Limits...
Visualization of system dynamics using phasegrams.

Herbst CT, Herzel H, Svec JG, Wyman MT, Fitch WT - J R Soc Interface (2013)

Bottom Line: A phasegram can be interpreted as a bifurcation diagram in time.In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known.Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

View Article: PubMed Central - PubMed

Affiliation: Department of Cognitive Biology, Laboratory of Bioacoustics, University of Vienna, Althanstrasse 14, 1090 Vienna, Austria. christian.herbst@univie.ac.at

ABSTRACT
A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

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Illustration of limitations of both spectrograms and phase space diagrams. Left column: I. deterministic chaos, created by a logistic map; middle column: II. stochastic signal created by additive synthesis with randomized phases; right column: III. signal created by a logistic map where the parameter a was gradually varied from 3.5375 to 3.6. (a) All 1000 data points of each time series. (b) Amplitude spectrum. (c) Spectrogram. (d) Phase space embedding. Note that both signals I. and II. look identical when analysed by Fourier series (b) and (c), whereas their different nature becomes apparent in phase space (d). The additional complexity of signal III. (period doubling cascade to chaos) is not apparent in the phase space diagram (d).
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RSIF20130288F1: Illustration of limitations of both spectrograms and phase space diagrams. Left column: I. deterministic chaos, created by a logistic map; middle column: II. stochastic signal created by additive synthesis with randomized phases; right column: III. signal created by a logistic map where the parameter a was gradually varied from 3.5375 to 3.6. (a) All 1000 data points of each time series. (b) Amplitude spectrum. (c) Spectrogram. (d) Phase space embedding. Note that both signals I. and II. look identical when analysed by Fourier series (b) and (c), whereas their different nature becomes apparent in phase space (d). The additional complexity of signal III. (period doubling cascade to chaos) is not apparent in the phase space diagram (d).

Mentions: The spectrogram's inability to distinguish deterministic chaos from a random process is illustrated in figure 1. A synthesized signal was generated from a logistic map with parameter a set to a value of 3.6 (please refer to case 1, later in the text, for a definition of the logistic map) at a sampling rate of 1000 Hz. The resulting irregular time-series data are shown in the left column of figure 1a (signal I). The signal's spectrum is displayed in figure 1b, and a spectrogram is shown in figure 1c. The phase space diagram3 (figure 1d) reveals that the system's trajectory is aligned along a parabola [4, p. 357].FigureĀ 1.


Visualization of system dynamics using phasegrams.

Herbst CT, Herzel H, Svec JG, Wyman MT, Fitch WT - J R Soc Interface (2013)

Illustration of limitations of both spectrograms and phase space diagrams. Left column: I. deterministic chaos, created by a logistic map; middle column: II. stochastic signal created by additive synthesis with randomized phases; right column: III. signal created by a logistic map where the parameter a was gradually varied from 3.5375 to 3.6. (a) All 1000 data points of each time series. (b) Amplitude spectrum. (c) Spectrogram. (d) Phase space embedding. Note that both signals I. and II. look identical when analysed by Fourier series (b) and (c), whereas their different nature becomes apparent in phase space (d). The additional complexity of signal III. (period doubling cascade to chaos) is not apparent in the phase space diagram (d).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4043161&req=5

RSIF20130288F1: Illustration of limitations of both spectrograms and phase space diagrams. Left column: I. deterministic chaos, created by a logistic map; middle column: II. stochastic signal created by additive synthesis with randomized phases; right column: III. signal created by a logistic map where the parameter a was gradually varied from 3.5375 to 3.6. (a) All 1000 data points of each time series. (b) Amplitude spectrum. (c) Spectrogram. (d) Phase space embedding. Note that both signals I. and II. look identical when analysed by Fourier series (b) and (c), whereas their different nature becomes apparent in phase space (d). The additional complexity of signal III. (period doubling cascade to chaos) is not apparent in the phase space diagram (d).
Mentions: The spectrogram's inability to distinguish deterministic chaos from a random process is illustrated in figure 1. A synthesized signal was generated from a logistic map with parameter a set to a value of 3.6 (please refer to case 1, later in the text, for a definition of the logistic map) at a sampling rate of 1000 Hz. The resulting irregular time-series data are shown in the left column of figure 1a (signal I). The signal's spectrum is displayed in figure 1b, and a spectrogram is shown in figure 1c. The phase space diagram3 (figure 1d) reveals that the system's trajectory is aligned along a parabola [4, p. 357].FigureĀ 1.

Bottom Line: A phasegram can be interpreted as a bifurcation diagram in time.In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known.Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

View Article: PubMed Central - PubMed

Affiliation: Department of Cognitive Biology, Laboratory of Bioacoustics, University of Vienna, Althanstrasse 14, 1090 Vienna, Austria. christian.herbst@univie.ac.at

ABSTRACT
A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

Show MeSH