Assumption Of Stationarity Research Articles

Introducing block-Toeplitz covariance matrices to remaster linear discriminant analysis for event-related potential brain–computer interfaces

Objective. Covariance matrices of noisy multichannel electroencephalogram (EEG) time series data provide essential information for the decoding of brain signals using machine learning methods. However, small datasets and high dimensionality make it hard to estimate these matrices. In brain–computer interfaces (BCI) based on event-related potentials (ERP) and a linear discriminant analysis (LDA) classifier, the state of the art covariance estimation uses shrinkage regularization. As this is a general covariance regularization approach, we aim at improving LDA further by better exploiting the domain-specific characteristics of the EEG to regularize the covariance estimates. Approach. We propose to enforce a block-Toeplitz structure for the covariance matrix of the LDA, which implements an assumption of signal stationarity in short time windows for each channel. Main results. An offline re-analysis of data collected from 213 subjects under 13 different event-related potential BCI protocols showed a significantly increased binary classification performance of this ‘ToeplitzLDA’ compared to shrinkage regularized LDA (up to 6 AUC points, p < 0.001) and Riemannian classification approaches (up to 2 AUC points, p < 0.001). In an unsupervised visual speller application, this improvement would translate to a relative reduction of spelling errors by 81% on average for 25 subjects. Additionally, aside from lower memory and reduced time complexity for LDA training, ToeplitzLDA proves to be robust against drastic increases of the number of temporal features. Significance. The proposed covariance estimation allows BCI researchers to improve classification rates and reduce calibration times of BCI protocols using event-related potentials and thus support the usability of corresponding applications. Its lower computational and memory needs could make it a valuable algorithm especially for mobile BCIs.

A Vine Copula‐Based Ensemble Projection of Precipitation Intensity–Duration–Frequency Curves at Sub‐Daily to Multi‐Day Time Scales

AbstractPrecipitation intensity–duration–frequency (IDF) curves play a crucial role in the design and planning of urban infrastructure to reduce the risk of urban flooding and rainfall‐triggered landslides. However, changing rainfall characteristics in a warming climate render conventional IDF curves inappropriate due to the statistical assumption of stationarity. In this study, we develop vine copula‐based projections of future IDF curves at sub‐daily to multi‐day time scales with a multi‐model ensemble of five regional climate simulations over China. Stochastic spatiotemporal downscaling of precipitation is achieved to generate extreme precipitation simulations at a high spatial (0.1°) and temporal (3 hourly) resolution. These downscaled simulations are combined by the vine copula to improve the reliability and accuracy of climate‐model‐based IDF curves relative to historical observations. Our findings reveal that climate‐model‐based stochastic downscaling of precipitation reproduces the IDF curves well based on historical observations in China. The vine copula multi‐model ensemble approach outperforms Bayesian model averaging by generating more accurate and reliable IDF curves. The urban areas of 196 Chinese cities are projected to experience an increase in extreme precipitation of up to 30% in intensity and nearly two times the frequency of historical events under a high emission scenario (RCP8.5). The current urban infrastructure of more than half of the 196 cities would thus be inadequate to prevent losses caused by rainfall‐triggered hazards if designed solely based on historical precipitation observations. Compared to the climate‐model‐based IDF curves, we find that statistical IDF curves based on a nonstationary time covariate (i.e., extrapolating historical trends) are likely to underestimate the risk of urban infrastructure failures under a warming climate. This work highlights that urban infrastructure design guidelines in China should be upgraded to adapt existing IDF curves to the changing climate.

