Probabilistic failure envelopes of strip foundations on soils with non-stationary characteristics of undrained shear strength

This paper investigates probabilistic failure envelopes of strip foundations on spatially variable soils with profiles of undrained shear strength su linearly increasing with depth using the lower bound random finite element limit analysis. The spatially variable su is characterised by a non-stationary random field with linearly increasing mean and constant coefficient of variation (COV) with depth. The deterministic uniaxial capacities and failure envelopes are firstly derived to validate numerical models and provide a reference for the subsequent probabilistic analysis. Results indicate that the random field parameters COVsu (COV of su) and Δ (dimensionless autocorrelation distance) have a considerable effect on the probabilistic normalised uniaxial capacities which alters the size of probabilistic failure envelopes. However, COVsu and Δ have an insignificant effect on the shape of probabilistic failure envelopes is observed in the V-H, V-M and H-M loading spaces, such that failure envelopes for different soil variabilities can be simply scaled by the uniaxial capacities. In contrast to COVsu and Δ, the soil strength heterogeneity index κ = μkB/μsu0 has the lowest effect on the probabilistic normalised uniaxial capacity factors but the highest effect on the shape of the probabilistic failure envelopes. A series of expressions are proposed to describe the shape of deterministic and probabilistic failure envelopes for strip foundations under combined vertical, horizontal and moment (V-H-M) loading.

Understanding change in hydrometeorological extremes with statistical models - the importance of model parametrization

<p>The potential for changes in hydrometeorological extremes is routinely investigated by fitting change-permitting extreme value models to long-term observations, allowing one or more distribution parameters to change as a function of time or some physically-motivated covariate. In most practical extreme value analyses, the main quantity of interest though is the upper quantiles of the distribution, rather than the parameters' values. This study focuses on the changes in quantile estimates under different change-permitting models. First, metrics which measure the impact of changes in parameters on changes in quantiles are introduced. The mathematical structure of these change metrics is investigated for several models based on the Generalised Extreme Value (GEV) distribution. It is shown that for the most commonly used models, the predicted changes in the quantiles are a non-intuitive function of the distribution parameters, leading to results which are difficult to interpret. Next, it is posited that commonly used change-permitting GEV models do not preserve a constant coefficient of variation, a property that is typically assumed to hold and that is related to the scaling properties of extremes. To address these shortcomings a new (parsimonious) model is proposed: the model assumes a constant coefficient of variation, allowing the location and scale parameters to change simultaneously. The proposed model results in more interpretable changes in the quantile function. The consequences of the different modelling choices on quantile estimates are exemplified using a dataset of extreme peak river flow measurements.</p>

Parametrisation of change-permitting extreme value models and its impact on the description of change

The potential for changes in environmental extremes is routinely investigated by fitting change-permitting extreme value models to long-term observations, allowing one or more distribution parameters to change as a function of time or some other covariate. In most extreme value analyses, the main quantity of interest is typically the upper quantiles of the distribution, which are often needed for practical applications such as engineering design. This study focuses on the changes in quantile estimates under different change-permitting models. First, metrics which measure the impact of changes in parameters on changes in quantiles are introduced. The mathematical structure of these change metrics is investigated for several change-permitting models based on the Generalised Extreme Value (GEV) distribution. It is shown that for the most commonly used models, the predicted changes in the quantiles are a non-intuitive function of the distribution parameters, leading to results which are difficult to interpret. Next, it is posited that commonly used change-permitting GEV models do not preserve a constant coefficient of variation, a property that is typically assumed to hold for environmental extremes. To address these shortcomings a new (parsimonious) model is proposed: the model assumes a constant coefficient of variation, allowing the location and scale parameters to change simultaneously. The proposed model results in changes in the quantile function that are easier to interpret. Finally, the consequences of the different modelling choices on quantile estimates are exemplified using a dataset of extreme peak river flow measurements in Massachusetts, USA. It is argued that the decision on which model structure to adopt to describe change in extremes should also take into consideration any requirements on the behaviour of the quantiles of interest.

Do estimates of numerosity really adhere to Weber’s law? A reexamination of two case studies

Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber’s law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate “number sense.” We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber’s law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber’s law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.

SQUARED COEFFICIENT OF VARIATION OF TAYLOR'S LAW FOR RANDOM ABSOLUTE DIFFERENCES

In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

Mission: Impossible? Modelling the Verbal Estimation of Duration

Three participants produced a large number of verbal estimates of tone durations in the range of 77–1183 ms. Data from this task were simulated by an ‘attractor model’, which used the idea of competition between ‘attractors’ (‘quantized’ values output as verbal estimates) which differed in weight, and distance from the stimulus duration to be estimated. To produce an estimate, all attractors competed for priority as output values, with the final value being decided probabilistically. The model embodied underlying scalar representations of time, in the form of mean accuracy and constant coefficient of variation. The model was able to reconcile such scalar properties of time with deviations from scalar properties often found in verbal estimation data, such as declining coefficients of variation with increasing duration value. The model furthermore showed that multiplicative and additive changes in underlying time representations should be translated veridically into behaviour, although the attractor competition process could distort patterns and absolute values of underlying variance.

Water sensitive papers simulation to assess deposits on targets

Aim of the study is to assess the possibility to use water sensitive papers to estimate, beside the superficial coverage, also the amount of deposit on the target at varying the spray features. To point out the main quantities influencing the deposit, the behaviour of the water sensitive papers was simulated by assuming some simplifying hypotheses: log-normal distribution of the diameter population of the drops and circular spots. Several images (630) of water sensitive papers, sprayed with drops of different mean diameter (from 100 up to 500 μm), constant coefficient of variation (0.50), and theoretical percentage of covered surface ranging from 10 up to 100%, were produced by means of simulation. These images were considered as effective water sensitive paper images and then analysed by means of an image processing software. The correlations between measured and effective values were studied and they allowed for an estimate of deposit and spray features from the image data. This implies that the analysis of the water sensitive paper images allows the determination of more complex parameters such as the unitary deposit and the impact density, all data strictly related to the efficacy of a phytosanitary treatment.