Figure 2c: Period of 2Pi. The confidence ellipse of the Serial Cosinor per Gouthière and Jacquin (The smaller the surface, the greater the precision in determination of the period)

Figure 2d: Period of 2Pi. The confidence ellipse of the "Population means Cosinor" per Nelson et al and Bingham et al
(The smaller the surface, the greater the precision in determination of the period)

Figure 2e : Initial period of 2Pi. The Reverse Elliptic Spectral plot per Gouthière, makes it possible to
detect periods. Here it shows a minimum at 6.28. The part in blue corresponds to the significance of the null hypothesis in the ellipse test.
By increasing probability, we constrain the interval of valid periods.

Figure 2f : Initial period of 2Pi. The Fisher Periodogram makes it possible to detect a period and its validity using a fixed
probability. It is a method derived from DFT. It tests the baseline (fundamental at 6.27 here) using as a null
hypothesis that this is characteristic of the phenomenon being studied at a given probability (This spectral analysis is less interesting
than the Lomb and Scargle periodogram or the Jenkins and Watts Autoperiodogram)

Figure 3c: Period of 2Pi. Representation of the confidence ellipse per Gouthière and Jacquin.

Figure 3d: Period of 2Pi. The confidence ellipse per Nelson et al. and Bingham et al. (The smaller the surface, the greater the precision in determination of the period)

Figure 3e : The elliptic inverse spectrum per Gouthière presents a minimum at 6.28. The blue part corresponds to the significance of the
null assumption of the ellipse test for a given probability.

Figure 3f : The periodogram of Lomb and Scargle allows us to highlight a period (6.28) identical to that obtained by the
previous method.

Figure 4a :
One of the detected periods is one of 2Pi (6.28) Representation of the model and the experimental points.

Figure 4b : The Reverse Elliptic Spectrum presents a minimum at 6.28 for a probability fixed at 0.95, means confidence interval = 95%, (α = 1p) corresponding the period (In
blue is the validity zone for the ellipse test, these are the confidence limits of the period)

Figure 4c : The periodogram per Jenkins and Watts provides the same result, (6.28), although the method is completely different
from regression and instead calculates the Fourier transforms of the autocorrelation function.

Figure 4d : Spectrum analysis per Blochner, but adapted to stationary phenomena, leads to the same result.

Figure 4e : Autospectrum analysis per Jenkins and Watts, consting of a classic spectrum analysis, (here with a Kaiser
window), gives a result of 6.28.

Figure 4f : The Lomb and Scargle periodogram confirms the previous results of 6.28 (2Pi).

Figure 4g : The Fisher periodogram tested a baseline or fundamental ray of 6.27.

Figure 4h : Confidence ellipse plot with a very small area of the ellipse indicating high accuracy in determination of the period.

Figure 5a: The Single Cosinor Model and the experimental points show one of the detected periods is 2Pi (6.28).

Figure 5b: Inverse Elliptic Spectral plot from the Single Cosinor. It determines the exact value of the period.

Figure 5c: Rapid Fourier Transforms. The spectral plot of rays shows a baseline or fundamental ray at 6.28 (2Pi).

Figure 5d: The Lomb and Scargle Periodogram found the exact period of 2Pi, shown as the highest peak.

Figure 6a: The linear trend is slowly increasing and highlights body temperature with time in alcoholic subjects.

Figure 6b: The Lomb and Scargle periodogram found a main period close to 27 hours in alcoholic subjects.

Figure 6c: The model for alcoholic subjects was then constructed using the previous period, 27.3 h, but without detrending the data.

Figure 6d: Confidence ellipse, per Nelson et al and Bingham et al, for the period of 27 hours. The acrophase is close to 6:40 PM.

Figure 6f: The linear trend is increasing body temperature followed by a slow decrease with time in control subjects. The main period of
just over 24 hours was found by spectral analysis.

Figure 6g: Confidence ellipse for the period of 24.4 hours in control subjects. The acrophase is close to 5:20 PM.

