pyMannKendall

What is the Mann-Kendall Test ?

The Mann-Kendall Trend Test (sometimes called the MK test) is used to analyze time series data for consistently increasing or decreasing trends (monotonic trends). It is a non-parametric test, which means it works for all distributions (i.e. data doesn't have to meet the assumption of normality), but data should have no serial correlation. If the data has a serial correlation, it could affect in significant level (p-value). It could lead to misinterpretation. To overcome this problem, researchers proposed several modified Mann-Kendall tests (Hamed and Rao Modified MK Test, Yue and Wang Modified MK Test, Modified MK test using Pre-Whitening method, etc.). Seasonal Mann-Kendall test also developed to remove the effect of seasonality.

Mann-Kendall Test is a powerful trend test, so several others modified Mann-Kendall tests like Multivariate MK Test, Regional MK Test, Correlated MK test, Partial MK Test, etc. were developed for the spacial condition. pyMannkendal is a pure Python implementation of non-parametric Mann-Kendall trend analysis, which bring together almost all types of Mann-Kendall Test. Currently, this package has 11 Mann-Kendall Tests and 2 sen's slope estimator function. Brief description of functions are below:

1. Original Mann-Kendall test (original_test): Original Mann-Kendall test is a nonparametric test, which does not consider serial correlation or seasonal effects.

2. Hamed and Rao Modified MK Test (hamed_rao_modification_test): This modified MK test proposed by Hamed and Rao (1998) to address serial autocorrelation issues. They suggested a variance correction approach to improve trend analysis. User can consider first n significant lag by insert lag number in this function. By default, it considered all significant lags.

3. Yue and Wang Modified MK Test (yue_wang_modification_test): This is also a variance correction method for considered serial autocorrelation proposed by Yue, S., & Wang, C. Y. (2004). User can also set their desired significant n lags for the calculation.

4. Modified MK test using Pre-Whitening method (pre_whitening_modification_test): This test suggested by Yue and Wang (2002) to using Pre-Whitening the time series before the application of trend test.

5. Modified MK test using Trend free Pre-Whitening method (trend_free_pre_whitening_modification_test): This test also proposed by Yue and Wang (2002) to remove trend component and then Pre-Whitening the time series before application of trend test.

6. Multivariate MK Test (multivariate_test): This is an MK test for multiple parameters proposed by Hirsch (1982). He used this method for seasonal mk test, where he considered every month as a parameter.

7. Seasonal MK Test (seasonal_test): For seasonal time series data, Hirsch, R.M., Slack, J.R. and Smith, R.A. (1982) proposed this test to calculate the seasonal trend.

8. Regional MK Test (regional_test): Based onHirsch (1982) proposed seasonal mk test, Helsel, D.R. and Frans, L.M., (2006) suggest regional mk test to calculate the overall trend in a regional scale.

9. Correlated Multivariate MK Test (correlated_multivariate_test): This multivariate mk test proposed by Hipel (1994) where the parameters are correlated.

10. Correlated Seasonal MK Test (correlated_seasonal_test): This method proposed by Hipel (1994) used, when time series significantly correlated with the preceding one or more months/seasons.

11. Partial MK Test (partial_test): In a real event, many factors are affecting the main studied response parameter, which can bias the trend results. To overcome this problem, Libiseller (2002) proposed this partial mk test. It required two parameters as input, where, one is response parameter and other is an independent parameter.

12. Theil-Sen's Slope Estimator (sens_slope): This method proposed by Theil (1950) and Sen (1968) to estimate the magnitude of the monotonic trend. Intercept is calculate using Conover, W.J. (1980) method.

13. Seasonal Theil-Sen's Slope Estimator (seasonal_sens_slope): This method proposed by Hipel (1994) to estimate the magnitude of the monotonic trend, when data has seasonal effects. Intercept is calculate using Conover, W.J. (1980) method.

Function details:

All Mann-Kendall test functions have almost similar input parameters. Those are:

• x: a vector (list, numpy array or pandas series) data
• alpha: significance level (0.05 is the default)
• lag: No. of First Significant Lags (Only available in hamed_rao_modification_test and yue_wang_modification_test)
• period: seasonal cycle. For monthly data it is 12, weekly data it is 52 (Only available in seasonal tests)

And all Mann-Kendall tests return a named tuple which contained:

• trend: tells the trend (increasing, decreasing or no trend)
• h: True (if trend is present) or False (if the trend is absence)
• p: p-value of the significance test
• z: normalized test statistics
• Tau: Kendall Tau
• s: Mann-Kendal's score
• var_s: Variance S
• slope: Theil-Sen estimator/slope
• intercept: intercept of Kendall-Theil Robust Line, for seasonal test, full period cycle consider as unit time step

sen's slope function required data vector. seasonal sen's slope also has optional input period, which by the default value is 12. Both sen's slope function return only slope value.

