## ASCI 896 Statistical Genomics

### Due date

Tuesday, April 27, 5pm

## Data

Continue from HW6 and use the same data.

## Question 1

Jiang and Reif (2015) (DOI: 10.1534/genetics.115.177907) showed that a Gaussian kerenel matrix can be decomposed into $$GK = \Lambda \tilde{H} \Lambda$$, where $$\Lambda = diag[\exp (- \theta \sum^m_{k=1} x^2_{1k} ) , \cdots, \exp (- \theta\sum^m_{k=1} x^2_{nk})]$$, $$\tilde{H} = 1_{n \times n} + \sum^{\infty}_{k=1} \frac{(2 \theta m)^k}{k!} G^{\#k}$$, and $$G$$ is the second genomic relationship matrix of VanRaden (2008). Confirm that this is indeed true. Use the bandwidth parameter $$\theta = 0.0001$$.

## Question 2

Create two Gaussin kernel matrices (GK1 and GK2) where means of lower triangler matrix are about 0.85 and 0.25, respectively.

## Question 3

Evaluate predictive performance of the reproducing kernel Hilbert spaces regression model by repeating three-fold cross-validation 5 times. Fit a multiple kernel method by using GK1 and GK2 simultaneously. Type set.seed(0403) at the begining of the code so that your analysis is reporducible. Compare the results with the ones obtained from the Bayesian ridge regression and Bayesian LASSO in HW6.

April 18, 2017