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.

Objective analysis of toric intraocular lens rotation and centration

PURPOSE: To assess the repeatability of an objective image analysis technique to determine intraocular lens (IOL) rotation and centration. SETTING: Six ophthalmology clinics across Europe. METHODS: One-hundred seven patients implanted with Akreos AO aspheric IOLs with orientation marks were imaged. Image quality was rated by a masked observer. The axis of rotation was determined from a line bisecting the IOL orientation marks. This was normalized for rotation of the eye between visits using the axis bisecting 2 consistent conjunctival vessels or iris features. The center of ovals overlaid to circumscribe the IOL optic edge and the pupil or limbus were compared to determine IOL centration. Intrasession repeatability was assessed in 40 eyes and the variability of repeated analysis examined. RESULTS: Intrasession rotational stability of the IOL was ±0.79 degrees (SD) and centration was ±0.10 mm horizontally and ±0.10 mm vertically. Repeated analysis variability of the same image was ±0.70 degrees for rotation and ±0.20 mm horizontally and ±0.31 mm vertically for centration. Eye rotation (absolute) between visits was 2.23 ± 1.84 degrees (10%>5 degrees rotation) using one set of consistent conjunctival vessels or iris features and 2.03 ± 1.66 degrees (7%>5 degrees rotation) using the average of 2 sets (P =.13). Poorer image quality resulted in larger apparent absolute IOL rotation (r =-0.45,P<.001). CONCLUSIONS: Objective analysis of digital retroillumination images allows sensitive assessment of IOL rotation and centration stability. Eye rotation between images can lead to significant errors if not taken into account. Image quality is important to analysis accuracy.

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

Independent detectors for expansion and rotation, and for orthogonal components of deformation

It is well known that optic flow - the smooth transformation of the retinal image experienced by a moving observer - contains valuable information about the three-dimensional layout of the environment. From psychophysical and neurophysiological experiments, specialised mechanisms responsive to components of optic flow (sometimes called complex motion) such as expansion and rotation have been inferred. However, it remains unclear (a) whether the visual system has mechanisms for processing the component of deformation and (b) whether there are multiple mechanisms that function independently from each other. Here, we investigate these issues using random-dot patterns and a forced-choice subthreshold summation technique. In experiment 1, we manipulated the size of a test region that was permitted to contain signal and found substantial spatial summation for signal components of translation, expansion, rotation, and deformation embedded in noise. In experiment 2, little or no summation was found for the superposition of orthogonal pairs of complex motion patterns (eg expansion and rotation), consistent with probability summation between pairs of independent detectors. Our results suggest that optic-flow components are detected by mechanisms that are specialised for particular patterns of complex motion.

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.

A study of the drying of single droplets in free-flight

The literature relating to evaporation from single droplets of pure liquids and the drying of solution and slurry droplets, and of droplet sprays has been reviewed. The heat and mass transfer rates for individual droplets suspended in free-flight, were investigated using a specially-designed vertical wind tunnel, to simulate conditions in a spray drier. The technique represented a unique alternative method for investigating evaporation from unrestrained single droplets with variable residence times. The experiments covered droplets of pure liquid allowbreak (water, isopropanol) allowbreak and of significantly different solutions (sucrose, potassium sulphate) over a range of temperatures of 37oC to 97oC, initial concentrations of 5 to 40wt/wt% , and initial drop sizes of 2.8 to 4.6mm. Drop behaviour was recorded photographically and dried particles were examined by Scanning Electron Microscopy. Correlations were developed for mass transfer coefficients for pure water droplets in free-flight; (i) experiencing oscillations, rotation and deformation, Sh = -105 + 3.9 [Ta - Td/Tamb]0.18Re0.5Sc033 for Re approx. > 1380 (ii) when these movements had ceased or diminished, Sh = 2.0 + 0.71 [Ta - Td/Tamb]0.18Re0.5Sc033 for Re approx. < 1060. Data for isopropanol drops were correlated resonably well by these equations. The heat transfer data showed a similar transition range. The drying rate curves for drops of sucrose and potassium sulphate solution exhibited three distinct stages; an initial increase in the drying rate as drop temperature reduced to the wet-bulb temperature, a short constant-rate period and a falling-rate period characterised by formation of a crust which controlled the mass transfer rate. Due to drop perturbation the rates in the high Re number region were up to 5 times greater than predicted from theory for spherical droplets. In the case of sucrose solution a `skin' formed over the drop surface prior to crust formation. This provided an additional resistance to mass transfer and resulted in extended drying times and a smooth crust of low porosity. The relevance of the results to practical spray drying operations is discussed.

The Kinematics of interference mechanisms in certain machining operations

The mechanism of "Helical Interference" in milled slots is examined and a coherent theory for the geometry of such surfaces is presented. An examination of the relevant literature shows a fragmented approach to the problem owing to its normally destructive nature, so a complete analysis is developed for slots of constant lead, thus giving a united and exact theory for many different setting parameters and a range of cutter shapes. For the first time, a theory is developed to explain the "Interference Surface" generated in variable lead slots for cylindrical work and attention is drawn to other practical surfaces, such as cones, where variable leads are encountered. Although generally outside the scope of this work, an introductory analysis of these cases is considered in order to develop the cylindrical theory. Special emphasis is laid upon practical areas where the interference mechanism can be used constructively and its application as the rake face of a cutting tool is discussed. A theory of rake angle for such cutting tools is given for commonly used planes, and relative variations in calculated rake angle between planes is examined. Practical tests are conducted to validate both constant lead and variable lead theories and some design improvements to the conventional dividing head are suggested in order to manufacture variable lead workpieces, by use of a "superposed" rotation. A prototype machine is manufactured and its kinematic principle given for both linear and non-linearly varying superposed rotations. Practical workpieces of the former type are manufactured and compared with analytical predictions,while theoretical curves are generated for non-linear workpieces and then compared with those of linear geometry. Finally suggestions are made for the application of these principles to the manufacture of spiral bevel gears, using the "Interference Surface" along a cone as the tooth form.

Multiwavelength fiber laser with fine adjustment, based on nonlinear polarization rotation and birefringence fiber filter

We have proposed and demonstrated a multiwavelength fiber laser based on nonlinear polarization rotation (NPR). The mechanism for stable room-temperature multiwavelength operation contributes to the ability of the intensity-dependent loss in NPR to effectively alleviate mode competition. In addition, through tuning the birefringence fiber filter, the lasing wavelength can be accurately tuned in the free spectrum range of the in-line periodic filter.