  Online monitoring is an effective validation approach for hybrid systems,
that, at runtime, checks whether the (partial) signals of a system satisfy a
specification in, e.g., Signal Temporal Logic (STL). The classic STL monitoring
is performed by computing a robustness interval that specifies, at each
instant, how far the monitored signals are from violating and satisfying the
specification. However, since a robustness interval monotonically shrinks
during monitoring, classic online monitors may fail in reporting new violations
or in precisely describing the system evolution at the current instant. In this
paper, we tackle these issues by considering the causation of violation or
satisfaction, instead of directly using the robustness. We first introduce a
Boolean causation monitor that decides whether each instant is relevant to the
violation or satisfaction of the specification. We then extend this monitor to
a quantitative causation monitor that tells how far an instant is from being
relevant to the violation or satisfaction. We further show that classic
monitors can be derived from our proposed ones. Experimental results show that
the two proposed monitors are able to provide more detailed information about
system evolution, without requiring a significantly higher monitoring cost.
