Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary widely across models. Besides some special cases where these integrals are easy to calculate, there exist no general analytical expressions, standard numerical methods or software for these integrals. Here we present mathematical results and open-source software that provide (i) the probability in any domain of a normal in any dimensions with any parameters, (ii) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector, (iii) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index and relation to the operating characteristic, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliably these methods may be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.
翻译:在作出不确定的决定时,常态和多变的正常概率分布被广泛使用。计算这些模型的性能需要将这些分布在特定领域,这些分布在不同的模型中可能有很大差异。除了这些整体体易于计算的一些特殊案例外,对这些整体体没有一般的分析表达、标准数字方法或软件。这里我们介绍数学结果和开放源软件,这些结果和软件提供了(一) 正常体任何层面中具有任何参数的概率,(二) 正常矢量的任何函数的概率密度、累积分布和反向累积分布,(三) 正常分布的任何数量、巴耶-最优不均度指数的分类错误,以及与操作特征的关系,(四) 这些问题的尺寸减少和可视化,以及(五) 测试在给定数据上使用这些方法的可靠性。我们用视觉研究应用在自然场景中探测occlud 对象和探测迷彩。