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Data-driven fault diagnosis of technical systems use training data from nominal and faulty operation to train machine learning models to detect and classify faults. However, data-driven fault diagnosis is complicated by the fact that training data from faults is scarce. The fault diagnosis task is often treated as a standard classification problem. There is a need for methods to design fault detectors using only nominal data. In model-based diagnosis, the ability construct fault detectors depends on analytical redundancy properties. While analytical redundancy is a model property, it describes the diagnosability properties of the system. In this work, the connection between analytical redundancy and the distribution of observations from the system on low-dimensional manifolds in the observation space is studied. It is shown that the intrinsic dimension can be used to identify signal combinations that can be used for constructing residual generators. A data-driven design methodology is proposed where data-driven residual generators candidates are identified using the intrinsic dimension. The method is evaluated using two case studies: a simulated model of a two-tank system and data collected from a fuel injection system. The results demonstrate the ability to diagnose abnormal system behavior and reason about its cause based on selected signal combinations.