New developments in penalized optimal scoring

在最优评分中，也称为多重对应分析和同质性分析，对多变量分类数据进行分析，并为类别和个体分配最优分数。在本研究中，我们扩展了最优评分，并开发了(A)一种用图形表示变量间非线性关系的方法和(B)一种用向量表示个体过渡趋势的方法。在(A)和(B)中，最小化目标函数定义为将现有最优得分的损失函数与惩罚函数相结合。接下来，我们详细介绍了(A)和(B)的研究。(A)我们开发了一种在低维构型中用非线性轨迹表示变量的方法。该方法将定量变量上的值作为标称类别，给出最优坐标，并将连接坐标的轨迹定义为自然三次样条函数。惩罚函数表示轨迹平滑度的损失，惩罚权重通过交叉验证程序选择。仿真研究和实际数据分析表明，该方法比现有的最优评分法和主成分分析法更能表征变量间的非线性关系。(B)我们提出了一种分析转换频率表的方法，并在低维配置中用向量表示转换趋势。这种方法找到个体的分数、类别的分数和趋势的向量，使个体的分数与所选类别的分数一致，趋势向量与个体分数在不同场合的变化一致。对实际数据的应用表明，所得到的趋势向量配置可以使我们很容易地掌握趋势。进一步，将这种方法推广到区别对待迁入者和滞留者。

In optimal scoring, also known as multiple correspondence analysis and homogeneity analysis, multivariate categorical data is analyzed and optimal scores are assigned to categories and individuals. In this study, we extend the optimal scoring and developed (A) a method for graphically representing inter-variable nonlinear relationships and (B) a method for representing transition trends of individuals by vectors. In both (A) and (B), an objective function to be minimized is defined as combing the loss function of the existing optimal scoring and a penalty function. Next, we detailed the studies on (A) and (B).(A)We developed a method for representing variables by nonlinear trajectories in a low-dimensional configuration. In this method, the values on quantitative variables are regarded as nominal categories to be given optimal coordinates, and the trajectories connecting the coordinates are defined as natural cubic spline functions. The penalty function expresses the loss of the smoothness of trajectories, and the penalty weight is chosen by a cross-validation procedure. Simulation study and real data analysis shows that the above method represents inter-variable nonlinear relations better than the existing optimal scoring and principal component analysis.(B)We proposed a method for analyzing transition frequency tables and representing transition trends by vectors in a low-dimensional configuration. This method finds scores of individuals, those of categories, and vectors for trends, in such a way that individuals' scores become homogeneous to the scores of chosen categories and trend vectors become homogeneous to the inter-occasion changes in individuals' scores. The applications to real data show that the resulting configurations of trend vectors allow us easily to grasp trends. Further, this method is extended to treat movers and stayers differently.