Figure 2 - uploaded by Harald Schmidbauer

Content may be subject to copyright.

Source publication

WaveletComp is an R package for continuous wavelet-based analysis of univariate and bivariate time series. Wavelet functions are implemented in WaveletComp such that a wide range of intermediate and final results are easily accessible. The null hypothesis that there is no (joint) periodicity in the series is tested via p-values obtained from simula...

## Contexts in source publication

**Context 1**

... absolute value less (larger) than π/2 indicates that the two series move in phase (anti-phase, respectively) referring to the instantaneous time as time origin and at the frequency (or period) in question, while the sign of the phase difference shows which series is the leading one in this relationship. Figure 2 (in the style of a diagram by Aguiar-Conraria and Soares [2]) illustrates the range of possible phase differences and their interpretation. In line with this style, phase differences are displayed as arrows in the image plot of cross-wavelet power. ...

**Context 2**

... 250000 (the latter was not found to have an adverse effect on file sizes). A grayscale can be convenient for black-and-white printout; it is easy to realize in WaveletComp (see Figure 20): ...

**Context 3**

... following code constructs a data frame with all weekdays of 2018 and a time series with weekly (low on Mondays, high on Fridays) and monthly (for simplicity, we let a month have 22 workdays) seasonality: The result is shown in Figure 23. So far, so good. ...

**Context 4**

... far, so good. Now suppose the company is closed for vacation from the beginning of July to mid-August, so that the time series is interrupted, retaining only what is outside of the shaded interval in Figure 22: The code above, with my.data.part substituted for my.data, then leads to the result shown in Figure 24. ...

**Context 5**

... suppose the company is closed for vacation from the beginning of July to mid-August, so that the time series is interrupted, retaining only what is outside of the shaded interval in Figure 22: The code above, with my.data.part substituted for my.data, then leads to the result shown in Figure 24. There are two problems here, due to the data interruption: ...

**Context 6**

... The plot looks as if there were a structural break -although the underlying data-generating model (and even the very realization to be transformed) is the same in Figures 23 and 24. ...

**Context 7**

... The time axis in Figure 24 pretends that data were available throughout July and August. To be sure, fixing the time axis won't make the plot more meaningful. ...

**Context 8**

... is, of course, true for all the tools of statistical analysis.) The idea here is that x and y (see Figure 25) represent hourly observations from a 96-day interval. The time axis labels of Figure 25 simply count through the observations; for certain applications, it may be more intuitive to display days (see also Section 2.7). ...

**Context 9**

... idea here is that x and y (see Figure 25) represent hourly observations from a 96-day interval. The time axis labels of Figure 25 simply count through the observations; for certain applications, it may be more intuitive to display days (see also Section 2.7). Series x and y have joint constant periods 1, 2, 4, 8, 16, but different amplitudes at period 16. ...

**Context 10**

... produces the plot in Figure 26. Horizontal arrows pointing to the right indicate that the two series x and y are in phase at the respective period with vanishing phase differences. ...

**Context 11**

... is also possible to define siglvl as a vector.) Figure 27 confirms that the rectified version of cross-wavelet power gives sound results for all periods. (Omitting rectification would severely underestimate the lower periods' power.) ...

**Context 12**

... time series x shares period 128 with series y within a certain range of time, and y is shifted: The result, which is somewhat counterintuitive, is displayed in Figure 29. Period 128 shows significance across the entire time interval, while one expects period 128 to be jointly important only in the middle, according to the construction of x and y. ...

**Context 13**

... 128 shows significance across the entire time interval, while one expects period 128 to be jointly important only in the middle, according to the construction of x and y. The arrows indicate that x and y are in phase in the middle, with x leading (see also Figure 2), and they tilt away off the middle. This example illustrates the dilemma of the cross-wavelet power (see also the comment in Section 1.2), which corresponds to the covariance - it can be large even if only one component swings widely. ...

**Context 14**

... Limit the area where arrows are drawn to the region where both individual wavelet transforms of x and y show significance (set which.arrow.sig = "wt"), and avoid the artifacts of the image in Figure 29 resulting from the steep power gradient by choosing color.key = "interval": ...

**Context 15**

... computation time, but no arrows indicating phase differences will be plotted.) Figure 32 results again from smoothing with Bartlett windows, but window.size.s = 1 now defines a window of length 101 (an even number will be increased by 1) and produces more blurring for low periods (high frequencies): This leads to less granularity in high frequency areas. ...

