The large number of markers in genome-wide prediction demands the use of methods with regularization and model comparison based on some hold-out test prediction error measure. In quantitative ...genetics, it is common practice to calculate the Pearson correlation coefficient (
r
2
) as a standardized measure of the predictive accuracy of a model. Based on arguments from the bias–variance trade-off theory in statistical learning, we show that shrinkage of the regression coefficients (i.e., QTL effects) reduces the prediction mean squared error (MSE) by introducing model bias compared with the ordinary least squares method. We also show that the LASSO and the adaptive LASSO (ALASSO) can reduce the model bias and prediction MSE by adding model variance. In an application of ridge regression, the LASSO and ALASSO to a simulated example based on results for 9,723 SNPs and 3,226 individuals, the best model selected was with the LASSO when
r
2
was used as a measure. However, when model selection was based on test MSE and coefficient of determination
R
2
the ALASSO proved to be the best method. Hence, use of
r
2
may lead to selection of the wrong model and therefore also nonoptimal ranking of phenotype predictions and genomic breeding values. Instead, we propose use of the test MSE for model selection and
R
2
as a standardized measure of the accuracy.
Genome-wide marker data are used both in phenotypic genome-wide association studies (GWAS) and genome-wide prediction (GWP). Typically, such studies include high-dimensional data with thousands to ...millions of single nucleotide polymorphisms (SNPs) recorded in hundreds to a few thousands individuals. Different machine-learning approaches have been used in GWAS and GWP effectively, but the use of neural networks (NN) and deep-learning is still scarce. This study presents a NN model for genomic SNP data.
We show, using both simulated and real pig data, that regularization is obtained using weight decay and dropout, and results in an approximate Bayesian (ABNN) model that can be used to obtain model averaged posterior predictions. The ABNN model is implemented in mxnet and shown to yield better prediction accuracy than genomic best linear unbiased prediction and Bayesian LASSO. The mean squared error was reduced by at least 6.5% in the simulated data and by at least 1% in the real data. Moreover, by comparing NN of different complexities, our results confirm that a shallow model with one layer, one neuron, one-hot encoding and a linear activation function performs better than more complex models.
The ABNN model provides a computationally efficient approach with good prediction performance and in which the weight components can also provide information on the importance of the SNPs. Hence, ABNN is suitable for both GWP and GWAS.
The genetic basis of phenotypic traits is highly variable and usually divided into mono-, oligo- and polygenic inheritance classes. Relatively few traits are known to be monogenic or oligogeneic. The ...majority of traits are considered to have a polygenic background. To what extent there are mixtures between these classes is unknown. The rapid advancement of genomic techniques makes it possible to directly map large amounts of genomic markers (GWAS) and predict unknown phenotypes (GWP). Most of the multi-marker methods for GWAS and GWP falls into one of two regularization frameworks. The first framework is based on formula omitted-norm regularization (e.g. the LASSO) and is suitable for mono- and oligogenic traits, whereas the second framework regularize with the formula omitted-norm (e.g. ridge regression; RR) and thereby is favourable for polygenic traits. A general framework for mixed inheritance is lacking. We have developed a proximal operator algorithm based on the recent LAVA regularization method that jointly performs formula omitted- and formula omitted-norm regularization. The algorithm is built on the alternating direction method of multipliers and proximal translation mapping (LAVA ADMM). When evaluated on the simulated QTLMAS2010 data, it is shown that the LAVA ADMM together with Bayesian optimization of the regularization parameters provides an efficient approach with lower test prediction mean-squared-error (65.89) than the LASSO (66.11), Ridge regression (83.41) and Elastic net (66.11). For the real pig data the test MSE of the LAVA ADMM is 0.850 compared to the LASSO, RR and EN with 0.875, 0.853 and 0.853, respectively. This study presents the LAVA ADMM that is capable of joint modelling of monogenic major genetic effects and polygenic minor genetic effects which can be used for both genome-wide assoiciation and prediction purposes. The statistical evaluations based on both simulated and real pig data set shows that the LAVA ADMM has better prediction properies than the LASSO, RR and EN. Julia code for the LAVA ADMM is available at: https://github.com/patwa67/LAVAADMM.
