Upper and lower bounds on the learning curve for Gaussian processes

In this paper we introduce and illustrate non-trivial upper and lower bounds on the learning curves for one-dimensional Gaussian Processes. The analysis is carried out emphasising the effects induced on the bounds by the smoothness of the random process described by the Modified Bessel and the Squared Exponential covariance functions. We present an explanation of the early, linearly-decreasing behavior of the learning curves and the bounds as well as a study of the asymptotic behavior of the curves. The effects of the noise level and the lengthscale on the tightness of the bounds are also discussed.

Fibre agglomerate transport in a horizontal flow

An experimental and theoretical study of the transport of mineral wool fibre agglomerates in nuclear power plant containment sumps is being performed. A racetrack channel was devised to provide data for the validation of numerical models, which are intended to model the transport of fibre agglomerates. The racetrack channel provides near uniform and steady conditions that lead to either the sedimentation or suspension of the agglomerates. Various experimental techniques were used to determine the velocity conditions and the distribution of the fibre agglomerates in the channel. The fibre agglomerates are modelled as fluid particles in the Eulerian reference frame. Simulations of pure sedimentation of a known mass and volume of agglomerations show that the transport of the fibre agglomerates can be replicated. The suspension of the fibres is also replicated in the simulations; however, the definition of the fibre agglomerate phase is strongly dependent on the selected density and diameter. Detailed information on the morphology of the fibre agglomerates is lacking for the suspension conditions, as the fibre agglomerates may undergo breakage and erosion. Therefore, ongoing work, which is described here, is being pursued to improve the experimental characterisation of the suspended transport of the fibre agglomerates.

A two-factor model of the U.K. yield curve

I model the forward premium in the U.K. gilt-edged market over the period 1982–96 using a two-factor general equilibrium model of the term structure of interest rates. The model permits the decomposition of the forward premium into separate components representing interest rate expectations, the risk premia associated with each of the underlying factors, and terms capturing the direct impact of the variances of the factors on the shape of the forward curve.

Factors determining the growth curve of the foliose lichen Parmelia conspersa

Growth curves of the foliose lichen Parmelia conspersa (Ehrh. Ex Ach.)Ach. Were obtained by plotting radial growth (RGR, mm yr-1) of the fastest measured lobe, the slowest measured lobe, a randomly selected lobe, and by averaging a sample of lobes from each thallus against thallus diameter. Growth curves derived from the fastest-growing lobe and by averaging lobes were asymptotic and could be fitted by the growth model of Aplin and Hill. Mean lobe width increased with thallus size, reaching a maximum at approx. 4.5 cm thallus diameter. In four out of six thalli, radial growth of lobes over four months was positively correlated with initial lobe width or area. The RGR of isolated lobes was unaffected until the base of the lobe was removed to within 1-2 mm of the tip. The concentration (micrograms mg-1 biomass) of ribitol, arabitol and mannitol was greater in the marginal lobes of large than in small thalli. The results suggested that the growth curve of P. conspersa is determined by processes that occur within individual marginal lobes and can be explained by the Aplin and Hill model. Changes in lobe width and in the productive capacity of individual lobes with thallus size are likely to be more important factors than the degree of translocation within the lobe in determining the growth curve.

Growth curve of the lichen Rhizocarpon geographicum

Data on the growth curve of the lichen Rhizocarpon geographicum were obtained by measuring the radial growth rates (mm per 1.5 years) of 39 thalli from 2 to 65 mm in diameter growing in the same environment. An Aplin and Hill plot (r2 – r1 against ln r2 – ln r1) of the data and regression analyses suggested an initial phase of growth (up to a diameter of about 7 mm) in which the relative growth rate increased rapidly. This was followed by a phase in which the relative growth rate fell but the radial growth rate continued to rise (7 to 20 mm in diameter). Radial growth was then relatively constant until about 45 mm diameter and then declined. The Aplin and Hill model did not fit the data as a whole but may apply for a transient period in thalli between about 7 and 16 mm in diameter. The curve shows some similarities to that suggested by lichenometric studies but differs in showing a less steep decline in growth rate after the ‘great’ period.

Statnote 19: non-linear regression: fitting an exponential curve to data

Non-linear relationships are common in microbiological research and often necessitate the use of the statistical techniques of non-linear regression or curve fitting. In some circumstances, the investigator may wish to fit an exponential model to the data, i.e., to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X. This type of model is straight forward to fit as taking logarithms of the Y variable linearises the relationship which can then be treated by the methods of linear regression.

Statnote 20: non-linear regression: fitting a general polynomial curve

In some circumstances, there may be no scientific model of the relationship between X and Y that can be specified in advance and indeed the objective of the investigation may be to provide a ‘curve of best fit’ for predictive purposes. In such an example, the fitting of successive polynomials may be the best approach. There are various strategies to decide on the polynomial of best fit depending on the objectives of the investigation.

Evaluating the environmental Kuznets curve

We examine the empirical evidence for an environmental Kuznets curve using a semiparametric smooth coefficient regression model that allows us to incorporate flexibility in the parameter estimates, while maintaining the basic econometric structure that is typically used to estimate the pollution-income relationship. This allows us to assess the sensitivity to parameter heterogeneity of typical parametric models used to estimate the relationship between pollution and income, as well as identify why the results from such models are seldom found to be robust. Our results confirm that the resulting relationship between pollution and income is fragile; we show that the estimated pollution-income relationship depends substantially on the heterogeneity of the slope coefficients and the parameter values at which the relationship is evaluated. Different sets of parameters obtained from the semiparametric model give rise to many different shapes for the pollution-income relationship that are commonly found in the literature.

