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2 : Study of a rectangular wave : sin(t) + sin(3t) + sin(5t)/5 with a period of 2Pi
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In this example, we use the Population Cosinor methodology (An overall model is calculated on the set of series by
vectorial mean of the models of each series) This type of modelling is used if you want to determine population
characteristics.
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Figure 2-a : One of the detected periods is a 2Pi (6.28) one. Representation of the model and the experimental points and the
interpolation curve (Modelling by "Population Mean Cosinor")
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Figure 2-b : Population Mean Cosinor and associated tests on the overall model (p fixed 0.95, alpha= 0.05) (abstract)
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Figure 2-c : Period of 2Pi. Representation of the confidence ellipse (p fixed 0.95, a = 0.05) of the Seriel Cosinor according to
Gouthière L. and Jacquin C. (The smaller the surface, the greater precision is in the determination of the period.
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Figure 2-d : Period of 2Pi. Representation of the confidence ellipse (p fixed 0.95, a = 0.05) of the "Population Mean Cosinor" according
to Nelson et al, Bingham et al (The smaller the surface, the higher precision is in the determination of the period)
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Figure 2-e : Initial period of 2Pi. The Inverse Elliptic Spectral plot according to Gouthière L. makes it possible to
detect periods. Here it shows in particular, a minimum at 6.28 for probability fixed at 0.95 (a = 1-p) 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.
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Figure 2-f : Initial period of 2Pi. The Fisher Periodogram (p fixed 0.95) makes it possible to detect a period and its validity using a fixed
probability. It is a method that is 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
much less interesting than the periodogram of Lomb N. and Scargle J. or even the Autoperiodogram according to Jenkins G.
and Watts D.)
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3 : Study of a saw tooth wave : sin(t) + sin(2t)/2 + sin(3t)/3 with a period of 2Pi
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In this example, we use the Single Cosinor methodology (An overall model is calculated on the set of all the points
belonging to the set of series) This type of modelling is used if you want to determine subject characteristics.
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Figure 3-a : One of the detected periods is a 2Pi (6.28) one. Representation of the model and the experimental points and the
interpolation curve.
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Figure 3-b : Single Cosinor and associated tests on the models (the period is 2Pi, p fixed 0.95, a = 1-p) (abstract)
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Figure 3-c : Period of 2Pi. Representation of the confidence ellipse according to Gouthière L. and Jacquin C. (p = 0.95, a = 1-p)
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Figure 3-d : Period of 2Pi. Representation of the confidence ellipse according to Nelson et al, Bingham et al (p fixed 0.95, a = 1-p)
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Figure 3-e : The elliptic inverse spectrum according to Gouthière L. presents a minimum at 6.28 for a probability level
of 0.95 (a = 1-p) (The blue part corresponds to the significance of the null assumption of the ellipse test for a given probability)
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Figure 3-f : The periodogram of Lomb N. and Scargle J. allows us to highlight a period (6.28) identical to that obtained by the
previous method.
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4 : Study of a triangular wave : sin(t) - sin(3t)/3² + sin(5t)/5² of period 2Pi
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In this example, we will show various methods to detect periods comparing them to the Reverse Elliptic Spectral (RES) plot
according to Gouthiere L. (The high quality of the data allows us to find the same results consistently. This is not always possible)
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Figure 4-a :
One of the detected periods is one of 2Pi (6.28) Representation of the model and the experimental points.
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Figure 4-b : The Reverse Elliptic Spectrum presents a minimum at 6.28 for a probability fixed at 0.95 (a = 1-p) corresponding the period (In
blue: the zone of validity for the ellipse test, this is the confidence limits of the period)
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Figure 4-c : The periodogram according to Jenkins G. and Watts D. allows us to find the previous result (6.28) although it is a method
that is completely different from regression (Indeed, we calculate the Fourier transforms of the autocorrelation function)
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Figure 4-d : Spectrum analysis according to Blochner likewise adapted to stationary phenomena leads to the same result.
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Figure 4-e : Autospectrum analysis according to Jenkins G. and Watts G. constituting classic spectrum analysis, (here with a Kaiser
window) allows us to find a result of 6.28.
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Figure 4-f : The periodogram of Lomb N. and Scargle J. confirms the previous results.
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Figure 4-g : The Fisher periodogram (p fixed 0.95, a = 1-p) will test a baseline (the fundamental ray) of 6.27.
