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eofunc_varimax_Wrap

Rotates EOFs using the using Kaiser row normalization and the varimax criterion and retains metadata.

Prototype

load "$NCARG_ROOT/lib/ncarg/nclscripts/csm/contributed.ncl"  ; not needed from 6.2.0 onward
                                                           ; contributed.ncl is automatically loaded

	function eofunc_varimax_Wrap (
		evec    : numeric,  
		optEVX  : integer   
	)

	return_val [dimsizes(evec)] :  numeric

Arguments

evec

A multi-dimensional array containing the EOFs calculated using eofunc.

optEVX

An integer which specifies options.

  1. 0 => do nothing; use the normalized eigenvectors directly.
  2. 1 => scale the normalized eigenvectors by the square root of the associated eigenvalue and return normalized eigenvectors.
  3. -1 => same as optEVX=1 but returns the scaled rotated eigenvectors.
The most common options are optEVX=1 and -1.

For versions through 4.2.0.a034, the optEVX argument had been of type logical. For backward compatibility, NCL will continue to accept the logical argument. If used, optEVX=False with default to 0 and optEVX=False with default to 1.

Return value

An array of the same size and type as evec. In addition, the percent variance [pcvar_varimax] and variance [variance] are returned.

Description

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. Missing values are ignored.

The Kaiser varimax rotation is a common rotation performed on atmospheric or oceanographic data. 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 there is some correlation between the time series expansion coefficients for each mode.

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:

  1. Don't use rotation unless you know what you are doing and why you are doing it!
  2. 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.
  3. If none of the patterns/coefficients are distinct then rotation may help reduce the noise and yield results that are more interpretable.
  4. If some are distinct and some are not then performing a rotation will mix the results.

NOTE: If the option is set to zero, the returned [ pcvar_varimax ] will all be equal. References on rotation:
J.C. Davis: Statistics and Data Analysis in Geology. John Wiley and
Sons, 2nd Ed, 1984.

See Also

eofunc, eofunc_n, eofunc_ts, eofunc_ts_n, eofunc_Wrap, eofunc_n_Wrap, eofunc_ts_n_Wrap

Examples

Example 1

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

  neval  = 3                         ; calculate 3 EOFs out of 7 
  ev     = eofunc(x,neval,False)   ; ev(neval,nvar)
  ev_rot = eofunc_varimax_Wrap(ev,1)
  
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     = eofunc(y(lat|:,lon|:,time|:),neval,False)   
  ; ev(neval,nlat,nlon)
  ev_rot = eofunc_varimax_Wrap(ev,1)
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     = eofunc(z,neval,False)      
; ev will be dimensioned neval, level, lat, lon
  ev_rot = eofunc_varimax_Wrap(ev,1)
Example 4

Calculate the EOFs at every other lat/lon grid point. 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     = eofunc(zTemp,neval,False)   ; ev(neval,nlat/2,mlon/2)
  ev_rot = eofunc_varimax_Wrap(ev, 1)
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     = eofunc(z(kl,:,:,:),neval,False)  
; ev will be dimensioned neval, lat, lon 
  ev_rot = eofunc_varimax_Wrap(ev, 1)
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     = eofunc(zTemp,neval,False)      
; ev will be dimensioned neval, lat, lon
  ev_rot = eofunc_varimax_Wrap(ev, 1)
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. 
; conform is used to make wgt the same size as pt
  ptw = pt*conform(pt, wgt, 0)        
                                      
  evec     = eofunc(ptw,neval,80.)   
  evec_rot = eofunc_varimax_Wrap(evec, 1)