Genome selection method for machine learning for additive and non-additive effects

Abstract

The invention discloses a genome selection method for machine learning aiming at additive and non-additive effects, which is characterized in that the non-additive effects are added into a GBLUP model, so that the prediction accuracy of the GBLUP model is greatly improved. By applying a kernel function in the GBLUP, the model can better simulate a nonlinear combination of a complex genetic relationship and a gene effect between genes, so that the explanation capability on genetic variation is improved. In addition, a machine learning strategy KPRR is constructed based on a polynomial kernel function and kernel ridge regression. In combination with the advantages of the polynomial kernel function and the kernel ridge regression method, the KPRR strategy allows remote data points to make contribution to the value of the model, so that interaction in genome data can be more effectively contributed, and the genome prediction effect can be greatly improved.

Description

Technical Field

[0001] The present invention relates to the technical field of genome selection, and in particular to a genome selection method for machine learning targeting additive and non-additive effects. Background Art

[0002] Genomic selection (GS) uses genome-wide single nucleotide polymorphism (SNP) marker information to predict the genomic estimated breeding value (GEBV) of individuals, thereby achieving efficient selection of important economic traits. The genomic relationship matrix measures the closeness of the relationship between individuals, which plays an important role in genomic selection. However, the genomic relationship matrix constructed by conventional methods currently has the following problems:

[0003] 1. Matrix singularity. When the number of loci is limited or two individuals have the same genotype on all markers, or the number of markers is less than the number of individuals, the genomic relationship matrix may be singular, which makes it impossible to solve the genomic selection model;

[0004] 2. Differences between genomic relationship matrices constructed by different methods may directly affect genomic selection decisions and genetic progress. For example, if there are different populations or subpopulations in the sample, it may lead to biased or misleading results in the genomic relationship matrix. Therefore, a suitable genomic relationship matrix is ​​needed to ensure the speed and direction of genetic improvement. However, the genomic relationship matrix constructed by existing methods is prone to affect the accuracy of genomic estimated breeding values.

[0005] In addition, traditional genomic selection methods, such as those based on Bayesian theory, have the following problems:

[0006] 1) Genomic selection relies on high-density genotypes to predict the genetic value of individuals. More markers mean that more accurate genomic breeding values ​​can be obtained, but more redundant information is also introduced, which will affect the accuracy of traditional genomic selection methods.

[0007] 2) There are certain limitations in the statistical models and algorithm selection used in genomic selection. For example, when the number of markers increases, the operating efficiency of the Bayesian method decreases significantly, which limits the further application of the Bayesian method.

[0008] 3) Traditional genomic selection methods are based on linear models and cannot take into account the interaction effects between genes. Therefore, the complex genetic background of quantitative traits cannot be fully considered.

[0009] In addition, in the early stages of genomic selection, many methods only considered additive effects in genomic data (the accumulation of genotype values ​​of multiple micro-effect genes that affect quantitative traits). In fact, non-additive effects (effects caused by interactions between alleles or non-allelic genes) play a core role in hybrid vigor, polymorphism, and genomic selection. Ignoring non-additive genetic effects will cause deviations in multiple genetic analysis processes. Therefore, genomic prediction models that integrate non-additive effects can obtain better prediction performance. However, how to integrate non-additive effects into the genomic selection process to improve the accuracy of genomic selection has become a technical problem that needs to be solved in this field. Summary of the invention

[0010] The present invention aims to improve the accuracy of genome selection and provides a genome selection method for machine learning targeting additive and non-additive effects.

[0011] To achieve this object, the present invention adopts the following technical solutions:

[0012] A method for genomic selection based on machine learning for additive and non-additive effects is provided, comprising the steps of:

[0013] S1, constructing the genome relationship matrix using the UKin unbiased kinship estimation method;

[0014] S2, adding non-additive effects to the linear model and constructing a non-additive genomic relationship matrix;

[0015] S3, captures the additive and non-additive effects of traits through machine learning, and predicts individual traits with the machine model that has completed learning and outputs the prediction results.

[0016] Preferably, in step S2, the statistical model for adding non-additive effects to the linear model is expressed by the following formula (2-10):

[0017] y=1μ+Z a g a +D d g d +e (2-10)

[0018] In formula (2-10), y represents the observed value of the target trait;

[0019] μ represents the group mean;

[0020] g a Indicates individual species value; Z a Indicates that it is a sports species value g a The correlation matrix of

[0021] g drepresents a dominant effect, Defined as dominant inheritance and normal distribution; is the dominant genetic variance; D is the marker-based dominant genomic relationship matrix;

[0022] d d Yes d The correlation matrix of

[0023] e represents the remaining residual.

[0024] Preferably, in step S2, the statistical model for adding the non-additive effect to the genome relationship matrix is ​​expressed by the following formula (2-11):

[0025] y=1μ+Z a g a +D ep g ep +e (2-11)

[0026] In formula (2-11), y is the observed value of the target trait;

[0027] μ is the population mean;

[0028] g a Indicates individual species value; Z a Indicates that it is a sports species value g a The correlation matrix of

[0029] d ep Yes ep The correlation matrix of

[0030] g ep For epistatic effect, It is defined as the epistatic genetic effect following a normal distribution; is the epistatic genetic variance; E is the epistatic genomic relationship matrix based on SNP markers.

[0031] Preferably, in step S1, the genome relationship matrix constructed by the UKin unbiased kinship estimation method is expressed by the following expression:

[0032]

[0033] From formulas (1-4)-(1-6), we can see that The expected value of is independent of the selected SNP, so ρ ii' The estimated value of can be expressed as:

[0034]

[0035] where ρ ii′is the true kinship coefficient between individuals i and i′; is ii′ An estimated value of Represents ρ ii′ The true calculated value of; k and l represent the individuals other than individuals i and i′, X ij and X i′j is the number of minor alleles; is the average minor allele frequency of the SNP, is the genotype variance; j represents the jth SNP; E represents the expected value; n represents the number of individuals in the population; m is the number of SNPs; X kj represents the jth SNP site of the kth individual; X lj represents the jth SNP site of the lth individual; express expected value.

[0036] 5. The genome selection method for machine learning of additive and non-additive effects according to claim 4, characterized in that in step S3, genome prediction is performed by polynomial kernel ridge regression, specifically comprising the steps of:

[0037] A1, construct a polynomial kernel ridge regression prediction model expressed as the following formula (2-1):

[0038]

[0039] In formula (2-4), y(x) represents the genome prediction value of the input individual x;

[0040] α i Represents the coefficient corresponding to the i-th training sample;

[0041] n represents the number of training samples;

[0042] K represents the kernel matrix composed of the kernel function values ​​of each training sample;

[0043] K(x,x i ) represents the i-th training sample x i The kernel function value between and x;

[0044] A2, input individual x into the polynomial kernel ridge regression prediction model, and the model predicts and outputs the genome prediction result of x.

[0045] Preferably, the loss function of the polynomial kernel ridge regression prediction model is expressed by the following formula (2-2):

[0046] L(w)=‖φ(X)wy‖ 2 +λ‖w‖ 2 (2-2)

[0047] In formula (2-2), L(w) represents the loss function;

[0048] φ(X) represents the result of mapping the original data X to a high-dimensional space through a kernel function;

[0049] w is the weight vector;

[0050] y is the target variable;

[0051] λ is the regularization parameter;

[0052] The goal of kernel ridge regression is to minimize the loss function expressed by formula (2-2).

[0053] Preferably, the optimal solution of the loss function expressed by formula (2-2) is obtained by solving the linear equation expressed by the following formula (2-3):

[0054] (K+λI)α=y (2-3)

[0055] In formula (2-3), I is the unit matrix;

[0056] α is the coefficient vector.

[0057] Preferably, genome prediction is performed by polynomial kernel support vector machine regression, the steps comprising:

[0058] B1, construct a support vector machine regression prediction model expressed as the following formula (2-8):

[0059]

[0060] In formula (2-8), f(x) represents the genome prediction value of the input individual x;

[0061] K(x,x i ) represents the i-th training sample x i The kernel function value between and input x;

[0062] It indicates the positive weight of the predicted individual;

[0063] α i represents the positive weight in the training set; n represents the number of training samples;

[0064] b means paranoia;

[0065] B2, input individual x into the support vector machine regression prediction model, and the model predicts and outputs the genome prediction result of x.

[0066] Preferably, the loss function of the support vector machine regression prediction model is expressed by the following formula (2-5):

[0067]

[0068] In formula (2-5), Loss ∈ represents the loss function;

[0069] y represents the true value of the genome corresponding to individual x;

[0070] ∈ is the threshold.

[0071] Preferably, the optimal solution of the loss function expressed by formula (2-5) is constrained by the following formulas (2-6) and (2-7):

[0072]

[0073] At the same time, the following conditions must be met:

[0074]

[0075] In formulas (2-6) and (2-7), ξ,ξ * is the slack variable;

[0076] ξ i , is the slack variable of the i-th training sample;

[0077] C is the regularization parameter;

[0078] w is the weight;

[0079] b is paranoia;

[0080] φ(x i ) is to transform the training sample x i Functions that map to high-dimensional spaces;

[0081] y i is the true value of the genome corresponding to the i-th training sample;

[0082] w T is the transpose of weight w.

