Chapter 2 Appendix The threshold for significant coherence depends on the degrees of freedom in the spectral analysis 12. For sufficiently large degrees of freedom, the estimated coherence can be approximated by a#2 distribution. Thus, equation A.1 merely applies an F-test with 2 and$-2 degrees of freedom, 2F (c ) i min ? 2,p /2 (A.1)2 p / 2 - 2F2,p /2 (c ) where 2min is the minimum level above which the squared coherence signifi- cantly differs from zero. With equation A.1 the coherence threshold for each combination can be calculated. Others 26 used the following calculation of the 2 coherence threshold:i min ? 1/ c p /2 5. This results in exactly the same cohe-2 rence thresholds as when applying the F-test criterion. The degrees of freedom are determined by the parameters used for spectral estimation and can be divided into two parts, one for the spectral averaging and one for the spectral smoothing. The total number of degrees of freedom for the combinations is the product of both numbers of degrees of freedom. The degrees of freedom for the spectral averaging is equal to the number of data segments and are proportional to N/L, where N is the total numbers of samples in the recor- ding and L is the number of samples per segment. For a 15-minute period this means that with L=256 N/Lð 17.6, resulting in$ = 17. For a 5-minute period this results in$ = 5 degrees of freedom. For combinations III and IV, L=2048 and so N/Lð 2.2 resulting in$ = 2 degrees of freedom. For the degrees of freedom for the spectral smoothing the triangular smoothing window defined in equation A.2 needs to be taken into account. 1 1 Wi ? h -1 / (h -1) 2 i (A.2) The degrees of freedom of this triangular smoothing window can be derived from equation A.3. 2 p ? (A.3) h -Wi 2 i?/h and for the different window widths the degrees of freedom are listed in table A1. 38
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