We introduce a rigid and comprehensive taxonomy and paradigm for characterizing the influence of the input feature space $X$ on the predictions $\hat{y}$ of a neural network (NN) used for event classification, based on a Taylor expansion of $\hat{y}$ in $X$. The complete process of introspection we refer to as Taylor Coefficient Analysis (TCA). Based on two simplistic example tasks, which can be easily understood and bencmarked, we illustrate the power of the TCA when it comes to revealing, what properties of $X$ have led to what value of $\hat{y}$, of a given NN model, building up intuition for the method. A more complex application is meant to represent $X$ of a typical classification task at a CERN LHC experiment. Based on this application, we play through the different levels of introspection that the TCA offers and discuss a number of practical aspects for a TCA application typical for the analysis of CERN LHC data. We conclude with a study to support the assumption that those properties of $X$ most relevant for tasks of the complexity typical for a CERN LHC experiment, are usually caught by a TCA up to the second order.

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