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Many real‐world scientific processes are governed by complex non‐linear dynamic systems that can be represented by differential equations. Recently, there has been an increased interest in learning, or discovering, the forms of the equations driving these complex non‐linear dynamic systems using data‐driven approaches. In this paper, we review the current literature on data‐driven discovery for dynamic systems. We provide a categorisation to the different approaches for data‐driven discovery and a unified mathematical framework to show the relationship between the approaches. Importantly, we discuss the role of statistics in the data‐driven discovery field, describe a possible approach by which the problem can be cast in a statistical framework and provide avenues for future work.

 

Poikilothermic animals comprise most species on Earth and are especially sensitive to changes in environmental temperatures. Species conservation in a changing climate relies upon predictions of species responses to future conditions, yet predicting species responses to climate change when temperatures exceed the bounds of observed data is fraught with challenges. We present a physiologically guided abundance (PGA) model that combines observations of species abundance and environmental conditions with laboratory-derived data on the physiological response of poikilotherms to temperature to predict species geographical distributions and abundance in response to climate change. The model incorporates uncertainty in laboratory-derived thermal response curves and provides estimates of thermal habitat suitability and extinction probability based on site-specific conditions. We show that temperature-driven changes in distributions, local extinction, and abundance of cold, cool, and warm-adapted species vary substantially when physiological information is incorporated. Notably, cold-adapted species were predicted by the PGA model to be extirpated in 61% of locations that they currently inhabit, while extirpation was never predicted by a correlative niche model. Failure to account for species-specific physiological constraints could lead to unrealistic predictions under a warming climate, including underestimates of local extirpation for cold-adapted species near the edges of their climate niche space and overoptimistic predictions of warm-adapted species. 

Statistical methods are required to evaluate and quantify the uncertainty in environmental processes, such as land and sea surface temperature, in a changing climate. Typically, annual harmonics are used to characterize the variation in the seasonal temperature cycle. However, an often overlooked feature of the climate seasonal cycle is the semi‐annual harmonic, which can account for a significant portion of the variance of the seasonal cycle and varies in amplitude and phase across space. Together, the spatial variation in the annual and semi‐annual harmonics can play an important role in driving processes that are tied to seasonality (e.g., ecological and agricultural processes). We propose a multivariate spatiotemporal model to quantify the spatial and temporal change in minimum and maximum temperature seasonal cycles as a function of the annual and semi‐annual harmonics. Our approach captures spatial dependence, temporal dynamics, and multivariate dependence of these harmonics through spatially and temporally varying coefficients. We apply the model to minimum and maximum temperature over North American for the years 1979–2018. Formal model inference within the Bayesian paradigm enables the identification of regions experiencing significant changes in minimum and maximum temperature seasonal cycles due to the relative effects of changes in the two harmonics.

 

There are multiple protocols for determining total nitrogen (TN) in water, but most can be grouped into direct approaches (TN‐d) that convert N forms to nitrogen‐oxides (NO_x) and combined approaches (TN‐c) that combine Kjeldahl N (organic N +NH₃) and nitrite+nitrate (NO₂+NO₃‐N). TN concentrations from these two approaches are routinely treated as equal in studies that use data derived from multiple sources (i.e., integrated data sets) despite the distinct chemistries of the two methods. We used two integrated data sets to determine if TN‐c and TN‐d results were interchangeable. Accuracy, determined as the difference between reported concentrations and the most probable value (MPV) of reference samples, was high and similar in magnitude (within 3.5–4.5% of the MPV) for both methods, although the bias was significantly smaller at low concentrations for TN‐d. Detection limits and data flagged as below detection suggested greater sensitivity for TN‐d for one data set, while patterns from the other data set were ambiguous. TN‐c results were more variable (less precise) by many measures, although TN‐d data included a small fraction of notably inaccurate results. Precision of TN‐c was further compromised by propagated error, which may not be acknowledged or detectable in integrated data sets unless complete metadata are available and inspected. Finally, concurrent measures of TN‐c and TN‐d in lake samples were extremely similar. Overall, TN‐d tended to be slightly more accurate and precise, but similarities in accuracy and the near 1 : 1 relationship for concurrent TN‐d and TN‐c measurements support careful use of data interchangeably in analyses of heterogeneous, integrated data sets.

 

Aquatic scientists require robust, accurate information about nutrient concentrations and indicators of algal biomass in unsampled lakes in order to understand and predict the effects of global climate and land‐use change. Historically, lake and landscape characteristics have been used as predictor variables in regression models to generate nutrient predictions, but often with significant uncertainty. An alternative approach to improve predictions is to leverage the observed relationship between water clarity and nutrients, which is possible because water clarity is more commonly measured than lake nutrients. We used a joint‐nutrient model that conditioned predictions of total phosphorus, nitrogen, and chlorophyll aon observed water clarity. Our results demonstrated substantial reductions (8–27%; median = 23%) in prediction error when conditioning on water clarity. These models will provide new opportunities for predicting nutrient concentrations of unsampled lakes across broad spatial scales with reduced uncertainty.