Concept Drift Monitoring and Diagnostics of Supervised Learning Models via Score Vectors

Supervised learning models are one of the most fundamental classes of models. Viewing supervised learning from a probabilistic perspective, the set of training data to which the model is fitted is usually assumed to follow a stationary distribution. However, this stationarity assumption is often violated in a phenomenon called concept drift, which refers to changes over time in the predictive relationship between covariates X and a response variable Y and can render trained models suboptimal or obsolete. We develop a comprehensive and computationally efficient framework for detecting, monitoring, and diagnosing concept drift. Specifically, we monitor the Fisher score vector, defined as the gradient of the log-likelihood for the fitted model, using a form of multivariate exponentially weighted moving average, which monitors for general changes in the mean of a random vector. In spite of the substantial performance advantages that we demonstrate over popular error-based methods, a score-based approach has not been previously considered for concept drift monitoring. Advantages of the proposed score-based framework include applicability to broad classes of parametric models, more powerful detection of changes as shown in theory and experiments, and inherent diagnostic capabilities for helping to identify the nature of the changes.

Identification and evaluation of soil moisture flash drought by a nonstationary framework considering climate and land cover changes

Soil moisture flash drought can cause extensive damage to agriculture, ecosystems, and economies due to its sudden onset. Previous research identified soil moisture flash drought using stationary methods, in which, stationary probability distributions were employed to derive cumulative percentages (CPs) of given soil moisture values, and then based on the CPs sequence, the run theory was used to identify soil moisture flash drought events. However, because changes in climate or land cover can induce significant variations in the underlying probability distributions of soil moisture, the method's usual assumption of stationarity should be questioned. In this study, a nonstationary framework based on the nonstationary frequency analysis method and the run theory was developed for identifying soil moisture flash drought events. This framework was applied to the study of the pentad average root-zone soil moisture (PRZSM) of the Pearl River basin (PRB) in South China based on the ERA5-Land and the GLEAM datasets from 1981 to 2020. The results of the ERA5-Land were general consistent with those of the GLEAM, and the major findings include: (1) without considering the nonstationarity of soil moisture, the onset rate of flash droughts may be underestimated. The average onset rate of flash droughts in nonstationary conditions is slightly greater by 1 %/pentad −2 %/pentad than that in stationary conditions; (2) without considering the nonstationarity of soil moisture, the severity of flash droughts may be overestimated. The average severity of flash droughts in nonstationary conditions is smaller by 10% ⋅ pentads–20% ⋅ pentads than that in stationary conditions; and (3) the trends of the characteristics of soil moisture flash drought are consistent between the stationary and the nonstationary conditions. In conclusion, the above findings contribute to a better understanding of the implications of soil moisture nonstationarity on flash drought identification.

71 Agent-Based Modeling: A Historical Perspective and Comparison to Other Modeling Techniques

Abstract The earliest known history of agent-based modeling (ABM) is traced back to Von Neumann’s self-reproducing cellular automata, designed in the 1940s. The usage of ABM in different scientific fields accelerated in the 1990s, supported by the phenomenal increase in computing capabilities. ABM is gaining acceptance in animal systems, as daily decision-making is becoming complex due to multiple competing outcomes of interest to food systems stakeholders. ABM typically replicates complex real-world systems, for example, a herd of animals that dynamically interact based on simple rules. ABM simulate the heterogenous, stochastic characteristics of agents observed in the real-world. The dynamic interaction of agents replicates the observable real-world complex system patterns. Application of ABM in animal systems ranges from modeling cell behavior, precision nutrition and herd management to predicting the spread of epidemics, and food animal supply chain optimization. Animal science's most widely used alternative modeling techniques include system dynamic models, differential equations-based models, and statistical modeling. The list of unrealistic assumptions that limit the utility of these modeling techniques includes the assumptions of linearity, homogeneity, normality, and stationarity. The advantageous characteristics of an agent in ABM that set it apart from other techniques include being identifiable, capable of existing in an environment where it interacts with other agents while being autonomous and self-directed, goal-oriented behavior, flexible learning, and capability of adaptations in its behavior over time, based on experience. Identifying the purpose of the model, the questions the model will answer, and the potential users are the key decision variables modelers should ponder upon before embarking on building ABM. The commonly used ABM software includes Repast, Swarm, Netlogo, and MASON. Developing an ABM has several highly interleaved stages: concept development, requirements definition, design, implementation, and operationalization, each of which will be illustrated during the hands-on training session on ABM.

