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.

The perceptual organization of sine-wave speech under competitive conditions

Speech comprises dynamic and heterogeneous acoustic elements, yet it is heard as a single perceptual stream even when accompanied by other sounds. The relative contributions of grouping “primitives” and of speech-specific grouping factors to the perceptual coherence of speech are unclear, and the acoustical correlates of the latter remain unspecified. The parametric manipulations possible with simplified speech signals, such as sine-wave analogues, make them attractive stimuli to explore these issues. Given that the factors governing perceptual organization are generally revealed only where competition operates, the second-formant competitor (F2C) paradigm was used, in which the listener must resist competition to optimize recognition [Remez et al., Psychol. Rev. 101, 129-156 (1994)]. Three-formant (F1+F2+F3) sine-wave analogues were derived from natural sentences and presented dichotically (one ear = F1+F2C+F3; opposite ear = F2). Different versions of F2C were derived from F2 using separate manipulations of its amplitude and frequency contours. F2Cs with time-varying frequency contours were highly effective competitors, regardless of their amplitude characteristics. In contrast, F2Cs with constant frequency contours were completely ineffective. Competitor efficacy was not due to energetic masking of F3 by F2C. These findings indicate that modulation of the frequency, but not the amplitude, contour is critical for across-formant grouping.

Binary sparse nonnegative matrix factorization

This paper presents a fast part-based subspace selection algorithm, termed the binary sparse nonnegative matrix factorization (B-SNMF). Both the training process and the testing process of B-SNMF are much faster than those of binary principal component analysis (B-PCA). Besides, B-SNMF is more robust to occlusions in images. Experimental results on face images demonstrate the effectiveness and the efficiency of the proposed B-SNMF.

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.

Polynomial function intervals for floating-point software verification

The focus of our work is the verification of tight functional properties of numerical programs, such as showing that a floating-point implementation of Riemann integration computes a close approximation of the exact integral. Programmers and engineers writing such programs will benefit from verification tools that support an expressive specification language and that are highly automated. Our work provides a new method for verification of numerical software, supporting a substantially more expressive language for specifications than other publicly available automated tools. The additional expressivity in the specification language is provided by two constructs. First, the specification can feature inclusions between interval arithmetic expressions. Second, the integral operator from classical analysis can be used in the specifications, where the integration bounds can be arbitrary expressions over real variables. To support our claim of expressivity, we outline the verification of four example programs, including the integration example mentioned earlier. A key component of our method is an algorithm for proving numerical theorems. This algorithm is based on automatic polynomial approximation of non-linear real and real-interval functions defined by expressions. The PolyPaver tool is our implementation of the algorithm and its source code is publicly available. In this paper we report on experiments using PolyPaver that indicate that the additional expressivity does not come at a performance cost when comparing with other publicly available state-of-the-art provers. We also include a scalability study that explores the limits of PolyPaver in proving tight functional specifications of progressively larger randomly generated programs. © 2014 Springer International Publishing Switzerland.

Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile

The fabrication of submicron-height sine-like relief of a trifocal diffractive zone plate using a nanoimprinting technique is studied. The zone plate is intended for use in combined trifocal diffractive-refractive lenses and provides the possibility to form trifocal intraocular lenses with predetermined light intensity distribution between foci. The optical properties of the designed zone plate having the optical powers 3 D, 0, -3D in the three main diffraction orders are theoretically and experimentally investigated. The results of the theoretical investigations are in good agreement with experimental measurements. The effects of the pupil size (lens diameter) as well as the wavelength-dependent behavior of the zone plate are also discussed.