Predicting expected progeny difference for marbling score in Angus

13 Pages·2013·1.1 MB·English

Okutetal.GeneticsSelectionEvolution2013,45:34 Genetics
http://www.gsejournal.org/content/45/1/34 Selection
Evolution
RESEARCH Open Access
Predicting expected progeny difference for
marbling score in Angus cattle using artificial
neural networks and Bayesian regression models
Hayrettin Okut1,2*, Xiao-Liao Wu1,3, Guilherme JM Rosa1,4, Stewart Bauck5, Brent W Woodward6,
Robert D Schnabel7, Jeremy F Taylor7 and Daniel Gianola1,3,4
Abstract
Background: Artificial neural networks (ANN) mimic the function of the human brain and are capable of
performing massively parallel computations for data processing and knowledge representation.ANN can capture
nonlinear relationships between predictors and responses and can adaptively learn complex functional forms, in
particular, for situations where conventional regression models are ineffective. In a previous study, ANN with
Bayesian regularization outperformed a benchmark linear model when predicting milk yield in dairy cattle or grain
yield ofwheat. Althoughbreeding values rely on theassumption of additive inheritance, the predictive capabilities
ofANN are ofinterest from the perspective of their potential to increase theaccuracyof prediction ofmolecular
breeding values used for genomic selection.This motivated thepresentstudy, in which the aim was to investigate
theaccuracy ofANN when predicting theexpectedprogeny difference (EPD) of marbling score inAngus cattle.
Various ANN architectures were explored, which involved twotraining algorithms,two types of activation functions,
and from 1to 4 neurons inhidden layers. For comparison, BayesCπ models were used to select a subset of optimal
markers (referred to as feature selection), under theassumption of additive inheritance, and then the marker effects
were estimatedusing BayesCπwith π set equalto zero. This procedure is referred to as BayesCpC and was
implemented on a high-throughput computing cluster.
Results: The ANN with Bayesian regularization method performed equally well for prediction ofEPD as BayesCpC,
based onprediction accuracyand sum of squared errors.With the 3K-SNP panel, for example,prediction accuracy
was 0.776 using BayesCpC,and ranged from 0.776to 0.807using BRANN. With the selected 700-SNP panel,
prediction accuracy was 0.863for BayesCpC and ranged from 0.842to 0.858 for BRANN. However,prediction
accuracy for the ANN with scaled conjugate gradient back-propagation was lower, ranging from 0.653 to 0.689with
the3K-SNP panel, and from 0.743to 0.793 with theselected 700-SNPpanel.
Conclusions: ANN with Bayesian regularization performed as well as linear Bayesian regression modelsin
predicting additive genetic values, supporting theidea thatANN are useful as universal approximatorsof functions
ofinterest inbreeding contexts.
*Correspondence:[email protected]
1DepartmentofAnimalSciences,UniversityofWisconsin,Madison,WI53706,
USA
2DepartmentofAnimalScience,BiometryandGeneticsBranch,Universityof
YuzuncuYil,Van65080,Turkey
Fulllistofauthorinformationisavailableattheendofthearticle
©2013Okutetal.;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreative
CommonsAttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,and
reproductioninanymedium,providedtheoriginalworkisproperlycited.
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Background complexfunctionalforms,throughaseriesoftransforma-
The availability of genome-wide dense marker panels for tions (i.e., activation functions) driven by parameters.
many species of plants and animals has opened doors Multilayer feed-forward is the most common architecture
for incorporating genomic information into practical used in ANN, which consists of neurons assembled into
breeding programs, an approach known as genomic se- layers. The first layer is called the input layer (the left-
lection[1].Itisnoweasytogenerategenome-widescans mostlayeroftheANN)thatacceptsdata(e.g.,SNPgeno-
with more than one million SNPs (single nucleotide types) from sources external to the network, and the last
polymorphisms) but these huge databases pose chal- layer (the right-most layer) is called the output layer that
lengesincomputationalcapacity, dataanalysisandinter- contains output units of the network. In between these
pretation of results for genomic selection [2]. For twolayersareso-calledhiddenlayers,becausetheir values
example, even with an initial screening to reduce the arenotobservedinthetrainingset.Hence,anANNarchi-
number of markers to less than ten thousand SNPs, it is tectureisspecifiedbythenumberoflayers,thenumberof
still not feasible for most computational platforms to neuronsineachlayer,andthetypeofactivationfunctions
evaluate all combinations of SNPassociations,evenwhen that are used. Neurons in each layer are connected to the
low-dimension interactions are explored [3]. Hence, re- neurons from the previous and the subsequent layer
duction of dimensionality and feature extraction arguably throughadaptablesynapticweights.
