A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

Abstract

Abstract:Since the prime days of stochastic hydrology back in 1960s, autoregressive (AR) and moving average (MA) models (as well as their extensions) have been widely used to simulate hydrometeorological processes. Initially, AR(1) or Markovian models with Gaussian noise prevailed due to their conceptual and mathematical simplicity. However, the ubiquitous skewed behavior of most hydrometeorological processes, particularly at fine time scales, necessitated the generation of synthetic time series to also reproduce higher-order moments. In this respect, the former schemes were enhanced to preserve skewness through the use of non-Gaussian white noise-a modification attributed to Thomas and Fiering (TF). Although preserving higher-order moments to approximate a distribution is a limited and potentially risky solution, the TF approach has become a common choice in operational practice. In this study, almost half a century after its introduction, we reveal an important flaw that spans over all popular linear stochastic models that employ non-Gaussian white noise. Focusing on the Markovian case, we prove mathematically that this generating scheme provides bounded dependence patterns, which are both unrealistic and inconsistent with the observed data. This so-called "envelope behavior" is amplified as the skewness and correlation increases, as demonstrated on the basis of real-world and hypothetical simulation examples.

Simulating Marginal and Dependence Behaviour of Water Demand Processes at Any Fine Time Scale

“…Essentially, the method aims to reproduce the target distributions which have assigned a priori to the variables under study. Tsoukalas et al [51,71] highlight the importance and benefits of this approach against the classical stochastic modelling of hydrometeorological processes, which typically focuses on the resemble of a series of specific statistical characteristics. As discussed in Section 1, this is also crucial in residential water demand modelling given that WDS applications require a proper reproduction of various characteristics (e.g., maximum demands) which cannot derive explicitly by the preservation of some low-order statistics.…”

Water Supply Risk Analysis Based on Runoff Sequence Simulation with Change Point under Changing Environment

“…Panjiakou reservoir evaporation loss refers to the conversion coefficient of water surface evaporation in the Luanhe river basin (Figure 2 [25]. e original T-F model is essentially a cyclostationary version of the classic stationary linear autoregressive model, in order to account for systematic changes and nonstationarities of statistical characteristics across seasons [26]. e marginal distributions of many hydrometeorological Advances in Meteorology processes are not Gaussian, which motivated omas and Fiering [27] propose to replace the Gaussian white noise with Gamma (G) or Pearson type-III (PIII) distributed white noise in order to account for the high skewness coefficient [28].…”

Exploring the Statistical and Distributional Properties of Residential Water Demand at Fine Time Scales

“…In the present study, taking into account the above along with recent advances in the stochastic simulation field [26][27][28][29][30], which alleviate the barriers [31] in the deployment of typical linear stochastic models under non-Gaussian assumptions, we treat and examine residential water demand as a random variable described by a mixed-type distribution. Therefore, our focus is on the study of the statistical properties and peculiarities of the positive magnitudes of the records (i.e., continuous part), but also on the discrete part of the process as it is expressed by probability of no demand.…”

“…This statistical analysis, which precedes the modeling phase, provides new insights and paves the ground for the development of concrete tools toward the probabilistic representation and simulation of water demand across different time scales. Towards this, the present study takes also into account recent advances in stochastic simulation field [26][27][28][29][30], which alleviate the barriers [31] in the deployment of typical linear stochastic models in cases of non-Gaussian and intermittent processes such as residential water demand at fine time scales, allowing the reproduction of the whole marginal distribution of the process.…”