Main Functions of Time Series Analysis - Cosinor

The following set of these software functions illustrates on the one hand, the Cosinor methodology (modelling, detection of periods and associated tests (according to Nelson et al, Bingham et al, etc.) and on the other hand, comparative spectrum methods of analysis of data (coming from the Exploratory Data Analysis for the most part) leading to the determination of periods, modelisation and forecasting models (read the Methodology and the References for more information)

Figure 1.2–a : Population Mean Cosinor Cosinor tests : Abstract of tests for a fixed period (Triangular function of period 2Pi)

Figure 1.5–a : Population Mean Cosinor : Inverse elliptic spectral plot for an interval of given period and a level of probability (Triangular function of period 2Pi, p=0.95 a=0.05)

Figure 1.6–a : Population Mean Cosinor : Spectral plot of Percent Rhythm for a given interval of time and a given probability (Triangular function of period 2Pi, p=0.95 a=0.05)

Figure 1.7–a : Population Mean Cosinor : Model, experimental points and interpolation curves (Body temperature / Alcoholized subjects)

Figure 1.8–a : Population Mean Cosinor - Residues Normal probability plot (Alcoholized subjects, period of 13.6, p=0.95 a=0.05)

Figure 1.9–a : Population Mean Cosinor - Chronogram and confidence intervals : phase and amplitude.

Figure 2.1–a : Single Cosinor : Abstract of Cosinor Single tests (Triangular function of period 2Pi, p=0.95 a=0.05)

Figure 2.2–a : Single Cosinor : Ellipse of confidence (Triangular function of period 2Pi, p=0.95 a=0.05)

Figure 2.3–a: Single Cosinor – Ellipse of Confidence according to Halberg et al. (Triangular function of period 2Pi, p=0.95 a=0.05)

Figure 2.4–a: Single Cosinor – Spectral plot of Percent Rhythm at an interval of a given period and at a given probability (Triangular function of period 2Pi p=0.95 a=0.05)

Figure 2.5–a: Single Cosinor – Model and experimental points curves (Cosine function with 3 periods: 6, 12, 24)

Figure 2.6–a: Single Cosinor – Inverse Elliptic Spectral plot (Triangular function, p=0.95 a=0.05)

Figure 2·13-a : Single Cosinor – Chronogram and confidence intervals: CI of the phase and CI of the amplitude (Hamsters activity/ Anti-depressant)

Figure 2.11 : Single Cosinor - Chronobiologic Window (Allows to detect the period by the study of the residues variance as a function of time) Example of a triangular function of period 2Pi.

Figure 2.7–a: Single Cosinor – Normal probability plot on the residues of the Single Cosinor Model (Alcoholized subjects, Period of 24.7, p=0.95 a=0.05)

Figure 2.8 : Single Cosinor – Complex Amplitude Demodulation plot (The aim is to verify the amplitude to be a constante)

Figure 2.9 : Single Cosinor – Complex Phase Demodulation plot (The aim is to verify the phase to be a constante)

Figure 2.10 : Single Cosinor – Variance Homogeneity plot.

Figure 3.1-a : Lag Plot (Triangular function of period 2Pi and Rectangular function of period 2Pi, p=0.95 a=0.05)

A lag plot checks whether a data set or time series is random or not. Random data should not exhibit any identifiable structure in the lag plot. Non-random structure in the lag plot indicates that the underlying data are not random ( EDA Graphical Techniques 1.3.3.15. Lag Plot, Engineering Statistics Handbook) A complementary test of Ljung and Box (Q test) check the assumption of "Independent Data".

Figure 3.2 : Autocorrelation plot (Triangular function of period 2Pi)

Autocorrelation plots (Box and Jenkins, pp. 28-32) are a commonly-used tool for checking randomness in a data set. This randomness is ascertained by computing autocorrelations for data values at varying time lags. If random, such autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero ( EDA Graphical Techniques 1.3.3.1. Autocorrelation Plot, Engineering Statistics Handbook) A confidence interval is calculated and shows on the plot the band of randmoness (close to the zero x axis)

Figure 3·14-1 : Partial Autocorrelation Plot (Triangular function of period 2Pi) The data are not random. The PACF allows to verify is data are randomn. It is also use to determine the order of an autoregressive model...

