1. 6. Kernel Regression

2. Framework Phenotype Genetic Value Model Residual
Ridge Regression / LASSO
Bayes A, Bayes B, Bayesian LASSO …
- Linear model:
- Reproducing Kernel Hilbert Spaces Regression
Neural Networks …
- Semi-parametric models:

3. RKHS Regressions (Background)
Uses:
Scatter-plot smoothing (Smoothing Splines) [1]
Spatial smoothing (‘Kriging’) [2]
Classification problems (Support vector machines) [3]
Animal model …
Regression setting
(it can be of any nature)
unknown function
[1] Wahba (1990) Spline Models for Observational Data.
[2] Cressie, N. (1993) Statistics for Spatial Data.
[3] Vapnik, V. (1998) Statistical Learning Theory.

4. RKHS Regressions (Background)
Non-parametric representation of functions
Reproducing Kernel:
Must be positive (semi) definite:
Defines a correlation function:
Defines a RKHS of real-valued functions [1]
[1] Aronszajn, N. (1950) Theory of reproducing kernels

5. Functions as Gaussian processes
K=A => Animal Model [1]
[1] de los Campos Gianola and Rosa (2008) Journal of Animal Sci.

6. RKHS Regression in BGLR1
ETA<-list(
list(K=K,model='RKHS') )
fm<-BGLR(y=y,ETA=ETA,nIter=...)
[1]: the algorithm is described in de los Campos et al. Genetics Research (2010)

7. Choosing the RK based on predictive ability
Strategies
Grid of Values of ө + CV
Fully Bayesian: assign a prior to ө
(computationally demanding)
Kernel Averaging [1]
[1] de los Campos et al. (2010) WCGALP & Genetics Research (In press)

8. Histograms of the off-diagonal entries of each of the three t kernels used (K1, K2, K3) in the RKHS models for the wheat dataset

9. How to Choose the Reproducing Kernel? [1]
Pedigree-models K=A
Genomic Models:
- Marker-based kinship -
Model-derived Kernel
Predictive Approach
Explore a wide variety of kernels
=> Cross-validation
=> Bayesian methods
[1] Shawne-Taylor and Cristianini (2004)

10. Example 2

11. Example 2

12. Example 2

13. Example 2

14. Example 3 Kernel Averaging

15. Kernel Averaging Strategies Grid of Values of  + CV
Fully Bayesian: assign a prior to 
(computationally demanding)
Kernel Averaging [1]
[1] de los Campos et al., Genetics Research (2010)

16. Kernel Averaging

17. Example 4 (100th basis function)

18. Example 4 (100th basis function, h=)

19. Example 4 (KA: trace plot residual variance)

20. Example 4 (KA: trace plot kernel-variances)

21. Example 4 (KA: trace plot kernel-variances)

22. Example 4 (KA: prediction accuracy)