Single spectral peaks are found only if the data cover an integer number of cycles. If this is not the case, the signal spreads over several spectral lines [ 10 ]. When this happens and the underlying signal was anticipated, it is possible to determine the period frequency corresponding to the maximal amplitude by applying the single cosinor procedure not only at the Fourier frequencies but at additional intermediary frequencies as well. Tapers such as a Hanning window [ 77 ] can be used to reduce sidelobes associated with the finite observation span, but this procedure also affects the estimation of the rhythm parameters.

It remains useful, however, for a macroscopic view of the time structure of the data. Effect of applying a Hanning taper on the least squares spectrum. Corresponding least squares spectra bottom indicate that while the spectral location of the two peaks remains the same, the amplitudes are reduced and the bandwidths are wider. Sidelobes are also greatly diminished. Simulation and original drawings from C.

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Least squares spectra can be very helpful in exploratory analyses, but it should be realized that assumptions underlying the use of the single cosinor notably independence and normality are violated more often than not. Population-mean cosinor spectra are a useful complementary approach not prone to this limitation.

This method is similar to the power spectrum obtained by smoothing the periodogram, which is more reliable for testing unknown periodicities [ 78 ], with the important difference, however, of retaining the phase information. The averaging smoothing can be done either in the frequency domain by averaging across consecutive Fourier frequencies, or in the time domain.

The population-mean cosinor spectrum uses averaging in the time domain. The method consists of subdividing the observation span T into several e.

## Chronobiotechnology And Chronobiological Engineering 1987

A least squares spectrum is computed for each interval, using the same common reference time. The population-mean cosinor procedure is then applied at each trial frequency to summarize results across the k intervals. The procedure can be repeated by using different values of k. Unknown signals consistently detected by this approach may thus be viewed with added confidence. Starting from an initial guess estimate for the period, all parameters can be estimated using iterations aimed at minimizing the residual sum of squares. Marquardt [ 79 ] developed an algorithm which performs an optimum interpolation between the Taylor series and gradient methods.

He also derived a way to approximate confidence intervals for all parameters, including the period [ 80 ]. For the particular case of single-component models, Bingham offers an easily understood approach [ 81 ]. For low-frequency signals, simulated annealing [ 82 ] is another suitable method that has the advantage of not requiring the specification of initial values for the periods.

This approach does not perform well, however, for very sharp signals in the higher frequency range of the spectrum. For signals with a symmetrical waveform, the nonlinear procedure can yield an acceptable estimate of the fundamental period on the basis of very short records not even covering a full cycle [ 84 ].

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This is not the case, however, when the waveform is asymmetrical. When data are equidistant or rendered equidistant by averaging and filling data gaps by interpolation, wavelets can be performed [ 85 ].

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This approach has been useful to uncover components not detected earlier [ 86 ]. Short-term Fourier transforms can be used to visualize changes in the spectral structure of the data as a function of time [ 87 ]. Alternatively, gliding spectral windows [ 75 ] can be computed. A least squares spectrum is computed over each interval over a specified frequency range. In such a display, time and frequency are the two horizontal axes and amplitude is shown on the vertical axis in a 3D plot or as different shadings in a surface chart.

One example relates to competing about Whereas the procedure can be performed on non-equidistant data, the interpretation of results is greatly helped when data are equidistant, as changes in sampling rate are also associated with changes in spectral structure appearing on the graph. A judicious choice of I and of the frequency range examined is important in order to minimize sidelobes.

The use of a Hanning taper [ 77 ] is also helpful in this kind of exploratory analysis. Gliding spectral window surface chart. Complementary nonlinear analyses not shown indicate the coexistence of about Changes in the most prominent circadian period as a function of time are apparent from the changes in amplitude shading and location along the vertical scale.

Whenever focus can be placed on a specified component with a given trial period, a chronobiologic serial section [ 21 ] can be performed. To data in each interval, the single-component single cosinor procedure is applied. To visualize the results, a chronogram is shown on top, followed by the sequence of MESORs, amplitudes and acrophases as they change as a function of time, provided with a measure of uncertainty. Corresponding P-values from the zero-amplitude test and the number of data per interval are also displayed to help interpret any change in the results.