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Figure 4-h : Confidence ellipse plot : The area of the ellipse (p fixed 0.95, a = 1-p) is weak because of the accuracy on the determination of the period.
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5 : Periodic wave: sin(t) + sin(3t) + sin(5t)/5 + sin(3t)/3² + sin(5t)/5² with a period of 2Pi
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In this example, we will compare the Inverse Elliptic Spectral plot resulting from the Single Cosinor with the spectrum
of rays obtained from the Rapid Fourier Transforms.
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Figure 5-a : One of the detected periods is one of 2Pi (6.28) Representation of the model and the experimental points.
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Figure 5-b : Inverse Elliptic Spectral plot resulting from the Single Cosinor. It allows to find the exact
value of the period.
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Figure 5-c : Rapid Fourier Transforms, Spectral plot of Rays. With a baseline (fundamental ray) at 6.28 (2Pi)
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Figure 5-d : Periodogram according to Lomb N. et Scargle J. found the exact period of 2Pi (see the more higher peak)
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6 : Study of temperature and alcoholization (Source: Dr T.Danel CHRU Lille France)
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In this study, we will look for periods corresponding to the temperature rhythms on the sample of data from alcoholic
subjects and control subjects respectively. The next stage will consist in model construction.
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Figure 6-a : Temperature/Alcoholization : The linear trend is slowly increasing and highlights an increase of the
body temperature.
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Figure 6-b : Temperature/Alcoholization : The Lomb and Scargle periodogram found a higher main period close to 27 hours.
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Figure 6-c : Temperature/Alcoholization : The model is then constructed with the previous value of period. The data have not been detrended.
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Figure 6-d : Temperature/Alcoholization : Confidence ellipse (Nelson et al, Bingham et al) for the period of 27 hours. The acrophase is close to 6:40 PM.
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Figure 6-e : Temperature/Alcoholization : Single Cosinor and associated tests for the periode of 27.3 hours.
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Figure 6-f : Temperature/Placebo : The linear trend is increasing then the body temperature slowly increases
in the course of time (A main period is found by spectral analysis)
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Figure 6-g : Temperature/Placebo : Confidence ellipse for the period of 24.4 hours. The acrophase is close to 5:20 PM.
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Figure 6-h : Temperature/Placebo : Single Cosinor and associated tests for a period of 24.5 hours
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7 : Study of plasma Cortisol
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In this study we will try to show the main periods of plasma Cortisol.
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Figure 7-a : Plasma Cortisol : The Autospectral plot (Jenkins and Watts) show us a main peak of period close to 22 hours.
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Figure 7-b : Plasma Cortisol : The Lomb and Scargle periodogram found main peak of period close to 24.9.
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Figure 7-c : Plasma Cortisol : The Fisher periodogram allows a fundamental period close to 24 hours.
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Figure 7-d : Plasma Cortisol : The Single Cosinor confidence ellipse for a period of 24 hours.
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8 : Study of plasma Melatonin
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In this study we will research the periods of plasma Melatonin on a data sample. It is difficult to consider
significantly periods higher than 24 hours. This can be partly explained by the variability and the specificity
of a sampling for a measurement taken on a group of subject under given conditions. The peak of the plasma Melatonin
usually admited is close to 27 hours or 3 hours.
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Figure 8-a : Plasma Melatonin : The Reverse Elliptic Spectrum (RES) allows us to determine the
main peak of period close to 25.9 hours (p = 0,95, a = 1-p)
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Figure 8-b : Plasma Melatonin : The Percent Rhythm spectrum allows us to determine the period close to 25.6 hours.
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Figure 8-c :
Plasma Melatonin : The Lomb and Scargle periodogram show us a main peak of period close to 24.8 hours.
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Figure 8-d :
Plasma Melatonin : The Fisher periodogram determines one fundamental period of 24 hours and a first harmonic
of 12 hours (H0 rejected means that que period of 24 hours is not confirmed)
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Figure 8-e :
Plasma Melatonin : Confidence ellipse plot (p = 0.95, a = 1-p) according to Nelson et al, Bingham et al (Period of 24 hours)
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Figure 8-f :
Plasma Melatonin : Confidence ellipse plot (p = 0.95, a = 1-p) according to Nelson et al, Bingham et al (Period of 24.8 hours)
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Figure 8-g :
Plasma Melatonin : Confidence ellipse plot (p = 0.95, a = 1-p) according to Nelson et al, Bingham et al (Period of 25.9 hours)
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10 : Study of the daily morning temperature of an adult woman
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The temperatures were recorded from day 10 (07/10) to day 69 (09/07) One finds a period close to 28 days (by spectral
analysis) The peak of day 43 seems to correspond to ovulation. The Scatter plot shows us that the first 4 or 5 points are
"outliers" and it is not necessary to consider them to calculate the periods of the temperature. These first
points have nevertheless a physiological interest.
