Estimating Periodicity and Strength of Rhythmicity

3.2.1. Estimates of Period

To estimate period, the concatenated data sets were analyzed using MESA and the results plotted using MATLAB software. Data were analyzed both with and without the use of a 4-h-cutoff low-pass two-pole Butterworth digital filter (see Note 10 and ref. 16). To report the best estimate, the three highest spectral peaks were read from the output files with MESPEAK. These results were then checked against the output plots and the autocorrelations to ensure that the peak best representing the periodicity was chosen.

The output of MESA analysis for the data file from SET I, containing 80% noise with no low-pass filtering, is shown in Fig. 4. The period, as retrieved from the output data file, is 24.62, corresponding closely to the known pro-grammed-in value. Despite the very low signal-to-noise ratio in the raw data, the MESA spectral peak is sharp and little of the noise appears in the region of the circadian periodicity. Table 1 contains the results for periodicity in SET I.

For the data in SET II, with the long-period trend, the data were either run as is, with and without the low-pass filter, or after first being conditioned by removal of all periodicity greater than 2 d using FILCON, set to null out the coefficients for periodicities greater than 2 d. Analysis proceeded as usual from that point on. Table 1 summarizes the output for these data. An improvement in the estimates is notable.

For SET III (with the circhoral component), the data were analyzed first untreated, and then again after conditioning with FILCON set to remove all periods greater than 10 h. Figure 5 depicts the spectrum derived from the data containing a circadian period of 24.75 h, a 1-h ultradian periodicity of much lower amplitude, and 80% noise after de-trending with FILCON. Table 1 shows the periodicities extracted by MESA before and after removal of the circadian periodicity. For data before filtering, both the primary and secondary peaks found are reported.

Fig. 4. The MESA spectrum from the data depicted in Fig. 1. No filtering was done, yet the spectrum is extremely clean, even given the high proportion of noise and attendant low signal-to-noise ratio.

Table 1

Primary Periods Found by MESA for Each Level of Noise

Table 1

Primary Periods Found by MESA for Each Level of Noise

% Noise

I

II U

II F

III U 1°

III U 2°

III F

10

24.78

25.26

24.81

24.8

1.00

1.00

20

24.78

25.34

24.81

24.72

1.00

1.00

30

24.78

25.34

24.88

24.72

1.00

1.00

40

24.78

25.43

24.88

24.80

1.00

1.00

50

24.78

25.43

24.88

24.72

1.00

1.01

60

24.78

25.51

24.97

24.80

1.00

1.01

70

24.78

25.43

24.97

24.80

1.00

1.01

80

24.62

25.10

24.88

24.80

1.00

1.01

The data set analyzed is noted by set number (see Subheading 3.2.1.). U = Raw (1° and 2° indicate primary and secondary peaks uncovered), and F = output after program FILCON was run on the data. Note that even before the strong circadian rhythm was removed, MESA found the hourly peak even in 80% noise.

The data set analyzed is noted by set number (see Subheading 3.2.1.). U = Raw (1° and 2° indicate primary and secondary peaks uncovered), and F = output after program FILCON was run on the data. Note that even before the strong circadian rhythm was removed, MESA found the hourly peak even in 80% noise.

Fig. 5. MESA plot produced from the data shown in Fig. 3 (A) before removal of the circadian component with FILCON and (B) after removal. The large circadian peak has been entirely eliminated, and the remaining ultradian periodicity is clear, constituting the only peak in the spectrum. The small bump at approx 6 h is an artifact of the sharp cutoff of the Fourier filtering.

Fig. 5. MESA plot produced from the data shown in Fig. 3 (A) before removal of the circadian component with FILCON and (B) after removal. The large circadian peak has been entirely eliminated, and the remaining ultradian periodicity is clear, constituting the only peak in the spectrum. The small bump at approx 6 h is an artifact of the sharp cutoff of the Fourier filtering.

Lag (hi

Fig. 6. Autocorrelation plot from the data set depicted in Fig. 1. The decay of the envelope in the function is a result of the large amount of noise added. Nonetheless, strong rhythmicity is evident and corroborates the periodicity reported by MESA.