play pivotal roles in current genomic studies [4]. The in- The network weights are evaluated in two steps: the
tensive computation inherent in these problems has al- feed-forward and the back-propagation steps. In the
teredthecourseofmethodologicaldevelopments, and the feed-forward step, information comes from the input
same is true for genomic selection [5]. layer and each unit evaluates its activation function,
In the genome-enabled prediction of genetic merit for which is transmitted to the units connected to the out-
breeding purposes, parametric statistical methods tend put layer. The back-propagation step consists of running
to make strong assumptions about functional forms and the whole network backwards to minimize the error
the statistical distribution of marker effects. On the one function in the space of weights using the method of
hand, ridge regression best linear unbiased prediction gradient descent. A set of weights that minimizes the
assumes that all markers have an effect on the trait of error function is considered to be a solution of the
interest, and that these effects share a common variance learning problemfor theANN.
intheirdistribution.Thissimpleassumptionisobviously Determination of the number of neurons to be placed
not true in reality. On the other hand, hierarchical linear in the hidden layer is a critical task in the design of
Bayesian regression models, such as BayesA and BayesB ANN. A network with an insufficient number of neurons
[1], allow marker effects to be estimated with differential maynotbecapableofcapturingcomplexpatterns.Incon-
shrinkage. However, posterior inference, in particular for trast, a network with too many neurons will suffer from
the variances of marker effects, depends heavily on the over-parameterization, leading to over-fitting and poor
prior assumptions used in these models [6]. Hence, predictive ability of yet to be observed data. Two popular
BayesCπ [7] was proposed to overcome some of the techniques to overcome the over-fitting problem in ANN
above mentioned drawbacks. A BayesCπ model postu- models are Bayesian regularization and cross-validated
lates an unknown probability π that a SNP has no effect early stopping (CVES). Bayesian regularization (BR) con-
at all on the trait, while all non-zero SNP effects are as- strains (shrinks) the magnitude of the networks weights,
sumedto berandomsamplesfrom anormaldistribution improvesgeneralizationandallowsbiasinparameteresti-
with null mean and a common variance. Recently, there mates towards values that are considered to be plausible,
has been interest in the use of non-parametric methods while reducing their variance; thus, there is a bias-
for the prediction of quantitative traits, such as reprodu- variance trade-off [13]. Unlike other back-propagation
cing kernel Hilbert space regressions [8,9], radial basis trainingalgorithmmethodsthatuseasinglesetofparam-
functionmodels[10]andnon-parametricBayesianmodels eters,BRconsidersallpossiblevaluesofnetworkparame-
with Dirichlet process priors [11]. These nonparametric tersweightedbytheprobabilityofeachsetofparameters.
methodsmakeweakerassumptionsandcanbemoreflex- In practice, Bayesian regularized ANN can be computa-
iblefordescribingcomplexrelationships[12]. tionally more robust and cost-effective than standard
Artificial neural networks (ANN), also known as back-propagationnetsbecausetheycanreduceorelimin-
neuro-computationalmodels,provide an appealing alter- atetheneedforlengthycross-validation.
native for genome-enabled prediction of quantitative With cross-validated early stopping (CVES) methods,
traits [13,14]. These machine learning methods can act the training data set is split into a new training and a
asuniversalapproximatorsofcomplexfunctions[15]be- validationsetandagradientdescentalgorithmisapplied
causetheyarecapableofcapturingnonlinearrelationships to the new training data set. The ANN performs an it-
betweenpredictorsandresponsesandcanadaptivelylearn erative process, first using the training data set to
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calculate the weight and bias estimates, and then applies BeadChip [16] was extracted. After data quality control
these parameter estimates in the validation data set to and screening, a total of 2421 polymorphic SNPs were
calculate the prediction errors. The process iterates re- retained for analysis. The target variable to be predicted
peatedly, substituting parameter estimates from the was EPD for marbling score, which had been computed
training data set into the validation data set to find the by the American Angus Association using BLUP (best
smallest possible average prediction errors for the valid- linear unbiased prediction) for each of these animals,
ation data set. Training ceases when the error in the val- based upon their pedigree data and progeny carcass and
idation data set increases in certain consecutive epochs ultrasound data [17]. In animal breeding, an EPD is de-
(iterations) in order to avoid the problem of over-fitting fined as the predicted performance of a future offspring
(the number of consecutive epochs is 6 by default in of an animal for a particular trait (marbling score), cal-
MATLAB). The ANN parameter estimates with the best culated from measurement(s) of the animal’s own per-
performanceinthe validationsetisthenusedonthetest- formance and/or the performance of its relatives under
ingdatatoevaluatethepredictiveabilityofthenetwork. the assumption of additive inheritance. Hence, EPD rep-
In a previous study, ANN with BR outperformed a resent a typical linear system since the EPD of an indi-
benchmark linear model when predicting milk yield in vidual can be represented as a linear function of the
dairy cattle or grain yield of wheat [14]. However, be- EPD of relatives and a residual term that reflects the fact
cause breeding values are defined in terms of linear that an individual inherits a random sample of the genes
functions based upon additive inheritance, the predictive present within its parents. The distribution of marbling
performance of ANN relative to linear additive systems score EPD in the Angus sample was symmetric and
is of some interest. This motivated the present study, in suggested a normal distribution (Figure 1), with mean
which the aim was to investigate the accuracy of ANN and standard deviation estimated at 0.0265 and 0.254,
for predicting expected progeny differences (EPD) for respectively [18].
