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Main Functions of Time Series Analysis - Cosinor Lab View

The following set of software functions illustrates Cosinor methodology such as modelling, period detection and associated tests (per Nelson et al., Bingham et al., etc.) and comparative spectrum methods of data analysis (mostly from the Exploratory Data Analysis) for the period determination, modelling and forecasting models (see  Methodology and References for more information). Other functions exist but are not described here, such as filters, data transformation, detrends, etc.

1.2: Population Mean Cosinor

Figure 1.2–a: Abstract of tests for a fixed period for a triangular function of a 2Pi period.


1.3: Population Mean Cosinor – Confidence ellipse plot per Gouthière and Jacquin

Population Mean Cosinor – Confidence Ellipse plot
Figure 1.3–a: Confidence ellipse plot with rectangular function of period 2Pi, p = 0.95 means CI = 95% α = 0.05.
Population Mean Cosinor - Confidence ellipse
Figure 1.3-b: Confidence ellipse plot Temperature / Alcoholic subjects with period of 13.6 hours, p = 0.95 means CI = 95%, α = 0.05)


1.4: Population Mean Cosinor – Confidence ellipse per Nelson et al., Bingham et al.

Population Mean Cosinor - Confidence Ellipse
Figure 1.4–a: Confidence ellipse plot with triangular function of period 2Pi, p = 0.95 means CI = 95%, α = 0.05.
Population Mean Cosinor – Ellipse of Confidence
Figure 1.4–c: Confidence ellipse (Temperature / Alcoholic subjects period of 13.6 hours, p = 0.95 means CI = 95%, α = 0.05)


1.5: Population Mean Cosinor – Reverse Elliptic Spectral plot per Gouthière

Population Mean Cosinor – Inverse Elliptic Spectral plot
Figure 1.5–a: Reverse elliptic spectral plot for an interval of given periods and a level of probability (Triangular function of period 2Pi, p = 0.95 means CI = 95%, α = 0.05)


1.6: Population Mean Cosinor – Percent Rhythm spectral plot

Population Mean Cosinor – Percent Rhythm spectral plot
Figure 1.6–a: Spectral plot of Percent Rhythm for a given interval of time and a given probability (Triangular function of period 2Pi, p = 0.95 means CI = 95%, α = 0.05)


1.7: Population Mean Cosinor - Model and experimental point curves

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


1.8: Population Mean Cosinor –Residues Normal probability plot per Chambers

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


1.9: Population Mean Cosinor - Chronogram and confidence intervals

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


2.1: Single Cosinor - Tests for one period

Figure 2.1–a: Single Cosinor: Abstract of Single Cosinor tests (Triangular function of period 2Pi, p = 0.95 means CI = 95%, α = 0.05)


2.2: Single Cosinor – Confidence ellipse per Gouthière and Jacquin

Figure 2.2-a Figure 2.2
Figures 2.2–a: Confidence ellipse (Triangular function of period 2Pi, p = 0.95 means CI = 95%, α = 0.05)


2.3: Single Cosinor – Confidence ellipse per Nelson et al. and Bingham et al.

Figures 2.3-a
Single Cosinor – Confidence ellipse plot
Figures 2.3–a: Confidence ellipse per Halberg et al. for a triangular function of period 2Pi (p = 0.95 means CI = 95%, α = 0.05)


2.4: Single Cosinor – Spectral plot of Percent Rhythm per Gouthière

Single Cosinor – Percent Rhythm spectral plot
Figure 2.4–a: Spectral plot of Percent Rhythm at an interval of a given periods and at a given probability for a triangular function of period 2Pi (p = 0.95 means CI = 95%, α = 0.05)


2.5: Single Cosinor – Model and experimental points curves

Single Cosinor  – Model and experimental points curves
Figure 2.5–a: Model and experimental points curves for a cosine function with 3 periods: 6, 12, 24.


2.6: Single Cosinor – Reverse Elliptic Spectral plot per Gouthière

Single Cosinor – Inverse Elliptic Spectral plot
Figure 2.6–a: Reverse Elliptic Spectral plot for a triangular function of 2Pi period (p = 0.95 means CI = 95%, α = 0.05)


2.13: Single Cosinor – Chronogram and confidence intervals

Figure 2.13-a
Figure 2.13-a: Chronogram and confidence intervals: CI of the phase and CI of the amplitude (rats)


2.11: Single Cosinor – Chronobiologic Window per Halberg et al.

