Precision-recall operating characteristic (P-ROC) curves in imprecise environments

Traditionally, machine learning algorithms have been evaluated in applications where assumptions can be reliably made about class priors and/or misclassification costs. In this paper, we consider the case of imprecise environments, where little may be known about these factors and they may well vary significantly when the system is applied. Specifically, the use of precision-recall analysis is investigated and compared to the more well known performance measures such as error-rate and the receiver operating characteristic (ROC). We argue that while ROC analysis is invariant to variations in class priors, this invariance in fact hides an important factor of the evaluation in imprecise environments. Therefore, we develop a generalised precision-recall analysis methodology in which variation due to prior class probabilities is incorporated into a multi-way analysis of variance (ANOVA). The increased sensitivity and reliability of this approach is demonstrated in a remote sensing application.

Learning curves for Gaussian processes models: fluctuations and universality

Based on a statistical mechanics approach, we develop a method for approximately computing average case learning curves and their sample fluctuations for Gaussian process regression models. We give examples for the Wiener process and show that universal relations (that are independent of the input distribution) between error measures can be derived.

A variational approach to learning curves

We combine the replica approach from statistical physics with a variational approach to analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive approximative relations between empirical error measures, the generalization error and the posterior variance.

Growth curves of four crustose lichens

The growth curves of four common species of crustose lichens, viz., Buellia aethalea (Ach.) Th. Fr., Lecidea tumida Massai., Rhizocarpon geographicum (L.) DC., and Rhizocarpon reductum Th. Fr. were studied at a site in south Gwynedd, north Wales, UK. Radial growth rates (RGR, mm 1.5 yr-1) were greatest in thalli of R. reductum and least in R. geographicum. Variation in RGR between thalli was greater in B. aethalea and L. tumida than in the species of Rhizocarpon. The relationship between growth rate and thallus diameter was not asymptotic; RGR increasing in smaller thalli to a maximum and then declining in larger diameter thalli. A polynomial curve was fitted to the data; the growth curves being fitted best by a second-order (quadratic) curve, the best fit to this model being shown by B. aethalea. A significant linear regression with a negative slope was also fitted to the growth of the larger thalli of each species. The data suggest that the growth curves of the four crustose lichens differ significantly from the asymptotic curves of foliose lichen species. A phase of declining RGR in larger thalli appears to be characteristic of crustose lichens and is consistent with data from lichenometric studies.

Optimizing measurement of subjective amplitude of accommodation with defocus curves

PURPOSE: To determine whether letter sequences and/or lens-presentation order should be randomized when measuring defocus curves and to assess the most appropriate criterion for calculating the subjective amplitude of accommodation (AoA) from defocus curves. SETTING: Eye Clinic, School of Life & Health Sciences, Aston University, Birmingham, United Kingdom. METHODS: Defocus curves (from +3.00 diopters [D] to -3.00 D in 0.50 D steps) for 6 possible combinations of randomized or nonrandomized letter sequences and/or lens-presentation order were measured in a random order in 20 presbyopic subjects. Subjective AoA was calculated from the defocus curves by curve fitting using various published criteria, and each was correlated to subjective push-up AoA. Objective AoA was measured for comparison of blur tolerance and pupil size. RESULTS: Randomization of lens-presentation order and/or letter sequences, or lack of, did not affect the measured defocus curves (P>.05, analysis of variance). The range of defocus that maintains highest achievable visual acuity (allowing for variability of repeated measurement) was better correlated to (r = 0.84) and agreed best with ( 0.50 D) subjective push-up AoA than any other relative or absolute acuity criterion used in previous studies. CONCLUSIONS: Nonrandomized letters and lens presentation on their own did not affect subjective AoA measured by defocus curves, although their combination should be avoided. Quantification of subjective AoA from defocus curves should be standardized to the range of defocus that maintains the best achievable visual acuity.

Multifocal intraocular lens differentiation using defocus curves

Purpose: To determine the most appropriate analysis technique for the differentiation of multifocal intraocular lens (MIOL) designs using defocus curve assessment of visual capability.Methods:Four groups of fifteen subjects were implanted bilaterally with either monofocal intraocular lenses, refractive MIOLs, diffractive MIOLs, or a combination of refractive and diffractive MIOLs. Defocus curves between -5.0D and +1.5D were evaluated using an absolute and relative depth-of-focus method, the direct comparison method and a new 'Area-of-focus' metric. The results were correlated with a subjective perception of near and intermediate vision. Results:Neither depth-of-focus method of analysis were sensitive enough to differentiate between MIOL groups (p>0.05). The direct comparison method indicated that the refractive MIOL group performed better at +1.00, -1.00 and -1.50 D and worse at -3.00, -3.50, -4.00 and -5.00D compared to the diffractive MIOL group (p

Size distribution curves of senile plaques in patients with Alzheimer’s disease: do plaques grow according to the log-normal model?