marbling score in Angus cattle relative to hierarchical
linear Bayesian regression models. Various ANN archi- Statisticalmethods
tectures were explored, which involved two training Artificialneuralnetworks
algorithms, two types of activation functions, and from 1 A schematic presentation of a multilayer perceptron
to4neuronsinhiddenlayers. (MLP) feed-forward of an ANN is presented in Figure 2.
This is a multi-layer feed-forward neural network with a
Methods single middle (hidden) layer and four neurons. Training
Datasets an ANN involves the minimization of an error function
The data contained 3079 registered Angus bulls, geno- that depends on the network’s synaptic weights (the w’s
typed with the Illumina BovineSNP50 BeadChip, from in Figure 2), and these weights are iteratively updated by
which the SNP content for the Illumina Bovine3K a learning algorithm, to approximate the target variable.
Mean 0.0265 Skewness 0.468
StdDev 0.254 Kurtosis 0.356
Range 1.920 Interquartile 0.330
Figure1Histogramanddensityplotofderegressedexpectedprogenydifferencesformarblingscorefor3079Anguscattle.
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b(1)
1
SNP1 X1 w1,1 ∑ f1
w2,1
b2(1) w1
SNP2 X2 .
∑ f2
SNP3 X3 . b3(1) w2 b(2)
. .
.
∑ f3 w3 ∑ g y
. b(1)
4
SNP2421 X2421 w4,2421 w4
∑ f4
Input layer Hidden layer, 4 neurons hyperbolic tangent Output layer, 1 neuron linear activation
Figure2Schematicrepresentationofathree-layerfeed-forwardneuralnetwork.Genotypesfor2421(or700)SNPswereusedasinputs
x={x|i=1,2,…,n},wherenisthenumberofindividualswithgenotypes;eachSNPwasconnectedtoupto4neuronsviacoefficientsw ,
j ij kj
wherekdenotesneuronandjdenotesSNP;here,w isaweightfromahiddenlayerunitstotheoutputunit,f isanactivationfunctionapplied
k k
tohiddenlayerunits(e.g.,thehyperbolictangent),gisanactivationfunctionappliedtotheoutputlayerunit(e.g.,linear),b(1)andb2arebiases
ofhiddenandoutputlayerunits,and^y isapredictedvalue.
Theupdatingisusuallyaccomplishedbyback-propagating Xp
bð1Þþ w x, where p=700 or 2421, and bð1Þ is the
the error, which is essentially a non-linear least-squares k kj j k
problem [19]. Back-propagation is a supervised learning j¼1
bias parameter defined in the hidden layer and this
algorithmbasedonasuitableerrorfunction,the valuesof
quantity is then transformed using some linear or
which are determined by the target (i.e., marbling EPD)
nonlinear activation function (f ) as:
and the mapped (predicted) outputs of the network (i.e., k
!
fitted values ofmarbling EPD).Weightsin anMLP archi- Xp
tecture are determined by a back-propagation algorithm f bð1Þþ w x : ð1Þ
k k kj j
to minimize the sum of squares of errors using gradient- j¼1
descent methods [13]. During the training, weights and
The above is the output of the hidden layer, which in
biases in the ANN are successively adjusted based on the
turn is delivered to the output layer (e.g., each neuron k
input and the desired output. Each iteration of feed-
in the hidden layer sums ANN input x after multiplying
forward in a MLP constitutes two sweeps: forward acti- j
them by the strengths of the respective connection
vation to produce a desired output, and backward
weights, w , and adds biases b and then computes its
propagation of the computed error to update the values kj k
outputasafunction ofthe sum),andiscollected as:
for the weights and biases. The forward and backward
!
