Statistical Background

Linear Trend

A simple linear regression model can be written as:

$$ \hat{y}_i = a + b \cdot x_i $$

where \(\hat{y}_i\) denotes the estimated values of the dependent variable, \(x_i\) the independent variable, \(a\) the intercept with the y-axis, and \(b\) the slope of the line. To determine the parameters \(a\) and \(b\), the sum of squared deviations is typically minimized.

Residuals

Residuals \(r_i = y_i - \hat{y}_i\) are defined as the difference between the measured values \(y_i\) and the estimated values \(\hat{y}_i\).

Assumptions

The simple linear model assumes that:

1. The independent variable \(x\) is deterministic (fixed)
2. For each \(x_i\), \(y_i\) is a random variable
3. The \(y_i\) are independent of each other (no autocorrelation) and identically distributed (each \(y_i\) has the same distribution)
4. The residuals \(r_i\) are normally distributed

LOESS Smoother

LOESS stands for locally estimated scatterplot smoothing, or locally weighted regression scatterplot smoother in German. Smoothers are used to obtain a fitted line through a noisy dataset, for example, for simple visualization of the “trend” in a chart, for sampling the average course, or even for estimating confidence intervals.

Mann-Kendall Test

The Mann-Kendall test is a test for monotonic trends in a time series, based on the Kendall rank correlation of the time series. It tests the strength of the monotonic relationship between a dependent and an independent variable. The test is particularly popular in environmental sciences because the test statistic \(S\) is nearly normally distributed for small \(n\).

References

Hipel, K.W. and McLeod, A.I., (2005). Time Series Modelling of Water Resources and Environmental Systems. Electronic reprint of our book orginally published in 1994. http://www.stats.uwo.ca/faculty/aim/1994Book/.

Kendall, M.G. (1976). Rank Correlation Methods. 4th Ed. Griffin.

Petzold, T. (2019). Datenanalyse mit R Ausgewählte Beispiele - Skript. https://wwwpub.zih.tu-dresden.de/~petzoldt/elements_de.pdf

Disclaimer

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Source code is licensed free of charge under the GNU General Public License 2.0 available from https://github.com/JFeldbauer/hydrobio

In case of questions, please consult the authors T. Petzoldt and J. Feldbauer.

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