Figure 7a: The Autospectral plot (Jenkins and Watts) shows a main period peak close to 22 hours.

Figure 7b: The Lomb and Scargle periodogram found a main period peak close to 24.9 hours.

Figure 7c: The Fisher periodogram finds a fundamental period close to 24 hours.

Figure 7d: The Single Cosinor confidence ellipse for a period of 24 hours.

Figure 8a: The Reverse Elliptic Spectrum (RES) determines the main peak of a period close to 25.9 hours (p = 0.95, means confidence interval 95%, α = 1p)

Figure 8b: The Percent Rhythm spectrum determines the period as close to 25.6 hours.

Figure 8c:
The Scargle periodogram shows a main period peak of period close to 24.8 hours.

Figure 8d:
The Fisher periodogram determines one fundamental period of 24 hours and a first harmonic of 12 hours. (“H0 rejected” means that the period of 24
hours is not confirmed)

Figure 8e:
Confidence ellipse plot for a period of 24 hours (p = 0.95, means confidence interval 95%, α = 1p) per Nelson et al and Bingham et al.

Figure 8f:
Confidence ellipse plot for a period of 24.8 hours (p = 0.95, means confidence interval 95%, α = 1p) per Nelson et al. and Bingham et al.

Figure 8g:
Confidence ellipse plot for a period of 25.9 hours (p = 0.95, means confidence interval 95%, α = 1p) per Nelson et al. and Bingham et al.

Figure 22a: The main periodicity is shown by the Reverse Elliptic Spectrum. As an example of the application of this new method,
we show here the cell proliferation rate in culture for human fibroblasts whose life was extended by the action of the TSV40 gene (Confidence interval:
Vertical dashed black lines, p = 0.95, means confidence interval 95%, α = 1p)

Figure 22b: The same main periodicity is shown by the Scargle periodogram plot.

Figure 10a: The Scatter plot reveal the behavior of the temperature curve with some "outliers"

Figure 10b: The outliers were removed and a linear detrend performed.

Figure 10c: Model with a period of 27 days. The second order period was revealed by substracting the signal
from the first part of the fundamental period.

Figure 10d:
Reverse Elliptic Spectrum per Gouthière, shows the 2nd order period of 27 days is found after treatement of the first signal.

Figure 11a: Scatter plot enables visualization of the general distribution of the points. A sinusoidal character seems present.

Figure 11b: A Percent Rhythm spectral plot found a peak at 125 months (10.4 years)

Figure 11c:
A Reverse Elliptic Spectrum, per Gouthière, found a maximum peak at 125 months (10.4 years).

Figure 11d:
A Fisher periodogram found and tests a fundamental ray at 126 months (10.5 years)

Figure 12a: Scatter plot enables visualization of the general distribution of the points. A sinusoidal character seems present.

Figure 12b:
Autospectral Spectrum per Jenkins and Watts found a maximum peak at 134 months (11.2 years)

Figure 12c:
The Fourier periodogram found a fundamental ray at 133 months (11.1 years)

Figure 12d:
A Reverse Elliptic Spectrum per Gouthière, found a mininmum peak at 136 months (11.3 years)

Figure 13a: Scatter plot enables visualization of the general distribution of the points. A sinusoidal character seems present.

Figure 13b: An autoperiodogram per Jenkins and Watts found a maximum peak at 11 years (132 months)

Figure 13c: A Fisher periodogram found one fundamental ray at 10 years (120 months) (p = 0.95, α = 1p).

Figure 13d:
A Reverse Elliptic Spectrum per Gouthière, found a minimum peak at 11 years (132 months)

Figure 14a: A Scatter plot enables us visualization of the general distribution of the points. A sinusoidal character is present,
and the linear trend is decreasing.

Figure 14b: An Autospectral analysis, per Jenkins and Watts shows a maximal peak of 127/12 (10.58), close to 11 years.