[0083] The present invention has the following beneficial effects:

[0084] 1. There is a deviation between the traditional genomic relationship matrix and the true kinship of individuals, and this deviation will affect a series of subsequent genetic analyses. This embodiment proposes an unbiased estimation Ukin method, which corrects this deviation by integrating genetic information from the entire population, and proves that this method can improve the estimation accuracy of heritability. Specifically, the prediction performance of the UKin method in the GBLUP model was evaluated in simulated data sets and multiple real data sets, and the results showed that the method had different degrees of changes in prediction accuracy and unbiasedness. In general, the UKin method has similar prediction accuracy to the traditional method in both simulated data sets and real data sets. These results show that the UKin method not only solves the problem of biased estimation of traditional methods, but also ensures the accuracy and unbiasedness of genomic prediction, and shows valuable application value in the prediction of complex traits. This good performance of the method may be due to two reasons. First, it can more accurately reflect the genetic relationship between individuals. Unbiased estimation can ensure that the estimated value of kinship does not deviate from the true value due to systematic bias, which is particularly important for the prediction of genomic estimated breeding values ​​in genomic prediction. Secondly, the UKin method effectively handles negative values ​​in the genomic relationship matrix, which may be the cause of the bias of traditional methods. By optimizing the handling of these negative values, the UKin method can more accurately capture genetic variation, thereby ensuring the accuracy and unbiasedness of the prediction.

[0085] 2. This embodiment introduces the concept of kernel function in machine learning, and studies the method of constructing the genomic relationship matrix between individuals. The kernel function of machine learning is used to construct the genomic relationship matrix between individuals, which is used to replace the traditional relationship matrix in GBLUP, and is studied for the first time in different types of data sets. The information required to construct the genomic relationship matrix using the kernel function is the same as the traditional method, which only requires the genotype data of the population. By applying the kernel function in GBLUP, the model can better simulate the complex genetic relationships between genes and the nonlinear combination of gene effects, thereby improving the ability to explain genetic variation. At the same time, the kernel function method provides a variety of kernel function options, which allows the model to flexibly select the most appropriate kernel function according to the characteristics of the data set and the research objectives.

[0086] 3. Non-additive effects were added to the GBLUP model through formulas (2-10) and (2-11), and the prediction accuracy of the GBLUP model was significantly improved.

[0087] 4. This application constructs a machine learning strategy K based on polynomial kernel function and kernel ridge regression PRR, first applied it to genome prediction research, and tested it in simulated and real datasets. The polynomial kernel function plays an important role in the prediction process. It maps the input genotype data into a kernel space, which is different from the linear model. This application combines the advantages of the polynomial kernel function and the kernel ridge regression method, K P The RR strategy allows distant data points to contribute to the value of the model, which enables interactions in genomic data to contribute more effectively and can significantly improve genomic prediction results. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments of the present invention. Obviously, the drawings described below are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0089] Figure 1 is a diagram of implementation steps of a machine learning genome selection method for additive and non-additive effects provided by an embodiment of the present invention;

[0090] Figure 2 It is a comparison chart of the performance of Ukin method and other genomic relationship matrix construction methods in GBLUP prediction;

[0091] Figure 3 It is a comparison of the unbiasedness of GBLUP prediction using different genomic relationship matrices in the simulated data set;

[0092] Figure 4 This is a comparison of the GBLUP prediction accuracy using different genomic relationship matrices in the wheat dataset;

[0093] Figure 5 This is a comparison chart of the GBLUP prediction accuracy using different genomic relationship matrices in the dairy cow dataset and the Large White pig dataset;

[0094] Figure 6 This is a comparison of the unbiasedness of GBLUP predictions using different genomic relationship matrices in real data sets;

[0095] Figure 7 It is a comparison of the GBLUP prediction accuracy using different kernel functions to construct the genomic relationship matrix in the simulated data set;

[0096] Figure 8 It is a comparison of the unbiasedness of GBLUP predictions using different kernel functions to construct the genomic relationship matrix in the simulated data set;

[0097] Fig. 9This is a comparison of the GBLUP prediction accuracy using different kernel functions to construct the genomic relationship matrix in the wheat dataset;

[0098] Fig.10 This is a comparison of the GBLUP prediction accuracy using different kernel functions to construct the genomic relationship matrix in the Holstein cattle and Large White pig datasets;

[0099] Fig.11 It is a comparison chart of the unbiasedness of GBLUP prediction using different kernel functions to construct the genomic relationship matrix in real data sets;

[0100] Fig.12 This is a comparison chart of the prediction accuracy of the wheat dataset using different prediction methods;

[0101] Fig.13 It is a comparison chart of the prediction accuracy of traits with epistatic genetic background;

[0102] Fig.14 It is a comparison chart of the prediction accuracy of traits with dominant effect genetic background;

[0103] Fig.15 This is a comparison chart of the computation time of all methods on all traits. DETAILED DESCRIPTION

[0104] The technical solution of the present invention is further described below with reference to the accompanying drawings and through specific implementation methods.

[0105] Among them, the drawings are only used for illustrative explanations, and they only represent schematic diagrams rather than actual pictures, and should not be understood as limitations on this patent; in order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of the actual product; for those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0106] The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "inner", "outer", etc. appear, the orientation or position relationship indicated is based on the orientation or position relationship shown in the drawings, which is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, the terms describing the position relationship in the drawings are only used for illustrative purposes and cannot be understood as a limitation on this patent. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

[0107] In the description of the present invention, unless otherwise clearly specified and limited, if the term "connection" or the like appears to indicate the connection relationship between components, the term should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, it can be the internal connection of two components or the interaction relationship between two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0108] The genome selection method based on machine learning of additive and non-additive effects provided by the embodiment of the present invention is as follows: Figure 1 As shown, the steps include:

[0109] S1, constructing the genome relationship matrix using the UKin unbiased kinship estimation method;

[0110] S2, adding non-additive effects to the linear model and constructing a non-additive genomic relationship matrix;

[0111] S3, captures the additive and non-additive effects of traits through machine learning, and predicts individual traits with the machine model that has completed learning and outputs the prediction results.

[0112] 1. Constructing a genome relationship matrix using the UKin method

[0113] One of the key steps of genomic selection is to construct a genomic relationship matrix between individuals. Although traditional genomic relationship matrix construction methods have been widely used and verified in a variety of biological backgrounds, the relationship matrices constructed by these methods have been proven to be biased estimates of true kinship. When the genetic structure of a population is complex, the deviations in the genomic relationship matrix constructed by the traditional method will affect the accuracy and reliability of the genomic selection model. In order to solve the above problems, this embodiment adopts an unbiased estimation method (unbiasedkinship, UKin) for the true kinship coefficient improved based on the traditional method, and corrects the deviations in the construction process of the traditional method by comprehensively utilizing the genetic information of the entire population. The application also tests the technical advantages of the genomic relationship matrix constructed by the Ukin unbiased kinship estimation method in the GBLUP (Genomic best linear unbiased prediction, genomic best linear unbiased estimation) model and compares it with the traditional construction method.

[0114] The following is a detailed description of the process of constructing a genome relationship matrix using the UKin unbiased kinship estimation method in this embodiment, as well as the process of testing its technical advantages through the GBLUP model and comparing it with the traditional construction method, which specifically includes the following technical links:

[0115] 1.1 Materials and methods

[0116] 1.1.1 Test methods

[0117] 1.1.1.1 Best linear unbiased estimate of the genome

[0118] The GBLUP method assumes that all markers follow the same normal distribution and contribute equally to the genetic variance. The marker-based genetic relationship matrix is ​​used to replace the pedigree-based relationship matrix in the mixed linear model. The GBLUP model can be written as:

[0119] y=1μ+Zg a +e (1-1)

[0120] where y is the vector of phenotypic values; μ is the population mean; and Z is the g a The correlation matrix of is the random additive genetic effect vector; is the residual effect vector, and are the additive genetic variance and the residual variance, respectively; I is the identity matrix; G is the marker-based additive genomic relationship matrix. The model was implemented using the R package “sommer”. The different relationship matrices constructed subsequently were all constructed using self-written code in the R language.

[0121] 1.1.1.2 Traditional genome relationship matrix construction method

[0122] At present, the traditional genome relationship matrix construction methods that are frequently used in the GBLUP model are expressed by the following formulas (1-2) and (1-3):

[0123]

[0124] In formulas (1-2) and (1-3), for the jth (1≤j≤m) SNP, is the estimated value of the kinship coefficient between individuals i and i'(1≤i,i'≤n); p j is the gene frequency of the jth SNP. m is the number of SNPs; n is the number of individuals; x ij is the genotype value of the j SNP loci of individual i; x i′j represents the genotype value of the j SNP sites of individual i'; Z represents the corrected genotype matrix, Z' represents the transpose of the Z matrix; G represents the kinship matrix. In this embodiment, the expression expressed by formula (1-2) is defined as the VanRaden method polynomial, and the expression expressed by formula (1-3) is defined as the Yang method polynomial.

[0125] 1.1.1.3 Unbiased Kinship Estimation Method

[0126] In this embodiment, the genome relationship matrix constructed by the unbiased kinship estimation method is expressed by the following expressions (1-4), (1-5), (1-6), and (1-7):

[0127]

[0128] where ρ ii' is the true kinship coefficient between individuals i and i'; k and l represent the individuals other than individuals i and i', X ij and X i'j is the number of minor alleles; is the average minor allele frequency of the SNP, is the genotype variance; j represents the jth SNP site; E represents the expected value. Simplifying the above formula, we get:

[0129]

[0130] From the above formula, we can see that The expected value of is independent of the selected SNP. ii' The estimated value of can be expressed as:

[0131]

[0132] From formulas (1-6) and (1-7), we can see that It is effective for any SNP. The expected value is still ρ ii' This means is ii' Unbiased estimate of . Combining the above equation, we can find In fact, it can be expressed as a Yang method polynomial. This means that the calculation of the unbiased estimate of the UKin matrix only requires the same genomic information as the Yang method. The mathematical relationship between the UKin method and the Yang method can be derived:

[0133]

[0134] It can be seen that the UKin method is a traditional method expression and a linear combination of constant terms, that is, its computational complexity will not increase significantly. Note that the introduction and derivation of the UKin method here are based on the Yang method. Similarly, the UKin method can also be based on the VanRaden method. Therefore, in the following study, this embodiment uses "Ukin+Yang" to represent the UKin unbiased method based on the Yang method; "Ukin+VanRaden" represents the UKin unbiased method based on the VanRaden method.