sweeps are repeatedly performed until the ANN solution XK Xp
agrees with the desired value to within a pre-specified bð2Þþ wkfk bð1Þþ wijxj ; ð2Þ
tolerance [20,21]. k¼1 j¼1
(cid:1) (cid:3)
Let x0j ¼ x1j;x2j::::::;xnj be a row vector that con- where wk is the weight specific to the kth neuron (k=1,
tains SNP genotypes for all i = 1,…,n individuals, for 2,…,K), and b(2) is the bias parameter defined in the
the jth SNP, where j = 1,…..,p, and p = 700 (referred output layer. Next, quantity (2) is activated with the fol-
to as the 700-SNP panel) or p = 2421 (referred to as lowing function:
" #
the 3K-SNP panel). In an ANN, SNP genotypes are XK
connected to each neuron in a single hidden layer via gð:Þ¼g w f ð:Þþbð2Þ : ð3Þ
k k
weights wkj, for k=1,…,K neurons, with each weight k¼1
defining a specific SNP-neuron connection with ap- The above yields the fitted marbling EPD, ^y , for each
propriate biases (intercepts), bð1Þ;bð1Þ; ::::::;bð1Þ, each individual in the training set. Finally, the prediicted value
1 2 k
pertaining to a specific neuron. The input into ofmarblingscoreEPD canbecomputedinthetestingset
neuron k, prior to activation is expressed linearly as using a formula similar to (3). We used the hyperbolic
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tangent sigmoid and linear (identity) activation functions minimize the following function, equal to a penalized
inthehiddenandoutputlayers,respectively. log-likelihood,
We used BR and scaled conjugate gradient (SCG)
F ¼βE ðDjw;MÞþαE ðwjMÞ; ð5Þ
back-propagation as training algorithms. The basic idea D W
of back-propagation algorithms is to adjust weights in
the steepest descent direction (negative of the gradient), where E (w|M) is the sum of squares of network
W
such that the objective function decreases most rapidly weights, M is the ANN architecture, and α and β are
[22]. However, in practice, this does not necessarily pro- positive regularization parameters that must be esti-
duce the fastest convergence, although the function mated. The second term on the right hand side of (5),
may decrease most rapidly along the negative of the known as weight decay, favors small values of w and de-
gradient. Several conjugate gradient algorithms have creases the tendency of a model to over-fit the data.
been proposed, in which a search is performed along Hence, training involves a tradeoff between model com-
conjugate directions, generally leading to faster conver- plexityandgoodnessoffit.Largevaluesofαleadtopos-
gence to a local function minimum than steepest des- terior densities of weights that are highly concentrated
cent directions [22,23]. On the one hand, with the around zero, so that the weights effectively disappear,
exception of SCG, the conjugate gradient algorithms the model discounts connections in the network [24,25],
use linear searches at each iteration and thus, are com- and complex models are automatically self-penalized.
putationally expensive as they require that the network From equation (5), if α << β, the fitting or training algo-
response to all training inputs be computed several rithm places the most weight on goodness of fit. If α >>
times for each search. On the other hand, the SCG al- β, emphasis is placed on reducing the magnitude of the
gorithm combines a model-trust region approach with weights at the expense of goodness of fit, while produ-
the conjugate gradient approach and uses a step size cinga smoothernetwork response[26].
scaling mechanism to avoid time-consuming linear In an empirical Bayesian framework, the “optimal”
searches [23]. Hence, SCG can significantly reduce the weights are those that maximize the conditional poster-
number of computations performed in each iteration, ior density P(w|D,α,β,M), which is equivalent to min-
but may require more iterations to converge than do the imizing the regularized objective function F in equation
other conjugate gradient algorithms [22]. SCG uses the (5). Minimization of F is identical to finding the (locally)
CVES technique to prevent over-fitting, which is com- maximum a posteriori estimates of w, denoted wMP,
monly used in neural networks and can improve their which minimize E using the back-propagation training
D
predictive ability (generalization). algorithms [24]. However, this is possible only if n > m,
Training proceeds as follows. Let the data set be D= where misthenumber ofparameterstobeestimated.
{y,{xi}i=1,…,n}, where xi is a vector of inputs (SNP geno- Bayestheoremyieldstheposteriordensityofαandβas:
types) forindividualiandyisavectoroftarget variables
(EPD). Once a set of weights, w, is assigned to the con- Pðα;βjD;MÞ¼PðDjα;β;MÞPðα;βjMÞ:
nections in the networks, this defines a mapping from PðDjMÞ
the input x to the output ^y . Let M denote a specific
i i If the prior density P(α,β|M) is uniform, maximization
network architecture (in terms of numbers of neurons
of P(α,β|D,M) with respect to α and β is equivalent to
and choice of activation functions), then the typical ob-
maximization of P(D|α,β,M). Bayesian optimization of
jective function used for training a neural network using
the regularization parameters requires computation of
CVESisthesum ofsquaredprediction errors(E ):
D the Hessian matrix of the objective function F evaluated
Xn at the optimum point wMP [27], but directly computing
E ðDjw;MÞ¼ ð^y−yÞ2 ð4Þ
D i i the Hessian matrix is not always necessary. As proposed
i¼1 by MacKay [28], the Gauss-Newton approximation to the
Hessian matrix can be used if the Levenberg-Marquardt
forninput-target pairsdefining D.
