wavelet basis. A model in the wavelet domain has both temporal (i.e. low and high frequencies) and spatial resolution, and penalizing small-scale coefficients effectively reduces the complexity of the model.</p><p>Comparing the novel scale-dependent wavelet-based regularization scheme with wavelet-based regularization with no scale-dependence, revealed significantly better results (Figure A and B) w.r.t. the true model. Comparing with standard Tikhonov regularization (Figure C and D) shows that our scheme can recover high amplitude anomalies in combination with globally smooth profiles. Furthermore, the adaptive nature of the inversion method  (due to the choice of wavelet) allows for high flexibility because the shape of the wavelet can be exploited to generate multiple representations (smooth, blocky or intermediate) of the inverse model.</p><p><img src="https://contentmanager.copernicus.org/fileStorageProxy.php?f=gnp.c279d29567ff54198400161/sdaolpUECMynit/12UGE&app=m&a=0&c=ae78587f05a9ca0f7486a4013a5ef551&ct=x&pn=gnp.elif&d=1" alt="" width="646" height="438"></p><p>We have introduced an alternative inversion scheme for EMI surveys that can be extended to any other 1D geophysical method. It involves a new model misfit or regularization term based on the wavelet transform and scale-dependent weighting which can easily be combined with the existing framework of deterministic inversion (gradient-based optimization methods, L-curve criterion for optimal regularization parameter). A challenge remains to select the optimal wavelet, however, the ensemble of inversion results with different wavelets can also be used to qualitatively assess uncertainty.</p>

On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.

Wavelet Screening: a novel approach to analysing GWAS data

SummaryWe present here an alternative method for genome-wide association study (GWAS) that is more powerful than traditional GWAS methods for locus detection. Single-variant GWAS methods incur a substantial multiple-testing burden because of the vast number of single nucleotide polymorphisms (SNPs) being tested simultaneously. Furthermore, these methods do not consider the functional genetic effect on the outcome because they ignore more complex joint effects of nearby SNPs within a region. By contrast, our method reduces the number of tests to be performed by screening the entire genome for associations using a sliding-window approach based on wavelets. In this context, a sequence of SNPs represents a genetic signal, and for each screened region, we transform the genetic signal into the wavelet space. The null and alternative hypotheses are modelled using the posterior distribution of the wavelet coefficients. We enhance our decision procedure by using additional information from the regression coefficients and by taking advantage of the pyramidal structure of wavelets. When faced with more complex signals than single-SNP associations, we show through simulations that Wavelet Screening provides a substantial gain in power compared to both the traditional GWAS modelling as well as another popular regional-based association test called ‘SNP-set (Sequence) Kernel Association Test’ (SKAT). To demonstrate feasibility, we re-analysed data from the large Norwegian HARVEST cohort.

Learning vector quantization and relevances in complex coefficient space

In this contribution, we consider the classification of time series and similar functional data which can be represented in complex Fourier and wavelet coefficient space. We apply versions of learning vector quantization (LVQ) which are suitable for complex-valued data, based on the so-called Wirtinger calculus. It allows for the formulation of gradient-based update rules in the framework of cost-function-based generalized matrix relevance LVQ (GMLVQ). Alternatively, we consider the concatenation of real and imaginary parts of Fourier coefficients in a real-valued feature vector and the classification of time-domain representations by means of conventional GMLVQ. In addition, we consider the application of the method in combination with wavelet-space features to heartbeat classification.