 Time Series analysis : Rhythms, Chronobiometry, Chronobiology, Chronoeconometry
 Analyse a rhythm and determine its characteristics
Detect various rhythms in a serial set of data,
Calculate the secondary periods starting from the basis period of a plurirhythmic phenomenon.
It allows the researcher to use the entire range of Cosinor methodology, and to perform different types of Spectral Analysis
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The
TSA-Seriel Cosinor© software also provide graphic tools for
the analysis of time series coming from the "Exploratory Data Analysis"
which is a complementary alternative to the classical statistical analysis.
The complete set of possibilities in studies and treatments gives a major point of interest to the TSA-Seriel Cosinor
software in different fields of application and any study in Chronobiology, Chronoeconometry or of a Periodic phenomenon.
(read methodology, references, full papers )
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Analysis of time series, rhythms : examples of methods with some tools of
Time Series Analysis Seriel Cosinor software
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Since the beginning of work on rhythms (in Chronobiology), the hypothesis that a rhythmic phenomenon is due to the existence of one or
several synchronisers has been formulated. You will remember that these synchronisers have often been considered to be
exogenous (take the classic example of the influence and the alternation of light and darkness). However these
synchronisers may be endogenous and so intracellular acting, for instance, as a sort of "Pacemaker". This has been
observed, for instance, in the motor activity of some protozoan life forms. The existence of a "Chronome" expressed at
the level of the genome has even been supposed. Rhythms in living matter are present from the start of life.
Various methods such as spectral analysis with or without a known period are used to study or reveal rhythms. However
the most famous is the Cosinor.
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Remember that among other things, the advantage of the Cosinor is that it is not sensitive to noise introduced into
the data, and that it does not require data to be equally distributed in time. Relative to spectral methods, in
particular those derived from Fourier analysis, this gives the Cosinor great intrinsic interest. The role of this
method from the beginning is to justify or not the existence of a given rhythm and to calculate its parameters
(Amplitude, Phase or Acrophase and average level or MESOR)
The TSA-Seriel Cosinor software, among other previously cited functions, has the possibility of detecting the
rhythmic period, of studying this period, of defining all its characteristics (See figure 1-a.1, figure 1-a.2,
figures 1-c.1, 1-c.2, 1-c-3, 1-c.4) TSA-Seriel Cosinor also makes it possible among other things to carry out the
"Single Cosinor" test (See figure 1-a.2)
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Figure 1-a.1 : Hamster activity/Anti-depressant. Period of 25.9 hours, probability 0.95 (a= 1 -p). Population Mean
Cosinor (according to Nelson et al, Bingham et al) and evaluation of the parameters relative to the detected rhythm.
Figure 1-a.2 : "Single Cosinor" test according to Nelson et al, Bingham et al. Same study (Period 25.3) probability 0.95 with a= 1-p. (Null amplitude test, ellipse test, analyse of the residues, and so on)
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TSA-Seriel Cosinor provides also a Cosinor of Population ("Population Mean Cosinor"). In other words, it brings together data from several subjects
in the form of a series; it deduces an overall periodic model (vectorial average) from the models in each series, etc.
It calculates overall the best sinusoidal model (harmonic regression by cosine function) that can best pass through the
set of experimental points (See figure 1-b.1, figure 1-a.1)
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Not only applicable to Circadian rhythms (24 +/- 4 h), it makes it possible to model lower as well as higher rhythms
(for instance ultradian (<24h), dian (24 +/- 2h), infradian (>24h), etc.)
It can also eliminate models of series of data that might be of limited interest as well as the disadvantage of
falsifying the results (through using an appropriate test with the Population Mean Cosinor)
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Figure 1-b.1 : Hamster activity/Anti-depressant. Chronogram : Period of 25.9 hours. Model from the Population Mean Cosinor (Seriel Cosinor),
experimental points, interpolated curve, average, variance and data standard deviation, confidence interval of the phase, phase or acrophase (according to Nelson et al, Bingham et al)
Figure 1-b.2 : Hamster activity/Anti depressant. Chronogram : Period of 25.2 hours. Model from the Single Cosinor, experimental points,
interpolated curve, variance and data standard deviation, phase or acrophase (according to Nelson et al, Bingham et al)
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We are used to taking into consideration the classic Cosinor method limited to mono-rhythmic modelization. The TSA-Seriel
Cosinor introduces the technique of reinjection of residues in order to reveal whether there is plurirhythmic
activity or not and to calculate its parameters.
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The geometrical representation of the various tests of the "Cosinor" is an ellipse that is covering or not the origin.