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Figure 10-a : Daily morning temperature : The Scatter plot reveal us the behavior of the temperature curve with some "outliers"
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Figure 10-b : Daily morning temperature : The outliers have been removed and a linear detrend
has been performed.
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Figure 10-c : Daily morning temperature : Model of period 27 days (Period of the second order found on the signal substracted from the first part of the fundamental period)
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Figure 10-d :
Daily morning temperature : Reverse Elliptic Spectrum (Gouthière L. et al) The period of order 2 (27) is found
(after the treatement of the first signal)
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11 : Study of the number of monthly sunspots from 1900 to 1983 (Source: Andrews & Herzberg 1985)
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This study will make it possible to seek the periods for the sunspots observed in Zurich from 1900 to 1983
(Source: Andrews & Herzberg - 1985) at a rate of a measurement of one observation per month. The fact that the data is
equispaced in time enables us to use any type of spectral analysis. One will use the "Scatter plot" in order to visualize the
general aspect of the distribution of the points. The sinusoidal character seems quite present. Two types of spectral
analysis on completely different principles are used. One is led to the determination of a "fundamental" ray of common
value. One can deduce from it that there is a periodicity from 123 to 125 months when one can observe the same type of
phenomenon in sunspots with frequency.
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Figure 11-a : Sunspots : Scatter plot enables us to visualize the general aspect of the distribution of the points. The sinusoidal
character seems present.
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Figure 11-b : Sunspots : Percent Rhythm spectral plot : A peak found at 125 months (10.4 years)
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Figure 11-c :
Sunspots : Reverse Elliptic Spectrum according to Gouthière L. : A maximum peak found at 125 months (10.4 years)
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Figure 11-d :
Sunspots : Fisher periodogram : a fundamental ray found and tested at 126 months (10.5 years)
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12 : Study of the number of monthly sunspots from 1749 to 1899 (Source: Andrews & Herzberg 1985)
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This study will make it possible to seek the periods for the sunspots observed in Zurich from 1749 to 1899
(Source: Andrews & Herzberg - 1985) at a rate of a measurement of one observation by month. The fact that the data is
equispaced in time enables us to use any kind of spectral analysis. One will use the "Scatter plot" in order to visualize the
general aspect of the distribution of the points. The sinusoidal character seems quite present. Two types of spectral
analysis of completely different principles are used. One is led to the determination of a "fundamental" ray of common
value. One can deduce from it that there is a periodicity from 133 to 136 months when one can observe the same type of
phenomenon of sunspots in frequency.
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Figure 12-a : Sunspots : Scatter plot enables us to visualize the general aspect of the distribution of the points. The sinusoidal
character seems present.
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Figure 12-b :
Sunspots : Autospectral Spectrum according to Jenkins G. and Watts D. : A maximum peak is found of 134 months (11.2 years)
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Figure 12-c :
Sunspots : The Fourier periodogram found a fundamental of 133 months (11.1 years)
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Figure 12-d :
Sunspots : Reverse Elliptic Spectrum according to Gouthière L. et al. : A mninmum peak is found of 136 months (11.3 years)
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13 : Study of yearly sunspots from 1700 to 1979 (Source: Tong 1983)
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This study will make it possible to seek the periods for the sunspots observed from 1700 to 1979
(Source: Tong - 1983) at a rate of a measurement of one observation by year. The fact that the data is equispaced in
time that enables us to use any kind of spectral analysis. We will use the "Scatter plot" in order to visualize the
general aspect of the distribution of the points. The sinusoidal aspect seems quite present. Two types of spectral
analysis with completely different principles are used. We are led to the determination of a "fundamental" ray of common
value. We can deduce from it that there is a periodicity of 10 to 10.8 years i.e. 120 to 130 months when one can observe the same type of
phenomenon of sunspots in frequency. These results confirm those obtained in examples 10 and 11, which is remarkable.
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Figure 13-a : Sunspots : Scatter plot enables us to visualize the general aspect of the distribution of the points. The sinusoidal
character seems present.
