In the regularizing procedures, several authors have studied finite-dimensional approximations obtained by projecting regularized approximations into finite-dimensional subspaces.
In many institutionalized religions there is a regularized process by which saints are officially recognized.
Then each of the 10 datasets could be used to calibrate a model with and without regularization, which would result 10 slightly different calibrated models for both the non-regularized and regularized calibration procedure.
Autocorrelations were analyzed by a multi-tau regularized-fit procedure to obtain the distribution of decay rates.
Finally, the bootstrap method was used [ 42] to assess the significance of the models selected by the L1-regularized regression procedure.
As a way of determining the likelihood of obtaining models that explain such a large fraction of variance by chance, we compare the machine learning L1-regularized regression procedure with random permutations of six and seven genes to determine what fraction of randomly generated data sets would explain as large or larger a fraction of birth weight variance as the L1 procedure.
We repeated the L1-regularized regression procedure using the larger data set and identified an additional set of seven genes (ATP6AP1, PRSS21, RCOR1, ANGPT4, CDK2, EVPL and NAT8L), whose methylation levels explained 70% of the variance in birth weight, independently (Table 3).
These data indicate that the L1-regularized regression procedure is a valuable method for identifying small groups of genes whose methylation levels are correlated with birth weight and that random groups of genes of the same size perform as well only rarely.
We used a conglomerate classifier based on three different regularized t-test procedures to identify DE genes: Cyber-T [19], limma [18], [109] and SAM [20].
A regularized Hotelling T2 procedure was used to identify sets of proteins, defined by biological pathways, that differ in concentrations between cases and controls for each study disease.
We developed a regularized Hotelling T2 procedure (Chen LS, Prentice RL, and Wang P, submitted for publication, 2009) to identify sets of proteins, defined by biologic pathways, that change concentration with E+P, or that change differentially in the E+P and E-alone project components.