Figure 14c: A Reverse Elliptic Spectrum, per Gouthière shows a minimal peak of 128/12 (10.7), or more than 10 years. Other, smaller peaks
indicate smaller periods.

Figure 14d: The Autocorrelation plot is characteristic of a periodic curve that is sinusoidal and with decreasing amplitude.

Figure 15a: A Scatter plot enables visualization of the general distribution of the points. A sinusoidal character seems present and the
linear trend is decreasing.

Figure 15b: A Fisher Periodogram shows a fundamental ray at 11 years (p = 0.95, α = 1p)

Figure 15c: A Reverse Elliptic Spectrum, per Gouthière highlights a fundamental ray at 10.5 years.

Figure 15d: A Cosine model (y=3.56E+02Cos((2Pit/10.50)6.34E01)+5.99E+03) with a period of 10.5 years.

Figure 15e: A Single Cosinor confidence ellipse with phase = 0.63 rd, and amplitude = 356, following international counterclockwise
units.

Figure 15f: A Single Cosinor confidence ellipse with phase = 0,63 rd, 36.,3 degree, and amplitude = 356, per Nelson et al., and Bingham et al.

Figure 16a: The Scatter plot displays the main distribution of the data. A sinusoidal character seems to be present.

Figure 16b: The Reverse Elliptic Spectrum allows us to determine several seasonal periods, including a marked one of 73.2 years.
(The data were detrended before searching for the main period)

Figure 16c: This plot is the forecasting model of monetary velocity, up to approximately 2000, showing the period of 73.2 years.

Figure 16d: The Autocorrelation plot helps to verify that the distribution is not random. In fact, the data are highly correlated.

Figure 17a: The distribution of the data is Normal and then seems to be stochastic.

Figure 17b: The Reverse Elliptic Spectrum (RES) highlights several periods including one more marked at 57 years.

Figure 17c: The sinusoidal model built with all the points from 1800 to 1997 and the selected period of 57.7 years.

Figure 17d: The sinusoidal model extended until 2100 with calculated predicted values 2000, 2010, 2020 and 2040.

Figure 19a: Before physical training : the confidence ellipse with associated chronobiological values shows
acrophase close to 10:25 PM. (p = 0.95, means confidence interval = 95%, α = 0.05)

Figure 19b: After physical training, amplitude increased, phase decreased and the confidence ellipse shows acrophase close to 3:10 AM.
(p = 0.95, means confidence interval = 95%, α = 0.05)

Figure 19c: Before physical training amplitude variations seems to be a constant and not a time function.

Figure 19d: Before physical training phase variations do not seem to be a constant rather the phase seems a complex function of time
and possibly even periodic.

Figure 20a: The Autocorrelation plot shows a strong autocorrelation.

Figure 20b: The Lag plot describes a linear form which excludes the assumption of data ranmdomness and is typical of an Autoregressive
process (AR). A Ljung Box test (Q test) verified the assumption of independent data.

Figure 20c: The Scatter plot illustrates an apparently exponential wage curve. There is no observable variation between scales nor
any outliers.

Figure 20d: The Box plot also illustrates exponential growth and shows increasing interquartile variation every decade. No outliers are seen.

Figure 20e: The PACF ("Partial Autocorrelation Function") graph shows a significant value for 1, suggesting an Autoregressive
process of order 1 or AR(1) as the best adapted model.

Figure 20f: The spectral analysis per Blochner shows a single high peak of frequency close to 0 that confirms AR process model as most suitable.

Figure 21a: The Fourier Transforms confirm the Fisher periodogram fundamental of 12 months period.

Figure 21b: The Fisher periodogram 12 months period is characteristic of one existing rhythm (p = 0.95, α = 1 p).

Figure 21c: The Halberg Chronobiologic Window highlights the 12 month period.

Figure 21d: The Reverse Elliptic Spectrum (RES) finds a single 12 month period.

Figure 21f: A theoretical forecast curve of the means temperatures in England in 2005 (January 2005 is the 1261st month from January 1900)