Using a process-based dendroclimatic proxy system model in a data assimilation framework: a test case in the Southern Hemisphere over the past centuries

Abstract. Currently available data-assimilation-based reconstructions of past climate variations have only used statistical proxy system models to make the link between climate model outputs and indirect observations from tree rings. However, the linearity and stationarity assumptions of the statistical approach may have limitations. In this study, we incorporate the process-based dendroclimatic model MAIDEN into a data assimilation procedure using the reconstruction of near-surface air temperature, precipitation and winds in the midlatitudes of the Southern Hemisphere over the past 400 years as a test case. We compare our results with a data assimilation approach including a linear regression as a proxy system model for tree-ring width proxies. Overall, when compared to instrumental data, the reconstructions using MAIDEN as a proxy system model offer a skill equivalent to the experiment using the regression model. However, knowing the advantages that a process-based model can bring and the improvements that can still be made with MAIDEN, those results are promising.

Impact of Climate Change on Cereal Crops Production in Ethiopia

Climate change adversely affected agricultural productivity in developing countries. This study aimed to explore the effects of this climate change, particularly on cereal crops production in Ethiopia. The study employed Autoregressive Distributed Lag (ARDL) model approach to the co-integration with an error correction term. ARDL technique was selected due to its stationarity assumption and unbiased estimates of its long-run coefficients. The estimated model justifies the existence of a long-run relationship between cereal crops production, climate change variables (temperature and precipitation), and other explanatory variables. Precipitation has a positive and significant effect on cereal crops production both in the long and short runs, while temperature change has a significant negative effect. In the long run, cereal crops production was positively and significantly affected by arable land, fertilizer consumption, and carbon dioxide emissions, while in the short run, labor force participation has a positive and significant effect on cereal crops production. The study results confirmed that there is a long-run relationship between cereal crops production and climate change variables. In agriculture, research and development should focus on varieties of cereal crops that can tolerate high temperatures. Climate Resilient Green Economy should have to strengthen in the country. All countries should have to work hand-in-hand to mitigate the effect of climate change.

Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton–Jacobi equation and shadow

The properties of light rays around compact objects surrounded by a plasma are affected by both strong gravitational fields described by a general-relativistic spacetime and by a dispersive and refractive medium, characterized by the density distribution of the plasma. We study these effects employing the relativistic Hamiltonian formalism under the assumption of stationarity and axisymmetry. The necessary and sufficient conditions on the metric and on the plasma frequency are formulated such that the rays can be analytically determined from a fully separated Hamilton–Jacobi equation. We demonstrate how these results allow us to analytically calculate the photon region and the shadow if they exist. Several specific examples are discussed in detail: the “hairy” Kerr black holes, the Hartle–Thorne spacetime metrics, the Melvin universe, and the Teo rotating traversable wormhole. In all of these cases, a plasma medium is present as well.

Policy Gradient for Continuing Tasks in Discounted Markov Decision Processes

Reinforcement learning aims to find policies that maximize an expected cumulative reward in Markov decision processes with unknown transition probabilities. Policy gradient (PG)-algorithms use stochastic gradients of the value function to update the policy. A major drawback of PG-algorithms is that they are limited to episodic tasks (multiple finite-horizon trajectories) unless stringent stationarity assumptions are imposed on the trajectories. Hence, they need restarts and cannot be fully implemented online, which is critical for systems need to adapt to new tasks and/or environments in deployment. Moreover, the standard stationary formulation ignores transient behaviors. This motivates our study of discounted MDPs of infinite horizon without restarts. However, it is unknown if in this case following stochastic PG-type estimates would improve the policy. The main result of this work is to establish that when policies belong to a reproducing kernel Hilbert space (RKHS), and the kernel is selected properly, then these PG-estimates are ascent directions for the value function conditioned to any arbitrary initial point. This allows us to prove convergence of our online algorithm to the local optima. A numerical example shows that an agent running our online algorithm learns to navigate and succeeds in a surveillance task that requires looping between two goal locations. This example corroborates our theoretical findings about the ascent directions of subsequent stochastic gradients. It also shows how our online algorithm guides the agent through a continuing cyclic trajectory that does not comply with the standard stationarity assumptions in the literature for non-episodic training.

