We present several methods for prediction of new observations in penalized regression using different methodologies, based on the methods proposed in: i) Currie et al. (2004), ii) Gilmour et al. (2004) and iii) Sacks et al. (1989). We extend the method introduced by Currie et al. (2004) to consider the prediction of new observations in the mixed model framework. In the context of penalties based on differences between adjacent coefficients (Eilers & Marx (1996)), the equivalence of the different methods is shown. We demonstrate several properties of the new coefficients in terms of the order of the penalty. We also introduce the concept memory of a P-spline, this new idea gives us information on how much past information we are using to predict. The methodology and the concept of memory of a P-spline are illustrated with three real data sets, two of them on the yearly mortality rates of Spanish men and other on rental prices.

中文翻译：

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惩罚回归预测的一般框架

我们提出了几种使用不同方法预测惩罚回归中新观察值的方法，基于以下中提出的方法：i) Currie 等人。(2004), ii) Gilmour 等人。(2004) 和 iii) Sacks 等人。(1989)。我们扩展了 Currie 等人介绍的方法。(2004) 考虑在混合模型框架中预测新的观察结果。在基于相邻系数之间差异的惩罚的背景下（Eilers & Marx (1996)），显示了不同方法的等效性。我们根据惩罚的顺序展示了新系数的几个属性。我们还介绍了 P 样条的概念记忆，这个新想法为我们提供了关于我们使用多少过去信息来预测的信息。P样条记忆的方法和概念用三个真实数据集来说明，