Measures of Dependence and Model Selection in Multiple Regression

多元回归中的相关性测度与模型选择 Kjell A Doksum，DMS-0505651 研究人员的做法的困境，估计曲线和曲面，如条件平均值几乎是不可能的高维数据，而不是集中在估计的措施ofdependence之间的响应和一组协变量。这些相关性的度量采用信号除以噪声的形式。它们用于选择要包含在模型中的协变量子集，并用于优化调整参数。信号测量Y对一组协变量的依赖强度。噪声是一个标准误差，也就是说，估计信号的标准偏差的估计。当模型中包含太多的变量时，以及当调整参数为了更小的偏差而牺牲精度时，它将是小的。选择变量和调整参数，使信噪比最小化，导致程序以传统的root-n速率收敛。研究人员使用渐近和蒙特卡罗方法来研究这些过程的性质。他们还将它们与传统的模型和调谐参数选择程序联系起来，并将传统程序与信噪比方法进行比较。一个很好的例子是人类基因组计划产生的数据，其中大量的基因需要被认为是某种疾病的可能贡献者。处理大量的变量是困难的，因为它们中的许多变量会产生可能淹没潜在的有趣关系（信号）的变化（噪声）。研究人员通过使用通用的灵活模型方程来表示变量之间的重要关系来解决这个问题。然后选择方程的变量和方面，以最小化信噪比。这个过程会自动剔除主要贡献噪声的变量，并选择一个强调数据中存在的信号的方程。

Measures of Dependence and Model Selection in Multiple Regression Kjell A Doksum, DMS-0505651 AbstractThe investigators approach the dilemma that estimation of curves andsurfaces such as the conditional mean is virtually impossible with highdimensional data by focusing instead on the estimation of measures ofdependence between a response and a set of covariates. These measures ofdependence take the form of signal divided by noise. They are used toselect the subset of covariates to include in the model, and to choosetuning parameters. The signal measures the strength of the dependence of Yon a set of covariates. The noise is a standard error, that is, anestimate of the standard deviation of the estimated signal. It will besmall when too many variables are included in the model and when thetuning parameter sacrifices precision for smaller bias. Choosing variablesand tuning parameters that minimize signal to noise leads to proceduresthat converge at the traditional root-n rate. The investigators useasymptotic and Monte Carlo methods to investigate the properties of suchprocedures. They also relate them to traditional model and tuningparameter selection procedures and compare traditional procedures with thesignal to noise approach.The last few years have seen the establishment of large databases'containing a large number of variables that are to be related andcompared. A good example is the data produced by the human genome projectwhere a great number of genes need to be considered as possiblecontributers to a certain disease. Dealing with a large number ofvariables is difficult because many of them will contribute variability(noise) that may drown out potential interesting relationships (signals).The investigators approach this problem by using general flexible modelequations to represent important relationships between variables. Thenvariables and aspects of the equations are selected to minimize the ratioof signal to noise. This procedure automatically weeds out the variablesthat contribute mostly noise and selects an equation that emphasizes thesignals present in the data.