[0135] 1.1.1.4Weir method

[0136] The Weir method uses the moment estimation method to estimate the kinship coefficient θ between individuals i and i' ii' This method is to construct a relationship matrix based on the pedigree for comparison. The method is expressed as follows:

[0137]

[0138] Among them, M ii' is the ratio of the same allele carried by individuals i and i' (1≤i, i'≤n). ii' It can be calculated as:

[0139]

[0140] Where X ij and X i'j is the reference allele dosage for individuals i and i' at SNP j. For a population of size n, it is the average pairwise value for all pairs of individuals:

[0141]

[0142] In fact, the ratio of the expected value of the numerator to the denominator is is the target parameter (θ ii' -θ S ) / (1-θ S ). This method has been shown to achieve lower mean square error and perform better when the number of SNPs is large.

[0143] 1.1.1.5 Polynomial Kernel Function Method

[0144] The polynomial kernel function performs a polynomial transformation on the data points in the original feature space, so that the model can capture the nonlinear relationship in the data.

[0145] The genomic relationship matrix constructed by the polynomial kernel function method is expressed by the following expression (1-12):

[0146]

[0147] In formula (1-12), γ, a, and d are adjustable parameters;

[0148] a is a positive real scalar that can affect the medium-high and low-order terms of the polynomial, where the medium-high order is the term with medium-high exponent (exponent greater than or equal to the exponent threshold) in the polynomial, and the low-order term is the term with low exponent (exponent less than the exponent threshold) in the polynomial;

[0149] d is a positive integer;

[0150] x i represents the genotype vector of the i-th individual;

[0151] x j represents the genotype vector of the jth individual;

[0152] K(x i ,x j ) represents x i and x j The kernel function value between , which can be controlled by adjusting the parameter d to control the scaling size and nonlinearity. The parameter a is used to adjust the generated polynomial, indicating the relative influence of high-order and low-order terms.

[0153] 1.1.1.6 Cosine kernel function method

[0154] The cosine kernel function evaluates the similarity between two vectors by measuring the cosine of the angle between them. When calculating the genomic relationship matrix, the genotype vectors of the two individuals are regarded as input vectors. The value of the kernel function is defined as the quotient of the dot product of the two vectors and their modulus.

[0155] The genome relationship matrix constructed by the cosine kernel function method is expressed by the following expression (1-13):

[0156]

[0157] In formula (1-13), x i ·x j Represents the genotype vector x of the i-th individual i and the genotype vector x of the jth individual j The dot product of

[0158] ‖x i ‖ and ||x j || are vectors x i and x j The Euclidean norm of ;

[0159] K(x i ,x j ) represents x i and x j The kernel function value between .

[0160] Cosine similarity does not involve parameter selection, making it relatively simple and efficient, especially when dealing with large-scale data.

[0161] 1.1.1.7 Gaussian kernel function method

[0162] The Gaussian kernel is also called the radial basis kernel. Its value depends on the Euclidean distance between the input vectors (i.e. the Euclidean norm of the difference) rather than the dot product. This kernel function is popular in machine learning models, but it is very sensitive to the choice of parameters and is prone to overfitting during training.

[0163] The genomic relationship matrix constructed by the Gaussian kernel function method is expressed by the following expression (1-14):

[0164]

[0165] In formula (1-14), the positive real scalar γ is the width parameter of the kernel function, which controls the decay speed of the function;

[0166] ||x i -x j || 2 are two vectors x i and x j The square of the Euclidean distance between them;

[0167] K(x i ,x j ) represents x i and x j The kernel function value between represents the genotype similarity between individuals.

[0168] 1.1.1.8 Hyperbolic tangent kernel function method

[0169] The sigmoid kernel is a commonly used kernel function for support vector machines and is also often used as an activation function for deep learning. If the parameters are adjusted correctly, the function can present a very complex nonlinear relationship. However, it can introduce model instability in some cases, so it is necessary to carefully evaluate and select appropriate parameters before application. At the same time, in some parameter settings, the sigmoid kernel function is similar to the Gaussian kernel function. Therefore, a large amount of computing resources may be required when calculating the genome individual similarity matrix.

[0170] The genomic relationship matrix constructed by the hyperbolic tangent kernel function method is expressed by the following expression (1-15):

[0171]

[0172] In formula (1-15), x i represents the genotype vector of the i-th individual;

[0173] x j represents the genotype vector of the jth individual;

[0174] tanh is the hyperbolic tangent function;

[0175] α is the slope parameter of the kernel function; b is the intercept parameter of the kernel function;

[0176] K(x i ,x j ) represents x i and x j The kernel function value between represents the genotype similarity between individuals, that is, the elements in the matrix.

[0177] 1.1.2 Test materials

[0178] 1.1.2.1 Simulated Dataset

[0179] The simulation process was based on the genetic background and statistical history of pigs. The existing QMSim software was used for simulation, and the parameter settings and breeding structure were based on existing studies. The entire simulation process was repeated 5 times and mainly included the following three parts: (1) Simulating historical ancestors. Linkage disequilibrium and mutation drift equilibrium were established from the historical ancestral population. There were 200 generations, and 420 animals were raised in each generation, including 20 males and 400 females. The number of offspring from each mating was 10, and the sex ratio was equal. (2) Simulating modern populations. The last generation of the historical ancestral population was selected as the founding animal. The litter size was 10 and the number of generations was 10. The female renewal rate was 0.8 and the male renewal rate was 1.0. The selection of animals was based on the breeding values ​​generated by the BLUP (Best linear unbiased prediction) method. As a result, a total of 40,421 offspring populations with pedigree, phenotypic and genomic information were obtained. The 4,000 animals of the last generation were used for genomic prediction. (3) Genomic parameters. The simulated genome consists of 18 pairs of chromosomes, each pair of chromosomes is 100cM in length. Two genome situations were simulated, one in which the trait is controlled only by minor polygenes, and the other in which the trait is controlled by major genes and minor polygenes, and the effect value follows a normal distribution. Each chromosome includes 25 QTLs (Quantitative trait locus) and 1,000 SNPs. All markers are biallelic and randomly distributed on all chromosomes. Three heritabilities are set in both cases: 0.1, 0.3, and 0.5. To facilitate subsequent analysis, these traits are represented by T1, T2, T3 and T4, T5, and T6, respectively. The genome parameters are shown in Table 2-1.

[0180] Table 1-1 Genomic information parameters of simulated data sets

[0181]

[0182] 1.1.2.2 Large White Pig Dataset

[0183] A total of 3,290 Shanghai Xinnong Large White pigs were selected, including 1,836 Canadian and 1,454 French pigs. All animals were tested for production performance at a body weight of 100 kg. Live weight and backfat thickness were obtained and corrected to age at 100 kg body weight (AGE100) and backfat thickness at 100 kg body weight (BF100). The phenotypic values ​​used are the corrected phenotypes. The Canadian and French lines were bred on two different farms, and the nutritional management levels of the two farms were basically the same. The two strains are not related. The pedigrees of these animals totaled 4,719 Large White pigs, recorded from 2012 to 2020. The average generation of genomes and phenotypes was 6.12 generations.

[0184] Genotyping was performed using the GeneSeek GGP Porcine HD chip. The chip was designed based on Susscrofa 10.2, so it was updated to Sus scrofa 11.1 before subsequent analysis. Quality control was performed using PLINK v1.90 software: (1) individual genotype detection rate > 90%; (2) marker genotype detection rate > 90%; (3) minimum allele frequency > 0.05; (4) only autosomal SNP markers were retained. After quality control, 35,172 SNPs and 3,290 individuals were retained. The traits used in the analysis are denoted as AGE100 and BF100.

[0185] 1.1.2.3 Wheat Dataset

[0186] This dataset includes 599 wheat lines from the Global Wheat Program of the International Maize and Wheat Improvement Center, which were tested in various wheat production environments, which were divided into 4 basic target environment sets. They were genotyped using the diversity array technology, which takes two values ​​to represent their presence or absence. Markers with minor allele frequencies below 0.05 were removed, and missing genotypes were interpolated with samples from the marginal distribution of marker genotypes. After these quality controls, there were 1,279 markers. The dataset is available at CROSSA et al. (2010). The phenotypic values ​​follow a normal distribution and are represented by W1, W2, W3, and W4, respectively, for the four traits of this dataset.

[0187] 1.1.2.4 German Holstein

[0188] The German Holstein cattle genomic prediction population includes 5,024 bulls, which were genotyped using the Inllumina cattle SNP50BeadChip. After quality control, 42,551 SNPs remained. Based on previous existing studies and their representative different genetic structures, the estimated breeding values ​​of three traits, milk yield (MY), milk fat percentage (MFP) and somatic cell score (SCS), were selected for further analysis. The trait MFP has a high heritability and is controlled by a major gene and a large number of genes with small effects. The trait MY has a medium heritability and is also controlled by a major gene and some genes with medium and small effects. The trait SCS is a low heritability trait controlled by a large number of genes with small effects. The phenotypic values ​​have been corrected to follow a normal distribution. Table 1-2 lists the statistical summary of all data sets in this embodiment.

