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# eofcor

Calculates empirical orthogonal functions via a correlation matrix (NCL's original function).

## Prototype

```	function eofcor (
data   : numeric,
neval  : integer
)

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.

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.

## 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:

• trace: A scalar value equal to the trace of the covariance/correlation matrix.
• eval: A one-dimensional array the same size as neval containing the eigenvalues in descending order.
• pcvar: A one-dimensional array the same size as neval containing the percent variance associated with each eigenvalue.
• eof_function: A scalar integer:
• 0 = eofcov was used to compute the EOFs.
• 1 = eofcor was used to compute the EOFs.
• 2 = eofcov_pcmsg was used to compute the EOFs.
• 3 = eofcor_pcmsg was used to compute the EOFs.
These attributes can be accessed using the @ operator:

```print(return_val@trace)
print(return_val@eval)
```

## Description

eofcor is the original NCL function for calculating EOFs. It can be slow if the input matrix is large. There is a faster function for calculating EOFs, eofunc. The answers may not match exactly because eofunc examines the input data array and may use a different correlation matrix than eofcor. If you do not want this feature, use eofcor.

Calculates empirical orthogonal functions via a correlation matrix. (It does not use the singular value decomposition approach.) This function computes the correlation matrix by removing the appropriate means and calculating the correlation matrix using anomalies. The eigenvectors are calculated using LAPACK's "dspevx" routine.

Use the eofcor_Wrap function if metadata retention is desired. The interface is identical.

Missing values are ignored, but each grid point must have at least some non-missing data. If grid points with all missing values exist, use eofcor_pcmsg

Note on weighting observations

Generally, when performing and 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.

## 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 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:

```  neval  = 3                         ; calculate 3 EOFs out of 7
ev     = eofcor(x,neval)   ; ev(neval,nvar)

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(x,neval)   ; ev(neval,nvar)
```
Example 2

Let x be three-dimensional with dimensions 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)

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(y(lat|:,lon|:,time|:),neval) ; 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     = eofcor(z,neval)
; ev will be dimensioned neval, level, lat, lon

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(z,neval)
```
Example 4

Calculate the EOFs at every other lat/lon 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     = eofcor(zTemp,neval)   ; ev(neval,nlat/2,mlon/2)

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(zTemp,neval)   ; ev(neval,nlat/2,mlon/2)
```
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

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(z(kl,:,:,:),neval)
```
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

; Use eofcor_Wrap if metadata retention is desired
; ev     = eofcor_Wrap(zTemp,neval)
```
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= eofcor(ptw,neval)

; Use eofcor_Wrap if metadata retention is desired
; evec= eofcor_Wrap(ptw,neval)
```