WAVOS: a MATLAB toolkit for wavelet analysis and visualization of oscillatory systems
© Harang et al; licensee BioMed Central Ltd. 2012
Received: 12 September 2011
Accepted: 26 March 2012
Published: 26 March 2012
Wavelets have proven to be a powerful technique for the analysis of periodic data, such as those that arise in the analysis of circadian oscillators. While many implementations of both continuous and discrete wavelet transforms are available, we are aware of no software that has been designed with the nontechnical end-user in mind. By developing a toolkit that makes these analyses accessible to end users without significant programming experience, we hope to promote the more widespread use of wavelet analysis.
We have developed the WAVOS toolkit for wavelet analysis and visualization of oscillatory systems. WAVOS features both the continuous (Morlet) and discrete (Daubechies) wavelet transforms, with a simple, user-friendly graphical user interface within MATLAB. The interface allows for data to be imported from a number of standard file formats, visualized, processed and analyzed, and exported without use of the command line. Our work has been motivated by the challenges of circadian data, thus default settings appropriate to the analysis of such data have been pre-selected in order to minimize the need for fine-tuning. The toolkit is flexible enough to deal with a wide range of oscillatory signals, however, and may be used in more general contexts.
We have presented WAVOS: a comprehensive wavelet-based MATLAB toolkit that allows for easy visualization, exploration, and analysis of oscillatory data. WAVOS includes both the Morlet continuous wavelet transform and the Daubechies discrete wavelet transform. We have illustrated the use of WAVOS, and demonstrated its utility for the analysis of circadian data on both bioluminesence and wheel-running data. WAVOS is freely available at http://sourceforge.net/projects/wavos/files/
Many real-world sources of data display suggestively periodic behavior, but with time-varying period, amplitude, or mean. This variation can lead to inaccurate results when the data is analyzed with standard Fourier techniques, as Fourier analysis assumes stationarity of the signal and its basis functions are unbounded in time . Wavelets, in contrast, are localized in both time and frequency. This in turn localizes the analysis, allowing the changes insignal properties to be tracked over time .
Wavelet analysis has proven to be invaluable in many problem domains, including ecological cycles , sunspot cycles , circadian cycles [5, 6], nfradian cycles associated with gene transcripts , blood-flow dynamics , and ECG signals [8–10]. Our work todate has focused on circadian data. This data typically displays many of the features that render a signal difficult to analyze via Fourier techniques, including changes in period length, sharp transients, and phase shifts [6, 11], as well as experimental artifacts such as loss of amplitude, shifting means , and "shot noise" when bioluminescence experiments are considered . These features may require significant pre-processing of the data before analysis . Recognizing the potential of wavelet methods for analysis of circadian data, Price et al.  developed the WAVECLOCK software, implementing the Morlet CWT in the R statistical programming environment. The WAVECLOCK software has enabled and inspired several investigations [15, 16], but requires familiarity with the R statistical programming language and command-line interface. Other wavelet implementations (e.g. WaveLab  or the MATLAB Wavelet Toolkit) exist, but are either general-purpose packages that assume significant familiarity with wavelets, proprietary software, or both.
The wavelet transform, in both continuous (Morlet) and discrete (Daubechies) versions, offers a set of tools for the analysis of nonstationary oscillators that can avoid many of the issues associated with techniques that assume stationarity. The Morlet continuous wavelet transform can nonparametricallydenoise, detrend, and analyze the local frequency content of a signal in a single operation [1, 15]. Application of the CWT enables such analyses as estimating the evolution of the period and phase of a signal across time, locating peaks and troughs even in the presence of large amounts of noise, tracking the evolution of the amplitude across time, and identifying multiple simultaneous oscillators. The discrete wavelet transform (DWT) enables multi-scale analysis of a signal using a sequence of compactly supported filters that decompose the signal into a set of component frequency bands . While it yields less precise frequency estimates than the CWT, it enables direct statistical significance testing of different frequency bands , and has better time localization properties than the CWT, allowing for more efficient removal of transients that are strictly time-limited. It also has reconstruction properties that are capable of preserving the mean of the signal . Thus the DWT can be used as a pre-processing step for other, non-wavelet analyses. Readers interested in learning more about the continuous wavelet transform are referred to ; for further information on discrete wavelets, see .
WAVOS is implemented in MATLAB to allow for easy modification. The convolutions required for both the discrete and continuous wavelet transforms are performed in the Fourier domain. Several options are provided for managing the implicit periodization and corresponding boundary effects that this produces in the CWT, including zero-padding, edge-reweighting, and truncation of affected CWT coefficients (as in ).
