# ASCI 896 Statistical Genomics

# Homework assignment 7

### 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.