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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. ...

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## Citations

... As such, the continuous wavelet analysis can detect variations in the dominant dive cycle periodicities across the time series (Cazelles et al. 2008), or intra-individual variability in the temporal patterns of diving behaviour. Moreover, another benefit of the continuous wavelet analysis is that it can be used to evaluate a customisable range of temporal periodicities, on the order of minutes to days depending on the length of the time-series intervals and the objectives (Rösch and Schmidbauer 2018). ...

... The continuous wavelet analysis was completed using the R package WaveletComp (ver. 1.1, A. Rösch and H. Schmidbauer, see https://CRAN.R-project.org/package=WaveletComp). Based on the temporal resolution of our data, periodicities of interest and related previous work (Thorburn et al. 2019;Burke et al. 2020) parameters were set as follows: a loess span of 0, a sampling resolution of 0.5 h, a frequency resolution of 1/250, a lower period for the wavelet function of 1 h, an upper period for the wavelet function of 128 h, and the number of simulations of 100 (details in Rösch and Schmidbauer 2018). The continuous wavelet analysis returns a plot of the wavelet power spectrum, which identifies dominant dive periodicities (h; y-axis) over the duration of the tag deployment (x-axis). ...

Context: For threatened marine species, data on their vertical habitat use patterns can reveal risk of interactions with fishing gear and can inform bycatch avoidance strategies. Such data are lacking for young porbeagles (Lamna nasus), which are captured as bycatch in northwestern Atlantic fisheries. Aims: We aimed to examine temporal patterns in diving and characterise vertical habitat use of young porbeagles during summer and autumn. Methods: We used data from short-term (28-day), high-resolution (5-min interval) pop-off satellite tags attached to 14 young (young-of-the-year and 1-year-old) porbeagles to model depth use. Key results: Occupied depths ranged from the sea surface to 679 m, with ambient water temperatures of −0.2 to 26°C. Diel period and season were factors related to depth use. Conclusions: Sharks exhibited a diel activity pattern characterised by more extensive use of the water column during the day while remaining primarily at the surface at night. Depth use differed between seasons, with summer characterised by greater affinity for surface waters (0-10 m) compared to autumn. Implications: Young porbeagles are at risk of interaction with active fisheries on the continental shelf, but interactions may be reduced by setting gear deeper at night or during summer.

... y CWT de la biblioteca R WaveletComp (https://cran.r-project. org/web/packages/WaveletComp/index.html)[30]. ...

La concentración de clorofila-a en los océanos es un indicador confiable de biomasa de fitoplancton que desempeña un papel importante en el control del ecosistema marino. El objetivo principal de este estudio es analizar la variabilidad de la concentración de clorofila-a (Chl-a) y temperatura superficial del mar (TSM) en el ecosistema de afloramiento peruano en la escala de tiempo interanual, utilizando información satelital del sensor MODIS a bordo del satélite Aqua en el periodo de 2003 hasta 2021. El área de estudio está delimitada por la isobata de 1mg m−3 de clorofila-a y las latitudes de 5°S y 20°S, esta área se divide en dos zonas, norte-centro (5°S-16°S) y sur (16°S-20°S). Iniciamos investigando las tendencias lineales, donde los resultados indicaron que se produjeron tendencias negativas de los dos parámetros en el área de estudio, notando que para la TSM es mayor este enfriamiento en la zona norte-centro (≈-0.076°C década−1), en el caso de la Chl-a también muestra un mayor decrecimiento en la zona sur (≈-0.07 mg m−3 década−1). Para determinar el periodo de variabilidad interanual de cada parámetro usamos el análisis del espectro de potencia de wavelet (EPW) y para determinar el periodo común de los dos parámetros usamos la transformada de ondas cruzadas (CWT), ambos análisis son basados en la transformada wavelet, aplicamos estos análisis a la componente principal (PC1) de cada parámetro. Los resultados permitieron identificar que en la zona norte-centro el periodo significativo de cada uno de los parámetros es mayor a 3 años y en la zona sur los periodos significativos también son mayores a 3 años. En cuanto al periodo común de variabilidad interanual de los parámetros en la zona norte-centro es de 3.4 años y en la zona sur el periodo común es de 3.3 años. Estos periodos de variabilidad interanual (>3 años) de la TSM y Chl-a en toda el área de estudio son principalmente relacionados al ENSO, estos periodos tienen mayor energía en la zona norte-centro y menor energía en la zona sur esto estaría en concordancia con la disminución en amplitud y energía de la onda de Kelvin a medida que se propaga a lo largo de la costa peruana.

... In the present study, only the positive significant signals (α = 0.01) against a thousand first-order autoregressive AR(1) surrogate time series were considered [61]. Periods of <20 h were highpassed. ...

... The maximum global wavelet power, which is the average cross-wavelet power in the frequency domain (averages over time) [61]; 3. ...

... The calculations were conducted in R 4.3.2 [63] using gstat for mapping and the WaveletComp package for wavelet spectrum and coherence analysis [61]. The grids were obtained with the sp and raster packages, and the mapping was performed in Golden Software Surfer 11 and CorelDraw 2021. ...

