Time Series analysis : Rhythmometry, Chronobiometry, Chronobiology, Chronoeconometry :

Powerful methods for the rhythms analysis

Powerful methods for the rhythms analysis

TSA-Cosinor© **(Time Series Analysis - Cosinor)** allows :

. Detection of various rhythms in a serial set of data.

. Calculation of secondary periods from the basis period of a plurirhythmic phenomenon.

It allows a researcher to use the entire range of Cosinor methodology, and to perform different types of **spectral analyses**.

Analysis of time series and rhythms : Examples of methods with tools of

Time Series Analysis Cosinor software

Time Series Analysis Cosinor software

Since the start of work on chronobiological rhythms, there has been the hypothesis that a rhythmic phenomena is due to
synchronizers often considered to be exogenous (the classic example is light and darkness). However a synchroniser may also be endogenous
and thus intracellular, as a sort of "Pacemaker" as has been observed, for instance, in the motor activity of some protozoa.
The existence of a "Chronome" expressed at the level of the genome has even been proposed. Rhythms in living beings are present from the start of life.
Assorted methods, such as spectral analysis with or without a known period are used to study or reveal rhythms. The most well known among these is Cosinor.

Among the primary advantages of Cosinor are an insensitivity to noise in the data, and no requirement for data to be equally distributed in time.
Relative to spectral methods, particularly those derived from Fourier analysis, this makes Cosinor very interesting. This method can justify, or not,
the existence of a given rhythm and it can calculate its parameters (Amplitude, Phase or Acrophase and average level or MESOR)
TSA-Cosinor software, can also detect the rhythmic periods, study these periods, and define all characteristics (See figure 1-a.1, figure 1-a.2,
figures 1-c.1, 1-c.2, 1-c-3, 1-c.4) TSA-Cosinor also makes it possible to carry out the "Single Cosinor" test (See figure 1-a.2)

TSA-Cosinor also provides a Cosinor for Population ("Population Mean Cosinor"). This brings together data from several subjects
in the form of a series and deduces an overall periodic model (vectorial average) and more from the models in each series.
It calculates the best sinusoidal model (via cosine function : harmonic regression) that can best pass through the
set of experimental points (See figure 1-b.1, figure 1-a.1)

Not only applicable to circadian rhythms (24 +/- 4 h), Cosinor makes it possible to model shorter and longer rhythms
(ultradian (<24h), dian (24 +/- 2h), infradian (>24h), and so on)
The software can also eliminate models of series of limited interest as well as the disadvantage of
falsified the results (via an appropriate test with the Population Mean Cosinor)

The classic Cosinor method is limited to mono-rhythmic model. The TSA-Cosinor surpasses this by introducing the technique of **reinjection of residues** to reveal whether there is plurirhythmic
activity (or not) and to calculate its parameters.

The geometrical representation of the various "Cosinor" tests is an ellipse that may cover the origin.
Primary periodic parameters are represented : the amplitude by a vector beginning at the origin and
ending to the center of the confidence ellipse and the phase by the angular position of the amplitude vector
on a graduated trigonometrical circle (see figure 1-c.1, 1-c.2, 1-c.3, 1-c.4)

TSA-Cosinor software also allows Spectral Analysis including :

. Those relating to **Cosinor modeling** (Spectrum of "Percent Rhythm" figure 1-d.1, 1-d.2, Inverse Elliptic
figure 1-d.3) and **not requiring equispaced time data**.
. Those derived from **Fourier analysis** or related analysis (Spectrum of spectral densisty, Autospectral
figure 1-e.1, Autoperiodogram, Periodogram, Amplitude Discrete Fourier Transforms Figure 1-e.2) **requiring equispaced
time data.**

In the following example, the Cosinor analysis is not recommended (due to the lack of results). We will use the Fourier periodogram here.

Other study tools are provided such as the Autocorrelation graphic, the "Normal" probability diagram, the Lag diagram (Lag plot), the Scatter plot,
the Run sequence plot, the Box plot, and so on These graphic tools from "Exploratory Data Analysis" (EDA) allows study of time series independently
of the data origin.

Fields of applications

. General Chronobiology.

. Chronopharmacology and Chronotherapy to investigate the time to administer a medicinal molecule or a treatment.

. Chronopsychological sociology to allow definition of profiles.

. Chronoeconomy or Chronoeconometry to forecast cyclical variation of economic factors.

. Chronopharmacology and Chronotherapy to investigate the time to administer a medicinal molecule or a treatment.

. Chronopsychological sociology to allow definition of profiles.

. Chronoeconomy or Chronoeconometry to forecast cyclical variation of economic factors.