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Parameters a and b in linear transformation are chosen so that the regression line for transformed data points aligns with the main diagonal A more advanced normalization alternative is the loess transformation It uses a scatterplot of log ratio of two channel intensities (log(cy3/cy5)) against average value of two channel intensities ((cy3 + cy5)/2) A locally weighted polynomial regression is used on this scatterplot to form a smooth regression curve Original data are then transformed using the obtained regression curve Loess normalization can also be used with single-channel microarrays where two arrays are observed as two channels and normalized together For data from more than two arrays, loess normalization can be iteratively applied on all distinct pairs of arrays, but this process has larger computational cost Some other forms of loess normalization are local loess [17], global loess, and two-dimensional loess [18] Several normalization methods make use of domain knowledge.



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All organisms have a subset of genes called housekeeping genes that maintain necessary cell activities, and, as a result, their expression levels are nearly constant under most biological conditions All the above-mentioned methods can be modi ed so that all transformation parameters are calculated based only on the expression levels of housekeeping genes 4232 Between-Chip Normalization Row column normalization [19] is applied to a data set comprised of several arrays, observed as a matrix with M rows (representing genes) and N columns (representing separate arrays and array channels) In one iteration, the mean value of a selected row (or column) is subtracted from all of the elements in that row (or column) This is iteratively repeated for all rows and columns of the matrix, until the mean values of all rows and columns approach zero This method xes variability among both genes and arrays.

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Proof: Let M be a Hermitian n x n matrix and X E cr(M) any of its eigenvalues. By Proposition A.2 there exists an eigenvector u of M such that M u = Xu. The ith row of this vector identity has the form

A major problem with this method is its sensitivity to outliers, a problem that can signi cantly increase computation time Outlier removal is thus crucial for the performance of this method The computation time can also be improved if standardization is rst applied to all individual arrays Distribution (quantile) normalization [20] is based on the idea that a quantile quantile plot is a straight diagonal line if two sample vectors come from the same distribution Data samples can be forced to have the same distribution by projecting data points onto the diagonal line For microarray data matrix with m rows.

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