The goal of genome-wide prediction (GWP) is to predict phenotypes based on marker genotypes, often obtained through single nucleotide polymorphism (SNP) chips. The major problem with GWP is ...high-dimensional data from many thousands of SNPs scored on several thousands of individuals. A large number of methods have been developed for GWP, which are mostly parametric methods that assume statistical linearity and only additive genetic effects. The Bayesian additive regression trees (BART) method was recently proposed and is based on the sum of nonparametric regression trees with the priors being used to regularize the parameters. Each regression tree is based on a recursive binary partitioning of the predictor space that approximates an unknown function, which will automatically model nonlinearities within SNPs (dominance) and interactions between SNPs (epistasis). In this study, we introduced BART and compared its predictive performance with that of the LASSO, Bayesian LASSO (BLASSO), genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space (RKHS) regression and random forest (RF) methods.
Tests on the QTLMAS2010 simulated data, which are mainly based on additive genetic effects, show that cross-validated optimization of BART provides a smaller prediction error than the RF, BLASSO, GBLUP and RKHS methods, and is almost as accurate as the LASSO method. If dominance and epistasis effects are added to the QTLMAS2010 data, the accuracy of BART relative to the other methods was increased. We also showed that BART can produce importance measures on the SNPs through variable inclusion proportions. In evaluations using real data on pigs, the prediction error was smaller with BART than with the other methods.
BART was shown to be an accurate method for GWP, in which the regression trees guarantee a very sparse representation of additive and complex non-additive genetic effects. Moreover, the Markov chain Monte Carlo algorithm with Bayesian back-fitting provides a computationally efficient procedure that is suitable for high-dimensional genomic data.
Genome-wide prediction (GWP) has become the state-of-the art method in artificial selection. Data sets often comprise number of genomic markers and individuals in ranges from a few thousands to ...millions. Hence, computational efficiency is important and various machine learning methods have successfully been used in GWP. Neural networks (NN) and deep learning (DL) are very flexible methods that usually show outstanding prediction properties on complex structured data, but their use in GWP is nevertheless rare and debated. This study describes a powerful NN method for genomic marker data that can easily be extended. It is shown that a one-dimensional convolutional neural network (CNN) can be used to incorporate the ordinal information between markers and, together with pooling and
-norm regularization, provides a sparse and computationally efficient approach for GWP. The method, denoted CNNGWP, is implemented in the deep learning software Keras, and hyper-parameters of the NN are tuned with Bayesian optimization. Model averaged ensemble predictions further reduce prediction error. Evaluations show that CNNGWP improves prediction error by more than 25% on simulated data and around 3% on real pig data compared with results obtained with GBLUP and the LASSO. In conclusion, the CNNGWP provides a promising approach for GWP, but the magnitude of improvement depends on the genetic architecture and the heritability.
Genome-wide prediction has become the method of choice in animal and plant breeding. Prediction of breeding values and phenotypes are routinely performed using large genomic data sets with number of ...markers on the order of several thousands to millions. The number of evaluated individuals is usually smaller which results in problems where model sparsity is of major concern. The LASSO technique has proven to be very well-suited for sparse problems often providing excellent prediction accuracy. Several computationally efficient LASSO algorithms have been developed, but optimization of hyper-parameters can be demanding.
We have developed a novel automatic adaptive LASSO (AUTALASSO) based on the alternating direction method of multipliers (ADMM) optimization algorithm. The two major hyper-parameters of ADMM are the learning rate and the regularization factor. The learning rate is automatically tuned with line search and the regularization factor optimized using Golden section search. Results show that AUTALASSO provides superior prediction accuracy when evaluated on simulated and real bull data compared to the adaptive LASSO, LASSO and ridge regression implemented in the popular glmnet software.
The AUTALASSO provides a very flexible and computationally efficient approach to GWP, especially when it is important to obtain high prediction accuracy and genetic gain. The AUTALASSO also has the capability to perform GWAS of both additive and dominance effects with smaller prediction error than the ordinary LASSO.