Chronobiologic serial section. Peak expiratory flow was self-measured several times a day by a year old man.

The data covering a month span are shown in row 1. Data in each interval are fitted with a hour cosine curve. From the P-values shown in row 2, it can be seen that the circadian rhythm was detected with statistical significance most of the time, except for two short spans, one coinciding with a transmeridian flight when fewer data were collected, row 5 and the other with influenza.

Whereas the hour acrophase remains relatively stable throughout the record row 4 , the MESOR row 3, lower curve and to a lesser extent the circadian amplitude row 3, distance between the two curves undergo sharp changes, notably in association with the influenza and earlier with a change in treatment timing. The procedure has been extended in two ways. First, a multiple-component instead of a single-component single cosinor model can be fitted in each interval. This procedure has been used for instance to illustrate that the prominence of an about 5-month component of heart rate self-measured over 4 decades by a clinically healthy man follows the about year change in solar flares in which this component had been documented [ 92 ].

In this analysis, the 5-month component was fitted together with 1. Multiple-component serial section. Heart rate, self-measured a few times each day by a healthy man over several decades, was averaged over consecutive weeks. The weekly averages are fitted with a 3-component model consisting of cosine curves with periods of 1. The time course of amplitudes of the 3 components is shown on top. Results for the 0. Nonlinear serial section. There are, of course, other important tools for the analysis of time series [ 95 — ].

Most of them, however, require that the data be equidistant. This overview focused specifically on the use of the cosinor and its different extensions.

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The method is fairly simple and its results lead to meaningful interpretations. Despite its several shortcomings related primarily to the difficulty of satisfying all assumptions underlying the use of regression techniques, its wide-ranging applications have played an important role in the development of chronobiology as a quantitative scientific discipline. Used with caution, results based on a combination of exploratory analyses with the different cosinor routines and other conventional statistical tests, progress has also been made in the field of chronomics which aims at mapping broad time structures from the high-frequency brain waves to the multi-decadal cycles characterizing space-terrestrial weather influencing human physiology and pathology [ 3 , ].

Despite its simplicity, some reluctance remains for some investigators to use the cosinor for estimating rhythm parameters or for considering more than a single test time in designing experiments. As discussed elsewhere, such practice can be misleading in missing an existing difference or even in finding a difference in mean when none exists [ 15 — 17 ]. To some extent, this status quo may be accounted for by the lack of dissemination of computer software offering chronobiologists tools for time series analysis applicable to non-equidistant data.

This situation is slowly changing, however. Personal computers have become more powerful and statistical packages have become more readily available for relatively easy use by investigators not necessarily versed in all statistical details underlying the programs included in the software packages. While professional statistical software packages remain somewhat expensive for individual users, several open-source packages such as Octave and R offer an attractive alternative, notably since some are platform-independent, running on PCs, Macs or Linux systems [ ].

Programmers have taken advantage of the tools available in these packages to write code to perform analytical tasks of interest to chronobiologists. Perhaps the most comprehensive package is that developed by Oehlert and Bingham [ ], offering a large array of procedures that can be applied by writing minimal coding instructions to call the different macros. Selected programs used in chronobiology have long been offered on the website of Refinetti [ ], with clear instructions on how to run the programs.

While not open-source, the Expert Soft Technologie website [ ] also offers an array of cosinor-based and other procedures, including techniques for the study of non-stationary signals. These programs have been used in the study of shift-workers [ ]. In summary, selected methods for the study of biologic time series have been reviewed and their relative merits have been discussed in the light of underlying assumptions.

Some illustrative applications have also been mentioned. When the choice of a model is justified, and it is functional and explicative, quantitative methods of data analysis are extremely valuable to specify the model and obtain estimates of its parameters. Even when underlying assumptions are not fully met, point estimates of the parameters can be very useful. More caution is needed, however, in deciding whether P-values and confidence intervals are trustworthy, since violation of underlying assumptions tends to yield results that are too liberal.

Once this limitation is taken into consideration, data analysis methods as described herein constitute extremely valuable tools for research in chronobiology and chronomics. Halberg F: Temporal coordination of physiologic function. Cold Spr Harb Symp quant Biol.

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