Mann-Kendall monotonicity test

The Mann-Kendall test is a non-parametric statistical test used to determine the presence or absence of a monotonic trend in a data series. It is commonly used in environmental studies, hydrology, and climate science to detect trends in time series data.

The Mann-Kendall test is based on the rank correlation between the data values and their order in time. The test calculates the number of upward and downward trends in the data series, and compares them to the expected number of trends under the null hypothesis of no trend.

The test statistic is calculated as follows:

S = ∑ (sgn(xi – xj))

where sgn is the sign function, xi and xj are the data values at times i and j, and the summation is over all pairs of times i and j.

The test statistic S is compared to its expected distribution under the null hypothesis using a standard normal distribution. A positive value of S indicates an increasing trend, while a negative value indicates a decreasing trend. The p-value of the test is then calculated based on the normal distribution.

If the p-value is less than the significance level (usually 0.05), the null hypothesis of no trend is rejected and a monotonic trend is detected in the data series. If the p-value is greater than the significance level, the null hypothesis cannot be rejected and no trend is detected.

The Mann-Kendall test is a powerful tool for detecting trends in time series data, but it has some limitations. It assumes that the data are independent and identically distributed, and it may not be suitable for detecting trends in data with complex patterns or non-monotonic trends.