Bayesian statistical methods based on simulation techniques have recently been shown to provide powerful tools for the analysis of genetic population structure. We have previously developed a Markov ...chain Monte Carlo (MCMC) algorithm for characterizing genetically divergent groups based on molecular markers and geographical sampling design of the dataset. However, for large-scale datasets such algorithms may get stuck to local maxima in the parameter space. Therefore, we have modified our earlier algorithm to support multiple parallel MCMC chains, with enhanced features that enable considerably faster and more reliable estimation compared to the earlier version of the algorithm. We consider also a hierarchical tree representation, from which a Bayesian model-averaged structure estimate can be extracted. The algorithm is implemented in a computer program that features a user-friendly interface and built-in graphics. The enhanced features are illustrated by analyses of simulated data and an extensive human molecular dataset. Availability: Freely available at http://www.rni.helsinki.fi/~jic/bapspage.html
We introduce a Bayesian method for estimating hidden population substructure using multilocus molecular markers and geographical information provided by the sampling design. The joint posterior ...distribution of the substructure and allele frequencies of the respective populations is available in an analytical form when the number of populations is small, whereas an approximation based on a Markov chain Monte Carlo simulation approach can be obtained for a moderate or large number of populations. Using the joint posterior distribution, posteriors can also be derived for any evolutionary population parameters, such as the traditional fixation indices. A major advantage compared to most earlier methods is that the number of populations is treated here as an unknown parameter. What is traditionally considered as two genetically distinct populations, either recently founded or connected by considerable gene flow, is here considered as one panmictic population with a certain probability based on marker data and prior information. Analyses of previously published data on the Moroccan argan tree (Argania spinosa) and of simulated data sets suggest that our method is capable of estimating a population substructure, while not artificially enforcing a substructure when it does not exist. The software (BAPS) used for the computations is freely available from http://www.rni.helsinki.fi/~mjs.
Multi-task learning (MTL) is a learning paradigm whose aim is to leverage information shared across related tasks to improve the generalization of models. Motivated by the success of proximal ...optimization algorithms and single-task learning regression models, sparse proximal multi-task learning (SPMTL) for genome-wide prediction (GWP) should be explored. This study investigates proximal gradient descent splitting algorithms with five non-smooth sparsity-inducing norm regularizers, including the novel <inline-formula> <tex-math notation="LaTeX">L_{2,\frac {1}{2}} </tex-math></inline-formula> norm for GWP. Additionally, two popular methods based on Markov chain Monte Carlo (MCMC) are examined. To improve the computational efficiency, parallel Bayesian optimization strategy is employed for efficient hyperparameter tuning. Evaluation is conducted on three different real-world genomic datasets from mice, pigs and wheat, each associated with two, five, and four traits, respectively. Performance is assessed using mean squared error (MSE) and correlation coefficient between predicted and observed trait values in test sets. Experimental results reveal that the <inline-formula> <tex-math notation="LaTeX">L_{2,\frac {1}{2}} </tex-math></inline-formula> regularizer consistently achieves the best out-of-sample prediction across all datasets, demonstrating the effectiveness of SPMTL in leveraging shared information for improved GWP accuracy. Furthermore, the influence of different regularizers on sparsity and other properties of the SPMTL model are also explored.
Runs of homozygosity (ROH) islands are stretches of homozygous sequence in the genome of a large proportion of individuals in a population. Algorithms for the detection of ROH depend on the ...similarity of haplotypes. Coverage gaps and copy number variants (CNV) may result in incorrect identification of such similarity, leading to the detection of ROH islands where none exists. Misidentified hemizygous regions will also appear as homozygous based on sequence variation alone. Our aim was to identify ROH islands influenced by marker coverage gaps or CNV, using Illumina BovineHD BeadChip (777 K) single nucleotide polymorphism (SNP) data for Austrian Brown Swiss, Tyrol Grey and Pinzgauer cattle.
ROH were detected using clustering, and ROH islands were determined from population inbreeding levels for each marker. CNV were detected using a multivariate copy number analysis method and a hidden Markov model. SNP coverage gaps were defined as genomic regions with intermarker distances on average longer than 9.24 kb. ROH islands that overlapped CNV regions (CNVR) or SNP coverage gaps were considered as potential artefacts. Permutation tests were used to determine if overlaps between CNVR with copy losses and ROH islands were due to chance. Diversity of the haplotypes in the ROH islands was assessed by haplotype analyses.
In Brown Swiss, Tyrol Grey and Pinzgauer, we identified 13, 22, and 24 ROH islands covering 26.6, 389.0 and 35.8 Mb, respectively, and we detected 30, 50 and 71 CNVR derived from CNV by using both algorithms, respectively. Overlaps between ROH islands, CNVR or coverage gaps occurred for 7, 14 and 16 ROH islands, respectively. About 37, 44 and 52% of the ROH islands coverage in Brown Swiss, Tyrol Grey and Pinzgauer, respectively, were affected by copy loss. Intersections between ROH islands and CNVR were small, but significantly larger compared to ROH islands at random locations across the genome, implying an association between ROH islands and CNVR. Haplotype diversity for reliable ROH islands was lower than for ROH islands that intersected with copy loss CNVR.
Our findings show that a significant proportion of the ROH islands in the bovine genome are artefacts due to CNV or SNP coverage gaps.
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