[0189] Table 1-2 Statistical summary of all data sets

[0190]

[0191]

[0192] 1.1.3 Evaluation indicators

[0193] In the present embodiment, the 5-fold cross validation procedure repeated 5 times was used for analysis. Through cross validation, each analyzed data set was randomly divided into 5 parts. That is, 80% of the individuals were used as training sets, and 20% of the individuals were used as validation sets. Prediction accuracy and unbiasedness were used as evaluation indicators of the model. Accuracy is the Pearson correlation coefficient between the phenotypic value and the genome estimated breeding value (i.e., predicted value). When evaluating the predictive performance of the model, the Pearson correlation coefficient provides a measure of linear consistency between the predicted value and the actual value. Unbiasedness is the regression coefficient of the phenotypic value to the individual genome estimated breeding value, which refers to whether the regression coefficient of the predicted value and the true value is close to 1.0. The difference between unbiasedness and 1.0 determines whether the predicted value has a systematic bias. It should be noted that there are both phenotypic values ​​and true breeding values ​​in the simulated data set. Therefore, when predicting simulated data, the present embodiment uses phenotype for prediction, and uses true breeding values ​​and genome estimated breeding values ​​(i.e., predicted values) to calculate accuracy and unbiasedness.

[0194] 1.2 Results

[0195] 1.2.1 Prediction performance of Ukin’s method in simulated datasets

[0196] Figure 2The following Tables 1-3 show the performance of the Ukin method (including UKin+Yang and UKin+VanRaden) and other genomic relationship matrix construction methods in GBLUP prediction. For each box plot, the middle line represents the median, the middle "×" symbol represents the mean, and the upper and lower ends of each box represent the maximum and minimum values. In the simulated data set, the prediction accuracy of all traits ranged from 0.466 to 0.629. The prediction accuracy of trait T1 ranged from 0.592 to 0.616, the prediction accuracy of trait T2 ranged from 0.613 to 0.629, the prediction accuracy of trait T3 ranged from 0.538 to 0.545, the prediction accuracy of trait T4 ranged from 0.505 to 0.535, the prediction accuracy of trait T5 ranged from 0.563 to 0.572, and the prediction accuracy of trait T6 ranged from 0.466 to 0.472. In general, Yang method and UKin+VanRaden method performed best in this dataset, with prediction accuracy of 0.616, 0.629, 0.545, 0.535, 0.572, 0.472 and 0.608, 0.627, 0.545, 0.525, 0.572, 0.471 for the six traits, respectively. The prediction accuracy of the other three methods was relatively close overall.

[0197] Table 1-3 GBLUP prediction accuracy using different genomic relationship matrices in simulated data sets

[0198]

[0199]

[0200] Figure 3 Tables 1-4 show the comparison of the unbiasedness of GBLUP prediction using different genomic relationship matrices. The predicted unbiasedness of all traits ranged from 0.953 to 1.598, which indicates that the overall GBLUP prediction value was overestimated relative to the true value. The predicted unbiasedness of trait T1 ranged from 1.038 to 1.107, the predicted unbiasedness of trait T2 ranged from 1.012 to 1.042, the predicted unbiasedness of trait T3 ranged from 0.953 to 0.955, the predicted unbiasedness of trait T4 ranged from 1.501 to 1.589, the predicted unbiasedness of trait T5 ranged from 1.004 to 1.020, and the predicted unbiasedness of trait T6 ranged from 1.070 to 1.087. Except for trait T3, the unbiasedness of all traits was greater than 1.0. The mean values ​​of prediction unbiasedness of all the methods for all traits were 1.11, 1.11, 1.13, 1.12, and 1.11, respectively, which shows that the prediction unbiasedness of all the methods is very close.

[0201] Table 1-4 Unbiasedness of GBLUP predictions using different genomic relationship matrices in simulated data sets

[0202]

[0203] 1.2.2 Prediction performance of UKin method in real datasets

[0204] The real datasets include wheat dataset, Holstein cattle dataset and Large White pig dataset. Figure 4 , Figure 5 Tables 1-5 show the prediction accuracy comparison results of the UKin method (including UKin+Yang and UKin+VanRaden) and other genomic relationship matrix construction methods in GBLUP in these data sets. In the wheat data set, the prediction accuracy of all traits ranged from 0.384 to 0.513. The prediction accuracy of trait W1 ranged from 0.490 to 0.513, the prediction accuracy of trait W2 ranged from 0.476 to 0.487, the prediction accuracy of trait W3 ranged from 0.384 to 0.391, and the prediction accuracy of trait W4 ranged from 0.450 to 0.466. Overall, the Yang method and the UKin+Yang method performed relatively well, with prediction accuracies of 0.513, 0.479, 0.391, 0.450 and 0.513, 0.478, 0.391, 0.451 for the four traits, respectively. In the Holstein cattle dataset, the prediction accuracy of all traits ranged from 0.738 to 0.816. The prediction accuracy of trait MY ranged from 0.768 to 0.772, the prediction accuracy of trait MFP ranged from 0.805 to 0.816, and the prediction accuracy of trait SCS ranged from 0.738 to 0.740. The results of the five methods in traits MF and trait SCS were very close, while the prediction accuracy of VanRaden, UKin+Yang, and Weir methods in trait MFP was 0.816, and the prediction accuracy of Yang and UKin+VanRaden methods was 0.805, which was quite different. In the Large White pig dataset, the prediction accuracy of trait AGE100 ranged from 0.742 to 0.746, and the prediction accuracy of trait BF100 ranged from 0.823 to 0.824. The accuracy results of the various genomic relationship matrix construction methods for these two traits were almost equal. In contrast, the average accuracy of the UKin+VanRaden method was slightly higher, with a mean of 0.785.

[0205] Figure 6Tables 1-6 show the comparison of GBLUP prediction unbiasedness using the UKin method and other genomic relationship matrices in all real data sets. For the wheat data set, the prediction unbiasedness of all traits ranged from 0.954 to 1.322. The prediction accuracy of trait W1 ranged from 1.018 to 1.234, the prediction accuracy of trait W2 ranged from 1.049 to 1.322, the prediction unbiasedness of trait W3 ranged from 0.954 to 1.069, and the prediction unbiasedness of trait W4 ranged from 1.002 to 1.157. In general, except for the VanRaden method, the average values ​​of the prediction unbiasedness of the other four methods were approximately equal to 1.0, which indicates that the prediction effect is relatively stable. For the Holstein cattle data set, the prediction unbiasedness of all traits ranged from 0.996 to 1.007, which indicates that the prediction effect of all methods on this data set is relatively stable. In the Large White Pig dataset, the prediction unbiasedness of all methods ranged from 0.988 to 0.991, and the performance of all methods was relatively stable. For these three datasets, the prediction unbiasedness of all traits ranged from 0.954 to 1.322. Except for the VaRaden method, which had an unbiasedness of 1.322±0.237 for the W2 trait in the wheat dataset, the unbiasedness ranges of the remaining traits and methods were very close to 1.0, indicating that the UKin method has the same stable prediction unbiasedness results as traditional methods in real datasets.

[0206] Table 1-5 GBLUP prediction accuracy using different genomic relationship matrices in real datasets

[0207]

[0208] Table 1-6 Unbiasedness of GBLUP predictions using different genomic relationship matrices in real datasets

[0209]

[0210] There is a deviation between the traditional genomic relationship matrix and the true kinship of individuals, and this deviation will affect a series of subsequent genetic analyses. This embodiment proposes an unbiased estimation Ukin method, which corrects this deviation by integrating genetic information from the entire population, and proves that this method can improve the estimation accuracy of heritability. Specifically, the prediction performance of the UKin method in the GBLUP model was evaluated in simulated data sets and multiple real data sets, and the results showed that the method had different degrees of changes in prediction accuracy and unbiasedness. In general, the UKin method has similar prediction accuracy to the traditional method in both simulated data sets and real data sets. These results show that the UKin method not only solves the problem of biased estimation of traditional methods, but also ensures the accuracy and unbiasedness of genomic prediction, and shows valuable application value in the prediction of complex traits. This good performance of the method may be due to two reasons. First, it can more accurately reflect the genetic relationship between individuals. Unbiased estimation can ensure that the estimated value of kinship does not deviate from the true value due to systematic bias, which is particularly important for the prediction of genomic estimated breeding values ​​in genomic prediction. Secondly, the UKin method effectively handles negative values ​​in the genomic relationship matrix, which may be the cause of the bias of traditional methods. By optimizing the handling of these negative values, the UKin method can more accurately capture genetic variation, thereby ensuring the accuracy and unbiasedness of the prediction.

[0211] The construction of the genomic relationship matrix is ​​a key step in genomic selection. This example explores the Ukin-based genomic relationship matrix construction method, and its application and performance in genomic prediction. Through research in simulated data sets and real data sets, the prediction effects of the Ukin method and traditional methods in GBLUP were systematically compared. The results show that the UKin method not only solves the problem of biased estimation in traditional methods, but also has similar prediction accuracy and unbiasedness as traditional methods in simulated data sets and three real data sets. The UKin method corrects the bias problem in the traditional genomic relationship matrix, provides new ways and means to improve the accuracy and unbiasedness of genomic prediction, and provides strong support for precise genetic improvement and breeding decisions.

[0212] 1.2.3 Prediction performance of kernel function method in simulated data set

[0213] In the analysis of the simulated data set, different kernel functions were used to construct the genomic relationship matrix to evaluate the GBLUP prediction performance. A total of four kernel function methods were analyzed, including polynomial kernel (Polynomial), cosine kernel (Cosine), Gaussian kernel (Rbf) and hyperbolic tangent kernel (Sigmoid).