For the discrete wavelet transform, the standard decimated transform may cause localization issues for particularly sharp transients, depending upon the place at which the signal is split during decimation. The transient may be completely contained in a particular coefficient of the transform, or it may be split between two adjacent coefficients. This has the undesirable effect of potentially allowing two different rotations of identical data to produce two different sets of DWT coefficients . To avoid this issue, we use a redundant version of the DWT known in various contexts as the "undecimated", "shift-invariant", or "maximum-overlap" DWT . This transform is invariant to rotation of the data. Statistical significance testing of the levels of the DWT decomposition is performed using a Chi-squared test, under the assumption of additive Gaussian noise as described in .
The default parameters in the GUI are geared towards analysis of circadian signals. In particular, the minimum and maximum periods default to 6 and 48, respectively, with a default sampling rate of 1. These presets assume hourly samples of data with periods of interest in the above range. Both the sampling rate and the periods of interest may be adjusted independently of each other. The tool will not attempt to analyse periodicities that exceed the minimum or maximum possible due to numberof samples and sampling rate, and will emit a warning message with the actual bounds used if such an analysis is attempted. For DWT denoising, a default p-value of 0.05 is used, as this is a generally accepted standard for statistical significance. Users may also select different levels of the decomposition by hand, in order to focus on periodicities of the data that interest them.
Due to the native MATLAB implementation, long time series (on the order of 10,000 or more observations) may result in analyses requiring several minutes per time series to complete. This may be mitigated by reducing the range of periods to be analyzed in the CWT, or down-sampling the series to a shorter one with a smaller number of samples per unit time. Data sets containing a large number of time series with a large number of samples may exhaust MATLAB's working memory, causing an error; this will depend strongly on the computer being used. Due to MATLAB's representation of data in matrix form, all time series in a single data set to be processed as a batch must contain an equal number of observations. Time series with differing numbers of observations may be processed and analyzed separately.
The CWT and DWT have different features that incline them toward different uses. The Morlet CWT is most useful for investigating local properties of a signal: its detailed decomposition of time and frequency allows for very close tracking of several statistics of interest, however it cannot efficiently reconstruct a signal from individual components. The DWT fills a complementary role; while its time-frequency decomposition is very coarse, it is capable of reconstruction of a signal and also permits statistical testing. This allows the DWT to be used for many familiar signal-processing tasks, such as denoising and detrending, that are difficult to accomplish with the CWT.
Using the Morlet CWT in WAVOS has enabled us to analyze the distributional properties of the period of rat SCN cells based on PER2::LUC bioluminescence data . Where previous studies had classified cells as either arrhythmic or circadian, our wavelet analysis revealed that individual cells, when removed from network interactions, intermittently express circadian and/or longer infradian periods. Results from our stochastic model suggest that the uncoupled cells may beswitching between two ocsillatory mechanism: the indirect negative feedback of protein complex PER-CRY on the expression of Per and Cry genes, and the negative feedback of CLOCK-BMAL1 on the expression of the Bmal1 gene.
WAVOS has many features to facilitate analysis with the DWT and CWT, including: both command-line and graphical user interfaces; automatic processing of multiple time series simultaneously; interfaces to multiple standard file formats (including .csv,.xls, .txt, and .mat); automated detection and extraction of multiple statistics of interest (period, phase, amplitude, peak-to-trough measurements, and Rayleigh synchronization measurements); visualization tools for both exploration and analysis; utilities for smoothing, detrending, and statistical significance testing using the DWT; and multiple ridge selection algorithms, including local maximum and simulated annealing (the "crazy climber" method of ). Complete details are provided in the WAVOS documentation (at the download link).
Future development of WAVOS will concentrate on feature extraction from circadian signals, in particular allowing end-users to select user-defined phase markers for further analysis.
Availability and requirements
• Project name: WAVOS - Wavelet Analysis and Visualization of Oscillatory Signals
• Project home page:http://sourceforge.net/p/wavos/home/Home/
• Operating Systems: Cross-platform with MATLAB 2007a or later
• Programming language: MATLAB
• Other requirements: MATLAB 2007a or later
• License: Modified BSD
We wish to thank the Herzog lab at Washington University in St. Louis for their generous help, the locomotor data presented above, and feedback in evaluating early versions of WAVOS.
This work was funded by: ICB grants through the Army Research Office (DAAD 19-03-D-0004) and NIH grants GM078993 and GM096873-01.
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