Figure 1

Analysis of numerically generated noisy signals (dimensionless units). (a) The raw signals derived from Eqs. (3) (black) and (4) (gray); and (b) the magnitude of their discrete FTs. (c) Wavelet transform amplitude $\left|W(\sigma ,t)\right|$ of the signal from (3). (d) Wavelet transform amplitude of the signal from (4). (e) Magnitude of the cross spectrum without windowing or averaging. (f) Wavelet phase coherence.Reuse & Permissions

Figure 2

Analysis of numerically generated noisy signals (dimensionless units) where one of them is an AAFT surrogate. (a) The raw signals, where the black curve is derived from Eq. (3) and the gray curve is an AAFT surrogate of the signal in Eq. (4). (b) The FTs of the signals in (a). (c) Their cross spectrum. (d) The wavelet phase coherence of the two signals in (a) (dashed black curve), that is, of a numerically generated signal with its surrogate, is compared with the wavelet phase coherence of the two original signals (solid black curve) from Fig. 1f. The gray curves indicate the distribution of thousand such surrogates: The solid gray curve is the mean, and the dashed and dotted gray curves represent, respectively, 1 and 2 standard deviations about the mean.Reuse & Permissions

Figure 3

Cumulative sum of phase difference phasors for three different types of bivariate data: random phase values (independent samples from a uniform distribution) in solid black, random phase differences (drawn from independent 0.3-Hz wavelet phase components) of Eq. (3) and Eq. (4) in gray, coherent phase difference values [drawn from matching 0.6-Hz wavelet phase components of Eqs. (3) and (4)] in dashed black.Reuse & Permissions

Figure 4

The windowed wavelet coherence with an adaptive window length to remove selfcorrelation biases introduced by the transform itself. The 0.6 Hz components of the artificial timeseries (dimensionless units) are identical. (a) The magnitude of the windowed wavelet coherence, i.e. the magnitude of the product of one wavelet transform and the complex conjugate of the corresponding terms from the other wavelet transform, averaged inside a time window at a given frequency. (b) The actual mean (full, green) of the windowed wavelet coherence, compared to the calculated expectation null mean (full, black), the AAFT null mean (dashed, gray), and the independent data null mean (dashed, black). (c) The actual standard deviation (st.d) (full, green) of the windowed wavelet coherence, compared to the calculated expectation null st.d. (full, black), the AAFT null st.d. (dashed, gray), and the independent data null st.d. (dashed, black). (d) The actual mean (full, green) of the windowed wavelet coherence, compared to a threshold set two standard deviations in the mean above the calculated expectation null mean (full, black), the AAFT null mean (dashed, gray), and the independent data null mean (dashed black). (e) The windowed wavelet coherence minus the calculated expectation null mean, divided by the expected null standard deviation in each window. Note the logarithmic ordinate scales in (b)–(d).Reuse & Permissions

Figure 5

The windowed wavelet coherence of ICP and ABP with an adaptive window length to remove self-correlation biases introduced by the transform itself. (a) A typical section (50–100 s) of the raw signals (ABP, gray; ICP, black). (b) Their Fourier transforms. (c) Magnitude of the windowed wavelet coherence (cf. Fig. 4a). (d) The actual mean (full, green) of the windowed wavelet coherence, compared to the expected null mean (full, black) and a threshold (dashed, black) set at twice the null standard deviation in the mean, above the mean (cf. Fig. 4d). (e) The windowed wavelet coherence minus the expected null mean, divided by the expected null standard deviation for each window (compare with Fig. 4e). Some low frequency features are more than 6 standard deviations above the expected null mean. (f) The magnitude of the phase difference between ABP and ICP oscillations in each window. Note the logarithmic ordinate scale in (d).Reuse & Permissions