The size frequency distributions of diffuse, primitive and cored senile plaques (SP) were studied in single sections of the temporal lobe from 10 patients with Alzheimer’s disease (AD). The size distribution curves were unimodal and positively skewed. The size distribution curve of the diffuse plaques was shifted towards larger plaques while those of the neuritic and cored plaques were shifted towards smaller plaques. The neuritic/diffuse plaque ratio was maximal in the 11 – 30 micron size class and the cored/ diffuse plaque ratio in the 21 – 30 micron size class. The size distribution curves of the three types of plaque deviated significantly from a log-normal distribution. Distributions expressed on a logarithmic scale were ‘leptokurtic’, i.e. with excess of observations near the mean. These results suggest that SP in AD grow to within a more restricted size range than predicted from a log-normal model. In addition, there appear to be differences in the patterns of growth of diffuse, primitive and cored plaques. If neuritic and cored plaques develop from earlier diffuse plaques, then smaller diffuse plaques are more likely to be converted to mature plaques.

A comparison of the growth curves of the foliose lichen Parmelia conspersa determined by a cross-sectional study and by direct measurement

Changes in the radial growth rate (RGR mm/yr) through life were studied in thalli of the foliose lichen Parmelia conspersa by two methods: (1) a cross-sectional study (Study A) in which the RGR was measured in 60 thalli from 0.2 to 13 cm in diameter, and (2) by radial growth measurements over 4.5 years of fragments, consisting of a single major lobe, which were removed from large thalli and glued to pieces of slate (Study B). Both studies suggested there was a phase of increasing RGR in small thalli followed by a more constant phase, the latter beginning at approximately a thallus radius of 6-8 mm. However, in Study B significantly increased RGR was observed during the second 6-month growth period. This phase of growth was more likely to be due to an increase in lobe width than to an effect of climate. In addition, a lobe in a large thallus with both adjacent lobes removed significantly increased in width over 1 year compared with control lobes. These results suggest that (1) mean lobe width in a thallus may be determined by the intensity of marginal competition between adjacent lobes, and (2) changes in lobe width during the life of a lichen thallus may be a factor determining the establishment of the linear phase of growth in foliose lichens. © 1992.

Neural population coding is optimized by discrete tuning curves

The sigmoidal tuning curve that maximizes the mutual information for a Poisson neuron, or population of Poisson neurons, is obtained. The optimal tuning curve is found to have a discrete structure that results in a quantization of the input signal. The number of quantization levels undergoes a hierarchy of phase transitions as the length of the coding window is varied. We postulate, using the mammalian auditory system as an example, that the presence of a subpopulation structure within a neural population is consistent with an optimal neural code.

Is randomisation necessary for measuring defocus curves in pre-presbyopes?

Purpose: Defocus curves are used to evaluate the subjective range of clear vision of presbyopic corrections such as in eyes implanted with accommodating intraocular lenses (IOLs). This study determines whether letter sequences and/or lens presentation order ought to be randomised when measuring defocus curves. Methods: Defocus curves (range +2.00DS to -2.00DS) were measured on 18 pre-presbyopic subjects (mean age 24.1 ± 4.2 years) for six combinations of sequential or randomised positive or negative lens progression and non-randomised or randomised letter sequences. The letters were presented on a computerised logMAR chart at 6 m. Results: Overall there was a statistically significant difference between the six combinations (ANOVA, p < 0.05) attributable to the combination of non-randomised letters with non-randomised lens progression from negative to positive defocus (p < 0.01). There was no statistically significant difference in defocus curve measurements if both letters and lens order were randomised compared to if only one of these variables was randomised (p > 0.05). Non-randomised letters, with a sequential lens progression from negative to positive, was significantly different to all other combinations when compared individually (Student's T-test, p < 0.003 on all comparisons), and was confirmed as the sole source of the overall significant difference. There was no statistically significant difference if both lens presentation order and letter sequences were randomised compared to if only one or the other of these variables was randomised. Conclusion: Non-randomised letters and non-randomised lens progression on their own did not affect the subjective amplitude of accommodation as measured by defocus curves, although their combination should be avoided. © 2007 British Contact Lens Association.

Approximation of Experimental Data by Bezier Curves

Very often the experimental data are the realization of the process, fully determined by some unknown function, being distorted by hindrances. Treatment and experimental data analysis are substantially facilitated, if these data to represent as analytical expression. The experimental data processing algorithm and the example of using this algorithm for spectrographic analysis of oncologic preparations of blood is represented in this article.

On the Automorphism Groups of some AG-Codes Based on Ca;b Curves

*Partially supported by NATO.

On The Brill-noether Theory of Spanned Vector Bundles on Smooth Curves

Here we study the integers (d, g, r) such that on a smooth projective curve of genus g there exists a rank r stable vector bundle with degree d and spanned by its global sections.