Given coherence-squared and the effective degrees-of-freedom, calculate the associated probability.
Available in version 6.4.0 and later.
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.
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.
Numeric (float or double) array containing probabilities of the same size and shape as cohsq.
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
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) )