[0214] From Table 1-7 and Figure 7From the results of accuracy, the prediction accuracy ranges of all traits are from 0.460 to 0.637. The prediction accuracy range of trait T1 is from 0.587 to 0.618, the prediction accuracy range of trait T2 is from 0.607 to 0.637, the prediction accuracy range of trait T3 is from 0.528 to 0.554, the prediction accuracy range of trait T4 is from 0.503 to 0.537, the prediction accuracy range of trait T5 is from 0.558 to 0.584, and the prediction accuracy range of trait T6 is from 0.460 to 0.489. Generally speaking, the accuracy of the GBLUP-Polynomial and GBLUP-Rbf methods is slightly better than other methods. Specifically, the average prediction accuracy of the GBLUP-Polynomial method is 0.570, followed by the GBLUP-Rbf method with an average prediction accuracy of 0.569. In contrast, the average prediction accuracies of the GBLUP-Cosine and GBLUP-Sigmoid methods are 0.550 and 0.541 respectively. This indicates that in the simulated dataset, the methods based on polynomial kernel and Gaussian kernel are relatively more effective in constructing the genomic relationship matrix.

[0215] Table 1-7 GBLUP prediction accuracy of constructing genomic relationship matrix using different kernel functions in the simulated dataset

[0216]

[0217] The results of prediction unbiasedness are as Figure 8 shown in Table 1-8. The prediction unbiasedness ranges of all traits are from 0.945 to 1.659. The prediction unbiasedness range of trait T1 is from 1.099 to 1.114, the prediction unbiasedness range of trait T2 is from 1.024 to 1.038, the prediction unbiasedness range of trait T3 is from 0.945 to 0.960, the prediction unbiasedness range of trait T4 is from 1.641 to 1.659, the prediction unbiasedness range of trait T5 is from 1.004 to 1.020, and the prediction unbiasedness range of trait T6 is from 1.064 to 1.077. Sorted by average prediction unbiasedness from low to high, it is GBLUP-Sigmoid < GBLUP-Rbf < GBLUP-Polynomial < GBLUP-Cosine. The average prediction unbiasedness of the GBLUP-Sigmoid method is 1.14, which is the lowest among the four methods. The unbiasedness of GBLUP-Cosine is 1.69, which is the highest among the four methods. This indicates that the prediction results of the Sigmoid kernel function are closer to the true values compared to other methods, while the prediction results of the cosine kernel function are relatively the farthest from the true values.

[0218] Table 1-8 GBLUP prediction unbiasedness of constructing genomic relationship matrix using different kernel functions in the simulated dataset

[0219]

[0220] 1.2.4 Prediction performance of kernel function method in real data sets

[0221] Fig. 9 , Fig.10 Tables 1-9 and 1-9 show the prediction accuracy of the GBLUP in the real data set using the kernel function method to construct the genomic relationship matrix. In the wheat data set, the prediction accuracy of all traits ranged from 0.378 to 0.548. Specifically, the prediction accuracy of trait W1 ranged from 0.474 to 0.548, the prediction accuracy of trait W2 ranged from 0.481 to 0.491, the prediction accuracy of trait W3 ranged from 0.378 to 0.401, and the prediction accuracy of trait W4 ranged from 0.451 to 0.497. In this data set, the GBLUP-Polynomial method performed best overall, followed by the GBLUP-Cosine method, and the GBLUP-Sigmoid method performed the worst. This shows that in the wheat data set, the polynomial kernel and the Gaussian kernel provide a better genomic relationship matrix, thereby enhancing the accuracy of the GBLUP prediction model. In the Holstein cattle data set, the prediction accuracy of all traits ranged from 0.737 to 0.816. The prediction accuracy of trait MY ranged from 0.768 to 0.775, that of trait MFP ranged from 0.802 to 0.816, and that of trait SCS ranged from 0.737 to 0.740. For the MFP trait, the GBLUP-Cosine method performed slightly better than the other methods. Overall, the differences between the methods were not large, indicating that the prediction accuracy of these kernel function methods was relatively balanced in the Holstein cattle dataset. In the Large White pig dataset, the prediction accuracy of trait AGE100 ranged from 0.742 to 0.782, and the prediction accuracy of trait BF100 ranged from 0.823 to 0.843. In this dataset, the GBLUP-Cosine and GBLUP-Polynomial methods provided relatively higher prediction accuracy, especially for the BF100 trait.

[0222] Table 1-9 GBLUP prediction accuracy using different genomic relationship matrices in real datasets

[0223]

[0224] Fig.11Tables 1-10 and 1-11 show the unbiasedness of GBLUP predictions using kernel functions to construct genomic relationship matrices in real data sets. In the wheat data set, the range of unbiasedness of all trait predictions is 0.929-1.077. The unbiasedness of trait W1 is 1.010-1.032, the unbiasedness of trait W2 is 1.052-1.077, the unbiasedness of trait W3 is 0.929-0.972, and the unbiasedness of trait W4 is 1.015-1.024. Overall, the average unbiasedness of all methods is close to or equal to 1.0, showing good prediction results. For the Holstein cattle data set, the range of unbiasedness of all trait predictions is 0.997-1.156. The unbiasedness of trait MY ranged from 0.997 to 1.066, that of trait MFP ranged from 0.998 to 1.156, and that of trait SCS ranged from 0.999 to 1.067. The unbiasedness of the GBLUP-Cosine method for the three traits was very close to 1.0, indicating that its prediction results were very close to the true values. The unbiasedness of the GBLUP-Polynomial and GBLUP-Rbf methods was slightly higher for some traits. This result emphasizes the excellent unbiased performance of the GBLUP-Cosine method in the Holstein cattle dataset, especially for the prediction of the MFP trait. For the Large White pig dataset, the unbiasedness of all trait predictions ranged from 0.989 to 1.027. The unbiasedness of trait AGE100 ranged from 0.992 to 1.015, and the unbiasedness of trait BF100 ranged from 0.991 to 1.027. In this dataset, all methods performed well overall. Considering the prediction unbiasedness results of wheat, Holstein cattle, and Large White pig datasets, the GBLUP-Cosine method showed excellent prediction unbiasedness on all datasets, especially in the Holstein cattle dataset. Similar to the results of the UKin method, except for the trait MFP, the unbiasedness of almost all methods in all traits is around 1.0.

[0225] Table 1-10 Unbiasedness of GBLUP predictions using different genomic relationship matrices in real datasets

[0226]

[0227] Although traditional genomic relationship matrix construction methods have been widely used in genomic selection, their potential to further improve selection performance is relatively limited. This embodiment introduces the concept of kernel function in machine learning and studies the method of constructing the genomic relationship matrix between individuals. The kernel function of machine learning is used to construct the genomic relationship matrix between individuals, which is used to replace the traditional relationship matrix in GBLUP, and is studied for the first time in different types of data sets. The information required to construct the genomic relationship matrix using the kernel function is the same as the traditional method, which only requires the genotype data of the population. By applying the kernel function in GBLUP, the model can better simulate the complex genetic relationship between genes and the nonlinear combination of gene effects, thereby improving the ability to explain genetic variation. At the same time, the kernel function method provides a variety of kernel function options, which allows the model to flexibly select the most appropriate kernel function according to the characteristics of the data set and the research objectives.

[0228] The kernel function maps the input space to a high-dimensional feature space through some nonlinear transformations, which means that the input data is easier to separate, thereby improving the accuracy and stability of regression and classification problems. In this embodiment, polynomial kernels, cosine kernels, Gaussian kernels and hyperbolic tangent kernels are used, and each kernel function has its specific application scenarios and advantages. In this embodiment, the GBLUP method using kernel functions is applied on different data sets. The results show that the polynomial kernel and Gaussian kernel perform better than the cosine kernel and hyperbolic tangent kernel in the overall performance of simulated data sets and real data sets, except in the Holstein cattle data set, where the Gaussian kernel has lower accuracy. These results also emphasize the applicability of different kernel function methods on different data sets and traits, and the importance of selecting the most appropriate kernel function to improve the performance of the GBLUP prediction model. Comparing the results of all kernel functions with the Yang method in the traditional genomic relationship matrix construction method, it can be found that the prediction accuracy of the polynomial kernel for all traits is improved by 0.4%, 1.2%, 2.1%, 0.0%, 2.0%, 3.7%, 8.1%, 0.5%, 4.2%, 7.7%, 0.3%, -1.7%, 0.0%, 5.2%, 2.3%, and the prediction accuracy of the Gaussian kernel is improved by 0.5%, 0.9%, 1.6%, 0.3%, 1.7%, 2.7%, 7.5%, 0.9%, 4.2%, 6.8%, 0.1%, 0.5%, 0.2%, 4.2%, 1.8%. Among these traits, especially the wheat dataset and the large white pig dataset, both kernel functions have shown certain advantages in improving the accuracy of genomic prediction. This means that polynomial kernels and Gaussian kernels have unique advantages in constructing genomic relationship matrices, which can solve some scientific problems in genetic evaluation and prediction in theory and practice, especially providing new perspectives and methods in analyzing and predicting the genetic mechanisms of complex traits.