**Context 16**

... resulting wavelet power spectrum is shown in Figure 42. A cluttered plot will appear with option plot.waves ...

**Context 17**

... plots in Figure 53 are not quite satisfactory: The three series' wavelet transforms have different maximum powers (as might have been guessed from Figure 52). The plots in Figure 53 thus cannot be directly compared. ...

**Context 19**

... on period 365 days, the arrows in the cross-wavelet power spectra in Figure 56 can then be interpreted with the help of Figure 2 (and remembering that a full circle corresponds to one year): ...

**Context 20**

... arrows (see also Figure 2) now reveal that both pairs under scrutiny are more or less in sync at the 24-hour period, but there is a very important difference between the pairs at the 12-hour period: Trump pos/neutral is leading over Clinton pos/neutral, while Clinton neg is leading over Trump neg most of the time. It looks like Trump supporters and Clinton opponents were eager to post media, while Trump opponents and Clinton supporters were sluggish. ...

## Citations

... The overall data processing and analysis was done in R environment. The wavelet analysis was done using WaveletComp package [11]. The time series was detrended before the application of wavelet transformation. ...

The characterization of the cyclical nature of semiconductor industry is a complex endeavor because of the presence of many interacting transient dynamics inherent in the industry's ecosystem. In this paper we present a methodology that addresses some of the issues, particularly the non-stationarity of the time series associated with the semiconductor industry. We use singular spectrum analysis to de-noise data before identifying the dominant pattern of the semiconductor stock market using singular value decomposition. By using continuous wavelet transformation and cross-wavelet coherence relation, the nexus between the dominant pattern of the stock market and the industrial production index of semiconductor is established. Using a bootstrap resampling method, statistically significant frequencies that characterize the cyclical nature of the semiconductor industry are identified.

... We therefore calculated a Fourier decomposition of each timeseries in addition to a time-averaged wavelet decomposition of each timeseries, meaning we obtained two semi-redundant frequency representations of our data. However, because the wavelet transform uses a differently-shaped "mother wavelet," and because it is not perfectly information-preserving when averaged across time, it will have some differences (Schmidbauer & Roesch, 2018;Torrence & Compo, 1998). We found that some information loss was advantageous because it reduced noise that otherwise obscured patterns in the data. ...

River flows change on timescales ranging from minutes to millennia. These vibrations in flow are tuned by diverse factors globally, for example, by dams suppressing multi‐day variability or vegetation attenuating flood peaks in some ecosystems. The relative importance of the physical, biological, and human factors influencing flow is an active area of research, as is the related question of finding a common language for describing overall flow regime. Here, we addressed both topics using a daily river discharge data set for over 3,000 stations across the globe from 1988 to 2016. We first studied similarities between common flow regime quantification methods, including traditional flow metrics, wavelets, and Fourier analysis. Across all these methods, the flow data showed low‐dimensional structure (i.e., simple and consistent patterns), suggesting that fundamental mechanisms are constraining flow regime. One such pattern was that day‐to‐day variability was negatively correlated with year‐to‐year variability. Additionally, the low‐dimensional structure in river flow data correlated closely with only a small number of catchment characteristics, including catchment area, precipitation, and temperature—but notably not biome, dam surface area, or number of dams. We discuss these findings in a framework intended to be accessible to the many communities engaged in river research and management, while stressing the importance of letting structure in data guide both mechanistic inference and interdisciplinary discussion.

... The luminescence intensities were quantified by continuous wavelet transformation techniques, which are implemented into Wavelet Comp package in R (Rösch and Schmidbauer, 2016) for detecting circadian rhythm (CR) and ultradian rhythm (UR). The rhythms were detected by periodic parameters of 2 to 6 hours for UR and 16 to 32 hours for CR. ...

Ultradian rhythms have been proved to be critical for diverse biological processes. However, comprehensive understanding of the short-period rhythms remains limited. Here, we discover that leaf excision triggers a gene expression rhythm with ~3-h periodicity, named as the excision ultradian rhythm (UR), which is regulated by the plant hormone auxin. Promoter–luciferase analyses showed that the spatiotemporal patterns of the excision UR were positively associated with de novo root regeneration (DNRR), a post-embryonic developmental process. Transcriptomic analysis indicated more than 4,000 genes including DNRR-associated genes were reprogramed toward ultradian oscillation. Genetic studies showed that EXCISION ULTRADIAN RHYTHM 1 (EUR1) encoding ENHANCER OF ABSCISIC ACID CO-RECEPTOR1 (EAR1), an abscisic acid signaling regulator, was required to generate the excision ultradian rhythm and enhance root regeneration. The eur1 mutant exhibited the absence of auxin-induced excision UR generation and partial failure during rescuing root regeneration. Our results demonstrate a link between the excision UR and adventitious root formation via EAR1/EUR1 , implying an additional regulatory layer in plant regeneration.