Single Cosinor – Chronobiologic Window Plot
Figure 2.11: 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.


2.7: Single Cosinor – Residues Normal Probability plot per Chambers

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


2.8: Single Cosinor – Complex Amplitude Demodulation plot per Gouthière.

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


2.9: Single Cosinor – Complex Phase Demodulation plot per Gouthière

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


2.10: Single Cosinor – Variance Homogeneity plot per Draper and Smith

Single Cosinor – Variance Homogeneity plot
Figure 2.10: Variance Homogeneity plot.


2.12: Single Cosinor – Comparison of rhythms, comparison of chronobiometric parameters per Bingham et al.

Table 2.12: Comparison of rhythms (MESOR, Amplitude and Acrophase) of different series for one period chosen (and a given probability) per Bingham et al (Study of the Cortisol, period of 24.4 hours, p = 0.95 means CI = 95%, α = 0.05)


3.1: Time series analysis – Lag plot

Lag plot
Lag plot
Figures 3.1-a: Lag Plot (triangular function of period 2Pi and rectangular function of period 2Pi, p = 0.95, α = 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".


3.2: Time series analysis – Autocorrelation plot

Autocorrelation plot
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)


3.14: Time Series Analysis – Partial Autocorrelation Plot (PACF)

Figure 3.14-a
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)


3.3: Time series analysis – Spectral Density plot per Blochner

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


3.4: Time series analysis – Autospectral plot per Jenkins and Watts

Spectral analysis
Figure 3.4.1: Autospectral plot (Hanning window) per Jenkins and Watts (Triangular function of period 2Pi)

Spectral analysis
Figure 3.4.2: Autospectral plot (Kaiser window) per Jenkins and Watts (Triangular function of period 2Pi)
Spectral analysis
Figure 3.4.3: Autospectral plot (Blackman-Harris window) per Jenkins and Watts (Triangular function of period 2Pi)


3.5: Time series analysis – AutoPeriodogram plot per Jenkins and Watts

Spectral analysis - Autoperiodogram
Figure 3.5: Spectral Autoperiodogram per Jenkins and Watts (Triangular function of period 2Pi)


3.6: Time series analysis – Periodogram plot per Scargle

Lomb and Scargle Periodogram analysis plot
Figure 3.6: Periodogram plot per Scargle (Triangular function of period 2Pi)


3.7: Time series analysis – Discrete Fourier Transforms - Amplitude

DFT-Amplitude
Figure 3.7.1: Discrete Fourier Transforms - Amplitude (Cosine function with periods 6, 12, 24, equispaced time data)
FFT-Amplitude
Figure 3.7.2: FFT - Amplitude (Cosine function with periods 6, 12, 24, equispaced time data)


3.8: Time series analysis – Fisher periodogram

Fisher periodogram
Figure 3.8: Fisher periodogram. Test of the fundamental ray (rats)


3.13: Time series analysis – Fourier periodogram and Cumulative Fourier Periodogram per Schuster

Figure 3.13.1
Figure 3.13.1: Fourier Periodogram (Sum of cosine functions of periods 8, 12, 24)


Figure 3.13.2
Figure 3.13.2: Cumulative Fourier Periodogram (Sum of cosine functions of periods 8, 12, 24)


3.15: Time series analysis – Maximum Entropy Spectral Analysis per Burg MESA

Figure 3.15.1
Figure 3.15.1: MESA spectrum of sum of cosine functions of periods 8, 12, 24.


Figure 3.15.2
Figure 3.15.2: MESA spectrum of plasma melatonin.


3.9: Normal probability plot per Chambers

Normal probability plot
Figure 3.9: Normal probability plot of 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 Komolgorov-Smirnov test (K-S test) is used to test this assumption with a level of probability p (or risk 1 -p)


3.10: Scatter plot per Chambers

Normal probability plot
Figure 3.10: Scatter plot of experimental data with a 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)


3.11: Run Sequence plot per Chambers

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


3.12: Box plot per Chambers

Box plot
Figure 3.12: Box plot per Chambers for the study of temperature in subjects with placebo (alcoholic study)

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 and is also effective for summarizing large quantities of information (EDA Graphical Techniques 1.3.3.7. Box Plot, Engineering Statistics Handbook)