[0229] 2. Using machine learning strategies to capture more non-additive effects in genomic prediction

[0230] With the rapid development of bioinformatics and statistical genetics, machine learning has become an important tool for analyzing complex biological data. Especially in genomic selection, the application of machine learning algorithms provides a new way to improve the efficiency and accuracy of genetic breeding. Compared with traditional genomic selection methods, kernel ridge regression with polynomial, K P The machine learning strategy of KRR combines the regularization advantages of ridge regression (RR) and the ability of kernel function to process nonlinear data, making it a powerful tool for processing high-dimensional genetic data. P RR can map the original feature space into a higher-dimensional space, thereby capturing the complex nonlinear relationships in genotype data, which is particularly important for understanding the polygenic control mechanism of genetic traits. P The theoretical basis of RR, its implementation method and how it can be applied to genomic data of different genetic backgrounds, including additive, dominant and epistatic effects. By comparing traditional genomic selection methods such as GBLUP and BayesB, K P The prediction performance and prediction efficiency of RR on multiple data sets. This example is expected to provide a more comprehensive perspective to demonstrate the application potential of machine learning methods in genomic selection and genetic breeding, in order to provide new tools and ideas for animal and plant breeding.

[0231] 2.1 Materials and methods

[0232] 2.1.1 Test methods

[0233] 2.1.1.1 Machine Learning Strategy Polynomial Kernel Ridge Regression

[0234] Ridge regression is a variant of linear regression that addresses the problem of collinearity of independent variables and avoids overfitting by adding a regularization term to the loss function. The loss function L(w) of ridge regression is defined as:

[0235] L(w)=‖Xw-y‖ 2 +λ‖w‖ 2 (2-1)

[0236] Where X is the data matrix; w is the weight vector; y is the target variable; and λ is the regularization parameter.

[0237] The polynomial kernel technique is a method that uses a kernel function to map data to a high-dimensional feature space so that the application of linear algorithms in this space implicitly handles the nonlinear relationship in the original space. Kernel ridge regression handles nonlinear problems by combining ridge regression and kernel techniques. The goal of kernel ridge regression is to minimize the following loss function:

[0238] L(w)=‖φ(X)wy‖ 2 +λ‖w‖ 2 (2-2)

[0239] Among them, φ(X) represents the result of mapping the original data X to a high-dimensional space through a kernel function. The optimal solution of this equation can be obtained by solving the following linear equations:

[0240] (K+λI)α=y (2-3)

[0241] Where K is the kernel matrix, where the element K ij =K(x i ,x j ) represents the kernel function value between training samples i and j; I is the identity matrix; α is the coefficient vector. For a new input x, its predicted value can be calculated by the following formula:

[0242]

[0243] Among them, α i is the coefficient obtained above, x i is a training sample. The machine learning strategy used in this embodiment is kernel ridge regression with polynomial, K P RR) is based on the above process and converts the polynomial kernel function The best parameters of the polynomial kernel are found by using the grid search method, and the K P The genomic prediction process of RR.

[0244] 2.1.1.2 Machine Learning Strategy Polynomial Kernel Support Vector Machine Regression

[0245] Support vector machine regression is an extension of support vector machine for regression problems. Its main idea is to minimize the error and maximize the margin. Some errors can be tolerated. Assume that when f(x) and y i The gap is greater than ò The loss is calculated only when , then the loss function can be expressed as:

[0246]

[0247] Based on the above loss function, the goal of support vector machine regression is to find a function f(x) = w T φ(x)+b, so as to minimize the following objective function:

[0248]

[0249] At the same time, the following conditions must be met:

[0250]

[0251] Here, φ(x) represents the function that maps the input x to a high-dimensional space, w and b are the weight and bias of the model respectively (w and b are solved iteratively), C is the regularization parameter (C is solved by grid search), ξ i and is the slack variable (ξ i and The solution is the difference method), which is used to deal with inevitable errors. Through the Lagrange multiplier method, the original optimization problem can be converted into its dual problem, which makes the problem more efficient, especially when using kernel functions to process high-dimensional data. After conversion and derivation, the prediction model of support vector machine regression can be expressed as:

[0252]

[0253] Where K is the kernel function; and α i is the positive weight of each observation, which is solved in the Lagrange multiplier method transformation process. Based on the above process, the polynomial kernel function is integrated with the support vector machine regression to obtain S P VR strategy. Similarly, this method also uses the grid search method to find the best parameters and uses the python package sklearn to implement S P The genome prediction process of VR.

[0254] 2.1.1.3 Bayesian Method

[0255] The BayesB method assumes that most markers have no effect, only a few markers have an effect, and follow a gamma distribution or exponential distribution (MEUWISSEN et al., 2009). Its statistical model can be expressed as:

[0256]

[0257] where y is the phenotypic value vector; m j is the genotype of the jth locus; α j is the effect size of the jth marker, subject to e is the random residual, αj The prior of depends on the marker effect method and the prior probability π, which determines the proportion of markers with effect values ​​to the total number of markers and is determined empirically. The R package "BGLR" is used to implement the genomic prediction process.

[0258] 2.1.1.4 Best linear unbiased prediction of genome

[0259] Same as 1.1.1.1, no further details.

[0260] 2.1.1.5 Best linear unbiased prediction of genomic dominance

[0261] The GDBLUP method uses additive and dominant genetic effects as random effects. The statistical model is:

[0262] y=1μ+Z a g a +D d g d +e (2-10)

[0263] Among them, D d Yes d The correlation matrix of It is defined as dominant inheritance. is the dominant genetic variance, D is the dominant genomic relationship matrix based on markers; y represents the observed value of the target trait; μ represents the mean value of the population; g a It is a sports value; Z a It is a sports species value g a The construction of the D matrix and the implementation of GDBLUP were both implemented using the R package “sommer”.

[0264] The construction method of D matrix is ​​briefly described as follows:

[0265] Where W ik =1-||x ik ||,x ik represents the genotype of k loci of the i-th individual.

[0266] Z a The construction method is briefly described as follows: Za represents the association matrix with breeding values, its dimension is m×n, m represents m phenotypes, and n represents n individuals.

[0267] 2.1.1.6 Best linear unbiased prediction of genomic epistasis

[0268] The GEBLUP method treats additive and epistatic genetic effects as random effects. The statistical model is:

[0269] y=1μ+Z a g a+D ep g ep +e (2-11)

[0270] Where D ep Yes ep The correlation matrix of Defined as epistatic genetic effects, is the epistatic genetic variance, E is the epistatic genomic relationship matrix based on SNP markers, y is the observed value of the target trait; μ is the mean value; g a It is a sports species value; Z a Is a sports species value g a The construction of the E matrix and the implementation of GEBLUP were both implemented using the R package “sommer”.

[0271] 2.1.2 Test materials

[0272] 2.1.2.1 Additive Effects Dataset

[0273] The additive effect data set uses the wheat data set and the Holstein cattle data set, which have been introduced in 1.1.2 Experimental Materials and will not be repeated here.

[0274] 2.1.2.2 Dominant Effect Dataset

[0275] (1) PIC Core Pig Dataset

[0276] This dataset comes from the core pig line of the pig improvement company (PIC), including 3,534 individuals and 5 traits. The quality control criteria for genotype data are: remove sites with no position information or located on sex chromosomes and repeated sites with Hardy-Weinberg equilibrium <10 -4 ; retain SNPs with minimum allele frequency>0.05 and site detection rate>0.95; delete individuals without phenotype. Finally, a total of 38,891 SNPs and 2,314 individuals were retained. Based on the heritability of these five traits, we selected trait T2 (h 2 =0.16), T3(h 2 =0.38) and T4(h 2 =0.58) for analysis. The dominance variances for these traits were 2%, 7%, and 1%, respectively. This data set has been used in several studies related to dominance effects. The phenotypes used in the analysis were either fixed effect corrected or weighted progeny average corrected phenotypes.

[0277] (2) Scottish Blackface Sheep Dataset

[0278] The data set consists of 752 Scottish blackface sheep. All animals had their paternal and maternal lines recorded at birth, as well as a complete pedigree since the flock was established in 1988. The three live body weight traits considered in this embodiment include body weight at 16 weeks (BW16), body weight at 20 weeks (BW20) and body weight at 24 weeks (BW24). Phenotypic values ​​were corrected by sex, year, age and group. After quality control, a total of 37,243 SNPs were retained for further analysis. The dominant variances of the three traits accounted for 38%, 6% and 30% of the total phenotypic variance, respectively.

[0279] 2.1.2.3 Epistatic effect dataset

[0280] (1) Backcross simulation data set

[0281] The dataset contains 600 individuals, 1 trait, and 121 molecular markers evenly distributed on one chromosome, of which 9 molecular markers overlap with the main effect QTL. There were 13 possible marker pairs with interaction effects. The fixed effects only included the overall mean, and the phenotypic variance was 106, of which 90 came from all marker effects (main effects and epistatic effects) and the rest came from the covariance term.

[0282] (2) Rice Dataset

[0283] The dataset includes four traits: yield, tiller, grain, and grain weight per kilogram (KWG). After quality control, 278 hybrid combinations and 1,619 genotypes remained. The phenotypic values ​​were corrected by year. The epistatic effect variances of the four traits were estimated to be 95%, 69%, 52%, and 23%, respectively.

[0284] Table 2-1 below lists the statistical summary of the above data set

[0285] Table 2-1 Statistical summary of dominant effect data set and epistatic effect data set

[0286]

[0287] 2.1.3 Evaluation indicators

[0288] As in 1.1.2, the same 5-fold cross validation procedure was used for analysis. Each analyzed data set was randomly divided into 5 parts. That is, 80% of the individuals were used as training sets and 20% of the individuals were used as validation sets. The performance of each method was evaluated using prediction accuracy and calculation time. The calculation method of accuracy has been introduced in 1.1.3 Evaluation Metrics. All calculations were performed on the same research server (Ubuntu 18.04.6, 72CPUs@2.30GHz), and the same CPU threads and memory were guaranteed.

