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An unsolved problem in creating diagnostic expert systems is generating a qualitative understanding of how the system is behaving from raw data, especially numerical data taken across time. Yet automating this critical step is necessary for building the next generation of expert systems. The theory described provides a means of interpreting observations made of a physical system across time in terms of qualitative theories. Importantly, the theory is ontology-independent as well as domain-independent in that it only requires a qualitative description of the domain capable of supporting envisioning and domain-specific techniques for providing an initial qualitative description of numerical measurements. The theory is illustrated step by step with two extended examples, one involving qualitative process theory and the other involving a qualitative state vector representation of motion. The performance of an implementation of the theory is also illustrated.