easyclimate.core.mk_test

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.

See also

• https://github.com/mmhs013/pymannkendall

• 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

Functions

original_test(...)	Original Mann-Kendall test is a nonparametric test, which does not consider serial correlation or seasonal effects.
hamed_rao_modification_test(...)	This modified MK test proposed by Hamed and Rao (1998) to address serial autocorrelation issues.
yue_wang_modification_test(...)	This is also a variance correction method for considered serial autocorrelation proposed by Yue, S., & Wang, C. Y. (2004).
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.
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.
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.
regional_test(...)	Based on Hirsch (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.
correlated_seasonal_test(...)	This method proposed by Hipel (1994) used, when time series significantly correlated with the preceding one or more months/seasons.
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.
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.

Module Contents

easyclimate.core.mk_test.original_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

Original Mann-Kendall test is a nonparametric test, which does not consider serial correlation or seasonal effects.

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.hamed_rao_modification_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, lag: int = None, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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.

See also

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: http://doi.org/10.1016/S0022-1694(97)00125-X

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

lag: int.

No. of First Significant Lags

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.yue_wang_modification_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, lag: int = None, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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.

See also

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: http://doi.org/10.1023/B:WARM.0000043140.61082.60

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

lag: int.

No. of First Significant Lags

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.pre_whitening_modification_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

This test suggested by Yue and Wang (2002) to using Pre-Whitening the time series before the application of trend test.

See also

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: http://doi.org/10.1029/2001WR000861

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.trend_free_pre_whitening_modification_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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.

See also

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: http://doi.org/10.1029/2001WR000861

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.seasonal_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, period: int = 12, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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

See also

Hirsch, R. M., Slack, J. R., & Smith, R. A. (1982). Techniques of trend analysis for monthly water quality data. Water resources research, 18(1), 107-121. doi: http://doi.org/10.1029/WR018i001p00107

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

Significance level (0.05 is the default).

period: int, default 12.

Seasonal cycle. For monthly data it is 12, weekly data it is 52.

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.regional_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

Based on Hirsch (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.

See also

Hirsch, R. M., Slack, J. R., & Smith, R. A. (1982). Techniques of trend analysis for monthly water quality data. Water resources research, 18(1), 107-121. doi: http://doi.org/10.1029/WR018i001p00107

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.correlated_seasonal_test(data_input: xarray.DataArray | xarray.Dataset, dim: str, alpha: float = 0.05, period: int = 12, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

This method proposed by Hipel (1994) used, when time series significantly correlated with the preceding one or more months/seasons.

See also

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

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

period: int, default 12.

Seasonal cycle. For monthly data it is 12, weekly data it is 52.

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.

easyclimate.core.mk_test.sens_slope(data_input: xarray.DataArray | xarray.Dataset, dim: str, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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.

See also

• 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).

• 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: http://doi.org/10.1080/01621459.1968.10480934

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

alpha: float, default 0.05.

significance level (0.05 is the default).

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• slope: Theil-Sen estimator/slope

• intercept: intercept of Kendall-Theil Robust Line, for seasonal test, full period cycle consider as unit time step.

easyclimate.core.mk_test.seasonal_sens_slope(data_input: xarray.DataArray | xarray.Dataset, dim: str, period: int = 12, returns_type: str = 'dataset_returns') → xarray.Dataset | easyclimate.core.datanode.DataNode

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.

See also

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

Parameters

data_inputxarray.DataArray or xarray.Dataset

The spatio-temporal data to be calculated.

dim: str

Dimension(s) over which to apply MK test.

period: int, default 12.

Seasonal cycle. For monthly data it is 12, weekly data it is 52.

returns_type: str, default dataset_returns.

Returns the sorting type of the value. Sort by variable (dataset_returns) or by return value (dataset_vars).

Returns

Mann-Kendall Test results (xarray.Dataset or easyclimate.DataNode).

• 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.