A method for detecting the non-stationarity during high flows under global change

The sustainability of existing water resources is influenced by extreme streamflow, and climate variability and human activities are generally the major factors controlling these dynamics. However, most of previously proposed methods to determine the effects of these factors have only been developed under the assumption of stationarity. Therefore, to overcome the existing research gap, an innovative method was proposed in this study to analyze and distinguish the effects of climate variability and human activities on extreme streamflow based on the non-stationarity theory. Accordingly, a rainfall-runoff model was developed using long-term hydrological data in the watersheds of Southeast China, which cover >75,000 km2. The model proposed in this study showed an acceptable performance, as indicated by the Nash-Sutcliffe efficiency coefficient (NSE), the Kling-Gupta efficiency (KGE), and percent bias (PBIAS). The NSE, KGE, and |PBIAS| were 0.67–0.75, 0.57–0.74, and 1.22–16.79 during the calibration periods, respectively. And the NSE, KGE, and |PBIAS| were 0.69–0.77, 0.65–0.76, and 0.98–17.51 during the calibration periods, respectively. The trends of the extreme streamflow were analyzed for these watersheds at different time scales. The streamflow extremes at short time scales were found to be more sensitive to changing environment than those at longer time scales. The major factor controlling streamflow extremes at short time scales was human activities and climate change may be the dominant factor influencing streamflow extremes at long time scales. The findings of this study could provide useful insights into water management under global change conditions.

Can 7000 Years of flood history inform actual flood risk management? A case study on Lake Mondsee, Austria

Flood risk models typically rely on discharge records of time series no longer than a few hundred years, which leads to high uncertainty in the estimates for major flood events. In this paper, we investigate the capability of geological flood records retrieved from lake sediment cores to prolong available data sets, lower the uncertainty of flood risk analysis, and detect changes in flood patterns by releasing the common assumption of stationarity. Paleoflood records of Lake Mondsee, Austria, covering 7000 years are linked to flood damage data of the considered catchment area to calculate (1) the average annual damage, i.e. the fair premium and (2) the damage at 99.5% Value at Risk, i.e. the Solvency II capital requirement. To illustrate our results, the performance of a hypothetical catastrophe fund for these 7000 years is studied at current vulnerability levels. Our findings show that high resolution paleoflood records with robust chronologies and flood information have significant impacts on flood model outcomes for disaster risk management and that damage potentials derived merely from recent damage data may overestimate actual risks.

An overall optimization and solution framework for urban historical and future DRIF under climate change

Abstract The estimation and application of Intensity-Duration-Frequency (IDF) curves depend on the assumption of stationarity of the rainfall series, which is that the intensity and frequency of extreme hydrological events remain unchanged in the future. Climate change will have a significant impact on the collection and utilization of rainwater and its spatial characteristics. When the Gray-Green infrastructure is designed, if only historical precipitation is adopted to calculate the urban design rainstorm intensity formula (DRIF) and the total annual runoff control rate, it may be difficult to meet the demand of future precipitation changes on the city's ability to accommodate rainfall. Therefore, it is very important to study the impact of climate change on the IDF curve. This study proposes an overall optimization solution framework for historical and future DRIF. The impact of the extreme value on the IDF curve during the historical period is analyzed. The calculation method of the IDF curve in the future period is established. The changes of the rainstorm intensity in the historical and future period (SSP1-2.6,SSP2-4.5,SSP3-7.0,SSP5-8.5) were analyzed for the 15 durations and eight return periods in Beijing, China. The results of this study show that the nondominated sorting and local search (NSLS) has the best accuracy in fitting the statistical samples of precipitation for different durations. The best methods to judge and process the extreme value of the statistical sample are Z-score and average value of series greater than critical value (AVG). Under the four SSP scenarios, the estimated IDF value is larger than the observed value in the historical period. The results of the equivalent return period calculated using the DRIF show that the the four SSP scenarios are smaller than the historical period for the return period greater than five years. Taking 120 min of short-duration precipitation as an example, the 100-year equivalent return periods of the observation under the four SSP scenarios are 35-, 20-, 54-, and 17-years, respectively. The research can provide valuable reference for the design and planning of the drainage facility under climate change.

