Wavelet analysis to describe biological cycles and signals of non-stationarity

By Allison Dawn, GEMM Lab Master’s student, OSU Department of Fisheries, Wildlife, and Conservation Sciences, Geospatial Ecology of Marine Megafauna Lab 

During my second term of graduate school, I have been preparing to write my research proposal. The last two months have been an inspiring process of deep literature dives and brainstorming sessions with my mentors. As I discussed in my last blog, I am interested in questions related to pattern and scale (fine vs. mesoscale) in the context of the Pacific Coast Feeding Group (PCFG) of gray whales, their zooplankton prey, and local environmental variables.

My work currently involves exploring which scales of pattern are important in these trophic relationships and whether the dominant scale of a pattern changes over time or space. I have researched which analysis tools would be most appropriate to analyze ecological time series data, like the impressive long-term dataset the GEMM lab has collected in Port Orford as part of the TOPAZ  project, where we have monitored the abundance of whales and zooplankton, as well as environmental variables since 2016. 

A useful analytical tool that I have come across in my recent coursework and literature review is called wavelet analysis. Importantly, wavelet analysis can handle non-stationarity and edge detection in time series data. Non-stationarity is when a dataset’s mean and/or variance can change over time or space, and edge detection is the identification of the change location (in time or space). For example, it is not just the cycles or “ups and downs” of zooplankton abundance I am interested in, but when in time or where in space these cycles of “ups and downs” might change in relation to what their previous values, or distances between values, were. Simply stated, non-stationarity is when what once was normal is no longer normal. Wavelet analysis has been applied across a broad range of fields, such as environmental engineering (Salas et al. 2020), climate science (Slater et al. 2021), and bio-acoustics (Buchan et al. 2021). It can be applied to any time series dataset that might violate the traditional statistical assumption of stationarity. 

In a recent review of climate science methodology, Slater et al. (2021) outlined the possible behavior of time series data. Using theoretical plots, the authors show that data can a) have the same mean and variance over time, or b) have non-stationarity that can be broken into three major groups – trend, step change, or shifts in variance. Figure 1 further demonstrates the difference between stationary vs. non-stationary data in relation to a given variable of interest over time. 

Figure 1. Plots showing the possible magnitude of a given variable across a time series: a) Stationary behavior, b) Non-stationary trend, step-change, and a shift in variance. [Taken from Slater et. al (2021)].

Traditional correlation statistics assumes stationarity, but it has been shown that ecological time series are often non-stationary at certain scales (Cazelles & Hales, 2006). In fact, ecological data rarely meets the requirements of a controlled experiment that traditional statistics require. This non-stationarity of ecological data means that while widely-used methods like generalized linear models and analyses of variances (ANOVAs) can be helpful to assess correlation, they are not always sufficient on their own to describe the complex natural phenomena ecologists seek to explain. Non-stationarity occurs frequently in ecological time series, so it is appropriate to consider analysis tools that will allow us to detect edges to further investigate the cause.

Wavelet analysis can also be conducted across a time series of multiple response variables to assess if these variables share high common power (correlation). When data is combined in this way it is called a cross-wavelet analysis. An interesting paper used cross-wavelet analysis to assess the seasonal response of zooplankton life history in relation to climate warming (Winder et. al 2009). Results from their cross-wavelet analysis showed that warming temperatures over the past two decades increased the voltinism (number of broods per year) of copepods. The authors show that where once annual recruitment followed a fairly stationary pattern, climate warming has contributed to a much more stochastic pattern of zooplankton abundance. From these results, the authors contribute to the hypothesis that climate change has had a temporal impact on zooplankton population dynamics, and recruitment has increasingly drifted out of phase from the original annual cycles. 

Figure 2. Cross-wavelet spectrum for immature and adult Leptodiaptomus ashlandi for 1965 through either 2000 or 2005. Plots show a) immatures and temperature, b) adults and temperature, c) immatures and phytoplankton, and d) adults and phytoplankton. Arrows indicate phase between combined time series. 0 degrees is in-phase and 180 degrees is anti-phase. Black contour lines show “cone of influence” or the 95% significance level, every value within the cone is considered significant. Left axis shows the temporal period, and the color legend shows wavelet frequency power, with low frequencies in blue and high frequencies in red. Plots show strong covariation of high common power at the 12-month period until the 1980s. This pattern is especially evident in plot c) and d). [Taken from (Winder et. al 2009)].

While wavelet and cross-wavelet analyses should not be the only tool used to explore data, due to its limitations with significance testing, it is still worth implementing to gain a better understanding of how time series variables relate to each other over multiple spatial and/or temporal scales. It is often helpful to combine multiple methods of analysis to get a larger sense of patterns in the data, especially in spatio-temporal research.

When conducting research within the context of climate change, where the concentration of CO2 in ppm in the atmosphere is a non-stationary time series itself (Figure 3), it is important to consider how our datasets might be impacted by climate change and wavelet analysis can help identify the scales of change. 

Figure 3. Plot showing the historic fluctuations of CO^2 and the recent deviation from normal levels. Source: https://globalclimate.ucr.edu/resources.html

When considering our ecological time series of data in Port Orford, we want to evaluate how changing ocean conditions may be related to data trends. For example, has the annual mean or variance of zooplankton abundance changed over time, and where has that change occurred in time or space? These changes might have occurred at different scales and might be invisible at other scales. I am eager to see if wavelet analysis can detect these sorts of changes in the abundance of zooplankton across our time series of data, particularly during the seasons of intense heat waves or upwelling. 

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References

Buchan, S. J., Pérez-Santos, I., Narváez, D., Castro, L., Stafford, K. M., Baumgartner, M. F., … & Neira, S. (2021). Intraseasonal variation in southeast Pacific blue whale acoustic presence, zooplankton backscatter, and oceanographic variables on a feeding ground in Northern Chilean Patagonia. Progress in Oceanography, 199, 102709.

Cazelles, B., & Hales, S. (2006). Infectious diseases, climate influences, and nonstationarity. PLoS Medicine, 3(8), e328.

Salas, J. D., Anderson, M. L., Papalexiou, S. M., & Frances, F. (2020). PMP and climate variability and change: a review. Journal of Hydrologic Engineering, 25(12), 03120002.

Slater, L. J., Anderson, B., Buechel, M., Dadson, S., Han, S., Harrigan, S., … & Wilby, R. L. (2021). Nonstationary weather and water extremes: a review of methods for their detection, attribution, and management. Hydrology and Earth System Sciences, 25(7), 3897-3935.

Winder, M., Schindler, D. E., Essington, T. E., & Litt, A. H. (2009). Disrupted seasonal clockwork in the population dynamics of a freshwater copepod by climate warming. Limnology and Oceanography, 54(6part2), 2493-2505.

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