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Figure 13-b :
Sunspots : Autoperiodogram according to Jenkins and Watts : A maximum peak found at 11 years (132 months)
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Figure 13-c :
Sunspots : Fisher periodogram : One fundamental ray found at 10 years (120 months) (p = 0.95, a = 1-p)
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Figure 13-d :
Sunspots : Reverse Elliptic Spectrum according to Gouthière L. : A minimum peak found at 11 years (132 months)
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14 : Study of the "American" number of monthly solar sunspots since 1945 to 2000 (Source: National Geophysical
Data Center US Department of Commerce)
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This study confirms the preceding studies. One finds a period close to 11 years corresponding to a cycle of solar activity. This
period is in conformity with what is commonly allowed as a period of the sunspots (see Fourier Analysis of Time
Series - Peter Bloomfield p 77)
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Figure 14-a : Solar sunspots : Scatter plot enables us to visualize the general aspect of the distribution of the points. The
sinusoidal character is present, the linear trend is decreasing.
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Figure 14-b : Solar sunspots : Autospectral analysis according to Jenkins G. et Watts D. : A maximal peak of 127/12 (10.58) close to 11 years found.
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Figure 14-c : Solar sunspots : Reverse Elliptic Spectrum according to Gouthière L. : A minimal peak of 128/12 (10.7) more than 10 years is
found. One can notice that there are some other peaks of small periods.
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Figure 14-d : Solar sunspots : The Autocorrelation plot is characteristic of a periodic curve (the curve is a sine curve with a
decreasing amplitude)
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15 : Study of the yearly average of cosmic rays from 1958 to 2002 in Kiel, Germany (Source: NGDC US Dep. of Com. 2003)
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We have previously shown (Examples 11 to 14) in the study of the number of sunspots that a cycle of one period
solar activity ranging between 10 and 12 years was probable. In this example we will check whether this period is
confirmed by the measurement of the cosmic rays which is a component of solar activity, measured here in Kiel, Germany.
It will be seen that one finds this period to be about 10.5 to 11 years.
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Figure 15-a : Cosmic rays : Scatter plot enables us to visualize the general aspect of the distribution of the points.
The sinusoidal character seems present, the linear trend is decreasing.
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Figure 15-b : Cosmic rays : Periodogram of Fisher makes it possible to highlight a fundamental ray of 11 years (p = 0.95, a = 1-p)
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Figure 15-c : Cosmic rays : Reverse Elliptic Spectrum according to Gouthière L. makes it possible to highlight a fundamental
ray of 10.5 years.
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Figure 15-d : Cosmic rays : Cosine model (y=3.56E+02Cos((2Pit/10.50)-6.34E-01)+5.99E+03) of the cosmic rays with a period of 10.5
years.
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Figure 15-e : Cosmic rays : Single Cosinor confidence ellipse (Phase = -0.63 rd, Amplitude = 356) "Single Cosinor" according
international units anticlockwise.
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Figure 15-f : Cosmic rays : Single Cosinor confidence ellipse (Phase = -0,63 rd, -36,3 degrés, Amplitude = 356) "Single Cosinor"
according to Nelson et al, Bingham et al.
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16 : Velocity of money, U.S. economy, 1869-1937, annual (Source: Friedman et Schwartz)
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In this example, we will search for the periodicity (seasonal) of the velocity of money, then we will plot the forecasting
model. We notice that in this case of econometric curves we can also modelize by an ARIMA(n, m) as we can see with the
strong dependent data shows by the Lag plot. The values of n and m can be determined by the autocorrelation and the PACF curves.
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Figure 16-a : Velocity of money : The Scatter plot enable us to display the main aspect of the distribution of the data.
The sinusoidal character stil seems to be present.
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Figure 16-b : Velocity of money : The Reverse Elliptic Spectrum allows us to determine several periods (seasonals) including a very marked one
of 73.2 years (The data have been detrended before the research of the main period)
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Figure 16-c : Velocity of money : This plot is the forecasting model of the monetary velocity (period of 73.2 years) of up to
approximately 2000.
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Figure 16-d : Velocity of money : The Autocorrelation plot helps us to verify that the distribution is not random. Data are highly correlated.
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17 : Variations of the U.S. copper price USA 1800-1997 (Source: Makridakis, Wheelwright and Hyndman 1998)
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After determination of the period, we build the model for 1800 to 1997 then a predictive model which will give us for
examples the values of the variations for 2000, 2010 etc.
