# eof_varimax

Rotates EOFs using the using Kaiser row normalization and the varimax criterion (deprecated version).

## Prototype

function eof_varimax ( evec : numeric ) return_val [dimsizes(evec)] : numeric

## Arguments

*evec*

A multi-dimensional array containing the EOFs calculated using
**eofcor** or **eofcov**.

## Return value

An array of the same size and type as *evec*. In addition, as
of version 4.3.0, the
percent variance is returned as an attribute of the returned value
called *pcvar_varimax*.

## Description

This function is deprecated and has been replaced by the faster
**eofunc_varimax**.

Rotates EOFs using the using Kaiser row normalization and the varimax criterion.
The results are identical to IMSL's "FROTA" routine with the parameters w=1.0 and
eps=0.0001. Currently, the percent variance explained after the rotation is not
returned by the function.

The Kaiser varimax rotation is a common rotation performed on atmospheric or
oceanographic data. Rotation of the spatial modes (i.e. EOFs) is called R-mode
while rotation of the amplitude time series (expansion coefficients) is called Q-mode.
The focus of Q-mode analysis is interobject relationships. Q-mode analysis is not
commonly used today due to the advent of cluster analysis. R-mode rotation focuses
upon intervariable relationships such as the covariance/correlation between stations
or grid points.

Generally, it is R-mode rotation that is performed on atmospheric/oceanographic data.
The objective of R-mode analysis is to derive simple structures. Under Kaiser
varimax rotation this is accomplished by performing an orthonormal rotation on a user
specified number of modes such that some values are near +/- 1 with many near 0 values.
As noted by Trenberth et al. (2004) [*title* Journal, Vol pp. , ]
the effect is to localize the main centers of action and maximize the regions of
small weightings.

The output of conventional eof analysis are spatial patterns (EOFs) and temporal
series (eof_ts) that are both orthogonal. The result of varimax rotation upon
standard EOFs are rotated EOFs that are orthonormal. However, the temporal patterns
derived by projecting the rotated spatial patterns onto the data will not be
orthogonal. This means that the there is some correlation between the time series
expansion coefficients for each mode. The reverse is the case for Q-mode analysis.

The results may be very dependent upon the user specified number of modes used in the
rotation. The "best" number of modes to use may have to be determined by experiment.

When to use rotation:

- Don't use rotation unless you know what you are doing and why you are doing it!
- If the EOF patterns/coefficients are sufficiently separated (see discussion of
North's 'rule of thumb' in Storch and Zwiers [
*Statistical Analysis in Climate Research*Cambridge Univ. Press. 1998] there may be no need to use rotation if the patterns can be interpreted in physical terms. - If none of the patterns/coefficients are distinct then rotation may help reduce the noise and yield results that are more interpretable.
- If some are distinct and some are not then performing a rotation will mix the results.

J.C. Davis: Statistics and Data Analysis in Geology. John Wiley and Sons, 2nd Ed, 1984.

## See Also

This function is deprecated use: **eofunc_varimax**

## Examples

**Example 1**

Let x be two dimensional with dimensions variables (size = nvar) and time:

neval = 3 ; calculate 3 EOFs out of 7 ev =eofcor(x,neval) ; ev(neval,nvar) option = True option@jopt = 1 ; use correlation matrix ev_cor =eofcor(x,neval) ; ev_cor(neval,nvar) ev_rot =eof_varimax(ev_cor)

**Example 2**

Let x be three-dimensional with dimensions of time, lat, lon. Reorder x so that time is the rightmost dimension:

y!0 = "time" ; name dimensions if not already done y!1 = "lat" ; must be named to reorder y!2 = "lon" neval = nvar ; calculate all EOFs ev =eofcor(y(lat|:,lon|:,time|:),neval) ; ev(neval,nlat,nlon) ev_rot =eof_varimax(ev)

**Example 3**

Let z be four-dimensional with dimensions lev, lat, lon, and time:

neval = 3 ; calculate 3 EOFs out of klev*nlat*mlon ev =eofcor(z,neval) ; ev will be dimensioned neval, level, lat, lon ev_rot =eof_varimax(ev)

**Example 4**

Calculate the EOFs at every other point rather. Use of a temporary array is NOT necessary but it avoids having to reorder the array twice in this example:

neval = 5 ; calculate 5 EOFs out of nlat*mlon zTemp = z(lat|::2,lon|::2,time|:) ; reorder and use temporary array ev =eofcor(zTemp,neval) ; ev(neval,nlat/2,mlon/2) ev_rot =eof_varimax(ev)

**Example 5**

Let z be four-dimensional with dimensions level, lat, lon, time. Calculate the EOFs at one specified level:

kl = 3 ; specify level neval = 8 ; calculate 8 EOFs out of nlat*mlon ev =eofcor(z(kl,:,:,:),neval) ; ev will be dimensioned neval, lat, lon ev_rot =eof_varimax(ev)

**Example 6**

Let z be four-dimensional with dimensions time, lev, lat, lon. Reorder x so that time is the rightmost dimension and calculate on one specified level:

kl = 3 ; specify level neval = 8 ; calculate 8 EOFs out of nlat*mlon zTemp = z(lev|kl,lat|:,lon|:,time|:) ev =eofcor(zTemp,neval) ; ev will be dimensioned neval, lat, lon ev_rot =eof_varimax(ev)

**Example 7**

Area weight the data prior to calculation. Let p be four-dimensional with dimensions lat, lon, and time. The array lat contains the latitudes.

; calculate the weights using the square root of the cosine of the latitude and ; also convert degrees to radians wgt =sqrt(cos(lat*0.01745329)) ; reorder data so time is fastest varying pt = p(lat|:,lon|:,time|:) ; (lat,lon,time) ptw = pt ; create an array with metadata ; weight each point prior to calculation. ;conformis used to make wgt the same size as pt ptw = pt*conform(pt, wgt, 0) evec = eofcor(ptw,neval) evec_rot =eof_varimax(evec)