Symbol sequences are generated from observed time series data to construct probabilistic finite state automata (PFSA) models that capture the evolution of the dynamical system under consideration. One of the key challenges here is to estimate the relevant history or depth (i.e., the size of temporal memory) of the symbol sequences; in this context, spectral decomposition of the one-step transition matrix has been recently proposed for depth estimation. This paper compares the performance of depth estimation by spectral analysis with that of other commonly used metrics (e.g., log-likelihood, entropy rate and signal reconstruction) for analysis of symbolic dynamic systems. For experimental validation of the proposed concept, time-series data of fatigue damage evolution in a polycrystalline alloy, collected on a laboratory apparatus, have been discretized to generate symbol sequences. The depths, estimated by the spectral decomposition method, are then compared with those obtained by other metrics, and the results are found to be in close agreement. Furthermore, unsupervised clustering of time-series data, obtained for a number of test specimens in the fatigue-test experiments, demonstrates the efficacy of the proposed depth estimation method as well as the accuracy of depth estimation via spectral analysis and PFSA model construction.