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`mann_kendall` documentation

`mann_kendall` performs a standard simple Mann-Kendall test to determine the presence of a significant trend. (Requires the Statistics Toolbox)

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## Contents

## Syntax

h = mann_kendall(y) h = mann_kendall(y,alpha) h = mann_kendall(...,'dim',dim) [h,p] = mann_kendall(...)

## Description

`h = mann_kendall(y)` performs a standard simple Mann-Kendall test on the time series `y` to determine the presence of a significant trend. If `h` is `true`, the trend is present; if `h` is `false`, you can reject the hypothesis of a trend. This function assumes `y` is equally sampled in time.

`h = mann_kendall(y,alpha)` specifies the alpha significance level in the range 0 to 1. Default `alpha is 0.05`, which corresponds to the 5% significance level.

`h = mann_kendall(...,'dim',dim)` specifies the dimension along which the trend is calculated. By default, if `y` is a 1D array, the trend is calculated along the first nonsingleton dimension of `y`; if `y` is a 2D matrix, the trend is calcaulated down the rows (dimension 1) of `y`; if `y` is a 3D matrix, the trend is calculated down dimension 3.

`[h,p] = mann_kendall(...)` also returns the p-value of the trend.

## Example 1: 1D array

Consider these two arrays, one with a trend and one without:

x = (1:1000)'; y0 = randn(size(x)) + 1000; % random data without trend y1 = randn(size(x)) + 1000 + x/500; % random data with trend of 1/500 plot(x,y0,'b'); hold on plot(x,y1,'r');

To help show the trends, use `polyplot`, which will plot the first-degree polynomial trend lines for `y0` and `y1`.

polyplot(x,y0,1,'b','linewidth',2) polyplot(x,y1,1,'r','linewidth',2)

And just to verify the magnitudes of the trends in `y0` and `y1`, use the `trend` function:

trend(y0)

ans = 7.8691e-05

...as expected, the trend in `y0` is about zero. Now check `y1`:

trend(y1)

ans = 0.0020

...and as expected, the trend in `y1` is 1/500 as we intentionally imposed. Both of these trend magnitudes may seem like small numbers that are close to zero, while not being exactly zero. So are either of these trends significant? Use `mann_kendall` to find out, starting with `y0`:

mann_kendall(y0)

ans = logical 0

The `false` (logical zero) confirms it: The `y0` time series contains nothing but noise. Now check `y1`, in which, recall, we imposed a trend of 1/500:

mann_kendall(y1)

ans = logical 1

The logical 1 or `true` answer confirms that although `y1` contains noise and its trend is small in terms of magnitude, the trend is nonetheless present at the default alpha = 5% significance level. Is 5% significance not strict enough for you? Tighten it up to 0.1% like this:

mann_kendall(y1,0.001)

ans = logical 1

And that confirms that the trend in `y1` is present at 0.1% significance.

## Example 2: Multiple time series

If you have multiple time series in a 2D matrix, the `mann_kendall` function can operate on them all at once. For example, we'll make a dataset `D` using the two 1D array time series from Example 1:

D = [y0 y1];

By default, if the input to `mann_kendall` is a 2D matrix, the function will operate down the rows, so testing for significance of trends in `y0` and `y1` is this easy:

mann_kendall(D)

ans = 1×2 logical array 0 1

The `0` and `1` answer means there no significant trend in the first column of `D`, but there is a significant trend in the second column of `D`.

If each time series is in its own row rather than in its own column, specify the dimension of operation like this:

Dt = D'; % transpose D to make time go across columns of Dt. mann_kendall(Dt,'dim',2)

ans = 2×1 logical array 0 1

It's also possible to specify the significance level alpha. Let's *really* relax our standards, and set it to 99.999%:

```
mann_kendall(Dt,0.99999,'dim',2)
```

ans = 2×1 logical array 1 1

Wow, if we relax our standards enough, even the `y0` noise contains what appears to be a significant trend!

## Example 3: 3D data

Have sea surface temperatures significantly changed in the past few decades? To answer that question, load the 60x55x802 pacific_sst dataset, which contains a grid of 802 monthly sea surface temperatures from 1950 to 2016:

```
load pacific_sst
```

With a sampling rate of 12 times per year (monthly data), use the `trend` function to calculate the SST trend in degrees per year, and use `imagescn` to make a map. Set the colormap with `cmocean`:

% Calculate the SST trend: tr = trend(sst,12); % Plot the trend: figure imagescn(lon,lat,tr) cb = colorbar; ylabel(cb,'sst trend (\circC/yr)') cmocean('balance','pivot')

The map above shows that in most places, the ocean seems to be warming. But is the trend significant? Use the `mann_kendall` function to find out, and plot the significant regions using the `stipple` function:

significant = mann_kendall(sst); % (may take a second) hold on stipple(lon,lat,significant)

The map above shows that the SST trend is significant most places, at the alpha = 5% level. Feel free to try it with stricter standards.

Curious about what effects the seasons might have on the calculation? Try using `deseason` to remove seasonal cycles from the sst dataset and re-calculate the trends and significance.

## References

Mann, H. B. (1945), Nonparametric tests against trend, Econometrica, 13, 245-259.

Kendall, M. G. (1975), Rank Correlation Methods, Griffin, London.

## Author Info

This function is part of the Climate Data Toolbox for Matlab. The function and supporting documentation were written by Chad A. Greene, adapted from the `Mann_Kendall` function by Simone Fatichi.