Groundwater flow systems are influenced by the changes in surface waters as well as climatic factors. These teleconnections significantly increase in cases of extreme weather conditions. To prepare and mitigate the effect of such phenomena, the background factors that create and influence natural processes must be recognized. In the present study, 94 shallow groundwater (SGW) wells’ water level time series were analyzed in the inner delta of the River Danube (Europe) the Szigetköz region to explore which factors contribute to the development of diurnal periodicity of SGW and what its drivers are. The relationship between surface meteorological processes and SGW dynamics in the Szigetköz region was investigated using hourly data from monitoring wells. Hourly water temperature data exhibited weak correlations with meteorological parameters. However, daily averaged data revealed stronger correlations, particularly between SGW levels and air temperature and potential evapotranspiration. Diurnal periodicity in SGW fluctuations correlated strongly with potential evapotranspiration. The study also demonstrated the role of capillary fringe dynamics in linking surface evapotranspiration with SGW fluctuations. Changes in groundwater levels, even small, can significantly affect soil moisture, vegetation, and ecosystem functioning, highlighting the sensitivity of the unsaturated zone to SGW fluctuations driven by surface processes.

... The wavelet analysis uses the implementation from package WaveletComp [31], where the Morlet function [32] is the basis function translated and scaled, given by ...

The evidence of seasonal patterns in malaria epidemiology in the Brazilian Amazon basin indicates the need for a thorough investigation of seasonality in this last and heterogeneous region. Additionally, since these patterns are linked to climate variables, malaria models should also incorporate them. This study applies wavelet analysis to incidence data from 2003 to 2020 in the Epidemiological Surveillance System for Malaria (SIVEP-Malaria) database. A mathematical model with climate-dependent parametrization is proposed to study counts of malaria cases over time based on notification data, temperature and rainfall. The wavelet analysis reveals marked seasonality in states Amazonas and Amapá throughout the study period, and from 2003 to 2012 in Pará. However, these patterns are not as marked in other states such as Acre and Pará in more recent years. The wavelet coherency analysis indicates a strong association between incidence and temperature, especially for the municipalities of Macapá and Manaus, and a similar association for rainfall. The mathematical model fits well with the observed temporal trends in both municipalities. Studies on climate-dependent mathematical models provide a good assessment of the baseline epidemiology of malaria. Additionally, the understanding of seasonality effects and the application of models have great potential as tools for studying interventions for malaria control.

... A 95% confidence level for wavelet analysis was conducted through a Monte Carlo Simulation. Wavelet analysis was done using the 'WaveletComp' package (Schmidbauer & Roesch, 2018) ...

... A 95% confidence level for the CWT was done through Monte-Carlo simulation using 1000 times. In this study, wavelet analysis was done using 'WaveletComp' package (Schmidbauer & Roesch, 2018) in R (R Core Team, 2019). ...

Increases in dissolved organic carbon (DOC) concentrations, or browning,
have occurred in many freshwater ecosystems across Europe and North
America in recent decades, giving the water a brownish color. Several
mechanisms have been proposed to explain aquatic browning, but
consensus regarding the relative importance of recovery from acid
deposition, climate change, and land management remains elusive.
Meanwhile, elevated CO2, longer growing seasons and permafrost thaw
induced by climate change have increased terrestrial productivity, which
may alter the export of DOC from terrestrial to aquatic ecosystems.
Although the link between terrestrial greening and aquatic browning has
recently gained more attention, it is not yet well established. Moreover,
heterogeneities of browning across space and time have been observed, but
the reasons behind these differences are still unclear.
The objectives of this thesis are to understand how varied factors affect
short–term variations and long–term trends of DOC concentrations in boreal
catchments, focusing especially on the nexus between terrestrial carbon (C) export and stream DOC concentrations, and the causes of spatial and
temporal heterogeneity in browning across boreal catchments. The thesis is
based on three studies. Study I developed methods to predict real–time
high-frequency DOC concentrations in catchments using in–situ UV–Vis
spectrophotometers, which can be applied especially in remote areas to
save time and money. Study II used the methods developed by Study I to
predict high–resolution time series DOC concentrations and analysed the
relationship between terrestrial productivity and short–term DOC variations
across boreal catchments. Study III extended the analysis to longer time
periods and wider range catchments by integrated mechanisms (recovery
from acid deposition, climate change, and site characteristics), quantified
the contributions of multiple drivers on long–term DOC trends and provided
an explanation for the spatiotemporal heterogeneity of browning.
Collectively, this thesis revealed that increased terrestrial productivity
induced by climate change can alter terrestrial DOC exports to aquatic
ecosystems through priming effect. Whereas no single mechanism can fully
explain long–term DOC trends; instead, recovery from sulfate deposition,
terrestrial productivity, discharge, and temperature jointly shaped DOC
trends. Site characteristics (catchment size and land cover type) can affect
the response rate of DOC to these drivers leading to the spatial
heterogeneity of browning across sub–catchments. Moreover, browning has
weakened in the last decade as sulfate deposition has fully recovered and
other current drivers are insufficient to sustain the long–term DOC trends.
My work improves our mechanistic understanding of surface water DOC
regulation in boreal catchments, confirms the importance of DOC fluxes in
regulating ecosystem C budgets and highlights the significance of
considering multifaceted, spatially structured, and nonstationary drivers
when predicting long–term DOC trends in the future.