optimization algorithm is employed to locate the mini-
Regularization produces shrinkage of parameter esti-
mum of F [13,29,30]. The Levenberg–Marquardt training
mates towards some plausible values, while at the same
algorithm [31] achieves second-order trainingspeed with-
time reducing their variance. In Bayesian models, re-
out computing the Hessian matrix. Briefly, the Hessian
gularization techniques involve imposing certain prior
matrixisapproximatedas:
distributions on the model parameters. In Bayesian
ANN (e.g., BRANN), the objective function that was H¼J0J; ð7Þ
specified in (4) has an additional term that penalizes
large weights, in the hope of achieving a smoother map- where J is the Jacobian matrix that contains first deriva-
ping. Gradient-based optimization is then used to tives of the network errors with respect to network
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parameters(theweightsand biases).Thegradientiscom- unknown and needs to be inferred, with the prior distri-
putedas: bution ofπtaken tobeuniform between0and1,
g¼J0e; ð8Þ πeUniformð0;1Þ: ð14Þ
andnetworkparametersareupdatedas: A Bernoulli indicator variable, δ, is introduced to fa-
j
cilitate sampling from the mixtures of the SNP effects,
wkþ1 ¼wk−ðJ0JþμIÞ(cid:2)J0e: ð9Þ asfollows:
Here, μ is Levenberg's damping factor, which is ad- pðδijπÞ¼π1−δið1−πÞδi:
justed at each iteration and guides the optimization
process. If the reduction of error sum of squares is Hence, unconditionally,(cid:1) the va(cid:3)riable αj follows a
rapid, a smaller value of μ is used to bring the algo- univariate-t distribution, t 0;S2α;υα , if δj=1, or is equal
rithm closer to the Gauss–Newton algorithm. Alterna- to zero otherwise [6]. Posterior inference of unknown
parameters in the Bayesian model via Markov chain
tively, the damping factor is increased to give a step
Monte Carlo (MCMC) implementation is described in
to gradient descent direction if an iteration provides
[7]. With a subset of, say k≤p, selected markers, the
insufficient reduction in the error sum of squares [32].
Optimal values of regularization parameters, α and β, statistical model takes the same form as (12), replacing p
with kforthenumberofmarkers.
in BRANN can be calculated as:
By assuming that all k of the selected SNPs (based on
γ
αMP ¼ ð10Þ the posterior model probability and including the fre-
2EwðwMPÞ quency of marker k during MCMC) have non-null
effects on the quantitative trait, we define a BayesCπ
and model with π=0, which was used for the statistical in-
n−γ ference and model cross-validation subsequent to selec-
βMP ¼ ; ð11Þ
2E ðwMPÞ tion of markers (referred to as post-selection hereafter).
D
So, posterior inference in BayesCπ with π = 0 was as for
where 0 ≤ γ=m−2αMPtr(HMP)−1≤m the number of BayesC π, except thatπ was fixedatzeroandhence sam-
effective parameters in the neural network, and m is plingoftheindicatorvectorδwasnolongerrelevant.
the total number of parameters. Here HMP is the max-
imum a posteriori estimate of H described in (7). Computationalimplementation
MATLAB [31] was used tofit the ANN. Each neural net-
BayesCπandBayesCwithπ=0 workconsistedofthreelayers(i.e.,input,hiddenandout-
Considerthefollowinglinearmodel: put layers). The number of neurons in a single hidden
layer variedfrom1to4.EachANNhad2421(or700)in-
Xp
y ¼μþ x α þe; ð12Þ puts (SNPs). Before processing, MATLAB automatically
i ij j i rescaled all input and output variables using the
j¼1
“mapminmax”functionsuchthattheyresidedintherange
where y is the marbling score EPD for the ith animal; μ [−1, +1], to enhance numerical stability. Two combina-
i
is the overall mean; α is the substitution effect associ- tions of activation functions were used: (i) a set of hyper-
j
ated with the jth SNP (j=1,…,p); x is an indicator vari- bolic tangent sigmoidal activation functions from the
ij
able corresponding to the genotype(cid:1)at the(cid:3) jth SNP (0, 1, input layer to the hidden layer, plus a linear activation
2) for the ith individual, and eeN 0;σ2 is a residual functionfromthehiddenlayertotheoutputlayer,and(ii)
i e
term,whereσ2 istheresidualvariance. a set of linearactivation functions from the input layerto
e
The BayesCπ model [7] assumes that each SNP effect the hidden layer and from the hidden layer to the output
is null with p(cid:1)roba(cid:3)bility π, or that it follows a normal dis- layer.Trainingwas stopped ifanyofthefollowingcriteria
tribution,N 0;σ2 ,with probability 1-π,i.e.: were met: (i) a maximum number (1000) of epochs was
α
reached, (ii) performance had met a pre-specified (the
(cid:5) (cid:4) (cid:1) (cid:3) performance function for feed-forward networks is the
αi(cid:5)(cid:5)(cid:5)π;σ2α ¼eN00;σ2α wwiitthhpprroobbaabbiilliittyyð1−ππÞ: ð13Þ msueitaanblesqtuaragreet,eorrro(riv))ltehveelL,e(viiei)nbthereg-gMraadriqenutarwdtapsabraemlowetear
exceeded1010.
