As an extension of the chronobiological serial section, gliding spectra illustrate the changing time structure (chronome) of physiological, physical and/or other variables in a given frequency range. For this purpose, least squares spectra are computed over a specified interval (much shorter than the observation span) that is progressively displaced by a given increment throughout the entire record. Results can be displayed either as 3D charts or as surface charts, displaying the estimated amplitudes, percentage rhythms or ordering P-values at each trial period for each interval. The procedure is illustrated for the record of Wolf numbers as a gauge of solar activity and for the number of marriages and divorces in Japan during the past century. Major components in these time series show deviations in period length and relative prominence over time. Particularly in the case of non-stationary time series, gliding spectra offer themselves as useful tools to examine changes in time structure beyond a specific spectral component.

Moving least squares spectra represent an extension of the chronobiological serial section (1, 2) and of serial sections on a serial section (3) to view changes as a function of time in the characteristics (MESOR, rhythm-adjusted mean value; amplitude, a measure of half the extent of predictable change within a cycle; and acrophase, a measure of the timing of overall high values recurring in each cycle) of an anticipated spectral component. The technique is useful to assess the consistency of the time structure of a given variable over time and to assess any changes when such occur, as in the case of variations in acrophase and/or frequency following a transmeridian flight (4) or in the case of variance transpositions in vascular time structures postnatally during child development (5).

A C++ program described earlier (2) was applied to several time series to illustrate the merits of the procedure, applicable to equidistant or non-equidistant data, time-coded in arbitrary units or, whenever possible, as century, year, month, day, hour, and minute of each value. The program is interactive and prompts the user to enter selections, such as the length of the interval to be used for the computation of the least squares spectrum, and the increment by which this interval should be moved throughout the time series. The program also offers many other choices between the computation of a least squares spectrum, wherein trial periods are in harmonic relation in a specified frequency range, and that of a chronobiological window (1), wherein trial periods are changed by a specific amount within a narrow frequency range. To examine a narrow frequency range, as for instance to check on free-running (6, 7), the sequence of the trial periods can be chosen in arithmetical progression, i.e., linearly in period.

The time structure of Wolf numbers, individual physiological time series, and social events such as marriages and divorces, all differ from exact periodicities. To analyse such non-stationary data, the moving spectra matrices and contour maps offer themselves as useful tools, beyond the chronobiological serial section, examining changes beyond a specific spectral component.