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This function is deprecated and has been replaced by the faster eofunc_varimax.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

References on rotation:

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

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.