Here, σ2α is a variance common to all non-zero SNP For each ANN architecture, eight replicates were run.
effects, which (cid:1)is assig(cid:3)ned a scaled inverse chi-square dis- Each replicate was independently initialized, in order to
tribution, χ−2 vα;s2α . Furthermore, the value of π is eliminate spurious effects caused by the starting values,
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and to improve predictive ability. The results were pre- results than prediction using a smaller panel, yet the op-
sented as averages across the eight replicates per ANN timal panel size may depend on many factors. In this
architecture. study, we empirically chose the 700-SNP subset as the
BayesCπ with π set equal to zero (referred to as the optimal panel. The fitting accuracy in the training set
BayesCpC procedure) was implemented via a high- and predictive accuracy in the testing set using the opti-
throughput computing pipeline to select SNPs for post- mal700-SNPsubset areillustrated inFigure3b.
selection statistical inference and cross-validation [5].
This pipeline ran multiple chains for both feature selec- DeterminationofanoptimalANNarchitecture
tion and cross-validation. A three-fold cross-validation The performance of the ANN architectures was exa-
approachwasemployed,inwhichthewholedatasetwas mined based on the sum of squared errors (SSE) of
divided into three approximately equal portions, with prediction with a 3K panel, averaged over eight inde-
two-thirdsusedfortrainingandone-thirdusedfortest- pendent replicates of each ANN, for both BRANN and
ing, and the portions used for training and testing were SCGANN. Each ANN had a distinct combination of
rotated three times. Each cross-validation experiment training algorithm, transformation method, and number
was randomly replicated eight times. Three parallel of neurons, but both, BRANN and SCGANN, had an
MCMC chains were run for each feature-selection ana- input of 3K SNPs. The average SSE ranged from 13.936
lysis, and each consisted of 50 000 iterations after a to 16.882 for BRANN, and from 36.531 to 39.571 for
burn-in of 5000 iterations, thinned every tenth iterate. SCGANN. Smaller SSE were produced by BRANN with
MCMC sampling for each cross-validation consisted of nonlinear activation functions and from 2 to 4 neurons,
100 000 iterations, with a burn-in of 10 000 iterations, and by SCGANN with nonlinear activation functions
thinnedeverytenthiterate. and from 1 to 4 neurons. There was no evidence that
more complex networks (e.g., with more neurons or a
Results non-linear transformation function) produced better
DeterminationofanoptimalSNPpanelsize predictions than the linear model, as the ANN were
The predictive performance of each ANN was examined similar in terms of their SSE. Possibly, this was because
usingeitherthe3K-SNPpaneloranoptimalsubsetof700 marbling score EPD is estimated under an additive
selectedSNPs.Thelatterwerederivedfromthe3K-panel, model of inheritance in which additive genetic merit has
selected using the BayesCpC procedure with three-fold a linear relationship with SNP effects. Nevertheless,
cross-validation.This was accomplishedbyexaminingthe BRANN performed as well as the linear models when
predictionperformanceof varyingpanelsizesfrom100to predicting this linear system. Also, BRANN consistently
2400 SNPs in 100-SNP increments, and the optimal set produced a more accurate prediction of marbling score
thatgavethebestpredictionincross-validationswasiden- EPD than did SCGANN. On average, SSE obtained from
tified. The reason for not choosing the optimal subset BRANN were about 40% to 50% of those obtained from
based on ANN models was because the selection tasks SCGANN (Figure 4). This is attributed to the use of
with a grid of 24 candidate SNP-panels of varying sizes Bayesian regularization, which handles over-fitting better
were too computationally intensive for BRANN. Never- than doestheSCG back-propagation.
