Clive W. Anderson

Case studies in Gaussian process modelling of computer codes

Abstract In this paper we present a number of recent applications in which an emulator of a computer code is created using a Gaussian process model. Tools are then applied to the emulator to perform sensitivity analysis and uncertainty analysis. Sensitivity analysis is used both as an aid to model improvement and as a guide to how much the output uncertainty might be reduced by learning about specific inputs. Uncertainty analysis allows us to reflect output uncertainty due to unknown input parameters, when the finished code is used for prediction. The computer codes themselves are currently being developed within the UK Centre for Terrestrial Carbon Dynamics.

Application of the Generalized Pareto Distribution to the estimation of the size of the maximum inclusion in clean steels

Abstract Predicting the maximum inclusion size in a large volume of clean steel from observations on a small volume is a key problem facing the steel industry. The maximum inclusion size controls fatigue behaviour and other mechanical properties. A statistical method based on the Generalized Pareto Distribution, which has not previously been applied to inclusions, is described and used to analyse results from two steels with different cleanness levels. The predictions of maximum size in larger volumes are compared with those from extrapolating the log-normal curve which fits the size distribution of inclusions observed on the surface of cold crucible remelted buttons of the steels. If the predicted maximum inclusion size is plotted against increasing volume of steel, there is a continuous increase for the log-normal extrapolation but for the Generalized Pareto Distribution method the curve tends towards an upper limit. This is of great significance to steelmakers because the existence of an upper limit is more in accord with the expectation from steelmaking practice.

The precision of methods using the statistics of extremes for the estimation of the maximum size of inclusions in clean steels

The maximum inclusion size in clean steels influences fatigue behaviour and other mechanical properties. Hence, its estimation and the uncertainties associated with the estimation are important issues for steel makers and users. Here, two methods based on the statistics of extremes, one termed the Statistics of Extreme Values (SEV) method and the other the Generalized Pareto Distribution (GPD) method, are used for the estimation. Both methods use data on the size of inclusions revealed on the surface of sampled areas. The influence of the number of sample areas and the way the sample areas are grouped on the estimated result and its confidence limits is determined and compared. For both the SEV and the GPD methods, the estimated largest inclusion size is relatively insensitive to the number of sample areas but, as might be expected, the width of the confidence interval decreases steeply as the number of sample areas increases. A key point is that the SEV method has a narrower confidence interval than the GPD method for a given number of sample areas, because the SEV method makes an extra assumption about the form of the distribution of large inclusions. The particular assumption is difficult to justify on the basis of the data alone, and leads to a potentially over-optimistic estimate of precision. For practical application of the GPD estimation procedure, the number of sample areas needed for estimation depends on the confidence interval required and the volume of steel of interest. It is suggested on the basis of the GPD size distribution that fatigue failure initiation in a component is unlikely to be caused by the single largest inclusion, but rather by more frequently occurring inclusions near the top of the size range. This provides the conceptual basis for a statistically based design approach in which the estimated distribution of inclusion sizes is used in defect tolerance design of steel components and in control of steel production processes.

Quantitative uncertainty analyses of ancient atmospheric CO2 estimates from fossil leaves

The relationship between atmospheric CO2 and ancient climate is of fundamental importance for gauging the climate sensitivity of the Earth system to a changing CO2 regime. One of the most widely adopted paleobiological CO2 proxies for reconstructing Earths atmospheric CO2 history exploits the inverse relationship between leaf stomatal index, the fraction of leaf epidermal cells that are stomatal structures, and atmospheric CO2. However, fossil leaf-based CO2 reconstructions make a priori assumptions about the form of the empirical relationship between SI and CO2 required for transfer functions and have failed to correctly propagate error terms. These effects can translate into erroneous interpretations that undermine the value of the proxy. Here we report the development and application of a rigorous generalized statistical framework overcoming these limitations that generates probability density functions for each atmospheric CO2 estimate. The utility of our statistical tools is demonstrated by showing how they revise earlier atmospheric CO2 estimates from fossil cuticles of Ginkgo and Metasequoia trees during the early Eocene and middle Miocene warm periods upwards by +150 to 250 ppm to 450 to 700 ppm. The revised CO2 reconstructions therefore help to resolve the paradox of warm Paleogene and Neogene “greenhouse” climates co-existing with near present-day levels of CO2 and support the emerging view from independent paleoclimate studies for a high climate sensitivity of the Earth system. The statistical tools presented are sufficiently versatile to permit their use in other investigations of paleoCO2 estimates from fossil leaves.

