平均2乗誤差指標によるリッジパラメータに関する影響力評価

Abstract

Several influence measures have been proposed to assess the influence of observations on ridge regression. Influence measures based on the case deletion procedure that are used in the case of linear regression, such as Cook's distance or the likelihood distance, have been extended to the case of ridge regression. Most of these influence measures are derived from benchmarks based on statistical or mathematical distance criteria. In the case of a typical linear regression, these measures are the primary methods for conducting such assessments. However, in the case of ridge regression, we must consider not only assessments of the influence of observations but also assessments of the influence of the ridge parameter, a feature unique to the case of ridge regression. Therefore, in the present paper, we consider an ancillary influence measure for the latter assessment. The present study is a detailed investigation of the properties of an ancillary influence measure derived from benchmarks based on the mean square error criterion. The mean square error criterion is a standard statistical benchmark used to assess the regression coefficients in ridge regression, and it is also frequently used to identify an optimal ridge parameter. To investigate the properties of this ancillary influence measure, we derive a derivative function by computing partial derivatives with respect to the ridge parameter. As a result of this investigation, we discovered that the derivative function may be expressed as the sum of the original function and functions associated with it. In addition, we plot the behavior of the derivative function for each observation in order to obtain a clear understanding of variations with respect to the ridge parameter. Our results demonstrate that assessments of the influence of the ridge parameter are a useful, albeit ancillary and supplemental, complement to the case deletion procedure for assessing the influence of observations.