[0289] 2.2 Results

[0290] 2.2.1 Prediction results in additive genetic background

[0291] Fig.11 , Fig.12 Table 2-2 shows the prediction accuracy results of the two data sets in the additive genetic background, using K P RR, S P Four methods, VR, GBLUP, and BayesB, predicted genomic estimated breeding values ​​for a total of seven traits. The middle line of the box plot represents the median, the upper and lower ends of each box represent the maximum and minimum values, and the "×" symbol represents the mean. In the wheat data set, the prediction accuracy of all traits ranged from 0.354 to 0.544. The prediction accuracy of trait W1 ranged from 0.502 to 0.544, the prediction accuracy of trait W2 ranged from 0.425 to 0.491, the prediction accuracy of trait W3 ranged from 0.354 to 0.389, and the prediction accuracy of trait W4 ranged from 0.440 to 0.483. The prediction accuracy of each trait ranged from 0.042, 0.066, 0.035, and 0.043, respectively, which shows that the prediction results are relatively stable. For all traits, K P RR has the highest prediction accuracy. In the Holstein cattle data set, the prediction accuracy of trait MY ranged from 0.756 to 0.791, the prediction accuracy of trait MFP ranged from 0.759 to 0.870, and the prediction accuracy of trait SCS ranged from 0.717 to 0.742. The mean prediction accuracy of each trait was 0.773, 0.814, and 0.736, and the results of different methods in the three traits varied slightly, except that the results of BayesB in trait MFP were significantly higher than those of other methods. Overall, BayesB had the highest prediction accuracy for traits MY and MFP, and K P RR and GBLUP performed similarly and stably in traits MY and MKG, while for trait SCS, the prediction accuracy of all methods was almost the same.

[0292] Table 2-2 Prediction accuracy of each trait in the additive genetic background dataset

[0293]

[0294] 2.2.2 Prediction results in the context of epistatic effects

[0295] Fig.13 Tables 2-3 show the prediction accuracy results of the two data sets in the epistatic genetic background. P RR, S P VR, BayesB, GBLUP and GEBLUP methods were compared and analyzed for all five traits. It can be seen that the prediction accuracy of the simulated traits ranged from 0.530 to 0.768, and the prediction accuracy of each method was ranked as K P RR>GEBLUP>BayesB>GBLUP>S P VR. In the rice dataset, the prediction accuracy of the trait Yield ranged from 0.372 to 0.417, the prediction accuracy of the trait Tiller ranged from 0.418 to 0.509, the prediction accuracy of the trait Grain ranged from 0.505 to 0.626, and the prediction accuracy of the trait KGW ranged from 0.759 to 0.830. For the simulated traits, trait Yield, and trait Tiller, K P RR performs best. For traits Grain and KWG, BayesB performs best. In general, S P VR performed the worst.

[0296] In addition to S P Except for VR and GBLUP, the prediction accuracy of other methods is not much different for the same trait. Comparing the results of GEBLUP and GBLUP, it is found that the prediction accuracy of different traits has changed after adding epistatic effects into the prediction model. The prediction accuracy of simulated traits, trait Yield and trait Grain increased by 24.75%, 6.64% and 2.24% respectively, the prediction accuracy of trait Tiller decreased by 1.83%, and the trait KWG remained unchanged. At the same time, K P The prediction accuracy of RR is improved by 25.42%, 12.01%, 2.42%, 3.07% and 0.23% respectively compared with GBLUP. P The results of RR and GEBLUP show that K P The prediction accuracy of RR is improved by 0.54%, 5.04%, 4.33%, 0.81% and -0.19% respectively compared with the latter. These comparison results show that K P RR does not need to calculate the epistasis matrix first, and can directly take epistasis into account in the model, while also achieving better prediction performance.

[0297] Table 2-3 Prediction accuracy of epistatic effect genetic background traits

[0298]

[0299] 2.2.3 Prediction results in the context of dominant effects

[0300] Fig.14 Tables 2-4 show the prediction accuracy results of the two data sets under the dominant effect genetic background. P RR, S P Five methods, including VR, BayesB, GBLUP, and GDBLUP, performed genomic predictions on all six traits. In the PIC core pig data set, the prediction accuracy of trait T2 ranged from 0.465 to 0.497, and the prediction accuracy of each method was ranked K P RR>GBLUP>GDBLUP>BayesB>S P VR. The prediction accuracy of trait T3 ranged from 0.320 to 0.323, and the prediction accuracy of each method was ranked S P VR>BayesB=GBLUP>GDBLUP>K P RR. The prediction accuracy of trait T4 ranged from 0.425 to 0.450, and the prediction accuracy of each method was ranked as BayesB = GBLUP > GDBLUP > K P RR>S P VR. GDBLUP and GBLUP, K P RR and GBLUP, K P Comparison between RR and GDBLUP revealed that the differences between them were all within ±0.01, which indicates that there was only a small difference in the prediction accuracy of different methods in this data set.

[0301] In the Scottish Blackface sheep dataset, the prediction accuracy of trait BW16 ranged from 0.189 to 0.261, and the prediction accuracy of each method was ranked S P VR>K P RR>BayesB>GDBLUP>GBLUP. The prediction accuracy of trait BW20 ranged from 0.297 to 0.360, and the prediction accuracy of each method was ranked S P VR>K P RR>BayesB>GBLUP>GDBLUP. The prediction accuracy of trait BW24 ranged from 0.133 to 0.212, and the prediction accuracy of each method was ranked as K P RR>BayesB>S P VR>GDBLUP>GBLUP. Overall, K P RR and S PVR has the highest prediction accuracy in this data set, while GBLUP has the worst. Comparing the results of GDBLUP and GBLUP, it is found that after adding the dominant effect matrix into the model, the prediction accuracy of the three traits increased by 5.78%, 0.60%, and 30.99%, respectively. P The results of RR and GDBLUP showed that K P The accuracy of RR was improved by 29.10%, 16.67% and 21.56% respectively compared with GDBLUP.

[0302] Table 2-4 Prediction accuracy of dominant effect genetic background traits

[0303]

[0304] 2.2.4 Comparison of running time

[0305] Fig.15 The running time of all traits and all methods on the same device was recorded. The x-axis represents different traits, and the y-axis represents the time consumed by different methods in seconds, and the logarithm with base 10 was calculated. The running time of wheat dataset traits ranged from 1.01 to 318.37, the running time of Holstein cattle dataset traits ranged from 154.19 to 33071.24, the running time of simulated traits of simulated datasets ranged from 0.50 to 1081.00, the running time of rice dataset traits ranged from 0.40 to 1099.30, the running time of PIC core pig dataset traits ranged from 149.30 to 227856.00, and the running time of Scottish Blackface sheep dataset traits ranged from 8.00 to 31985.00. It can be observed that the running time of the method increases with the increase in the number of individuals or markers. Among all traits, BayesB takes the longest time, K P RR takes the shortest time. In some cases, such as rice traits, K P RR is observed to be approximately 2500 times faster than BayesB. In both the simulated and rice datasets, S P VR is computed faster than GBLUP, while in the pig and sheep datasets, S P VR has slower computing speeds.

[0306] 2.3 Discussion

[0307] 2.3.1K P Prediction performance of RR in additive datasets

[0308] The performance of genomic selection methods depends largely on the genetic structure of the trait and the population structure. Although no one method can consistently outperform other methods in all data sets, the results of this example reveal the unique advantages of different methods. In this example, we first tested the machine learning strategy K P Performance of RR in datasets controlled only by additive effects, including the wheat dataset and the Holstein cattle dataset. P RR, S P The prediction results of genomic estimated breeding values ​​by VR, BayesB and GBLUP in these two datasets showed that K P The RR method performed best in all four traits of the wheat dataset, and showed the same prediction accuracy as GBLUP in the dairy cow dataset. In previous studies, the accuracy of genomic prediction of wheat datasets using GBLUP was 0.512, 0.483, 0.401, and 0.463, and the results of this embodiment were 0.507, 0.487, 0.385, and 0.461, respectively, which are basically consistent. In previous studies, the prediction accuracy of the three traits estimated by the GBLUP method in the German Holstein cattle dataset was 0.774, 0.816, and 0.738, respectively, and the results of this embodiment were 0.772, 0.816, and 0.740, respectively, which are basically consistent.

[0309] 2.3.2K P Prediction performance of RR in non-additive datasets

[0310] In the early stages of genomic selection, many methods only considered additive effects in genomic data. In fact, non-additive effects play a central role in hybrid vigor, polymorphism, and genomic selection. Some studies have shown that models that integrate non-additive effects can achieve better prediction performance than genomic selection models that only consider additive effects. For example, in existing studies, the prediction performance of the two-step method and the non-additive effect model was evaluated in 7 traits of the cassava population, and the results showed that the prediction performance of the latter was improved by an average of 10%. Therefore, it is very necessary to consider non-additive effects in genomic selection models. Therefore, in this embodiment, the epistatic effect matrix and the dominant epistatic matrix are respectively integrated into the GBLUP model to obtain the GEBLUP model and the GDBLUP model. The results show that in the data set containing non-additive effects, the prediction accuracy of the GEBLUP and GDBLUP models is significantly improved compared with GBLUP, which also demonstrates the above viewpoint.

