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Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
Rule Based Kernels For Semi Parametric Mixed Models
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Rule Based Kernels For Semi Parametric Mixed Models

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Rule ensembles based kernels used in semi parametric mixed models

Rule ensembles based kernels used in semi parametric mixed models

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  • 1. Rule Based Kernels in Semi-Parametric Mixed ModelsRule Based Kernels in Semi-Parametric MixedModelsDeniz AkdemirPostdoctoral ResearcherCornell UniversityDepartment of Plant Breeding and GeneticsIthaca, NY
  • 2. Rule Based Kernels in Semi-Parametric Mixed ModelsExtraction of RulesISLE AlgorithmAlgorithm 1.1: ISLE(M, ν, η)F0(x) = 0.for j=1 to Mdo(cj, θj) = argmin(c,θ)i∈Sj(η) L(yi, Fj−1(xi) + cf(xi, θ))Tj(x) = f(x, θj)Fj(x) = Fj−1(x) + νcjTj(x)return ({Tj(x)}Mj=1 and FM (x).)Here L(., .) is a loss function, Sj(η) is a subset of the indices{1, 2, . . . , n} chosen by a sampling scheme η, 0 ≤ ν ≤ 1 is amemory parameter.
  • 3. Rule Based Kernels in Semi-Parametric Mixed ModelsPost ProcessingLASSO Post-ProcessingThe final ensemble models considered by the ISLE framework havean additive form:F(x) = w0 +Mj=1wjf(x, θj) (1)where {f(x, θj)}Mj=1 are base learners selected from F. ISLE uses atwo-step approach to produce F(x). The first step involvessampling the space of possible models to obtain {θj}Mj=1. Thesecond step proceeds with combining the base learners by choosingweights {wj}Mj=0 in (1).Friedman and Popescu [1] recommend learning the weights{wj}Mj=0 using lasso [2].
  • 4. Rule Based Kernels in Semi-Parametric Mixed ModelsPost ProcessingTrees to RulesFigure: A simple regression tree which can be represented asy = 20I(x < 0)(z < 1) + 15I(x < 0)I(z ≥ 1) + 10I(x ≥ 0). Each leafnode defines a rule which can be expressed as a product of indicatorfunctions of half spaces. Each rule specifies a ’simple’ rectangular regionin the input space.
  • 5. Rule Based Kernels in Semi-Parametric Mixed ModelsKernel Learning and Clustering With RulesSimilarities in Phenotype and Genotype spacesSimilarities in phenotype space: Y Y ′,Similarities in genotype space: MM′,Y Y ′ = MM′.We want f(M) = K ≈ Y Y ′ : Kernel Learning.
  • 6. Rule Based Kernels in Semi-Parametric Mixed ModelsKernel Learning and Clustering With RulesSemi-Supervised Importance Sampling ClusteringAlgorithm (SS-ISCA)Algorithm 3.1: SS-ISCA(X, Y, M, m, ν)R1 : A random projection of Yfor j = 1 to MdoGenerate m logic rules {lℓ(x)}mℓ=1 to estimate Rj from X:Sj(x) ⇐ {lℓ(x)}mℓ=1T(X) ⇐ {Sj(xi)}ni=1jd=1Rj+1 : A random projection of YRj+1 ⇐ (I − νPT(X))Rj+1return (T(X))
  • 7. Rule Based Kernels in Semi-Parametric Mixed ModelsKernel Learning and Clustering With RulesSemi-Parametric Mixed ModelSelection in animal or plant breeding is usually based onestimates of genetic breeding values (GEBV) obtained withsemi-parametric mixed models (SPMM).A SPMM for the n × 1 response vector y is expressed asy = Xβ + Zg + e (2)where X is the n × p design matrix for the fixed effects, β is ap × 1 vector of fixed effects coefficients, Z is the n × q designmatrix for the random effects; the random effects (g′, e′)′ areassumed to follow a multivariate normal distribution withmean 0 and covarianceσ2gK 00 σ2e Inwhere K is a q × q kernel matrix.
  • 8. Rule Based Kernels in Semi-Parametric Mixed ModelsKernel Learning and Clustering With RulesFHB Data Set1 210203040ISCAp−val=01 210203040SS−ISCAp−val=01 210203040RFp−val=0.5671 210203040PAMp−val=0.1671 210203040Mclustp−val=0.725Figure: (FHB Data Set, Semi-Supervised Clustering) p values from the ttests corresponding to different clustering approaches indicate that theSS-ISCA and ISCA produce groups that are different from each other interms of the mean FHB.
  • 9. Rule Based Kernels in Semi-Parametric Mixed ModelsKernel Learning and Clustering With RulesFHB Data SetTable: (FHB Data Set, Semi-Supervised Clustering) SS-ISCA and ISCAclusterings outperform other clusterings.silhouette dunn connectSS-ISCA 0.108 0.419** 44.450**ISCA 0.114* 0.344* 67.763*RF 0.092 0.344* 111.410PAM 0.126** 0.160 206.743Mclust 0.114* 0.317 126.918
  • 10. Rule Based Kernels in Semi-Parametric Mixed ModelsRule Based Similarity Matrix in Mixed ModelRule Based Similarity Matrix in Mixed Model-FHB DataSetTarget variables DON and FHB are used in SS-ISCA.Linear Gaussian ISCA SS−ISCA0.50.60.70.8Figure: (Vertical axis: Correlation between trait and GEBVs in the test
  • 11. Rule Based Kernels in Semi-Parametric Mixed ModelsRule Based Similarity Matrix in Mixed ModelRule Based Linear Gaussian0.890.900.910.920.930.940.95Grincore2012Aberdeen_2011, trait: Heading DateFigure: Grincore2011Aberdeen Heading Date (Vertical axis: Correlationbetween trait and GEBVs in the test data)
  • 12. Rule Based Kernels in Semi-Parametric Mixed ModelsRule Based Similarity Matrix in Mixed ModelYieldRule Based Linear Gaussian0.50.60.70.80.9MN_SP2_NormN_2011_Crookston, trait: yieldFigure: Crookston 2011 Yield (Vertical axis: Correlation between traitand GEBVs in the test data)
  • 13. Rule Based Kernels in Semi-Parametric Mixed ModelsRule Based Similarity Matrix in Mixed ModelHeightRule Based Linear Gaussian0.50.60.70.80.9MN_SP2_NormN_2011_Crookston, trait: yieldFigure: Crookston 2011 Height (Vertical axis: Correlation between traitand GEBVs in the test data)
  • 14. Rule Based Kernels in Semi-Parametric Mixed ModelsBibliographyJ.H. Friedman and B.E. Popescu.Importance sampled learning ensembles.Journal of Machine Learning Research, 94305, 2003.R. Tibshirani.Regression shrinkage and selection via the lasso.Journal of the Royal Statistical Society. Series B (Methodological), pages 267–288, 1996.

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