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Figure 17-a : Variations in price of copper : The distribution of the data is Normal then seems to be stochastic.
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Figure 17-b : Variations in price of copper : The Reverse Elliptic Spectrum (REI) highlights several periods including one more
marked at 57 years.
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Figure 17-c : Variations in price of copper : The sinusoidal model built on all the points from 1800 to 1997 (the selected period is 57.7 years)
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Figure 17-d : Variations in price of copper : The sinusoidal model is extended until 2100. The predicted values of the variations
planned for 2000, 2010, 2020 and 2040 are calculated.
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19 : Temperature, physical training and night shift work (Source: Dr B.Mauvieux STAPS CRAPS University of Caen, PSA 2003)
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In this example we study temperatures coming from two samples of the same population. The first one is a sample of
sedentary subjects who have a night shift work, the second is a sample of the same subjects who underwent a physical
training of a few weeks. We notice that the physical training appreciably modifies the temperature curves more
particularly the amplitude and the phase change (the amplitude becomes more marked, the phase decreases) A complementary
study of the variations of the amplitude and the phase make it possible to notice that the amplitude remains constant
and that the phase seems to be a time function.
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Figure 19-a : Temperature of a group of technicians before physical training : Ellipse of confidence with the associated
chonobiological values (p = 0.95, a = 0.05) The acrophase is close to 10:25 PM.
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Figure 19-b : Temperature of a group of technicians before physical training : Ellipse of confidence. The amplitude increases, the
phase decreases (p = 0.95, a = 0.05) The acrophase is close to 3:10 AM.
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Figure 19-c : Temperature of a group of technicians before physical training : Study of the amplitude variations with the
spectrum of amplitude complex demodulation. The variations seems to be a constant: the amplitude is not a time function.
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Figure 19-d : Temperature of a group of technicians before physical training : Study of the phase variations with the
spectrum of phase complex demodulation. The variations do not seem to be a constant: the phase is a complex function of
time possibly even periodic.
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20 : Annual wages U.S. 1900 to 1970 (Source: Hipel and Mcleod 1994)
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This example of time series illustrates the strong autocorrelation of the data. In the second time we will try to
identify which type of model will be the best adapted.
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Figure 20-a : Autocorrelation plot shows a strong autocorrelation.
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Figure 20-b : The Lag plot describes a linear form which excludes the assumption of data ramdomness.
A Ljung Box test (Q test) allows to verify the assumption of independent data. The linear form is typical of
an Autoregressive process (AR)
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Figure 20-c : The Scatter plot illustrates the apparently exponential wages curve.
There is no observable variation between scales nor any outliers.
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Figure 20-d : The Box plot also illustrates this exponential growth and shows the
increasing inter-quartile variation every decade. No outliers are seen.
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Figure 20-e : The graph of the PACF ("Partial autocorrelation Function") shows a significant value for 1. One
can suppose an Autoregressive process of order 1 or AR(1) would be the best adapted model.
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Figure 20-f : The spectral analysis according to Blochner shows us a high single peak of frequency close to 0
that confirms us that the choice of an AR process model seems the best suitable one.
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21 : Monthly temperature in England from 1900 to 1970 (Source: Hipel et Mcleod 1994)
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This example makes it possible to highlight one inconstestable period about 12 months (circannual, seasonal change) with various methods.
All the methods give unanimously this value of period. Single Cosinor tests makes it possible to give the
characteristic parameters of this rhythmic phenomenon. We deduce from it the forecasting curve of temperatures
for the year 2005 (The data have been detrended before the analysis to keep only the periodic part of the signal,
the linear trend has a weak slope)
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Figure 21-a : The Fourier Transforms as the Fisher peridogram confirm this fundamental of 12 months.
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Figure 21-b : Fisher periodogram. The period of 12 months is characteristic of one existing rhythm
of this previous period (p = 0.95, a = 1 -p)
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Figure 21-c : The Halberg Chronobiologic Window highlights the 12 months period.
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Figure 21-d : In the same way the Reverse Elliptic Spectrum (RES) finds this single 12 months period.
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Figure 21-e : The tests of Single Cosinor make it possible to characterize this rhythmic phenomenon
of 12 months period (abstract)
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Figure 21-f : Curve of theoretical forecast of the mean temperatures in England for 2005 (January 2005 is the
1261st month from January 1900)
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Examples of various experimental or fictitious series