theless, the parallel-BayesCpC pipeline handled this task
easily, because all jobs were submitted to run in parallel Predictiveperformanceusingthe3K-SNPpanel
onaclusterofdedicatedcomputers. BRANN and BayesCpC performed very similarly with
As shown in Figure 3a, the correlation between marb- the 3K-SNP panel and both methods yielded higher pre-
ling score EPD and their fitted values in the training set diction accuracies than did SCGANN (Figure 5). On
(referred to hereafter as fitting accuracy) increased al- average, the correlation in the training set was 0.941
most monotonically with panel size, until reaching a with BayesCpC. This correlation ranged from 0.942 to
plateau at a panel size of 1400 SNPs. However, the 0.967 with BRANN, and from 0.796 to 0.897 with
correlation between marbling score EPD and their pre- SCGANN. The average correlation (in the testing set)
dicted values in the testing set (referred to hereafter as was 0.776 with BayesCpC, and ranged from 0.776 to
predictive accuracy) reached its peak (0.863) with a 0.807 with BRANN, and from 0.653 to 0.689 with
panel size of 700 SNPs, and decreased thereafter. The SCGANN. In general, these correlations increased
decrease in prediction accuracy with > 700 selected slightly with the number of neurons, but no consistent
SNPs possibly reflects over-fitting in the training set, pattern wasobserved(Figure 5).
which, inthis case, happened much before the panel size Withthe3K-SNPpanel,thenumberofSNPs(i.e.,2421)
exceeded the training set size (i.e., approximately 2000 exceeded the number of animals (~2000 animals) in the
animals). Hence, with Bayesian regression models, pre- training set. This means that there were more parameters
dictionusingmoreSNPsmaynotnecessarilyyieldbetter to be estimated than data points, even when all SNPs
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a Training Testing
0.96
0.94
0.92
0.9
0.88
s
n 0.86
o
ati 0.84
rel 0.82
r
o 0.8
C
0.78
0.76
0.74
0.72
0.7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Number of Selected Markers X 100
b
Figure3SelectionofanoptimalSNPpanelsizeforpredictingmarblingscoreexpectedprogenydifference(EPD)usingtheBayesCpC
approach.(a)CorrelationsbetweenmarblingscoreEPDandtheirfitted(predicted)valuesinthetrainingandtestingsets,wheretheSNPpanel
consistedof100,200,…,2400markers,respectively;(b)correlationbetweenEPD(inputvalues)andfittedvaluesinthetrainingset(upper)and
correlationbetweenEPD(inputvalues)andpredictedvaluesinthetestingsetfortheoptimalsetof700SNPs(lower).
entered the model linearly. This explains the better pre- the posterior mean (standard deviation) of π was 0.621
dictive performance of BRANN over SCGANN: Bayesian (0.0830), based on 2421 polymorphic SNPs; this means
regularization imposes a penalty for model complexity that, on average, 918 SNPs were fitted in the model at
and prevents over-fitting, while SCGANN captures both each MCMC iteration. Hence, over-fitting was not a
signalandnoiseinthetrainingset,leadingtopoorpredic- concern. Posterior densities for π are shown in Figure 6
tion. The results illustrate that ANN with Bayesian and were computed using posterior samples obtained
regularization can behave better than ANN with SCG from each of the three parallel chains and from all
back-propagation. BRANN penalize the estimation of pooled samples of the three chains. The density plots for
largeweightstoachieveasmoothermapping,whichleads the three parallel chains highly resembled each other
toeffectivelymoreparsimoniousmodels. and had similar means, indicating that convergence and
In the BayesCpC procedure, the BayesCπ model pos- length of the MCMC chains were appropriate for π. We
tulates that a portion, π, of all SNPs have no effect on noticed that the estimate of π obtained from the 3K
marblingEPD.Inahigh-densitySNPpanel,πistypically panel wassmallerthanthatbased on BovineSNP50geno-
expected to be large, meaning that the portion of “sig- types, because the latter has typically been greater than
nal” SNPs, 1 - π, is small, and the chance of over-fitting 0.95[5].Thus,itwouldseemthat,withalow-densitySNP
is reduced. Using the Illumina Bovine3K SNP genotypes, panel, the interpretation of SNPs having putatively non-
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Figure4Sumofsquarederrors(y-axis)inthetestingsets,computedasaveragesfromeightreplicatesforBayesianneuralnetworks
(BRANN)andANNcoupledwiththeSCGalgorithm(SCGANN),forANNwithdifferentnumbersofneurons.
zero effects on the target traits should be taken with cau- thebestpredictionswhengeneralizedbeyondthetraining
tion, because many could be distant from the functional set.Thisphenomenonisreferredtoaspoorgeneralization
genes or quantitative trait loci. We also observed that the inmachinelearningmethods[12].
selected number of SNPs having non-zero effects based
on BayesCπ in the training set did not correspond to the Predictiveperformanceusingthe700-SNPpanel
number of SNPs in the optimal SNP panel for prediction Unlike the case with 3K-SNPs, prediction using the
(918 vs 700 SNPs). We suspect that parameter π in a selected 700-SNPs was not challenged by over-fitting.