[0311] K P RR, S PVR, BayesB, GBLUP, and GEBLUP methods were first applied to a dataset containing epistatic effects. In the simulated dataset, 13 possible marker pairs had interactive effects and a high proportion of additive and epistatic variance was present. The method with the best prediction accuracy was K P RR, followed by GEBLUP, BayesB, GBLUP, and S P VR. Nine markers in this dataset overlapped with QTLs for main effects, which may be the reason why BayesB performed better than GBLUP and S P The reason for VR. Because the BayesB method has been observed to perform well in predicting traits controlled by large effect genes or QTLs, this actually explains the excellent performance of BayesB in the trait MFP in the Holstein cattle dataset. For the rice dataset, the epistatic effects of its four traits accounted for 95%, 69%, 52% and 23% respectively. The results of this dataset show that when there is more epistatic variance in the trait, the machine learning strategy K P RR has better prediction effect.

[0312] Afterwards, K P RR, S P VR, BayesB, GBLUP and GDBLUP methods were applied to datasets with dominant effects. For the PIC core pig dataset, the proportion of dominant variance in the three traits was relatively small, 2%, 7% and 1%, respectively. The low proportion of dominant variance may affect K P The result of RR, namely K P RR does not perform better than other methods in this dataset, which is why GDBLUP does not show significant improvement over GBLUP. In the Scottish Blackface sheep dataset, the dominance variance of the three traits accounts for 38%, 6%, and 30% of the total phenotypic variance, respectively. P RR consistently outperforms the other methods. It can be observed that as the proportion of non-additive effects increases, K P The RR prediction performance also shows an increasing trend. From the above results and discussions, it can be seen that the machine learning strategy K P RR can effectively capture non-additive effects in genomic data and make them contribute to genomic prediction models.

[0313] 2.3.3 Model Computational Efficiency

[0314] In this example, the study counted the running time of all methods and traits in the same device. The results showed that BayesB usually takes the longest computing time because it uses a specific Markov chain Monte Carlo algorithm for solving. In contrast, the machine learning strategy K PRR has the shortest computation time in all analyzed datasets, sometimes even thousands of times faster than other methods. This apparent speed advantage may be related to the P The RR algorithm is related to its own characteristics. First, kernel ridge regression can use closed solutions to estimate model parameters, which means that the model can directly obtain the optimal solution without iterative optimization. Second, the regularization term in kernel ridge regression makes the model more inclined to use low-dimensional feature subspaces, thereby reducing the complexity of complex calculations. In the results, we also noticed that S P VR's running time is always longer than K P RR is longer. Although both are based on kernel function training data, they solve the optimization problem in different ways. P VR is based on the principle of support vector machine. It minimizes the prediction error by constructing an optimal hyperplane. The process of finding this hyperplane involves solving a convex quadratic programming problem, that is, while optimizing the loss function, a series of inequality constraints must be satisfied. The solution of convex quadratic programming problems is usually complicated, so when dealing with large-scale data sets, S P VR runtimes can be extremely long.

[0315] Ignoring non-additive genetic effects will cause bias in multiple genetic analysis processes. In this embodiment, a machine learning strategy K based on kernel ridge regression and polynomial kernel function is proposed. P RR. Using this strategy and traditional genomic selection methods, we conducted genomic prediction studies in data sets with different genetic backgrounds. The results showed that (1) in an additive genetic background, the machine learning strategy K P RR has a similar prediction accuracy of genomic estimated breeding values ​​as the traditional genomic selection methods GBLUP and BayesB. When the trait is controlled by a large effect gene, the BayesB method has better prediction accuracy. (2) In a non-additive genetic background, adding non-additive effects to the GBLUP model significantly improved the prediction accuracy. Compared with the traditional genomic selection methods GBLUP and BayesB, the machine learning strategy K P RR has higher or similar prediction accuracy, which illustrates the advantages of machine learning in transforming high-dimensional feature space and capturing non-additive effects. (3) Machine learning strategy K P RR takes the shortest running time overall, while BayesB takes the longest.

[0316] It should be noted that the above specific implementations are only preferred embodiments of the present invention and the technical principles used. Those skilled in the art should understand that various modifications, equivalent substitutions, changes, etc. can be made to the present invention. However, as long as these changes do not deviate from the spirit of the present invention, they should be within the scope of protection of the present invention. In addition, some terms used in the specification and claims of this application are not restrictive, but are only for the convenience of description.

Claims

1. A machine learning genomic selection method for additive and non-additive effects, characterized in that Includes steps: S1, constructing the genome relationship matrix using the UKin unbiased kinship estimation method; S2, adding non-additive effects to the linear model and constructing a non-additive genomic relationship matrix; S3, captures the additive and non-additive effects of traits through machine learning, and predicts individual traits with the machine model that has completed learning and outputs the prediction results.

2. The method of genomic selection for machine learning of additive and non-additive effects according to claim 1, characterized in that: In step S2, the statistical model for adding non-additive effects to the linear model is expressed by the following formula (2-10): y=1μ+Z a g a +D d g d +e (2-10) In formula (2-10), y represents the observed value of the target trait; μ represents the group mean; g a Indicates individual species value; Z a Indicates that it is a sports species value g a The correlation matrix of g a represents a dominant effect, Defined as dominant inheritance and normal distribution; is the dominant genetic variance; D is the marker-based dominant genomic relationship matrix; D d Yes d The correlation matrix of e represents the remaining residual.

3. The method of genomic selection for machine learning of additive and non-additive effects according to claim 1, characterized in that: In step S2, the statistical model of adding the non-additive effect to the genome relationship matrix is ​​expressed by the following formula (2-11): y=1μ+Z a g a +D ep g ep +e (2-11) In formula (2-11), y is the observed value of the target trait; μ is the population mean; g a Indicates individual species value; Z a Indicates that it is a sports species value g a The correlation matrix of D ep Yes ep The correlation matrix of g ep For epistatic effect, It is defined as the epistatic genetic effect following a normal distribution; is the epistatic genetic variance; E is the epistatic genomic relationship matrix based on SNP markers.

4. The method for genomic selection based on machine learning for additive and non-additive effects according to any one of claims 1 to 3, characterized in that: In step S1, the genome relationship matrix constructed by the UKin unbiased kinship estimation method is expressed by the following expression: From formulas (1-4)-(1-6), we can see that The expected value of is independent of the selected SNP, so ρ ii’ The estimated value of can be expressed as: where ρ ii′ is the true kinship coefficient between individuals i and i′; is ii′ An estimated value of Represents ρ ii′ The true calculated value of; k and l represent the individuals other than individuals i and i′, X ij and X i′j is the number of minor alleles; is the average minor allele frequency of the SNP, is the genotype variance; j represents the jth SNP; E represents the expected value; n represents the number of individuals in the population; m is the number of SNPs; X kj represents the jth SNP site of the kth individual; X lj represents the jth SNP site of the first individual; express expected value.

5. The method of genomic selection for machine learning of additive and non-additive effects according to claim 4, characterized in that: In step S3, genome prediction is performed by polynomial kernel ridge regression, which specifically includes the following steps: A1, construct a polynomial kernel ridge regression prediction model expressed as the following formula (2-1): In formula (2-4), y(x) represents the genome prediction value of the input individual x; α i Represents the coefficient corresponding to the i-th training sample; n represents the number of training samples; K represents the kernel matrix composed of the kernel function values ​​of each training sample; K(x,x i ) represents the i-th training sample x i The kernel function value between and x; A2, input individual x into the polynomial kernel ridge regression prediction model, and the model predicts and outputs the genome prediction result of x.

6. The method of genomic selection for machine learning of additive and non-additive effects according to claim 5, characterized in that: The loss function of the polynomial kernel ridge regression prediction model is expressed by the following formula (2-2): L(w)=||φ(X)w-y|| 2 +λ||w|| 2 (2-2) In formula (2-2), L(w) represents the loss function; φ(X) represents the result of mapping the original data X to a high-dimensional space through a kernel function; w is the weight vector; y is the target variable; λ is the regularization parameter; The goal of kernel ridge regression is to minimize the loss function expressed by formula (2-2).

7. The method of genomic selection for machine learning of additive and non-additive effects according to claim 6, characterized in that: The optimal solution of the loss function expressed by formula (2-2) is obtained by solving the linear equation expressed by the following formula (2-3): (K+λI)α=y (2-3) In formula (2-3), I is the unit matrix; α is the coefficient vector.

8. The method of genomic selection for machine learning of additive and non-additive effects according to claim 4, characterized in that: Genome prediction is performed by polynomial kernel support vector machine regression, the steps include: B1, construct a support vector machine regression prediction model expressed as the following formula (2-8): In formula (2-8), f(x) represents the genome prediction value of the input individual x; K(x,x i ) represents the i-th training sample x i The kernel function value between and input x; It indicates the positive weight of the predicted individual; α i represents the positive weight in the training set; n represents the number of training samples; b means paranoia; B2, input individual x into the support vector machine regression prediction model, and the model predicts and outputs the genome prediction result of x.

9. The method of genomic selection for machine learning of additive and non-additive effects according to claim 8, characterized in that The loss function of the support vector machine regression prediction model is expressed by the following formula (2-5): In formula (2-5), Loss ∈ represents the loss function; y represents the true value of the genome corresponding to individual x; ∈ is the threshold.

10. The method of genomic selection for machine learning of additive and non-additive effects according to claim 9, characterized in that: The optimal solution of the loss function expressed in formula (2-5) is subject to the following constraints (2-6) and (2-7): At the same time, the following conditions must be met: In formulas (2-6) and (2-7), ξ, ξ* are relaxation variables; ξ i , is the slack variable of the i-th training sample; C is the regularization parameter; w is the weight; b is paranoia; φ(x i ) is to transform the training sample x i Functions that map to high-dimensional spaces; y i is the true value of the genome corresponding to the i-th training sample; w T is the transpose of weight w.

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