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.

Testing term structure estimation methods:evidence from the UK STRIPs market

Prices and yields of UK government zero-coupon bonds are used to test alternative yield curve estimation models. Zero-coupon bonds permit a more pure comparison, as the models are providing only the interpolation service and also not making estimation feasible. It is found that better yield curves estimates are obtained by fitting to the yield curve directly rather than fitting first to the discount function. A simple procedure to set the smoothness of the fitted curves is developed, and a positive relationship between oversmoothness and the fitting error is identified. A cubic spline function fitted directly to the yield curve provides the best overall balance of fitting error and smoothness, both along the yield curve and within local maturity regions.

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.

Computer aided design and manufacture for three dimensional milling

Advances in both computer technology and the necessary mathematical models capable of capturing the geometry of arbitarily shaped objects has led to the development in this thesis of a surface generation package called 'IBSCURF' aimed at providing a more economically viable solution to free-form surface manufacture. A suit of computer programs written in FORTRAN 77 has been developed to provide computer aids for every aspect of work in designing and machining free-form surfaces. A vector-valued parametric method was used for shape description and a lofting technique employed for the construction of the surface. The development of the package 'IBSCURF' consists of two phases. The first deals with CAD. The design process commences in defining the cross-sections which are represented by uniform B-spline curves as approximations to give polygons. The order of the curve and the position and number of the polygon vertices can be used as parameters for the modification to achieve the required curves. When the definitions of the sectional curves is complete, the surface is interpolated over them by cubic cardinal splines. To use the CAD function of the package to design a mould for a plastic handle, a mathematical model was developed. To facilitate the integration of design and machining using the mathematical representation of the surface, the second phase of the package is concerned with CAM which enables the generation of tool offset positions for ball-nosed cutters and a general post-processor has been developed which automatically generates NC tape programs for any CNC milling machine. The two phases of these programs have been successfully implemented, as a CAD/CAM package for free-form surfaces on the VAX 11/750 super-minicomputer with graphics facilities for displaying drawings interactively on the terminal screen. The development of this package has been beneficial in all aspects of design and machining of free form surfaces.

From cardinal spline wavelet bases to highly coherent dictionaries

Wavelet families arise by scaling and translations of a prototype function, called the mother wavelet. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a dictionary, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterize the correlation of the dictionary elements by measuring their 'coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation.

Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval

Non-uniform B-spline dictionaries on a compact interval are discussed in the context of sparse signal representation. For each given partition, dictionaries of B-spline functions for the corresponding spline space are built up by dividing the partition into subpartitions and joining together the bases for the concomitant subspaces. The resulting slightly redundant dictionaries are composed of B-spline functions of broader support than those corresponding to the B-spline basis for the identical space. Such dictionaries are meant to assist in the construction of adaptive sparse signal representation through a combination of stepwise optimal greedy techniques.

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.

Laminar distribution of the pathological changes in sporadic frontotemporal lobar degeneration with transactive response (TAR) DNA-binding protein of 43kDa (TDP-43) proteinopathy:a quantitative study using polynomial curve fitting

Aims: Previous data suggest heterogeneity in laminar distribution of the pathology in the molecular disorder frontotemporal lobar degeneration (FTLD) with transactive response (TAR) DNA-binding protein of 43kDa (TDP-43) proteinopathy (FTLD-TDP). To study this heterogeneity, we quantified the changes in density across the cortical laminae of neuronal cytoplasmic inclusions, glial inclusions, neuronal intranuclear inclusions, dystrophic neurites, surviving neurones, abnormally enlarged neurones, and vacuoles in regions of the frontal and temporal lobe. Methods: Changes in density of histological features across cortical gyri were studied in 10 sporadic cases of FTLD-TDP using quantitative methods and polynomial curve fitting. Results: Our data suggest that laminar neuropathology in sporadic FTLD-TDP is highly variable. Most commonly, neuronal cytoplasmic inclusions, dystrophic neurites and vacuolation were abundant in the upper laminae and glial inclusions, neuronal intranuclear inclusions, abnormally enlarged neurones, and glial cell nuclei in the lower laminae. TDP-43-immunoreactive inclusions affected more of the cortical profile in longer duration cases; their distribution varied with disease subtype, but was unrelated to Braak tangle score. Different TDP-43-immunoreactive inclusions were not spatially correlated. Conclusions: Laminar distribution of pathological features in 10 sporadic cases of FTLD-TDP is heterogeneous and may be accounted for, in part, by disease subtype and disease duration. In addition, the feedforward and feedback cortico-cortical connections may be compromised in FTLD-TDP. © 2012 The Authors. Neuropathology and Applied Neurobiology © 2012 British Neuropathological Society.