Mann-Kendall trend test

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Mann-Kendall trend test is used to detect statistically significant decreasing or increasing trends in long-term temporal data. It is based on two hypotheses: one is null hypothesis (H₀), which specifies the absence of a trend, and the other is alternate hypothesis (H₁), which expresses a significant increasing or decreasing trend in data over a time period.

The Mann-Kendall trend test is a non-parametric test, meaning that this test is applicable to all types of distributions.

It can be applied to any data set containing a number of data points greater than four, but sometimes with a smaller number of samples, the test has a higher chance of not finding a trend, even if one exists, if more data points were considered.

Yearly rainfall data of state Assam and Meghalaya, India.

This test is widely used on real-world data like hydrological data, climate data, and environmental data.

The idea behind the test is that it looks for all possible differences between the relative magnitude of one sample and other successive samples, and if the differences consistently increase or decrease, it signifies the presence of a trend.

Mathematically, this idea is expressed with the Mann-Kendall statistic (S).

Initially, it is assumed that there is no trend, meaning the value of S is assumed to be equal to 0. If the value of the data point at the next time period is greater than the value of the data point at the earlier time period, 1 is added to the value of S. Otherwise, if it is lower, 1 is subtracted from the value of S.

Mann-Kendall statistic

The Mann-Kendall statistic is denoted by ‘S’ and is defined as

$S=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}sig(xj-xi)$

Where, n is the total number of data points in the dataset, and x₁, x₂, x₃, x₄, x₅, …………. x_n are the values of the data points.

$sig(xj-xi)=-1,ifxj<xi$

$sig(xj-xi)=0,ifxj=xi$

$sig(xj-xi)=1,ifxj>xi$

For an increasing trend, the value of the Mann-Kendall statistic (S) should be a high positive value. And for a decreasing trend, the value of S should be a low negative value. To statistically express the significance of the trend, it is necessary to calculate the probability associated with the Mann-Kendall statistic.

Calculation of p-value associated with Mann-Kendall statistic(S)

The distribution of the dataset can be assumed to be a normal distribution if the number of data points in the dataset is greater than 10 and the number of tied values inside the dataset is low.

Variance of the Mann-Kendall statistic is given as

$variance(s)=\frac{1}{18}\ast \{n(n-1)(2n+5)-\sum _{i=1}^p{t}_i(t_i-1)(2t_i+5)\}$

Where,

n is the total number of data points in a dataset, p is the number of tied groups (a set of sample data having equal value), and t_i is the total number of data points in the i^th tied group.

Z-value associated with S

$Z=(S-1)/\sqrt{\left(VARIANCE\left(S\right)\right)}IFS>0$

$Z=0IFS=0$

$Z=\frac{S+1}{\sqrt{VARIANCE\left(S\right)}}IFS<0$

Probability (p-value) can be calculated using the z-table.

Based on a 5% significance level, if the p-value is less than or equal to 0.05, then the alternate hypothesis is accepted, which signifies the presence of a trend, and if the p-value is greater than 0.05, then the null hypothesis is accepted, which indicates the absence of a trend in the data.

Limitations

1. This test is not suitable for data having a seasonal effect. So, to make the test more effective, it is recommended to remove the seasonal effect before applying the test.
2. Often, this test gives negative results for time series having fewer data points.

References

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