Open challenges in environmental data analysis and ecological complex systems^(a)

Environmental data are distributed over spatial domains of varying size and span different time periods. Hence, suitable statistical models should include both space and time dependence. In addition, they should account for the inherent uncertainty of the data and the a priori unknown complex variability of the generating process. A space-time random field $\{ X(\mathbf{s},t, \omega)$: $\mathbf{s} \in \mathcal{D} \subset \mathbb{R}^{d}, t \in T \subset \mathbb{R} \}$, where s denotes the spatial position in the domain $\mathcal{D}$ and t the time instant in the interval T, is a scalar, real-valued random function defined on a probability space $(\Omega, F, P)$; ${\Omega}$ is the sample space, $F$ is the σ-field of subspaces of ${\Omega}$, and $P$ is a probability measure [55].

Random fields were introduced in fluid turbulence studies by Kolmogorov and his students. Due to their flexible space-time (ST) dependence, random fields are widely used in various disciplines including statistical field theory, hydrology, and ST data modeling, among others [11,15,17,52–54,56,57].

The construction of Gaussian random fields follows different paths in statistical physics and in statistics: in the former the field is often formulated in the Boltzmann-Gibbs (B-G) representation by means of a suitable energy function, while in the latter the field is defined in terms of its expectation (mean) and covariance function. The B-G framework provides sparse representation and computational speed for lattice data, e.g., [58,59]. These advantages stem from explicit expressions which can be obtained for the precision (inverse covariance) matrix. On regular grids the B-G representation leads to Gauss-Markov random fields [60–62]. On the other hand, in continuum systems the B-G representation gives rise to Gaussian field theories [57,63,64]. If the latter admit closed-form solutions, they can lead to new spatial covariance functions, e.g., [65,66]. However, space-time field theories do not yield easily explicit expressions for the covariance function [67].

Random field theory provides a powerful toolbox for the interpolation, forecasting and simulation of complex ST processes. The best linear unbiased predictor (BLUP), also known as kriging, is the key tool for prediction purposes. The predictive equations at an unobserved ST point require solving an $N\times N$ linear system (where N is the number of data) which involves the field's covariance function. The computational time required for solving such systems with dense covariance matrices scales as $\mathcal{O}(N^3)$ and the memory storage requirements as $\mathcal{O}(N^2)$ [17,53,68,69]. Hence, approximations or different approaches are necessary in order to handle large datasets.

Open research questions of relevance to statistical physics include the following: 1) The development of ST covariance functions which are not only mathematically permissible but also physically meaningful, e.g., [54,67,70–72]. 2) New interpolation and simulation approaches for big datasets that reduce the computational cost, e.g., methods based on sparse precision matrices [42,44,59]. 3) The construction of more flexible B-G random fields for continuum and lattice spaces. 4) Novel, computationally tractable models for non-Gaussian dependence. For example, one possibility is offered by using the kappa exponential and logarithm functions [73] to derive κ-lognormal random (KLN) fields by transforming a latent Gaussian random field [17]. These modified lognormal fields have a probability density function (pdf) whose right tail is lighter than the lognormal's and its decay is controlled by the κ parameter. A κ-deformed Weibull marginal pdf [74] has been successfully applied to fit the long (power-law) tail of earthquake recurrence times data [75] and Covid-19 mortality data [76]. A joint pdf generalizing the κ-Weibull marginal for joint "many-body" dependence correlations would be a welcome addition. The topics outlined above are also pertinent for machine learning (see below).

ML research has captured the interest of physicists [15,69,77,78]. ML methods can "learn" from the data almost automatically (i.e., with minimal or no interaction with the modeler) and easily adapt (generalize) to new data. Such features are very appealing in the era of big data. ML methods can successfully perform complex classification tasks, and they also provide new methods for the solution of partial differential equations that represent physical processes [78–80].