3.2.2. Autocorrelation Analysis

To test for significance of periodicity and provide a crosscheck on the period estimates, the program AUTOCO was run on all data as above. All data were treated as had been done with MESA with regard to filtering and conditioning with FILCON. Plotted output was compared with the spectral analysis results to check for agreement. The autocorrelogram was shown in Fig. 6. computed for the same data set that was analyzed by MESA in Subheading 3.2.1. For all figures in this section, correlograms shown will be for the same files as the MESAs in the previous section. Note that the peaks of autocorrelation are robust and repeat regularly, clearly verifying the periodicity reported by MESA in Fig. 4.

The autocorrelation for the data initially containing a trend are shown before and after operation of FILCON (Fig. 7). Note that the peaks of autocorrelation are superimposed on the strong trend, but that this disappears after its removal. It is substantially easier to verify the MESA peak using correlograms from de-trended data.

An autocorrelation of the data from SET III, containing the ultradian rhythm in the presence of the strong circadian rhythm and 80% noise, is depicted in

Fig. 7. Autocorrelation analysis of the data depicted in Fig. 2 with a strong monotonic trend shown before (A) and after (B) de-trending with FILCON.

Fig. 8. Autocorrelation produced from data containing a 1-h ultradian periodicity, as depicted in Fig. 3, after the circadian component had been removed. Before removal, there was no visible evidence of this period. The scale has been shortened to ±10 h of lags. This corroborates the MESA spectrum for both the filtered and unfil-tered data.

Fig. 8. Autocorrelation produced from data containing a 1-h ultradian periodicity, as depicted in Fig. 3, after the circadian component had been removed. Before removal, there was no visible evidence of this period. The scale has been shortened to ±10 h of lags. This corroborates the MESA spectrum for both the filtered and unfil-tered data.

Fig. 8. Here, the correlograms before and after removal of the circadian rhythm are shown to illustrate the dramatic change in the character of the signal. The hourly rhythm is unequivocally verified by the regular peaks in the autocorrelation function (see Note 11).

3.2.3. Assessing Robustness of Rhythmicity

The output files from the autocorrelation analysis were further scrutinized for RI with a BASIC program set to find the height of the third peak (counting the peak at lag zero as 1). As has been discussed elsewhere (7-11), the decay envelope of the autocorrelation function is a measure of the long-range regularity of the rhythmicity, as well as its robustness. These numbers could then be compared with the known (programmed-in) signal-to-noise ratios.

Table 2 contains the extracted RIs for SETS I through III. Note that the RI covaries with the amount of noise added. Filtering the data produces a striking increase in the RI at the higher noise levels. For the data containing the ultradian component, the RIs before filtering are representative of the circadian periodicity, whereas numbers after filtering are for the remaining ultradian rhythm.

Table 2

Rhythmicity Index (RI) for Each Percentage of Noise

Rhythmicity Index (RI) for Each Percentage of Noise

Table 2

% Noise

SET I

SET II U

SET II F

SET III U

SET III F

10

0.79

0.73

0.79

0.79

0.66

20

0.79

0.71

0.78

0.79

0.66

30

0.78

0.68

0.77

0.79

0.65

40

0.77

0.65

0.76

0.79

0.63

50

0.74

0.59

0.73

0.79

0.59

60

0.70

0.50

0.68

0.79

0.52

70

0.60

0.38

0.58

0.78

0.39

80

0.41

ARR

0.40

0.77

0.21

Data set numbers are indicated (see text). U = raw data, F = output after FILCON program operated on the data. ARR means the signal was deemed arrhythmic. For data in SET III, the RI for the circadian component is reported for the raw data, and for the data treated with FILCON in the F group.

Data set numbers are indicated (see text). U = raw data, F = output after FILCON program operated on the data. ARR means the signal was deemed arrhythmic. For data in SET III, the RI for the circadian component is reported for the raw data, and for the data treated with FILCON in the F group.

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