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# dtrend_msg_n

Estimates and removes the least squares linear trend of the dim-th dimension from all grid points (missing values allowed).

## Prototype

```	function dtrend_msg_n (
x           [*] : numeric,
y               : numeric,
remove_mean [1] : logical,
return_info [1] : logical,
dim         [1] : integer
)

return_val [dimsizes(y)] :  numeric
```

## Arguments

x

One-dimensional array containing the coordinate of the dim-th dimension of y. [eg: time].

y

A multi-dimensional array or scalar value equal to the data to be detrended. The dimension from which the trend is calculated needs to be the dim-th dimension. This is usually time.

remove_mean

A logical scalar indicating whether or not the mean is removed from return_val. True = remove mean, False = do not remove mean.

return_info

A logical scalar controlling whether attributes corresponding to the y-intercept and slope are attached to return_val. True = attributes returned. False = no attributes returned.

dim

A scalar integer indicating which dimension of y to do the calculation on. Dimension numbering starts at 0.

## Return value

An array of the same size as y. Double if y is double, float otherwise.

Two attributes (slope and y_intercept) may be attached to return_val if return_info = True. These attributes will be one-dimensional arrays if y is one-dimensional. If y is multi-dimensional, the attributes will be the same size as y minus the dim-th dimension but in the form of a one-dimensional array. e.g. if y is 45 x 34 and dim is 1, then the attributes will be a one-dimensional array of size 45*34. This occurs because attributes cannot be multi-dimensional. Double if return_val is double, float otherwise.

You access the attributes through the @ operator:

```  print(return_val@slope)
print(return_val@y_intercept)
```

## Description

Estimates and removes the least squares linear trend of the dim-th dimension from all grid points. The mean is optionally removed. Missing values are allowed. Optionally returns the slope (i.e., linear trend per unit time interval) and y-intercept for graphical purposes.

## Examples

Example 1

Let x be one-dimensional with dimension time and y be three-dimensional with dimensions lat,lon, and time. The return_val will be three-dimensional with dimensions lat,lon, time. The mean is removed.

```    yDtrend = dtrend_msg_n (x,y,True,False,2)
```
Example 2

Same as example 1 but with the optional attributes. Let y be temperatures in units of K and the time dimension have units of months.

```    yDtrend = dtrend_msg_n (x,y,True,True,2)
;   yDtrend@slope is a one-dimensional array of size nlat x nlon elements.
```

Since attributes cannot be returned as two-dimensional arrays, the user should use onedtond to create a two-dimensional array for plotting purposes:

```
yDtrend = dtrend_msg_n (x,y,False,True,2)

slope2D = onedtond(yDtrend@slope, (/nlat,mlon/) )
delete (yDtrend@slope)
slope2D = slope2D*120        ; would give [K/decade]

yInt2D  = onedtond(yDtrend@y_intercept, (/nlat,mlon/) )
delete (yDtrend@y_intercept)
```
Example 3

Let y be a three-dimensional array with dimensions time, lat, lon.

```  yDtrend = dtrend_msg_n(y&time,y,True,False,0)
; yDtrend will be three-dimensional with dimensions lat, lon, time.
```
Example 4

This example shows how to calculate the significance of trends by evaluating the incomplete beta function using betainc. Let z be a three-dimensional array with dimensions named lat, lon, time.

```  dimz = dimsizes(z)   ; retrieve dimension sizes of z
zDtrend = dtrend_msg_n(ispan(0,dimz(2)-1,1),z,True,True,2)

tval = new((/dimz(0),dimz(1)/),"float")  ; preallocate tval as a float array and
df = new((/dimz(0),dimz(1)/),"integer")  ; df as an integer array for use in regcoef

rc = regcoef(ispan(0,dimz(2)-1,1),z,tval,df)   ; regress z against a straight line to
; return the tval and degrees of freedom

df = equiv_sample_size(z,0.05,0)  ; If your data may be significantly autocorrelated
; it is best to take that into account, and one can
; do that by using equiv_sample_size. Note that
; in this example df (output from regcoef) is
; overwritten with the output from equiv_sample_size.
; If your data is not significantly autocorrelated one
; can skip using equiv_sample_size.

df = df-2          ; regcoef/equiv_sample_size return N, need N-2
beta_b = new((/dimz(0),dimz(1)/),"float")    ; preallocate space for beta_b
beta_b = 0.5       ; set entire beta_b array to 0.5, the suggested value of beta_b
; according to betainc documentation
z_signif = (1.-betainc(df/(df+tval^2), df/2.0, beta_b))*100. ; significance of trends
; expressed from 0 to 100%
```