Gaussian processes generalize Gaussian random fields; they represent functions $\Phi(\mathbf{x}; \omega)$, where $\mathbf{x} \in \mathbb{R}^{D}$ is an input vector in a D-dimensional space, not necessarily restricted to the space-time domain. GP regression (GPR) is an ML procedure which provides an optimal estimate of the output variable $y_{\ast}=\Phi(\mathbf{x}_{\ast}; \omega)$ based on a set of input vectors and their respective outputs $\{ \mathbf{x}_{n}, y_{n} \}_{n=1}^{N}$, where $\mathbf{x}_{\ast} \neq \mathbf{x}_{n}, \forall n =1, \ldots, N$. GPR predictive equations are similar to BLUP. The main difference is that the ST covariance used in BLUP is replaced by a kernel function in GPR; the latter measures output correlations in terms of the distance between the input vectors. The Bayesian framework allows including informed (non-flat) prior guesses for GP parameters. Hence, the mathematical machinery developed for random fields carries over nicely to GPs. Open research questions for random fields are also pertinent for GPs. In particular, the development of scalable GP models that can handle massive data is a current priority [45,81–84]. Sparse GPs are approximations which can improve the computational cost of GPs to $\mathcal{O}(N^{2}M)$ and the memory requirements to $\mathcal{O}(NM)$, where M < N [85,86]. A somewhat different approach is the development of sparse GPs based on local inverse covariance (precision) operators [17,44,66].

NN are popular ML tools for classification and regression tasks. NN are thus ideal candidates for environmental data modeling. For example, a GP regression network applied to ground pollution data gave improved statistical validation measures compared to other methods [87]. Strong links exist between NN and GP (and therefore random fields), which are little known outside the ML community. For example, it has been shown that single-layer, feed-forward Bayesian NN with an infinite number of hidden units (i.e., an infinitely wide NN) and independent, identically distributed priors over the parameters are equivalent to GPs [88]. The kernel (covariance function) of the equivalent GP can be obtained in closed form in this case [17,69]. More recently, it was also shown [43,89] that deep, infinitely wide NN are also equivalent to GPs. A computationally efficient recipe for computing the GP covariance corresponding to wide neural networks with a finite number of layers was formulated, and NN accuracy was found to approach the respective GP's accuracy with increasing layer width. Another recent contribution shows that the output of a (residual) convolutional NN with an appropriate prior over the weights and biases is equivalent to a GP in the limit of infinite depth, and that the equivalent kernel can be computed exactly [90].

ML applications in environmental data modeling will multiply in coming years. Harnessing connections between ML, statistical physics and statistical learning will lead to new advances in all of these fields.

Population dynamics is a specific branch of the dynamics of complex ecological systems, which can be considered as a foundational subfield of non-equilibrium statistical physics. Recently, population dynamics has become a crucial tool for investigating the fundamental puzzle of the emergence and stabilization of biodiversity [123]. Ecological complex systems are open, subject to random environmental perturbations, and involve nonlinear interactions between constituent parts. These systems are very sensitive to the initial conditions, deterministic external perturbations and random fluctuations. The study of far-from-equilibrium stochastic processes is crucial for modeling ecological dynamics and understanding the mechanisms which govern the spatiotemporal dynamics of ecosystems, see ref. [47]. Even low-dimensional systems exhibit a huge variety of noise-driven phenomena, ranging from less to more ordered system dynamics [47,50,51,124,125].

In systems of interacting populations, even small random disturbances can cause opposite effects, such as extinction or explosive population growth. The identification of general laws governing such stochastic phenomena and the development of constructive methods for their mathematical modeling and analysis are important tasks. The diverse and complex behavior of population systems is associated with the nonlinear nature of interacting biological factors: limited ecological niche, age and gender differences, dependence of fertility on population size, interactions with other populations, and environmental influences [126].

Random environmental fluctuations represent a major source of risk for wild populations. Hence, identifying general mechanisms linking global environmental changes with population dynamics is an important topic of research. In connection to this, there is an urgent need to understand and predict the temporal and spatial autocorrelation patterns of environmental noise.