Dependencies

For the installation of pyMannKendall, the following packages are required:

• numpy
• scipy

Installation

You can install pyMannKendall using pip. For Linux users

sudo pip install pymannkendall

or, for Windows user

pip install pymannkendall

or, you can use conda

conda install -c conda-forge pymannkendall

or you can clone the repo and install it:

git clone https://github.com/mmhs013/pymannkendall
cd pymannkendall
python setup.py install

Tests

pyMannKendall is automatically tested using pytest package on each commit here, but the tests can be manually run:

pytest -v

Usage

A quick example of pyMannKendall usage is given below. Several more examples are provided here.

import numpy as np
import pymannkendall as mk

# Data generation for analysis
data = np.random.rand(360,1)

result = mk.original_test(data)
print(result)

Output are like this:

Mann_Kendall_Test(trend='no trend', h=False, p=0.9507221701045581, z=0.06179991635055463, Tau=0.0021974620860414733, s=142.0, var_s=5205500.0, slope=1.0353584906597959e-05, intercept=0.5232692553379981)

Whereas, the output is a named tuple, so you can call by name for specific result:

print(result.slope)

or, you can directly unpack your results like this:

trend, h, p, z, Tau, s, var_s, slope, intercept = mk.original_test(data)

Citation

If you publish results for which you used pyMannKendall, please give credit by citing Hussain et al., (2019):

Hussain et al., (2019). pyMannKendall: a python package for non parametric Mann Kendall family of trend tests.. Journal of Open Source Software, 4(39), 1556, https://doi.org/10.21105/joss.01556

@article{Hussain2019pyMannKendall,
	journal = {Journal of Open Source Software},
	doi = {10.21105/joss.01556},
	issn = {2475-9066},
	number = {39},
	publisher = {The Open Journal},
	title = {pyMannKendall: a python package for non parametric Mann Kendall family of trend tests.},
	url = {http://dx.doi.org/10.21105/joss.01556},
	volume = {4},
	author = {Hussain, Md. and Mahmud, Ishtiak},
	pages = {1556},
	date = {2019-07-25},
	year = {2019},
	month = {7},
	day = {25},
}

Contributions

pyMannKendall is a community project and welcomes contributions. Additional information can be found in the contribution guidelines.

Code of Conduct

pyMannKendall wishes to maintain a positive community. Additional details can be found in the Code of Conduct.

References

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2. Conover, W.J., (1980). Some methods based on ranks (Chapter 5), Practical nonparametric statistics (2nd Ed.), John Wiley and Sons.

3. Cox, D. R., & Stuart, A. (1955). Some quick sign tests for trend in location and dispersion. Biometrika, 42(1/2), 80-95. doi:10.2307/2333424

4. 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

5. Helsel, D. R., & Frans, L. M. (2006). Regional Kendall test for trend. Environmental science & technology, 40(13), 4066-4073. doi:10.1021/es051650b

6. Hipel, K. W., & McLeod, A. I. (1994). Time series modelling of water resources and environmental systems (Vol. 45). Elsevier.

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8. Jacquelin Dietz, E., (1987). A comparison of robust estimators in simple linear regression: A comparison of robust estimators. Communications in Statistics-Simulation and Computation, 16(4), pp.1209-1227. doi: 10.1080/03610918708812645

9. Kendall, M. (1975). Rank correlation measures. Charles Griffin, London, 202, 15.

10. Libiseller, C., & Grimvall, A. (2002). Performance of partial Mann-Kendall tests for trend detection in the presence of covariates. Environmetrics: The official journal of the International Environmetrics Society, 13(1), 71-84. doi:10.1002/env.507

11. Mann, H. B. (1945). Nonparametric tests against trend. Econometrica: Journal of the Econometric Society, 245-259. doi:10.2307/1907187

12. 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

13. Theil, H. (1950). A rank-invariant method of linear and polynominal regression analysis (parts 1-3). In Ned. Akad. Wetensch. Proc. Ser. A (Vol. 53, pp. 1397-1412).

14. Yue, S., & Wang, C. (2004). The Mann-Kendall test modified by effective sample size to detect trend in serially correlated hydrological series. Water resources management, 18(3), 201-218. doi:10.1023/B:WARM.0000043140.61082.60

15. Yue, S., & Wang, C. Y. (2002). Applicability of prewhitening to eliminate the influence of serial correlation on the Mann-Kendall test. Water resources research, 38(6), 4-1. doi:10.1029/2001WR000861

16. Yue, S., Pilon, P., Phinney, B., & Cavadias, G. (2002). The influence of autocorrelation on the ability to detect trend in hydrological series. Hydrological processes, 16(9), 1807-1829. doi:10.1002/hyp.1095