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 5).

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 19. 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 19. 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 19.

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.

The Morlet CWT may be viewed as a modification of the windowed Fourier transform (WFT) that scales the size of the window to a constant number of wavelengths for any analysis frequency 2, resulting in good frequency detection across a wider range of periods than many other methods. For example, Figure 4 displays a synthetic signal with a period that is shifting from 24 to 18 hours in length. Figure 5 shows the CWT heatmap of that signal with Gaussian edge correction applied and the edge-affected coefficients both included and excluded (all options that can be set within the WAVOS CWT module, see Figures 1 and 2). Note that the ridge of the heatmap (highlighted in green) tracks the changing period of oscillation. The changing period appears in more detail in Figure 6, where the instantaneous period is estimated using several different options within WAVOS. The Fourier-transform-like nature of the Morlet CWT allows for estimation of phase but localizes the signal, thus improving the accuracy of locating associated features such as the signal peak and trough. Figure 7 shows the peak and trough points of the signal, as well as the instantaneous amplitude measurements, obtained from the phase along the ridge identified in Figure 5.

The CWT is excellent for examining local features of a signal. On the other hand, it is not well-suited for signal processing tasks such as decomposing the signal into different wavelengths, discarding or shrinking some coefficients related to particularwavelengths, and reconstructing the signal. Fortunately, the Discrete Wavelet Transform is well-suited for these tasks. Figure 8 illustrates the use of the DWT in two different modes provided by WAVOS (see Figure 3) to process the signal and remove either statistically insignificant noise, or portions of the signal outside of a particular range of frequencies. An analysis of a signal with missing data is provided in Figures 9, 10, 11, 12, 13; these figures illustrate the robustness of the CWT and DWT to significant amounts of missing data. All figures were generated using WAVOS.

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 6. 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.

As another example, we have also used WAVOS to analyze wheel-running data for wild-type mice in a 12:12 light:dark cycle (unpublished data, courtesy of the Herzog lab). The data contains wheel revolutions per minute sampled every 15 minutes for 10 days. Figure 14 shows the DWT decomposition plot; the main periodicity of roughly 24 hours is clearly visible in the 16-32 hour band, and the noise associated with waking behavior is visible in the highest frequency/shortest period bands at the bottom. The same data is analyzed in Figure 15 via CWT; the circadian periodicity of 24 hours is clearly visible, highlighted via the ridge, along with intermittent ultradian rhythms. Edge effects have been removed (black masking) to avoid potential artifacts in the analysis. Finally, Figure 16 shows peak and trough phase markers inferred automatically from the CWT (green) overlaid on the original data (blue). As above, edge-effect masking has been performed. Despite the noisy and non-sinusoidal nature of the data, the CWT accurately selects the appropriate phase markers. Figures 14, 15, 16 were created using the WAVOS GUI.

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 20). 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.