Clogging in horizontal subsurface flow constructed wetlands

Horizontal Subsurface Flow Treatment Wetlands (HSSF TWs) are used by Severn Trent Water as a low-cost tertiary wastewater treatment for rural locations. Experience has shown that clogging is a major operational problem that reduces HSSF TW lifetime. Clogging is caused by an accumulation of secondary wastewater solids from upstream processes and decomposing leaf litter. Clogging occurs as a sludge layer where wastewater is loaded on the surface of the bed at the inlet. Severn Trent systems receive relatively high hydraulic loading rates, which causes overland flow and reduces the ability to mineralise surface sludge accumulations. A novel apparatus and method, the Aston Permeameter, was created to measure hydraulic conductivity in situ. Accuracy is ±30 %, which was considered adequate given that conductivity in clogged systems varies by several orders of magnitude. The Aston Permeameter was used to perform 20 separate tests on 13 different HSSF TWs in the UK and the US. The minimum conductivity measured was 0.03 m/d at Fenny Compton (compared with 5,000 m/d clean conductivity), which was caused by an accumulation of construction fines in one part of the bed. Most systems displayed a 2 to 3 order of magnitude variation in conductivity in each dimension. Statistically significant transverse variations in conductivity were found in 70% of the systems. Clogging at the inlet and outlet was generally highest where flow enters the influent distribution and exits the effluent collection system, respectively. Surface conductivity was lower in systems with dense vegetation because plant canopies reduce surface evapotranspiration and decelerate sludge mineralisation. An equation was derived to describe how the water table profile is influenced by overland flow, spatial variations in conductivity and clogging. The equation is calibrated using a single parameter, the Clog Factor (CF), which represents the equivalent loss of porosity that would reproduce measured conductivity according to the Kozeny-Carman Equation. The CF varies from 0 for ideal conditions to 1 for completely clogged conditions. Minimum CF was 0.54 for a system that had recently been refurbished, which represents the deviation from ideal conditions due to characteristics of non-ideal media such as particle size distribution and morphology. Maximum CF was 0.90 for a 15 year old system that exhibited sludge accumulation and overland flow across the majority of the bed. A Finite Element Model of a 15 m long HSSF TW was used to indicate how hydraulics and hydrodynamics vary as CF increases. It was found that as CF increases from 0.55 to 0.65 the subsurface wetted area increases, which causes mean hydraulic residence time to increase from 0.16 days to 0.18 days. As CF increases from 0.65 to 0.90, the extent of overland flow increases from 1.8 m to 13.1 m, which reduces hydraulic efficiency from 37 % to 12 % and reduces mean residence time to 0.08 days.

Scale and scope economies and the efficient vertical and horizontal configuration of the water industry:a survey of the literature

This paper surveys the literature on scale and scope economies in the water and sewerage industry. The magnitude of scale and scope economies determines the cost efficient configuration of any industry. In the case of a regulated sector, reliable estimates of these economies are relevant to inform reform proposals that promote vertical (un)bundling and mergers. The empirical evidence allows some general conclusions. First, there is considerable evidence for the existence of vertical scope economies between upstream water production and distribution. Second, there is only mixed evidence on the existence of (dis)economies of scope between water and sewerage activities. Third, economies of scale exist up to certain output level, and diseconomies of scale arise if the company increases its size beyond this level. However, the optimal scale of utilities also appears to vary considerably between countries. Finally, we briefly consider the implications of our findings for water pricing and point to several directions for necessary future empirical research on the measurement of these economies, and explaining their cross country variation.

Vertical and horizontal scope economies in the regulated US electricity power industry

This paper investigates vertical economies between generation and distribution of electric power, and horizontal economies between different types of power generation in the U.S. electric utility industry. Our quadratic cost function model includes three generation output measures (hydro, nuclear and fossil fuels), which allows us to analyze the effect that generation mix has on vertical economies. Our results provide (sample mean) estimates of vertical economies of 8.1% and horizontal economies of 5.4%. An extensive sensitivity analysis is used to show how the scope measures vary across alternative model specifications and firm types. © 2012 Blackwell Publishing Ltd and the Editorial Board of The Journal of Industrial Economics.

Measuring economies of horizontal and vertical integration in the US electric power industry:how costly is unbundling?

This research employs econometric analysis on a cross section of American electricity companies in order to study the cost implications associated with unbundling the operations of integrated companies into vertically and/or horizontally separated companies. Focusing on the representative sample average firm, we find that complete horizontal and vertical disintegration resulting in the creation of separate nuclear, conventional, and hydro electric generation companies as well as a separate firm distributing power to final consumers, results in a statistically significant 13.5 percent increase in costs. Maintaining a horizontally integrated generator producing nuclear, conventional, and hydro electric generation while imposing vertical separation by creating a stand alone distribution company, results in a lower but still substantial and statistically significant cost penalty amounting to an 8.1 % increase in costs relative to a fully integrated structure. As these results imply that a vertically separated but horizontally integrated generation firm would need to reduce the costs of generation by 11% just to recoup the cost increases associated with vertical separation, even the costs associated with just vertical unbundling are quite substantial. Our paper is also the first academic paper we are aware of that systematically considers the impact of generation mix on vertical, horizontal, and overall scope economies. As a result, we are able to demonstrate that the estimated cost of unbundling in the electricity sector is substantially influenced by generation mix. Thus, for example, we find evidence of strong vertical integration economies between nuclear and conventional generation, but little evidence for vertical integration benefits between hydro generation and the distribution of power. In contrast, we find strong evidence suggesting the presence of substantial horizontal integration economies associated with the joint production of hydro generation with nuclear and/or conventional fossil fuel generation. These results are significant because they indicate that the cost of unbundling the electricity sector will differ substantially in different systems, meaning that a blanket regulatory policy with regard to the appropriateness of vertical and horizontal unbundling is likely to be inappropriate.