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# dtrend

Estimates and removes the least squares linear trend of the rightmost dimension from all grid points.

## Prototype

```	function dtrend (
y               : numeric,
return_info [1] : logical
)

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

## Arguments

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 rightmost dimension. This is usually time. Missing values (_FillValue) are not allowed. Use dtrend_msg or dtrend_msg_n if missing values are present.

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.

## 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 rightmost dimension but in the form of a one-dimensional array. e.g. if y is 45 x 34, 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 rightmost dimension from all grid points. Missing values are not allowed, use dtrend_msg or dtrend_msg_n if missing values exist. The mean is also removed. Optionally returns the slope (i.e., linear trend per unit time interval) and y-intercept for graphical purposes.

Assumes y is equally spaced. If this is not the case, then use dtrend_msg even if the data do not contain missing values.

Use dtrend_n if the dimension to do the calculation on is not the rightmost dimension and reordering is not desired. This function can be significantly faster than dtrend.

## Examples

Example 1

y is three-dimensional with dimensions lat, lon, and time. The returned array will have the same size. Remember that the mean is also removed.

```    yDtrend = dtrend(y,False)
```
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(y,True)
; yDtrend@slope = a one-dimensional array of nlat * nlon elements.
; the units are K/month
```

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

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

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

Let y be a three-dimensional array with dimensions time, lat, lon. Reorder y so that time is the rightmost dimension.

```  yDtrend = dtrend(y(lat|:,lon|:,time|:),False)
; yDtrend will be three-dimensional with dimension lat, lon, time.

; In V5.2.0 or later, you can use dtrend_n to avoid reordering:
;  yDtrend = dtrend_n(y,False,0)
```
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.

```  zDtrend = dtrend(z,True)
dimz = dimsizes(z)   ; retrieve dimension sizes of z

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%
```