BayesCπmodeldoesnotfullyinformonthesizeofanop- Hence, we observed a smaller difference in prediction
timal SNP panel for prediction; a model that describes performance between BRANN and SCGANN because
variation in a training set well does not necessarily yield regularization was not a decisive issue in the prediction.
1.2
Training Testing
1
0.8
0.6
0.4
0.2
0
ayes CpC Linear 1_Neuron _Neurons _Neurons _Neurons Linear 1_Neuron _Neurons _Neurons _Neurons
B 2 3 4 2 3 4
BRANN SCGAN
Figure5Correlationsbetweenmarblingscoreexpectedprogenydifferencesinthetraining(testing)setsandtheirfitted(predicted)
valuesusingBayesCpCandBRANNandSCGANNwithdifferentnumbersofneuronsinthehiddenlayerandusingthe3K-SNPpanel.
1training=correlationsinthetrainingsets;testing=correlationsinthetestingsets;2BayesCpC=Bayesianregressionmodel,wheretheBayesCπ
modelisusedforfeatureselectionandBayesCπwithπ=0isusedforpost-selectionstatisticalinferenceandcross-validation;BRANN=artificial
neuralnetworkwithBayesianregularization;SCGANN=artificialneuralnetworkwithscaledconjugategradientback-propagation.
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5 5
4 4
sity 3 sity3
n n
De 2 De2
1 1
0 0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Chain #1 Chain #2
5 5
4 4
sity 3 sity3
n n
De 2 De2
1 1
0 0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Chain #3 pooled
Figure6PosteriordensityforparameterπintheBayesCπmodelobtainedfromthreeparallelchainsandfrompooledsamplesofthe
threechains.
With the 700-SNP panel, the predictive performance of inheritance, and this smoothing may mask non-linear
BayesCpC was slightly better than that of BRANN, variation in the response variable. A better way to
possibly because the subset of SNPswasoptimal, at least analyze this trait would be to remove variation due to
as selected by BayesCpC. As expected, BayesCπ and contemporary groups from the field data and then
BRANN outperformed SCGANN in the prediction of analyze individual marblingphenotypes, but this was not
marbling score EPD but differencesin performance were possible here because we did not have access to the raw
smaller with the 700-SNP panel than with the 3K-SNP data, which were, in general, collected on progeny of the
panel. Average prediction accuracy was 0.863 with genotyped bulls. Theaccuracy ofEPD varies between in-
BayesCpC and ranged from 0.843 to 0.858with BRANN, dividuals, which suggests that the residuals may be
whereas the average prediction accuracy with SCGANN heteroscedastic due to unequal prediction error vari-
ranged from 0.743 to 0.793. For both BRANN and ances. An alternative is to use deregressed EPD as the
SCGANN, the difference in predictive accuracy between target variable, for which the parent averages are re-
the linear and non-linear activation functions was negli- moved and appropriate weights can be applied to ac-
gible (Figure 7). count for heteroscedastic residuals [33]. Some reports
suggest that training on deregressed EBVcould generate
Discussion higher accuracy of genome-enhanced breeding values
We investigated the predictive performance of ANN in than training on EBV [34]. However, the main purpose
comparison to a hierarchical Bayesian regression model, of the present research was to investigate the predictive
using 3K-SNP and 700-SNP panels. The 700 SNPs were performance of ANN in comparison with Bayesian re-
preselected from the former based upon their power to gressionmodelsinalinearsystem.WeusedEPDinstead
predict marbling EPD. Various ANN architectures to of deregressed proofs because correct deregression in-
predict marbling score EPD were examined, in conjunc- volves matrix operations (the data available were incom-
tionwithtwotrainingalgorithms,twotypesofactivation plete for correct deregression), or approximations. Both
functions, and from 1 to 4 neurons in the hidden layer. EPD and deregressed EPD are heteroscedastic because
We did not observe significant differences in predictive of an unequal amount of information, but the neural
accuracy between linear and non-linear models, prob- network software useddidnot allowfor incorporation of
ably because the relationship between marbling score heteroscedasticityina straightforwardmanner.
EPD and SNP effects is theoretically linear. An EPD In genomic selection, the joint estimation of a genome-
produces a smoothed data point based on additive wide SNP set is challenging due to the so-called “large p,