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eofunc_Wrap

Computes empirical orthogonal functions (aka: Principal Component Analysis, Eigen Analysis) and retains metadata.

Prototype

load "$NCARG_ROOT/lib/ncarg/nclscripts/csm/contributed.ncl"

	function eofunc_Wrap (
		data    : numeric,  
		neval   : integer,  
		optEOF  : logical   
	)

	return_val  :  numeric

Arguments

data

A multi-dimensional array in which the rightmost dimension is the number of observations. Generally, this is the time dimension.

If your rightmost dimension is not time, then see eofunc_n_Wrap.

Commonly, the data array contains anomalies from some base climatology, however, this is not required.

neval

A scalar integer that specifies the number of eigenvalues and eigenvectors to be returned. This is usually less than or equal to the minimum number of observations or number of variables, and is typically 3 to 5.

optEOF

A logical variable to which various optional arguments may be assigned as attributes. These optional arguments alter the default behavior of the function. Must be set to True prior to setting the attributes which are assigned using the @ operator:

```
optEOF      = True
optEOF@jopt = 1
```
optEOF@jopt = 1: uses correlation matrix to compute EOFs. The default is to use a covariance matrix (optEOF@jopt = 0).
```
optEOF       = True
optEOF@pcrit = 85
```
optEOF@pcrit = %: a float value that indicates the percentage of non-missing points that must exist at any single point in order to be calculated. The default is 50%. Points that contain all missing values will automatically be set to missing.

Return value

A multi-dimensional array of the same size as data with the rightmost dimension removed and an additional leftmost dimension of the same size as neval added. Double if data is double, float otherwise.

Will contain the following attributes:

pcvar: a one-dimensional array of size neval equal to the percent variance associated with each eigenvalue.
pcrit: The same value and type of optEOF@pcrit if the user changed the default.
matrix: A string indicating the type of matrix used, "correlation" or "covariance".
method: A string indicating the status of the matrix, "transpose" or "no transpose"
sig: (v6.3.0 onward) A one-dimensional logical array of size neval indicating if each eigenvalue is significantly separated from adjacent eigenvalues. See eofunc_north

These attributes can be accessed using the @ operator:

print(return_val@pcvar)
print(return_val@pcrit)
print(return_val@sig)      ; v6.3.0 onward

Description

Computes Empirical Orthogonal Functions (EOFs) via an anomaly covariance matrix or, optionally, via a correlation matrix and retains metadata. This is also known as Principal Component Analysis or Eigen Analysis. The eigenvectors are calculated using LAPACK's "dspevx" routine. The returned values are normalized such that the sum of squares for each EOF pattern equals one. To denormalize the returned EOFs multiply by the square root of the associated eigenvalue (aka,the singular value).

If data does NOT have time as the rightmost dimension, then use eofunc_n_Wrap to avoid having to reorder the data.

Most commonly, the input data consists of anomalies. Missing values are ignored when computing the covariance or correlation matrix.

Note on weighting observations

Generally, when performing an EOF analysis on observations over the globe or a portion of the globe, the values are weighted prior to calculating. This is usually required to account for the convergence of the meridions (area weighting) which lessens the impact of high-latitude grid points that represent a small area of the globe. Most frequently, the square root of the cosine of the latitude is used to compute the area weight. The square root is used to create a covariance matrix that reflects the area of each matrix element. If weighted in this manner, the resulting covariance values will include quantities calculated via:

[x*sqrt(cos(lat(x)))]*[y*sqrt(cos(lat(y)))] = x*y*sqrt(cos(lat(x)))*sqrt(cos(lat(y)))

Note that the covariance of a grid point with itself yields standard cosine weighting:

[x*sqrt(cos(lat(x)))]*[x*sqrt(cos(lat(x)))] = x^2 * cos(lat(x)).

Note on standard EOF analysis

Conventional EOF analysis yields patterns and time series which are both orthogonal. The derived patterns are a function of the domain. The calculated patterns may resemble physical modes of the system. However, the procedure is strictly mathematical (not statistical) and is not based upon physics.

If desired, EOF spatial patterns may be tested for orthogonality by using the dot product:


  d01 = sum(eof(0,:,:)*eof(1,:,:))
  d12 = sum(eof(1,:,:)*eof(2,:,:))
  d02 = sum(eof(0,:,:)*eof(2,:,:))
  print("d01="+d01+"  d12="+d12+"  d02="+d02)  ; may be +/- 1e-8

References:


Quadrelli, Roberta, Christopher S. Bretherton, John M. Wallace, 2005:
On Sampling Errors in Empirical Orthogonal Functions. 
J. Climate, 18, 3704-3710 
    
North, G. R., T. L. Bell, R. F. Cahalan, and F. J. Moeng, Sampling
errors in the estimation of empirical orthogonal functions, Mon.
Wea. Rev., 110, 699-706, 1982.

Dawson, A.: EOF Analysis

Acknowledgement: The code used is a modified version of David Pierce's Fortran code.

Examples

In the following, the attribute pcvar can be output via:

  print(ev@pcvar)             ; 1D vector of length "neval"

This attribute could also be used in graphics. For example, it is it could be used in a title.

  title = "%=" + ev@pcvar(1)

sprintf can be used to format the title more precisely:

  title = "%=" + sprintf("%5.2f", ev@pcvar(1) )

Example 1

Let x be two-dimensional with dimensions variables (size = nvar) and time. Commonly, this is an anomaly array.

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

Example 2

Let x be three-dimensional with dimensions of time, lat, lon.

With NCL versions 6.3.0 and earlier, you need to reorder x so that time is the rightmost dimension. With NCL versions 6.4.0 and later, use eofunc_n_Wrap to avoid having to reorder.

  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_Wrap(y(lat|:,lon|:,time|:),neval,False)   
;;ev     = eofunc_n_Wrap(y,neval,False,0)      ; NCL V6.4.0 and later
  ; ev(neval,nlat,nlon)

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_Wrap(z,neval,False)      
; ev will be dimensioned neval, level, lat, lon

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_Wrap(zTemp,neval,False)   ; ev(neval,nlat/2,mlon/2)
;;ev     = eofunc_n_Wrap(z,neval,False,0) ; NCL V6.4.0 and later, no reordering

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_Wrap(z(kl,:,:,:),neval,False)  
; ev will be dimensioned neval, lat, lon

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_Wrap(zTemp,neval,False)      
;;ev     = eofunc_n_Wrap(z,neval,False)  ; no reordering needed
; ev will be dimensioned neval, lat, lon

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.