Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.

翻译：可区分性及其扩展概念——可观测性，是动力系统的关键属性。建立这些属性具有挑战性，尤其是在没有解析模型可用且需直接从测量数据推断时。噪声的存在进一步复杂化了分析，因为标准的可区分性概念是针对确定性系统设计的。我们基于分布可区分性展开研究，该方法通过比较随机系统输出的分布来扩展确定性概念。我们首先证明，对于包含线性系统在内的一类系统，这两种概念是等价的。随后，我们提出一种从输出数据评估和量化分布可区分性的方法。具体而言，我们的量化方法测量了区分两个初始状态所需的数据量，从而导出一个连续的可区分性谱。我们提出了一种统计检验方法，用于确定一个阈值，超过该阈值可以高置信度地认为两个状态是可区分的。我们通过仿真计算状态空间上的可区分性图谱来演示这些工具，随后利用该检验方法在硬件上比较传感器配置。