Which statistical tests to use for identifying trends in multiple time series?

I'm processing several time series in an effort to identify meaningful trends over time. More specifically, I'm computing several time- and frequency-domain metrics (e.g. RMS, Peak-to-Peak, Integral, Median and Peak frequencies) over sequential same-length windows/segments (no overlap) and performing a 1st order polynomial fit (using least-squares method). However, I'm struggling to understand how to objectively quantify the statistical significance of the polynomials I've obtained. For instance, a specific metric, computed for 4 separate series, over 31 intervals varies like:

How to I check if these trends are statistically significant? 1 and 3 are visually increasing, but the original data (from which metrics are computed) is inherently noisy, and therefore the evaluation metrics fluctuate between subsequent repetitions, which may influence the slope of these lines.

1. What kind of preliminary tests should I perform on either the original time series or the metric value distribution (regarding normality and autocorrelation of the time series and resulting metrics distribution)?

2. Which statistical test(s) are indicated to evaluate the existence (or not) of significant trends in the metric values? I've looked into the Mann-Kendall test (which, in the example above, results in very low p-values for sets 1 and 3, moderate for 2, and high for 4), which tests for monotonic trends without making any assumptions on its normality, but requires the data to not have any autocorrelation.

In addition, the original 4 time series are in truth rows in the coefficient matrix obtained after factorization (NNMF is used) of higher dimensional (10) sensor data, and I'm unsure as to what implications this has on the independence of each curve/trend.

• Note that in general the theory behind tests does not allow for tests to be chosen dependently of the data, This means that looking at the data first and then trying to find a test that tests the specific thing you're seeing is invalid. People may recommend something here, but you will need to apply this to new data of the same kind for it to be valid. May 18, 2023 at 23:53
• @ChristianHennig note that 'm not trying to cherry-pick tests that lead to any conclusion or bias I may have about the data. If any test shows that these variations are not statistically significant, then that's all the conclusions I take from this. The image I provided is simply an example of a subset of my data - I'm simply trying to understand which statistical tools exist for the type of analysis I'm trying to conduct, since I'm far from an expert on this and claiming "it looks like it's increasing" is not exactly objective and scientifically robust. May 19, 2023 at 13:17