A procedure to derive optimal discrimination rules is formulated for binary functional classification problems in which the instances available for induction are characterized by random trajectories sampled from different Gaussian processes, depending on the class label. Specifically, these optimal rules are derived as the asymptotic form of the quadratic discriminant for the discretely monitored trajectories in the limit that the set of monitoring points becomes dense in the interval on which the processes are defined. The main goal of this work is to provide a detailed analysis of such optimal rules in the dense monitoring limit, with a particular focus on elucidating the mechanisms by which near-perfect classification arises. In the general case, the quadratic discriminant includes terms that are singular in this limit. If such singularities do not cancel out, one obtains near-perfect classification, which means that the error approaches zero asymptotically, for infinite sample sizes. This singular limit is a consequence of the orthogonality of the probability measures associated with the stochastic processes from which the trajectories are sampled. As a further novel result of this analysis, we formulate rules to determine whether two Gaussian processes are equivalent or mutually singular (orthogonal).

中文翻译：

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均方和异方差情况下高斯过程的最优分类

针对二元功能分类问题制定了导出最佳判别规则的过程，其中可归纳的实例由从不同高斯过程中采样的随机轨迹表征，具体取决于类别标签。具体来说，这些最佳规则是作为离散判别轨迹的二次判别式的渐近形式而得出的，其极限是在定义过程的间隔中，监测点集变得密集。这项工作的主要目的是对密集监视范围内的此类最佳规则进行详细分析，尤其着重于阐明产生近乎完美分类的机制。在一般情况下，二次判别式包含在此限制中为单数的项。如果这些奇异点没有消除，则将获得近乎完美的分类，这意味着对于无限大的样本量，误差渐近地接近零。这个奇异极限是与随机过程相关的概率测度正交的结果，从随机过程中采样轨迹。作为该分析的另一新颖结果，我们制定了规则来确定两个高斯过程是等效的还是互为奇异的（正交的）。