(Read EDA Graphical Techniques 6.4.4.6.3. Partial Autocorrelation Plot, Engineering Statistics Handbook)

Figure 3.3 : Spectral density plot (Triangular function of period 2Pi)

Figure 3.4.1 : Autospectral plot (Hanning window) according to Jenkins and Watts (Triangular function of period 2Pi)

Figure 3.4.2 : Autospectral plot (Kaiser window) according to Jenkins and Watts (Triangular function of period 2Pi)

Figure 3.4.3 : Autospectral plot (Blackman-Harris window) according to Jenkins and Watts (Triangular function of period 2Pi)

Figure 3.5 : Spectral Autoperiodogram according to Jenkins and Watts (Triangular function of period 2Pi)

Figure 3.6 : Time series analysis – Periodogram plot according to Lomb et Scargle (Triangular function of period 2Pi)

Figure 3.7.1 : Time series analysis – Discrete Fourier Transforms - Amplitude (Cosine function with periods 6, 12, 24, equispaced time data)

Figure 3.7.2 : Time series analysis – FFT - Amplitude (Cosine function with periods 6, 12, 24, equispaced time data)

Figure 3.8 : Time series analysis – Fisher periodogram. The fundamental ray is tested (Hamster activity/Antidepressant)

Figure 3·13.1 : Time series analysis – Fourier Periodogram (Sum of cosine functions of periods 6, 12, 24)

Figure 3·13.2 : Time series analysis – Cumulative Fourier Periodogram (Sum of cosine functions of periods 6, 12, 24)

Figure 3·15.1 : Time series analysis – MESA spectrum (Sum of cosine functions of periods 6, 12, 24)

Figure 3·15.2 : Time series analysis – MESA spectrum (Plasma Melatonin)

Figure 3·7.1 : Time series analysis – Discrete Fourier Transforms - Amplitude (Cosine function of periods 6, 12, 24)

Figure 3·7.2 : ;Time series analysis – Fast Fourier Transforms- Amplitude (Cosine function of periods 6, 12, 24)

Figure 3.9 : Normal probability plot of the experimental data (Temperature/Alcoholic subjects) The normal probability plot (Chambers 1983) is a graphical technique for assessing whether or not a data set is approximately normally distributed. The data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from this straight line indicate departures from normality... (EDA Graphical Techniques 1.3.3.21. Normal Probability Plot, Engineering Statistics Handbook) A complementary test of Komolgorov-Smirnov (K-S test) is used to test this assumption with a level of probability p (or risk 1 -p)

Figure 3.10 : Scatter plot of the experimental data with the linear trend (Temperature/Alcoholic subjects) A scatter plot (Chambers 1983) reveals relationships or association between two variables. Such relationships manifest themselves by any non-random structure in the plot (EDA Graphical Techniques 1.3.3.26. Scatter Plot, Engineering Statistics Handbook)

Figure 3.11 : Run Sequence plot. With run sequence plots, shifts in location and scale are typically quite evident. Also, outliers can easily be detected (EDA Graphical Techniques 1.3.3.25. Run Sequence Plot, Engineering Statistics Handbook)

Figure 3·12 : Box plot according to Chambers (Study of temperature with placebo). The Box plot can provide answers to the following questions : 1) Is a factor significant ?, 2) Does the location differ between subgroups ?, 3) Does the variation differ between subgroups ?, 4) Are there any outliers ? The Box plot is an important EDA tool for determining if a factor has a significant effect on the response with respect to either location or variation. The Box plot is also an effective tool for summarizing large quantities of information (EDA Graphical Techniques 1.3.3.7. Box Plot, Engineering Statistics Handbook)