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# trend_manken

Calculates Mann-Kendall non-parametric test for monotonic trend and the Theil-Sen robust estimate of linear trend. Available in version 6.3.0 and later.

## Prototype

```	function trend_manken (
x       : numeric,
opt  : logical,
dims    : integer
)

return_val  :  float or double
```

## Arguments

x

Series to be tested. If there is a strong seasonal cycle, the data should be deseasonalized, prior to input. The successive input values are assummed to be independent and evenly spaced. The series must be at least 10 values. Currently, missing values are not allowed.

dims

The dimension(s) of x on which to calculate the average. Must be consecutive and monotonically increasing.

opt

In NCL V6.4.0, the return_trend attribute was added such that if opt=True and opt@return_trend=False, only the Mann-Kendall trend significance is returned.

This argument was inactive in NCL V6.3.0 and earlier.

## Return value

The Mann-Kendall trend significance and the Theil-Sen estimate of linear trend (aka, Sen trend estimate) will be returned for series.

NOTE: for large arrays with very long time series, calculating the linear trend estimate will result in a large memory requirement and slower execution. This is because internally the slope estimates must be stored [N*(N-1)/2, where N is the length of the time series] and subsequently sorted into ascending order. To turn off this calculation, set opt=True and opt@return_trend=False.

The output will be double if x is double, and float otherwise.

The output dimensionality is best described via examples: The dimension rank of the input variable will be reduced by the rank of dims and an 'extra' dimension of size 2 will be prepended. Let NT refer to the 'time' dimension; KZ,NY,MX refer to spatial dimensions and NE refer to an ensemble dimension.

```
x(NT)            : p=trend_manken(x,False,0) => p(2) where p(0)=probability,  p(1)=trend
x(NT,NY,MX)      : p=trend_manken(x,False,0) => p(2,NY,MX)
x(NY,MX,NT)      : p=trend_manken(x,False,2) => p(2,NY,MX)
x(NT,KZ,NY,MX)   : p=trend_manken(x,False,0) => p(2,KZ,NY,MX)
x(NE,NT,KZ,NY,MX): p=trend_manken(x,False,1) => p(2,NE,KZ,NY,MX)
```

## Description

The Mann-Kendall (M-K) test is a non-parametric (ie, distribution free) test used to detect the presence of linear or non-linear trends in time series data. The M-K test assesses if a series is steadily increasing/decreasing or unchanging.

The null hypothesis is that there is no trend. The three alternative hypotheses are that there is a negative, non-null, or positive trend.

The series should be pre-whitened if serial correlation exits and/or deseasonalized if an annual cycle is present.

The M-K test is based on the relative ranking of the data values. Ties are allowed and appropriate corrections are made to the variance. The S statistic used to estimate the significance is calculated via:

```                                           = 1 (X(k)-X(i)) > 0
S = SUM[ SIGN((X(k)-X(i)) ]       = 0  X(k)=X(i)
=-1 (X(k)-X(i)) < 0
```

The Theil-Sen trend estimate is a robust estimate of linear trend. It is the median of slopes between all data pairs.

```         trend = MEDIAN[ (X(k)-X(i))/(k-i) ]  ; all i < k
```
The Theil-Sen trend estimation method "is insensitive to outliers; it can be significantly more accurate than simple linear regression for skewed and heteroskedastic data, and competes well against non-robust least squares even for normally distributed data in terms of statistical power. It has been called 'the most popular nonparametric technique for estimating a linear trend'."

A nice description for the Theil-Sen estimate and simple linear regression is here.

The underlying code follows the first part of:

```       http://vsp.pnnl.gov/help/Vsample/Design_Trend_Mann_Kendall.htm
```
The following illustrates usage:
```
Mondal et al (2012): RAINFALL TREND ANALYSIS BY MANN-KENDALL TEST:
A CASE STUDY OF NORTH-EASTERN PART OF CUTTACK DISTRICT, ORISS

International Journal of Geology, Earth and Environmental Sciences ISSN: 2277-2081 (Online)
An Online International Journal Available at http://www.cibtech.org/jgee.htm
2012 Vol. 2 (1) January-April,  pp.70-78/Mondal et al.
```

## Examples

Please see: Mann-Kendall Regression & Trend Examples 1 and 2

Example 1

Some pathological examples with the probability (significance) and Theil-Sen trend estimate indicated to the right:

```     a  = ispan(1,20,1)
pa = trend_manken(a, False, 0)     ; pa(0)=1.0   ,   pa(1)=1.0

b  = (/1,2,3,4,5,6,5,4,3,2,1/)
pb = trend_manken(b, False, 0)     ; pb(0)=0.0   ,   pb(1)=0.0

c  = (/-1,2,-3,4,-5,6,-7,8,-9,10,-11,12/)
pc = trend_manken(c, False, 0)     ; pc(0)=0.2683,   pc(1)=1.0
```

For comparison, on the last example R (after loading the 'Kendall' library) returns:

```    < library(Kendall)
> q  <- c(-1,2,-3,4,-5,6,-7,8,-9,10,-11,12)
> x  <- c(1,2,3,4,5,6,7,8,9,10,11,12)
> qx <-Kendall(x,y)
> qx

tau = 0.0909, 2-sided pvalue =0.7317    (Note: 1-0.7317= 0.2683 which matches NCL)
```
Example 1a

In NCL V6.4.0 and later, if opt=True and opt@return_trend=False, then only the probability is returned:

```     a  = ispan(1,20,1)
opt = True
opt@return_trend = False
pa = trend_manken(a, opt, 0)     ; pa=1.0
```

Example 2

Read a time series of global anomalies and calculate the significance and the Theil-Sen trend estimate. The data are annual means and reside in column 2 of the input file. The file structure is described here.

```   diri = "./"
fili = "TA_Globe.1850-2014.txt"
pthi = diri+fili

nrow = numAsciiRow(pthi)

ta   = data(:,1)   ; for clarity

opt  = False
pt   = trend_manken(ta, opt, 0)   ; ===> pt(0)=1.000  pt(1)= 0.004

```
The overall trend is 0.00443 per year (~0.04 C/decade) for 1850-2014. However, the series is perhaps better analyzed by partitioning the series into two periods (1850-1929) and (1930-2014). In the 1st period, the trend was ~0.0 while in the latter half the trend was 0.08 C/decade.

Example 3

Consider q(time,lat,lon) Calculate the probability level and Theil-Sen linear trend estimates at every grid point.

```     opt = False
pq  = trend_manken(q, opt, 0)  ; ===> px(2,nlat,mlon)
; if meta data is desired
copy_VarCoords(q(0,:,:),pq(0,:,:))
pq!0= "prob_trend"               ; ===> size 2
printVarSummary(pq)            ; ===> pq(prob_trend,lat,lon)
```