... The coherence relationships outside or overlapping the cone of influence should be approached prudently, though falling within black contour lines. Because they may be influenced by edge effects that stem from the fact that wavelet is not completely localized in time [132][133][134][135][136]. In wavelet plots, horizontal and vertical axes illustrate frequency (scale) and time (years). ...

... We have used the continuous wavelet transform method for our study, as it can reveal the features under the multi-temporal scale (Sang, 2013). In the study we used the WaveletComp package in R (Schmidbauer and Roesch, 2018) apply the Morlet (1982) wavelet to transform the discharge signals: ...

The overall theme of the 40th International School of Hydraulics is ADVANCES IN HYDRAULIC RESEARCH. The conference intends to provide a forum for scientists and engineers working in the field of broadly understood hydraulics. Bringing together experts (academics and practitioners) and young scientists creates a very good atmosphere for scientific debate and learning.
The book contains the abstract of the research work prepared for presentation during the 40th International School of Hydraulics, including lectures, oral presentations and posters.

... Data were tested for seasonality by Edward's model, [13] using Episheet, [14] which is freely available Excel sheet for various statistical calculations. Time series data were decomposed using local polynomial regression, Wavelet analyses was performed, and a wavelet power spectrum was plotted using the package WaveletComp [15] in R Software version 3.1. [16] ...

Background: Bacterial meningitis is one of the most feared infectious diseases, being an important cause of death and
long-term neurological disability. Various studies have observed seasonality of occurrence of bacterial meningitis most
commonly in dry and cold seasons. Studies of seasonal variation contribute to understanding the etiology of infections,
health-care planning for prevention, and rational use of hospital resources. Objectives: The objectives of the study were to
know the seasonal variations and trend in the admissions of community-acquired acute bacterial meningitis in Karnataka
Institute of Medical Sciences (KIMS) Hospital. Methods: A time series analysis by compiling monthly data of bacterial
meningitis cases admitted in KIMS, Hubli, from the reports of medical records department for a period of 5 years from
January 2014 to December 2018. Results: The peaks for hospitalizations were predominantly in March–May and even
the seasonality for aggregate years occurs as peak effect in summer. Hospitalizations for bacterial meningitis showed a
periodicity of 3–4 months in each year along with cyclical trend over the years studied. Conclusion: This study found
evidence of seasonality for bacterial meningitis, thereby suggesting that hospitals should be prepared for providing clinical
services round the year; however, more resources may be required during the peak months of March–May.

... However, the coherence relationships outside or overlapping the cone of influence should be approached prudently, though falling within black contour lines. Because they may be influenced by edge effects that stem from the fact that wavelet is not completely localized in time [86][87][88][89][90]. ...

... The coherence relationships outside or overlapping the cone of influence should be approached prudently, though falling within black contour lines. Because they may be influenced by edge effects that stem from the fact that wavelet is not completely localized in time [86][87][88][89][90]. In wavelet plots, horizontal and vertical axes illustrate frequency (scale) and time (years). ...

... Null distributions for correlations were generated by permuting the starting point of the wavelet-transformed climate time series while maintaining periodic boundaries (i.e., adding climate values from before the randomly chosen staring points to the end of the permuted climate series) and calculating the Pearson correlation between the randomized climate wavelet and the WMR (n = 1000, with 70 or 50 unique possible values for Yasuní and Cocha Cashu, respectively, and ignoring observations within one period of the boundary). The wavelet transform of climate was done in the WaveletComp R package (Rösch & Schmidbauer, 2016). ...