Interrelationship between statistical methods for estimating the size of the maximum inclusion in clean steels

Abstract Two types of approach based on the statistics of extremes have been developed recently to estimate the sizes of large inclusions in clean steel. The first type (termed here “threshold” approaches) includes methods based on the Generalized Pareto distribution (GPD) and the Exponential distribution (EXPGPD). Both of these methods use measurements of the sizes of all inclusions larger than a certain threshold size in a sample. The second type of approach (termed here the “extreme values” type) includes the Statistics of Extreme Values (SEV) method and the Generalized Extreme Values (GEV) method. In these, only the size of the largest inclusion in each of a set of samples is measured. This paper compares the four methods and describes their inter-relationship. The distribution of large sizes depends on a shape parameter ξ. The influence of ξ on confidence intervals for the characteristic size of the maximum inclusion is studied by considering the shape of the likelihood function. The value of ξ is found to have a considerable effect on the precision of estimation. In the methods based on the GPD and GEV the value of ξ is not specified in advance. In such a case the GPD method gives more precise estimation of the characteristic size of the maximum inclusion than the GEV method. On the other hand in the EXPGPD and SEV methods ξ is assumed to be zero. In this case the EXPGPD method gives more precise estimation than the SEV method. The choice of a method of estimation of the characteristic size of the maximum inclusion is discussed in the light of these findings.

Comparison of extreme value statistics methods for predicting maximum inclusion size in clean steels

AbstractThe prediction of the maximum inclusion size in a large volume of clean steel from data on small specimens is a key issue for steelmakers and users. The statistics of extremes has recently emerged as a powerful tool for this purpose. Murakami and coworkers have applied one branch of the theory to steels, based on measuring the maximum inclusion size in a series of areas on the polished surface of the specimen. The present authors have recently reported on the application to steels of another branch of the theory, using the Generalised Pareto Distribution (GPD), and in this paper the two methods are compared using data on oxide inclusions obtained by quantitative image analysis on polished cross-sections. The most important feature of the GPD method is that it predicts an upper limit to the inclusion size, in contrast to the method of Murakami and coworkers and indeed the more basic route of simply extrapolating the log- normal distribution where, as the volume of steel is increased, the size of th...

Ordered multivariate extremes

Multivariate extreme value models and associated statistical methods are developed for vector observations whose components are subject to an order restriction. The approach extends the multivariate threshold methodology of Coles and Tawn, Joe and co-workers and Smith and co-workers. The results are illustrated by an analysis of extreme rainfalls of different durations, and by a study of the problem of linking a long series of daily rainfall extremes with a partially overlapping shorter series of hourly extremes.

The Largest Inclusions in a Piece of Steel

Fatigue properties of steels are strongly influenced by the presence of microscopic particles of oxides or foreign material known as inclusions. The size of the largest inclusion is an important determinant of fatigue strength. This paper studies the problem of estimating the sizes of large inclusions from measurements made on a two-dimensional section of the steel. The approach combines traditional stereological ideas with more recent extreme value modeling. It is shown that both classical likelihood and Bayesian approaches are useful in the inference.

Computer simulation of the estimation of the maximum inclusion size in clean steels by the generalized pareto distribution method

Abstract The Generalized Pareto Distribution (GPD) method has recently been applied to the estimation of the characteristic size of the maximum inclusion in clean steels for the first time. This allows data on inclusion sizes in small samples of steel to be used to predict the size of the maximum inclusion in a large volume of steel, a parameter of importance to steel users. The methodology for finding the confidence limits for the estimate has also been developed, again using data from real experimental samples. Here, computer simulation of data (using the Monte Carlo method) allows a much wider range of data sets to be explored quickly and efficiently. The relationship between the GPD parameters ( ξ and σ ′), the number of simulated inclusions, the volume of steel used for the prediction, the predicted characteristic size and the width of the associated confidence intervals on size has been determined using simulated data. The characteristic size and width of confidence intervals increase with an increase of ξ and σ ′, ξ being the dominant parameter. Small negative ξ values give bigger values for the characteristic size and confidence intervals than more negative ξ values. The information given here allows an experimentalist to determine how many inclusions to measure for a desired precision on the estimation to be obtained.

A model for extreme wind gusts

Estimates of the largest wind gust that will occur at a given location over a specified period are required by civil engineers. Estimation is usually based on models which are derived from the limiting distributions of maxima of stationary time series and which are fitted to data on extreme gusts. In this paper we develop a model for maximum gusts which also incorporates data on hourly mean speeds through a distributional relationship between maxima and means. This joint model is closely linked to the physical processes which generate the most extreme values and thus provides a mechanism by which data on means can augment those on gusts. It is argued that this increases the credibility of extrapolation in estimates of long period return gusts. The model is shown to provide a good fit to data obtained at a location in northern England and is compared with a more traditional modelling approach, which also performs well for this site.