# regCoef

Calculates the linear regression coefficient between two variables.

## Prototype

function regCoef ( x : numeric, y : numeric ) return_val : float or double

## Arguments

*x*

An array of any dimensionality. Missing values should be indicated by
*x*@_FillValue.
If *x*@_FillValue is not set, then the NCL
default (appropriate to the type of *x*) will be assumed.

*y*

An array of any dimensionality. The last (rightmoast) dimension of *y* must
be the same as the last dimension of *x*. Missing values
should be indicated by *y*@_FillValue. If *y*@_FillValue
is not set, then the NCL default (appropriate to the type of
*y*) will be assumed.

## Return value

If either *x* or *y* are of type double, then the return
array is returned as double. Otherwise, the returned variable is returned as
type float. The dimensionality is a bit more complicated; see the
description and examples below.

## Description

**regCoef** computes the linear regression coefficient
via least-squares.
**regCoef** is designed to work with multi-dimensional
x and y arrays. If the regression information for a single best fit
line for 1-dimensional x and y data is desired, then **regline**
is the appropriate choice. Missing data are allowed.

The **regCoef** function returns the following attributes:
*yintercept* (y-intercept),
*tval* (t-statistic), *rstd* (standard error of the estimated
regression coefficient) and *nptxy* (number of elements used).
These will be returned
as scalars for one-dimensional *x*/*y*; otherwise, as
one-dimensional attributes of the return variable (call it *rc*). The
type of *tval* and *rstd* will be the same as *rc*
while *nptxy* will be of type integer.
These attributes may be used for statistical testing
(see examples).

The dimensions of *rc* are illustrated as follows:

x(N), y(N) rc, tval, mptxy are scalars x(N), y(K,M,N) rc, tval, mptxy are arrays of size (K,M) x(I,N), y(K,M,N) rc, tval, mptxy are arrays of size (I,K,M) x(J,I,N), y(L,K,M,N) rc, tval, mptxy are arrays of size (J,I,L,K,M)There's a special case when

*all*dimensions of

*x*and

*y*are identical:

x(J,I,N), y(J,I,N) rc, tval, mptxy are arrays of size (J,I)Note on the units of the returned regression coefficient(s): if

*x*has units of, say, degrees Kelvin (K), and

*y*has units of, say, meters (M), then the units of the regression coefficient are M/K. The function does not standardize

*x*(or

*y*) prior to calculating the regression coefficient. If this is desired, then it is the user's responsibility do so. The NCL function

**dim_standardize**can be used.

**Use regCoef_n if the dimensions to do the
calculation on is not the rightmost dimension and reordering is not
desired. This function can be significantly faster
than regCoef.**

## See Also

## Examples

In the code snippets below, there are some examples of both
**regcoef** and **regCoef**, so you can
see how they both can be utilized. The choice of procedure or
function versions is strictly a matter of personal choice.

**Example 1**

In the example below, the regression coefficient for the case with no
missing data is 0.97 with standard error (rc@rstd) of 0.025.
There are 16 degrees of freedom (=*nptxy*-2). A test of the
null hypothesis (i.e., that the regression coefficient is zero) yields
a t-statistic (=*tval*) of 38.7, which is distributed as t(16). Obviously, the
null hypothesis (ie: rc=0.0) would be rejected. [data source: * Statistical Theory
and Methodology*, K.A. Brownlee, 1965, John Wiley & Sons, pp 342-345.]

The **betainc** function can be used to evaluate
a t-statistic distributed appoximately as a Student-t distribution.
[See: *Numerical Recipes: Fortran*, Cambridge Univ. Press, 1986.]

begin x = (/ 1190.,1455.,1550.,1730.,1745.,1770. \ , 1900.,1920.,1960.,2295.,2335.,2490. \ , 2720.,2710.,2530.,2900.,2760.,3010. /) y = (/ 1115.,1425.,1515.,1795.,1715.,1710. \ , 1830.,1920.,1970.,2300.,2280.,2520. \ , 2630.,2740.,2390.,2800.,2630.,2970. /) rc =The (edited) output is:regCoef(x,y)

Variable: rc Type: float Total Size: 4 bytes 1 values Number of Dimensions: 1 Dimensions and sizes: [1] Coordinates: Number Of Attributes: 5 _FillValue : 9.96921e+36 nptxy : 18Now use the information to calculate the probability.rstd: 0.02515461yintercept: 15.35228tval: 38.74286 (0) 0.9745615

; for clarity only, explicitly assing to a new variable df = rc@nptxy-2 ; degrees of freedom tval = rc@tval ; t-statistic prob =For illustration, add some missing values.betainc(df/(df+tval^2),df/2.0,0.5)

x(2) = x@_FillValue ; arbitrarily set msg values y(8) = y@_FillValue rc =regCoef(x,y) df = rc@nptxy-2 ; degrees of freedom tval = rc@tval ; t-statistic b = 0.5 ; b must be same size as tval (and df) prob =betainc(df/(df+tval^2),df/2.0,b)Note 1:Some users prefer to express probability as prob = (1 -betainc(df/(df+tval^2),df/2.0,b) )Note 2:To construct 95% confidence limits for the regression coefficient: [a] Thetfor 0.975 and 16 degrees of freedom is 2.120 [table look-up]. [b] rc@rstd * 2.12 = 0.053. This yields 95% confidence limits of (0.97-0.053) < 0.97 < (0.97+0.053) or (0.92 to 1.03).

**Example 2**

Assume *x* is a one dimensional array (1D) array of size
*ntim* and type float. Assume *y* is a three-dimensional
array (3D) array of size *nlat* x *nlon* x *ntim*.

Attributes returned by NCL functions must be one-dimensional arrays.
To transform to a multidimensional array use **onedtond**
and **dimsizes**.

rc =Obviously, user defined attributes may be assigned any time after a variable's creation. For example:regCoef(x,y) ; rc(nlat,nlon) ; rc@tval and rc@nptxy will be 1D arrays of ; size nlat*nlonprintVarSummary(rc) ; variable overview tval =onedtond(rc@tval ,dimsizes(rc)) df =onedtond(rc@nptxy,dimsizes(rc)) - 2 b = tval ; b must be same size as tval (and df) b = 0.5 prob =betainc(df/(df+tval^2),df/2.0,b) ; prob(nlat,nlon)

rc@long_name = "regression coefficient" prob@long_name = "probability"

**Example 3**

Same as Example 2 but *y* has
named dimensions,
"time", "lat" and "lon" with the following ordering *y*(time,lat,lon). Here, use
NCL's dimension reordering
to make "time" the rightmost dimension.

rc =regCoef(x, y(lat|:,lon|:,time|:) )

Note: with NCL V6.2.1 or later, you can use **regCoef_n**
to avoid having to reorder the arrays first:

rc =IfregCoef_n(x, y, 0, 0)

*y*has coordinate variables these may readily be assigned via NCL syntax:

rc!0 = "lat" ; name dimensions rc!1 = "lon" rc&lat = y&lat ; assign coordinate values to named dimensions rc&lon = y&lon