Precursor Detection of Charge Density Wave Phase Transitions Using CUSUM Filters and Explosiveness Tests

The critical temperature ($T_c$) in charge density waves remains an open problem from an experimental perspective, as there is no sharp signal, whether a Raman peak or X-ray intensity, to indicate a transition from a noisy signal. Five different approaches to signal detection from noisy data are compared here, namely, a simple amplitude-based approach, a CUSUM filter, a wavelet-based change point detection, manual fitting and an SADF explosiveness test. The SADF test detects the transition approximately 8.6(17) K before $T_c$ after optimizing the parameters. This early detection is a result of growing fluctuations before a true ordering transition. Manual fitting also gives similar results, namely, 8.6(17) K before $T_c$, verifying that indeed our algorithm detects what a skilled human would also judge to be a signal transition. Wavelet detection does not perform well for a gradual transition such as this one. Finite size scaling gives us a scaling exponent $\nu =-0.020\pm 0.023$, showing detection accuracy to be largely size independent within our simulated range.