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Chi-square (Sokolove-Bushell) Periodogram

Source: R/circadian_chisq.R
chi.sq.periodogram.Rd

Estimates the dominant period of an activity time series with the Sokolove-Bushell (1978) chi-square periodogram, the periodogram most widely used in chronobiology and actigraphy. It pairs with the Lomb-Scargle estimator in circadian.period: Lomb-Scargle is the least-squares spectral method for unevenly sampled data; the chi-square periodogram is the analysis-of-variance method for regularly sampled data and reports a significance threshold.

Usage

chi.sq.periodogram(
  counts,
  timestamps,
  from = 18,
  to = 30,
  alpha = 0.05,
  epoch_length = NULL
)

Arguments

counts

Numeric vector of activity counts on a regular epoch grid. NA values (e.g. non-wear) are handled by removal from the phase and grand means.

timestamps

A POSIXct vector (or numeric seconds) the same length as counts; used to infer the epoch length.

from, to

Period search window in hours (default 18 to 30).

alpha

Significance level for the chi-square threshold (default 0.05).

epoch_length

Epoch length in seconds. If NULL (default) it is inferred from the median spacing of timestamps.

Value

A named list with period (hours, the \(Q_P\) peak), Qp_peak, p_value (family-wise Sidak p-value of the peak across the scanned periods), significant (logical, p_value < alpha), scanned (trial periods in hours), Qp (the periodogram aligned to scanned), critical (the per-period chi-square threshold), epoch_length and alpha. On insufficient/degenerate data the same shape is returned with period/Qp_peak NA and empty vectors; the function never throws.

Details

For each trial period \(P\) (an integer number of epochs) the series is folded into \(P\) phase bins over \(K = \lfloor N/P \rfloor\) complete cycles and the statistic $$Q_P = \frac{K\,N \sum_{h=1}^{P}(\bar{A}_h - \bar{A})^2}{\sum_{i=1}^{N}(A_i - \bar{A})^2}$$ is computed, where \(\bar{A}_h\) is the mean at phase \(h\) and \(\bar{A}\) the grand mean over the \(N = KP\) retained points. Under the null hypothesis of no rhythm at \(P\), \(Q_P\) follows a chi-square distribution with \(P-1\) degrees of freedom, giving the significance line \(\chi^2_{P-1,\,1-\alpha}\). The estimated period is the \(P\) maximising \(Q_P\) within the search window.

References

Sokolove PG, Bushell WN (1978). “The chi square periodogram: its utility for analysis of circadian rhythms.” Journal of Theoretical Biology, 72(1), 131–160. doi:10.1016/0022-5193(78)90022-X .

Refinetti R, Cornelissen G, Halberg F (2007). “Procedures for numerical analysis of circadian rhythms.” Biological Rhythm Research, 38(4), 275–325. doi:10.1080/09291010600903692 .

Sidak Z (1967). “Rectangular confidence regions for the means of multivariate normal distributions.” Journal of the American Statistical Association, 62(318), 626–633. doi:10.1080/01621459.1967.10482935 .

See also

circadian.period for the Lomb-Scargle estimator.