Significance of synchrosqueezed (and other) scalograms against autocorrelated noise

Tests which parts of a synchrosqueezed transform (analyse_wavelet_sst, analyse_superlet_sst, analyse_conceft) are unlikely to be produced by noise. The noise of the record is modelled by an autoregressive process (order chosen by AIC, ar_order_max); optionally its variance may change along the record (a moving standard deviation of the innovations, as in the time-varying AR bootstrap of Chen and Wu, 2025). n_sim noise records are simulated and transformed with exactly the same settings as the original. For every period the (1 - alpha) quantile of the noise power, pooled over all positions, gives the threshold; with local_variance = TRUE it is scaled by the local noise variance. Coefficients below the threshold are set to zero in the returned object, so only significant ridges remain in plots and tracking.

Under the null hypothesis (no cycles) the record itself is used as the noise model; this is conservative when strong cycles are present, because they increase the variance attributed to noise.

Usage

sst_significance(
  wavelet,
  n_sim = 50,
  alpha = 0.05,
  ar_order_max = 5,
  local_variance = FALSE,
  window = NULL,
  seed = 1,
  verbose = FALSE
)

Arguments

wavelet

Transform to test (required, no default): the list returned by analyse_wavelet_sst, analyse_superlet_sst or analyse_conceft (it must contain the transform and settings elements, which are used to transform the noise records with exactly the same settings).

n_sim

Number of simulated noise records (integer, > 0); more simulations give a more stable threshold but take proportionally longer. Default=50.

alpha

Significance level (numeric, between 0 and 1): the threshold of every period is the (1 - alpha) quantile of the pooled noise power. Default=0.05.

ar_order_max

Maximum order of the autoregressive noise model (integer, >= 1); the order is chosen by AIC with ar (Yule-Walker) up to this value. 1 gives the classical red noise (AR1). Default=5.

local_variance

Let the noise variance change along the record (logical): the simulated noise is scaled by the moving standard deviation of the AR innovations and the threshold by the local noise variance. Default=FALSE.

window

Length of the moving window of the local variance, in depth/time units (numeric, > 0); only used with local_variance = TRUE. NULL uses twice the longest period of the transform. Default=NULL.

seed

Seed of the random number generator (integer) used for the noise simulations, so that the result can be reproduced; the previous random state is restored afterwards. Default=1.

verbose

Print a message after every simulation (logical, TRUE or FALSE). Default=FALSE.

Value

The input object with Power, Wave, Ampl and Power.avg restricted to the significant coefficients, and the extra elements significant (logical matrix), Power_all (the unmasked power), threshold_period (noise power threshold per period) and alpha.

References

Chen, Z., Wu, H.-T., 2025. Uncertainty quantification of synchrosqueezing transform under complicated nonstationary noise. arXiv:2506.00779.

Torrence, C., Compo, G.P., 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society 79, 61-78.

Examples

# \donttest{
z <- 0:999
x <- sin(2 * pi * z / 40) + as.numeric(arima.sim(list(ar = 0.7), 1000))
s <- analyse_wavelet_sst(data.frame(z, x), lowerPeriod = 10, upperPeriod = 200, dj = 1 / 40)
sig <- sst_significance(s, n_sim = 20)
plot_wavelet(sig)

# }