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cohsq_c2p

Given coherence-squared and the effective degrees-of-freedom, calculate the associated probability.

Available in version 6.4.0 and later.

Prototype

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

	function cohsq_c2p (
		cohsq  : numeric,  ; float or double
		edof   : numeric   
	)

	return_val  :  An array of the same size, shape and shape  as cohsq.

Arguments

cohsq

A scalar or array containing coherence-squared values (0 to 1.0)

edof

A scalar or array containing the effective degrees-of-freedom. If an array, it must match the size and shape as cohsq.

Return value

Numeric (float or double) array containing probabilities of the same size and shape as cohsq.

Description

The coherence-squared is a statistic that can be used to examine the degree of linear association between two series. It allows identification of significant frequency-domain correlation between the two time series. It is analogous to the coefficient of determination (square of the correlation coefficient) between two series.

NOTE: Phase estimates in the cross spectrum are only useful where significant frequency-domain correlation exists.

NCL's specxy_anal returns the total (FFT-based) degrees of freedom. The effective degrees-of-freedom (edof) is half that number. (See Example 4)

References: 

Data Analysis Methods in Physical Oceanography /
  William J. Emery; Thomson, Richard E.
  Elsevier, 2001 (2nd Edition); ISBN: 0444507566 (hardbound); 0444507574 (paperback).

Course Notes
  Dennis Hartmann (Univ. Washington)
  See Table 6.2 , page 187 and the associated caption on page 186
 

Comments on the Determination of Significance Levels of the Coherence Statistic
   Paul R. Julian
   J. of Atm. Sci. (1975), Volume 32, pp 836-837. 

Coherence Significance Levels
   Rory O. R. Y. Thompson
   Journal of the Atmospheric Sciences, 1979
   Volume 36, pp 2020-2021

-------------------------------------------------------------------------------
Tables of the Distribution of the Coefficient of Coherence 
          for Stationary Bivariate Gaussian Processes
   Amos, D.  E., and L.  H.  Koopmans (1963) 
   Washington, Office of Technical  Services, Dept.  of Commerce. 

On the Joint Estimation of the Spectra, Cospectrum and Quadrature Spectrum of a 
          Two-dimensional Stationary Gaussian Process 
   Goodman, N.R. (1957)
   New York University

See Also

cohsq_p2c, specxy_anal

Examples

The following match the numbers of Hartmann's Table 6.2. Note: The examples use nice round degrees of freedom but, they may be fractional (eg: 20.37):

Example 1: Both coherence-squared and edof are scalars

    c2   = 0.283
    edof = 10
    p    = cohsq_c2p(c2, edof)    ; p=0.95

Example 2: The coherence-squared is an array and edof is a scalar:

    c2   = (/0.036, 0.112, 0.146, 0.215, 0.305 /)                     ; c2(5)
    edof = 20                                                         ; scalar
    p    = cohsq_c2p(c2, edof)    ; p(5) ==> (/0.50, 0.90, 0.95, 0.99, 0.999 /)

Example 3: Both coherence-squared and edof are arrays which have the same shape and size:

    c2   = (/0.159, 0.226, 0.146, 0.090, 0.068 /)                     ; c2(5)
    edof = (/  5  ,  10  ,  20  ,  50  ,  100  /)                     ; edof(5)
    p    = cohsq_c2p(c2, edof)    ; p(5) ==> (/0.50, 0.90, 0.95, 0.99, 0.999 /)

Example 4: Calculate the probability level of each coherence-squared returned by specxy_anal.

    d      = 0                    ; detrending opt: 0=>remove mean 1=>remove mean and detrend
    sm     = 7                    ; smoothing periodogram: should be at least 3 and odd
    pct    = 0.10                 ; percent tapered: 0.10 common. 

                                  ; calculate the cross-spectrum
    sxydof = specxy_anal(x,y,d,sm,pct)   ; sdof is a scalar quantity
    printVarSummary(sxydof)     ; look at the returned variable

    edof   = sxydof/2             ; effective degrees of freedom

    c2     = spec@coher           ; c2(N)
    p      = cohsq_c2p(c2, edof)  ; p(N)

    print(sprintf("%6.3f", c2) +"  "+ sprintf("%6.3f", p) )