The mains periodic parameters are represented there : the amplitude by a vector beginning at the origin
ending to the center of the confidence ellipse, the phase is represented by the angular position of the amplitude vector
on a graduated trigonometrical circle (see figure following 1-c.1, 1-c.2, 1-c.3)
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Figure 1-c.1 : Hamster activity/Anti depressant. Confidence ellipse (Population Mean Cosinor) with probability of 0.95 (a = 1-p = 0.05) according to Nelson et al, Bingham et al (clockwise, no rhythm for the period 25.3 hours) and
according to Gouthière L. and Jacquin C. Rhythm. This last one type of representation (not clockwise)
uses international units. Calculations are geometric.
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Figure 1-c.2 : Hamster activity/Anti depressant. Confidence ellipse (Population Mean Cosinor) with probability of 0.95 (a= 1-p) according to Nelson et al, Bingham et al.
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Figure 1-c.3 : Temperature/Alcoholization (Cf Dr Danel T. CHRU Lille France): Confidence ellipse ("Single Cosinor")
with a probability of 0.95 (a = 1-p) according to Nelson et al, Bingham et al.
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TSA-Seriel Cosinor software also allows Spectral Analysis among which :
Those relating to Cosinor modeling (Spectrum of "Percent Rhythm" figure 1-d.1, 1-d.2, Inverse Elliptic
figure 1-d.3) not requiring equispaced time data.
Those derived from the 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.
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That which is noticeable and comes from regression and the Fourier Analysis like the Lomb N. and Scargle J.
Periodogram (figure 1-f1) not requiring equispaced time data.
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Figure 1-d.1 : Hamster activity/Anti depressant: Spectrum of "Percent Rhythm" according to the Population Mean
Cosinor. This allows detection of the Period, the interval of periods detected (in blue the confidence interval) by the
ellipse test and valid with a probability level of 0.95 ( a= 1-p). (-> Frequencies spectrum)
Figure 1-d.2 : Hamster activity/Anti-depressant. Spectrum of "Percent Rhythm" according to the "Single Cosinor". This
allows detection of the Period, the interval of periods detected (in blue) by the ellipse test and valid with a
probability level of 0.95 with a= 0.05. ( a= 1-p). (-> Frequencies spectrum)
Figure 1-d.3 : Hamster activity/Anti-depressant. Inverse Elliptic Spectrum from the Seriel Cosinor or Population Mean
Cosinor according to Gouthière L. This allows detection of periods by testing the surface of the ellipse, the interval
of detected periods in blue (Confidence Interval), valid at the given probability p with a= 1-p.
Figure 1-e.1 : Hamster activity/Anti-depressant. Spectral analysis according to Jenkins and Watts, the main peak
corresponds to the detected period (Confirmation of the above methods). The disadvantage of this type of analysis is
that it requires equispaced time data. (-> Frequencies spectrum)
Figure 1-e.2 : Hamster activity/Anti-depressant. Amplitude Discrete Fourier Transforms and spectrum of lines
showing the baseline of maximum amplitude (The fundamental ray) of about 24 hours here and the harmonics. (-> Periods spectrum)
Figure 1-e.3 : Hamster activity/Anti depressant. The periodogram according to Fisher is a method derived from the DFT.
The baseline (fundamental ray) is tested to verify whether it is characteristic of the periodic phenomenon being
studied at a given probability (a = 1-p).
Figure 1-f.1 : Hamster activity/Anti-depressant. Spectral periodogram according to Lomb N. and Scargle J., the main peak
corresponds to the period detected. The advantage of the Periodogram is that it does not require equispaced time data
unlike the methods derived from Fourier transforms (Confirmation of the above methods) The level of probability : 0.95 with ( a = 1-p) allows
to check the randomness of the signal. (-> Frequencies spectrum)
Figure 1-g.1 : Activity night shift work: The "Scatter periodogram" allows to analyse the period variations
due to a sliding window. On the other hand this kind of graphic is adapted to the study of the "jet lag".
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Other tools of study are provided like the autocorrelation graphic, the "Normal" probability diagram, the diagram of
Lag ("Lag plot"), the "Scatter plot", the "Run sequence plot", etc. These graphic tools coming from the "Exploratory Data Analysis" (EDA) allows to study the characteristics of the time series independently of the origin of data.
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The complete set of possibilities in studies and treatments gives a major point of interest to the TSA-Cosinor Seriel
software in any study in Chronobiology, Chronoeconometry or of Periodic phenomenon.
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The TSA Seriel Cosinor is connected to various fields of application that study rhythmic phenomena :
General Chronobiology,
Chronopharmacology and Chronotherapy that makes it possible to decide on the time to administer a medicinal molecule or a a treatement.
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Chronopsychological sociology with definition of profiles.
Chronoeconomy or Chronoeconometry that allows forecasts from cyclical variation of data in the economy field.
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