Time-series analysis in monitoring - trends, seasonality, anomalies

Knowledge base · July 15, 2026

A single reading means little. Only a time series shows whether something is rising, falling, returning each year, or has suddenly “shot up”. This article says what to look for in such a chart and with which tools.

Three things we read from a series

• Trend - the long-term direction (e.g. a slowly rising phosphorus concentration over the years). It answers the question “are things getting better or worse”.
• Seasonality - a repeating annual or daily rhythm (e.g. summer minima of oxygen, night-time drops, spring blooms). It has to be recognised so as not to confuse it with a trend.
• Anomalies - single, unusual deviations (e.g. a spike in turbidity after heavy rain). Some are real events, some are measurement errors.

Interpretation tools - with names and conditions of use

The limits of an I-MR chart are not computed from the standard deviation of the whole series but from the moving range: σ̂ = MR̄ / 1.128 (the d₂ constant for a two-element window), and the limits are the mean ± 3σ (warning limits ± 2σ). This detail matters: a standard deviation taken from the whole series contains the trend and the season, so the limits would become too wide and let some anomalies through. The moving range has a weakness of its own, though: under positive autocorrelation neighbouring readings resemble each other, so it understates σ and the limits come out too tight - some ordinary points then look like anomalies.

⚠️ A trap few people write about: environmental data are correlated in time. A probe measuring every 15 minutes does not give 96 independent observations per day - the next reading largely repeats the previous one. Tests that assume independent observations (including Mann-Kendall and the significance test of a correlation) understate p under positive autocorrelation, that is, they show significance where there is none. (Negative autocorrelation would work the other way, but it almost never occurs in environmental series.) The Hamed-Rao correction for Mann-Kendall or an effective sample size n_eff restore the right scale: a trend that is “significant” on the raw data can stop being significant after correction, and a month of daily measurements may be worth only a handful of independent observations, not 31 - with denser sampling the gap is even larger. Check the test’s assumptions before you accept a trend.

The most common mistakes

• Confusing seasonality with a trend - “rising” in July does not mean “rising year on year”.
• Uneven intervals / gaps - they affect averages and charts; it is worth marking them, not hiding them.
• Reacting to a single point - an anomaly is only confirmed by the next measurement or by context.
• A “categorical” time axis - dates must be treated as time (proportionally), not as labels.
• Mixing stations in one series - data from several sites glued together describe a spatial gradient, not a process in time; it can even reverse the sign of a relationship.
• Treating dense sampling as a larger sample - see above: the number of rows is not the number of independent pieces of information.

Sources

• Mann-Kendall test: Mann H.B. (1945), Nonparametric tests against trend, “Econometrica” 13(3): 245-259, doi:10.2307/1907187; Kendall M.G. (1975), Rank Correlation Methods.
• Sen’s slope: Sen P.K. (1968), Estimates of the regression coefficient based on Kendall’s tau, “Journal of the American Statistical Association” 63(324): 1379-1389, doi:10.1080/01621459.1968.10480934.
• Correction for autocorrelation: Hamed K.H., Rao A.R. (1998), A modified Mann-Kendall trend test for autocorrelated data, “Journal of Hydrology” 204(1-4): 182-196, doi:10.1016/S0022-1694(97)00125-X.
• Control charts and the d₂ constant: classic SPC methodology (Shewhart); the constant 1.128 applies to a moving range with a window of 2.

In practice

Good analysis starts with an orderly, continuous series at fixed points. In LimnoLog measurements immediately form charts over time (with a season comparison). You can export the full dataset to Excel or prepare it for statistical analysis when you need deeper methods.