... The difference in phase between two time-series data depicts an in-phase oscillation pattern of two time-series data if they are matched with each other; in contrast, the difference in phase depicts an out-of-phase pattern if the two time-series data do not match (Grinsted et al., 2004). The lagging or the leading tendencies of the variables are indicated through the arrows (Schmidbauer and Roesch, 2018). The coherence between the variables was evaluated at the p-value of 0.05 and was incorporated in the cone of influence (Pal and Devara, 2012). ...

... The coherence between the variables was evaluated at the p-value of 0.05 and was incorporated in the cone of influence (Pal and Devara, 2012). The entire process of wavelet analysis and its graphical expression was performed in RStudio using the WaveletComp package (Schmidbauer and Roesch, 2018). To detect the presence or absence of multicollinearity among climatic variables, the variance inflation factor (VIF) was obtained in RStudio using the faraway package (Faraway, 2022). ...

... The maximum CWP level was observed after 2010 and in the period segment of the 11th to the 12th month period in a discrete manner. The arrows in the segments toward the right (fifth to the seventh month period) and top-right (ninth to the 12th month period) directions indicate the in-phase cyclicity of evapotranspiration and NDVI (Schmidbauer and Roesch, 2018;Liu et al., 2023). In the case of the ninth to the 12th month period, the segment was associated with the situation when the mean NDVI was leading and evapotranspiration was lagging. ...

An arid climate is a unique condition that has a significant impact on the growth of crops and natural vegetation. The normalized difference vegetation index (NDVI) is a crucial remotely sensed measurement of greenness due to its strong correlation with crop and vegetation growth and productivity. In the present study, the spatiotemporal dynamics of NDVI were analyzed from 2000 to 2021 in the segment of the arid western plain zone of Rajasthan, India. NDVI time-series data, as well as data related to climatic factors, viz., precipitation, soil moisture, evapotranspiration, and 2-m air temperature, were collected from Giovanni, the Goddard Earth Science dataset. The Mann–Kendall (MK) trend test and Sen’s slope depicted the long-term continuous time–frequency trend, while Karl Pearson’s correlation analysis depicted the significant relationship between all the factors except 2-m air temperature. The seasonal and mean monthly results of all the factors except 2-m air temperature showed considerable coherence with NDVI. The multiscale time–frequency decomposition or wavelet analysis depicted the fifth to the seventh month and the ninth to the 15th month of the cycle, showing the significance of the cropping pattern and the natural vegetation growth cycle. The cross-wavelet analysis further depicted important coherence, leading, and lagging phases among climatic factors and NDVI. Our research provided significant insights into the long-term variability and coherence of various climatic factors with NDVI that are applicable on regional and global scales.

... For this analysis, we use the R-packages WaveletComp [37] that analyze the frequency structure of uni-and bivariate time series using the Morlet mother wavelet [33,35] ...

... This leads to a continuous complex-valued wavelet transform that can be separated into its real and imaginary parts, providing information on both local amplitude and instantaneous phase of any periodic process across time, a prerequisite for investigating multiple time series coherence [37]. ...

... The concepts of cross-wavelet analysis provide a tool for (i) comparing the frequency of twotime series, (ii) concluding the series' synchronicity at specific frequencies and across certain ranges of time [37]. Its modulus can be interpreted as cross-wavelet power. ...

Dengue transmission poses significant challenges for public health authorities worldwide due to its susceptibility to various factors, including environmental and climate variability, affecting its incidence and geographic spread. This study focuses on Costa Rica, a country characterized by diverse microclimates nearby, where dengue has been endemic since its introduction in 1993. Using wavelet coherence and clustering analysis, we performed a time-series analysis to uncover the intricate connections between climate, local environmental factors, and dengue occurrences. The findings indicate that multiannual dengue frequency (3 yr) is correlated with the Oceanic Niño Index and the Tropical North Atlantic Index. This association is particularly prominent in cantons located along the North and South Pacific Coast, as well as in the Central cantons of the country. Furthermore, the time series of these climate indices exhibit a leading phase of approximately nine months ahead of dengue cases. Additionally, the clustering analysis uncovers non-contiguous groups of cantons that exhibit similar correlation patterns, irrespective of their proximity or adjacency. This highlights the significance of climate factors in influencing dengue dynamics across diverse regions, regardless of spatial closeness or distance between them. On the other hand, the annual dengue frequency was correlated with local environmental indices. A persistent correlation between dengue cases and local environmental variables is observed over time in the North Pacific and the Central Region of the country’s Northwest, with environmental factors leading by less than three months. These findings contribute to understanding dengue transmission’s spatial and temporal dynamics in Costa Rica, highlighting the importance of climate and local environmental factors in dengue surveillance and control efforts.

... 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.

... WMT was used by R Wavelet package [21], in which the Morlet mother function is ...

... 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.