@TechReport{	  it:2002-005,
  author	= {Kajsa Ljungberg and Sverker Holmgren and {\"O}rjan
		  Carlborg},
  title		= {Efficient Kernel Algorithms for QTL Mapping Problems},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2002},
  number	= {2002-005},
  month		= feb,
  abstract	= {The advent of sophisticated and powerful methods for
		  molecular genetics pushes the need for efficient methods
		  for data analysis. Advanced algorithms are necessary for
		  extracting all possible information from laboriously
		  obtained data sets. We present a general linear algebra
		  framework for QTL mapping, applicable to many commonly used
		  methods, using both linear regression and maximum
		  likelihood estimation. The formulation simplifies future
		  comparisons between and analyses of the methods. We show
		  how the common structure of QTL analysis models can be used
		  to improve the kernel algorithms, drastically reducing the
		  computational effort while retaining the original analysis
		  results. We have evaluated our new algorithms on data sets
		  originating from two large F$_2$ populations of domestic
		  animals. Using an updating approach, we show that 1-3
		  orders of magnitude reduction in computational demand can
		  be achieved for matrix factorizations. For interval
		  mapping/composite interval mapping settings using a maximum
		  likelihood model, we also show how to use the original EM
		  algorithm instead of the ECM approximation, significantly
		  improving the convergence and introducing an additional
		  reduction in the computational time. The algorithmic
		  improvements makes it feasible to perform analyses
		  previously deemed impractical or even impossible. For
		  example, using the new algorithms it is reasonable to
		  perform permutation testing using exhaustive search on
		  populations of 200 individuals for fully epistatic two-QTL
		  models with a large number of parameters. }
}