Reference [1] shows experimentally that the environmental autocorrelation has significant impact on population dynamics and extinction rates, and the latter can be accurately predicted if the memory of the past environment is accounted for. The experiment exposed nearly 1000 lines of the microalgae Dunaliella salina to randomly fluctuating salinity, with autocorrelation ranging from negative to strongly positive values. The authors observed lower population growth and greater extinction rates for the lower autocorrelation values, thus demonstrating that non-genetic inheritance is potentially a major driver of population dynamics in randomly fluctuating environments.

Stochastic population dynamics models in spatially extended systems demonstrate the crucial role of noise and correlations in biological systems [123]. Theoretical approaches typical of non-equilibrium statistical physics capture the noisy kinetics of complex many-body biological systems and have led to unexpected new and intriguing behavior in simple paradigmatic model systems. This behavior ranges from persistent population oscillations stabilized by intrinsic noise and strong renormalization of the associated kinetic parameters induced by correlations to the emergence of continuous out-of-equilibrium phase transitions, as well as the spontaneous formation of rich spatiotemporal patterns. The spatial degrees of freedom can drastically extend extinction times through the emergence of noise-stabilized structures, and hence promote ecological stability and species diversity. For example, spatially extended predator-prey systems display noise-stabilized activity fronts that generate persistent correlations, so that the critical steady-state and non-equilibrium relaxation dynamics at the predator extinction threshold are governed by the directed percolation universality class [123]. Ecosystems display complex spatial organization. However, spatial models still present several open problems, thus limiting the quantitative understanding of spatial biodiversity and different time scales. Indeed, the connection between spatially extended ecological models and the physics of non-equilibrium phase transitions represents an open problem of paramount importance. In this context, translating concepts developed in statistical physics into language and tools applicable to population dynamics is crucial for progress [123,127]. Moreover, changes of external conditions strongly influence the dynamics and the organization of biological systems. In fact, the characteristic timescales of environmental variations as well as their correlations play a fundamental role in how living systems adapt and respond to the variability of environmental parameters. Relevant questions include the stationary population density and the role of random fluctuations on the extinction dynamics within an ecosystem [24].

The crucial role of random fluctuations is evident in cell biology. It is known that the presence of noise in intracellular processes can limit the performance of cells by preventing optimal concentration of the cell's molecular components. In contrast, noise can be used to create diversity in a clonal population, providing the basis for bet-hedging strategies in fluctuating environments. As a result, noise optimization can lead to substantial selective forces acting on genome evolution [128].

Experimental population dynamics data sampled in natural systems are in general modeled as multiplicative white Gaussian noise [23,129]. Aiming to provide a more realistic description of stochastic dynamics of natural systems, some recent works investigate intrinsically non-Gaussian noise signals, characterized by sudden random variations [130]. In particular, the environmental fluctuations were modeled by using the archetypical pulse noise source, i.e., a sequence of rectangular pulses with the following properties: i) fixed width; ii) height h distributed according to a certain probability function w_h ; iii) times t of occurrence distributed according to a certain probability function w_t . The impact of a pulse noise source, modeled as Poisson white noise, on population dynamics was studied in [131]. More recently, the stability conditions for the dynamics of termite populations have been investigated in the presence of a noise source with different statistical properties, ranging from sub- to super-Poisson process, in two different cases: i) positive-defined pulses; ii) negative-defined pulses [130]. The effect of noise correlations has been evaluated by means of a stochastic differential equation with a noise source modeled as a renewal process with suitable statistics. This work extended previous studies (see ref. [130] and references therein), in which the stability properties of such a model were investigated in the presence of a multiplicative, positive-defined, sub-Poisson pulse process.

This Perspective paper poses as an urgent open problem the development of new and effective tools for describing and modeling the dynamics that underlie environmental and biological data. This requires theoretical approaches in which non-equilibrium statistical physics and stochastic processes play a key role in the construction of realistic and effective models, which can capture the intrinsically nonlinear and noisy dynamics of natural systems.