Nonstationary flood and low flow frequency analysis in the upper reaches of Huaihe River Basin, China, using climatic variables and reservoir index as covariates

Conventional water infrastructure designs for flood and low flows are usually based on the assumption of stationarity of extreme events. However, recent evidence suggests that the influences of climate variability and human activities have made the hypothesis of stationarity questionable. In this study, we used the generalized additive models for location, scale, and shape (GAMLSS) to construct a nonstationary model in which the parameters of the selected distributions were modelled as a function of climatic variables (i.e., climate indices and precipitation) and/or the reservoir index (RI). The nonstationary models were then used to analyse annual flood and low flow frequency at four hydrological stations in the upper reaches of the Huaihe River Basin, including Dapoling (DPL), Changtaiguan (CTG), Zhuganpu (ZGP), and Xixian (XX) stations. Annual floods were represented by the maximum daily streamflow in each year, and low flows were represented by the 95th quantile of the daily streamflow (Q95) in each year. The change point and trend analysis revealed that the flood series of the ZGP station and the low flow series of the DPL and XX stations exhibited significant downward and upward trends (p < 0.1)), respectively. The low flow series of the ZGP station showed a significant change point in 1980 (p < 0.1). GAMLSS modelling results showed that, in comparison with stationary models, nonstationary models that included precipitation and the Arctic Oscillation climate index as covariates for the gamma distribution location parameter provided a superior description of the flood series at the four stations. Nonstationary models that incorporated precipitation and/or RI as covariates for the Weibull distribution parameters fit the low flow series better than stationary models at all stations. Furthermore, we found that nonstationary models outperformed stationary models in terms of flood frequency analysis, covering all flood observation points and capturing the generally decreasing trend in flood series, as well as a decrease in the scatter of estimated flood value magnitudes. For the low flow frequency analysis, the comparison results showed that the nonstationary and stationary models performed identically for the DPL, CTG, and XX stations, where no significant change point was detected. However, for the ZGP station, where a significant change point was detected, the nonstationary models performed better than the stationary models and could accurately capture the changes in the magnitude of the estimated low flow values before and after the change point. Overall, the proposed nonstationary model can serve as a tool for nonstationary frequency analysis of flood and low flow series under the influence of climate variability and reservoir regulations, thus providing a reference for regional water infrastructure design.

Bayesian Spatial Modeling of Diabetes and Hypertension: Results from the South Africa General Household Survey

Determining spatial links between disease risk and socio-demographic characteristics is vital in disease management and policymaking. However, data are subject to complexities caused by heterogeneity across host classes and space epidemic processes. This study aims to implement a spatially varying coefficient (SVC) model to account for non-stationarity in the effect of covariates. Using the South Africa general household survey, we study the provincial variation of people living with diabetes and hypertension risk through the SVC model. The people living with diabetes and hypertension risk are modeled using a logistic model that includes spatially unstructured and spatially structured random effects. Spatial smoothness priors for the spatially structured component are employed in modeling, namely, a Gaussian Markov random field (GMRF), a second-order random walk (RW2), and a conditional autoregressive (CAR) model. The SVC model is used to relax the stationarity assumption in which non-linear effects of age are captured through the RW2 and allow the mean effect to vary spatially using a CAR model. Results highlight a non-linear relationship between age and people living with diabetes and hypertension. The SVC models outperform the stationary models. The results suggest significant provincial differences, and the maps provided can guide policymakers in carefully exploiting the available resources for more cost-effective interventions.