Phenology has long been hypothesized as an avenue for niche partitioning or interspecific facilitation, both promoting species coexistence. Tropical plant communities exhibit striking diversity in reproductive phenology, but many are also noted for large synchronous reproductive events. Here we study whether the phenology of seed fall in such communities is non‐random, what are the temporal scales of phenological patterns, and ecological factors that drive reproductive phenology. We applied multivariate wavelet analyses to test for phenological synchrony versus compensatory dynamics (i.e. anti‐synchronous patterns where one species’ decline is compensated by the rise of another) among species and across temporal scales. We used data from long‐term seed rain monitoring of hyperdiverse plant communities in the western Amazon. We found significant synchronous whole‐community phenology at multiple time scales, consistent with shared environmental responses or positive interactions among species. We also observed both compensatory and synchronous phenology within groups of species (confamilials) likely to share traits and seed dispersal mechanisms. Wind‐dispersed species exhibited significant synchrony at ~6 mo scales, suggesting these species might share phenological niches to match seasonality of wind. Our results suggest that community phenology is shaped by shared environmental responses but that the diversity of tropical plant phenology may partly result from temporal niche partitioning. The scale‐specificity and time‐localized nature of community phenology patterns highlights the importance of multiple and shifting drivers of phenology.

... Over the course of one year (May 2021-May 2022), we examined four of the most representative ponds within the DNP and analyzed their hydrological functioning and relation with the aquifer. The main method used to interpret data collected in the field was through wavelet analysis [19,20], a statistical technique that has been increasingly used in a hydrogeologic context to help better understand groundwater and surface-water interactions under non-stationary assumptions [15,21]. This is because wavelet transforms can be used to discern minute changes presented throughout time, indicating the variability and complexity of aquifer systems. ...

... Measurements were recorded from May 2021 to May 2022. Analysis of the hydraulic time series was performed using wavelet analysis [32][33][34], specifically by using the WaveletComp package in R [20], which perform these calculations. Wavelet analysis utilizes the Morlet-wavelet transform, which calculates the frequency structure across time, producing an optimal time-frequency-resolution spectrogram [19,20,23]. ...

... Analysis of the hydraulic time series was performed using wavelet analysis [32][33][34], specifically by using the WaveletComp package in R [20], which perform these calculations. Wavelet analysis utilizes the Morlet-wavelet transform, which calculates the frequency structure across time, producing an optimal time-frequency-resolution spectrogram [19,20,23]. ...

The Doñana National Park (DNP) is a protected area with water resources drastically diminishing due to the unsustainable extraction of groundwater for agricultural irrigation and human consumption of a nearby coastal city. In this study, we explore the potential of wavelet analysis applied to high-temporal-resolution groundwater-and-surface-water time series of temporary coastal ponds in the DNP. Wavelet analysis was used to measure the frequency of changes in water levels and water temperature, both crucial to our understanding of complex hydrodynamic patterns. Results show that the temporary ponds are groundwater-dependent ecosystems of a through-flow type and are still connected to the sand-dune aquifer, regardless of their hydrological affection, due to groundwater withdrawal. These ponds, even those most affected by pumping in nearby drills, are not perched over the saturated zone. This was proven by the evidence of a semi-diurnal (i.e., 6 h) signal in the surface-level time series of the shallow temporary ponds. This signal is, at the same time, related to the influence of the tides affecting the coastal sand-dune aquifer. Finally, we detected other hydrological processes that affect the ponds, such as evaporation and evapotranspiration, with a clear diurnal (12 h) signal. The maintenance of the ecological values and services to the society of this emblematic wetland is currently in jeopardy, due to the effect of the groundwater abstraction for irrigation. The results of this study contribute to the understanding of the behavior of these fragile ecosystems of DNP, and will also contribute to sound-integrated water-resource management.

... WT has been used as a decomposition tool for analyzing streamflow time series in many studies [49]; [12,28,50,51]. The Morlet wavelet was applied to the time series, and the package WaveletComp in R [52] was used in this study. Fig. 2 The overall processes of the multi-model methodologies applied in this study. ...

Reliable long-term (decadal scale) streamflow prediction would provide significant planning information for water resources management, particularly in areas marked by significant variability at those time scales. In this study, a multi-model for prediction using four models that incorporate preprocessing methods along with data-driven forecast models coupled using the least absolute shrinkage and selection operator (LASSO) regression method is proposed. Models utilized complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and wavelet transform (WT) as the decomposition methods and autoregressive (AR) and hidden Markov models (HMM) as the predictive method. The model is evaluated in a comparative analysis with a variety of models previously proposed for hydrological time series prediction. We compare the predictive skill of alternative data-driven models for average annual streamflow (3 ~ 15 years) prediction. Results indicate that the multi-model performed better than the other models, presenting lower values of MAE and RSME. This multi-model can be a reliable tool for forecasting, which can be explored for hydrological data that have remarkably nonlinear and nonstationary features.