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.

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.

A shape-based decomposition of the yield adjustment term in the arbitrage-free Nelson and Siegel (AFNS) model of the yield curve

The appealing feature of the arbitrage-free Nelson-Siegel model of the yield curve is the ability to capture movements in the yield curve through readily interpretable shifts in its level, slope or curvature, all within a dynamic arbitrage-free framework. To ensure that the level, slope and curvature factors evolve so as not to admit arbitrage, the model introduces a yield-adjustment term. This paper shows how the yield-adjustment term can also be decomposed into the familiar level, slope and curvature elements plus some additional readily interpretable shape adjustments. This means that, even in an arbitrage-free setting, it continues to be possible to interpret movements in the yield curve in terms of level, slope and curvature influences. © 2014 © 2014 Taylor & Francis.

Integer Points Close to a Smooth Curve

∗ This research is partially supported by the Bulgarian National Science Fund under contract MM-403/9

Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.

Structural Analysis of Contours as the Sequences of the Digital Straight Segments and of the Digital Curve Arcs

Recognition of the object contours in the image as sequences of digital straight segments and/or digital curve arcs is considered in this article. The definitions of digital straight segments and of digital curve arcs are proposed. The methods and programs to recognize the object contours are represented. The algorithm to recognize the digital straight segments is formulated in terms of the growing pyramidal networks taking into account the conceptual model of memory and identification (Rabinovich [4]).

Key Agreement Protocol Using Elliptic Curve Matrix Power Function

* Work is partially supported by the Lithuanian State Science and Studies Foundation.

On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

2000 Mathematics Subject Classification: 33D60, 26A33, 33C60

Cauchy-Type Problem for Diffusion-Wave Equation with the Riemann-Liouville Partial Derivative

2000 Mathematics Subject Classification: 35A15, 44A15, 26A33

On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials

Mathematics Subject Classification: 26A33

Inversion Formulas for the q-Riemann-Liouville and q-Weyl Transforms Using Wavelets

Mathematics Subject Classification: 42A38, 42C40, 33D15, 33D60

Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.