Should Multivariate Bias Corrections of Climate Simulations Account for Changes of Rank Correlation Over Time?

AbstractInter‐variable dependencies are key properties to characterize many climate phenomena—such as compound events—and their future changes. Yet, climate simulations often have statistical biases. Hence, univariate (1dBC) and multivariate bias correction (MBC) methods are regularly applied. Inter‐variable properties (e.g., correlations) can be altered by BC corrections. Then, it is necessary to assess how underlying assumptions of BC methods on climate change affect the adjustments. This can lead to better choices of BC methods. Here, we investigate whether an MBC method should try reproducing, preserving or modifying the changes in rank correlations between daily temperature and precipitation over Europe. An original “perfect model experiment” is set up and applied to two different climate simulation ensembles over 2001–2100: 40 runs from the Community Earth System Model (CESM) global climate model and 11 runs from the Coupled Models Intercomparison Project (CMIP6) exercise. The results highlight that, within the multi‐run single Global Climate Models ensemble (CESM), accounting for correlation changes brings valuable information for long‐term projections but that a stationarity assumption provides less biased correlations, up to medium‐term projections (2060). For the multi‐model ensemble (CMIP6), the non‐stationarity assumption provides larger biases than the stationarity approach, up to the end of the century. Not correcting the model rank correlations (1dBC) provides the worst results. Whenever an ensemble is available, the best results come from accounting for the “robust” part of the change signal (i.e., average change from different runs). This pleads for using ensembles and their robust information, in order to perform robust bias corrections.

The decomposition-based nonstationary flood frequency analysis

The nonstationary flood frequency analysis (NS-FFA) is conducted when the assumption of stationarity in hydrologic extremes is violated. The commonly used approach is the performance-based NS-FFA, which jointly determines the distribution and the nonstationary structure based upon the model performance. However, this approach is challenged from both theoretical and practical perspectives. An alternative is the herein named decomposition-based NS-FFA, which determines the two NS-FFA model components separately and explicitly uses the available knowledge of the nonstationarity. However, it has been barely implemented in practice. This paper proposed a novel decomposition procedure that strictly follows the theoretical decomposition of nonstationary stochastic processes to advance the decomposition-based NS-FFA. The proposed decomposition procedure was compared with a previously reported method in both an analytical deduction and a simulation study. The proposed decomposition-based NS-FFA was further compared to the performance-based NS-FFA using both synthetic and real datasets from North America, which exhibit different patterns of nonstationarity. The Particle Filter was adopted for uncertainty quantification and parameter estimation. The results revealed that the proposed decomposition-based approach was advantageous in preserving the moments of underlying stochastic component, particularly the higher-order moment (i.e., skewness). In addition, the comparison of the two NS-FFA approaches demonstrated the superiority of the proposed decomposition-based approach in capturing the underlying (known) nonstationary stochastic process and being competitive with the performance-based approach from the performance perspective in real applications. The results from the simulation study and the real application also revealed several caveats of the performance-based approach, including the potential overfitting and equifinality problems as well as the selection of distinct models when adopting different performance metrics. In addition, differing from the performance-based approach, the decomposition-based NS-FFA avoided/alleviated the ergodicity violation. All these results demonstrated the advancements of the proposed decomposition-based NS-FFA and advocated its application in the NS-FFA.

Extraction Of Temperature Fields From PIV Data Of Turbulent Rayleigh-Bénard Convection Using DNS

