For two random vectors whose dimensions are both proportional to the sample size, we in this paper propose two ridge ratio criteria to determine the number of canonical correlation pairs. The criteria are, respectively, based on eigenvalue difference-based and centered eigenvalue-based ridge ratios. Unlike existing methods, the criteria make the ratio at the index we want to identify stick out to show a visualized “valley-cliff” pattern and thus can adequately avoid the local optimal solutions that often occur in the eigenvalues multiplicity cases. The numerical studies also suggest its advantage over existing scree plot-based method that is not a visualization method and more seriously underestimates the number of pairs than the proposed ones and the AIC and \(C_p\) criteria that often extremely over-estimate the number, and the BIC criterion that has very serious underestimation problem. A real data set is analyzed for illustration.

对于两个维度都与样本大小成比例的随机向量，我们在本文中提出了两个脊线比率标准来确定规范相关对的数量。该标准分别基于基于特征值差和基于中心特征值的脊比率。与现有方法不同，该标准使我们要识别的索引处的比率突出以显示可视化的“山谷-悬崖”模式，因此可以充分避免特征值多重性案例中经常出现的局部最优解。数值研究还表明，它相对于现有的基于碎石图的方法（不是可视化方法）具有优势，比拟议的对数和AIC和\（C_p \）更严重地低估了对数经常会高估数字的准则，以及BIC准则却存在严重低估问题的准则。分析实际数据集以进行说明。