To gain insights into the dynamical processes that govern the problem of turbulent Rayleigh-Bénard convection, information about the temperature field---next to the velocity field---is required. In the present study, we exploit the governing flow equations to extract the temperature field from a velocity field measured via tomographic particle image velocimetry. The methods used within this study are derived from algorithms originally used in the framework of direct numerical simulations. First, we enforce the divergence-free condition on the measured velocity field using a fractional-step method based on a Poisson equation. Second, we extract the temperature field from the velocity field via the momentum equation. Third, we extract the temperature field from the velocity field via the solution of the stationary convection-diffusion equation. The extracted velocity field---together with the corresponding divergence-free velocity field---can serve as an initial condition for a direct numerical simulation of the problem. Validating the extraction method based on the stationary convection-diffusion equation with data from direct numerical simulations shows a good correlation coefficient between the extracted and the simulated temperature field (0.86). Although the third method incorporates the ambitious assumption of stationarity, temperature fields are successfully extracted from the measured velocity fields. However, with a correlation coefficient of 0.57, the agreement between the average extracted temperature field and the average temperature field measured in a subvolume of the flow domain is somewhat less than the one found in the simulated data. The latter is presumably attributed not only to the extraction method, but also to the underlying measurement data. Thus, an increase in measurement accuracy in future applications is likely to improve the accuracy of the extraction methods as well.

Tempered stable autoregressive models

In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. It is shown that the distribution of the error term is infinitely divisible. Parameter estimation of the introduced TAR(1) process is done by adopting the conditional least square and method of moments based approach and the performance of the proposed methods are evaluated on simulated data. Also, we study an autoregressive model of order one with tempered stable innovations. Using appropriate test statistic it is shown that the model fits very well on real and simulated data. Our models generalize the inverse Gaussian and one-sided stable autoregressive models existing in the literature.

Distributed Order Estimation of ARX Model under Cooperative Excitation Condition

In this paper, we consider the distributed estimation problem of a linear stochastic system described by an autoregressive model with exogenous inputs when both the system orders and parameters are unknown. We design distributed algorithms to estimate the unknown orders and parameters by combining the proposed local information criterion with the distributed least squares method. The simultaneous estimation for both the system orders and parameters brings challenges for the theoretical analysis. Some analysis techniques, such as double array martingale limit theory, stochastic Lyapunov functions, and martingale convergence theorems are employed. For the case where the upper bounds of the true orders are available, we introduce a cooperative excitation condition, under which the strong consistency of the estimation for the orders and parameters is established. Moreover, for the case where the upper bounds of true orders are unknown, a similar distributed algorithm is proposed to estimate both the orders and parameters, and the corresponding convergence analysis for the proposed algorithm is provided. We remark that our results are obtained without relying on the independency or stationarity assumptions of regression vectors, and the cooperative excitation conditions can show that all sensors can cooperate to fulfill the estimation task even though any individual sensor cannot.

Diagnostic Framework for Evaluating How Parametric Uncertainty Influences Agro‐Hydrologic Model Projections of Crop Yields Under Climate Change

AbstractDespite the prevalence of climate change assessments of crop yields, there are significant limits to our understanding of how parametric uncertainty in the underlying agro‐hydrologic models as well as the stationarity assumptions tacit to their commonly employed calibration procedures are influencing projections. This study addresses this knowledge gap by clarifying how parametric uncertainty in agro‐hydrologic models influences yield projections under changing future climate. We focus on rain‐fed winter wheat systems in the drylands of United States Pacific Northwest. We use a tightly coupled agro‐hydrologic model, VIC‐CropSyst, as a representative of this class of models. Our contributed diagnostic global sensitivity analysis framework identifies differences in how influential factors (e.g., temperature during early growth stages or the growing degree‐day required to reach peak leaf area index) vary across zones during historical and future periods. Our results show that the dominant parametric controls for yield projection and their sensitivities change subject to agro‐climatic zones and differences in the specific temperature‐precipitation trends in future climate scenarios. Our results also indicate that the stationarity assumptions tacit to using historical observations to calibrate agro‐hydrologic model parameters and their subsequent use in future yield projections may introduce significant bias. Employing the stationarity assumption in future projections problematically ignores how shifts in climate influence the relative dominance of underlying agro‐hydrologic processes in the model. This study's contributed diagnostic exploratory modeling framework has promise for advancing our understanding of how calibration, parametric uncertainties, and climate induced changes in the dominance of model biophysical processes shape crop yield projections.