scholarly journals Local Convexity Shape-Preserving Data Visualization by Spline Function

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd Majid ◽  
Mohd Nain Hj Awang ◽  
Jamaludin Md Ali
Keyword(s):  
Error Bound ◽  
Data Visualization ◽  
Free Parameter ◽  
Spline Function ◽  
Convex Curve ◽  
Local Convexity ◽  
Shape Preserving ◽  
User Choice ◽  
Free Parameters ◽  
Cubic Function

The main purpose of this paper is the visualization of convex data that results in a smooth, pleasant, and interactive convexity-preserving curve. The rational cubic function with three free parameters is constructed to preserve the shape of convex data. The free parameters are arranged in a way that two of them are left free for user choice to refine the convex curve as desired, and the remaining one free parameter is constrained to preserve the convexity everywhere. Simple data-dependent constraints are derived on one free parameter, which guarantee to preserve the convexity of curve. Moreover, the scheme under discussion is, C1 flexible, simple, local, and economical as compared to existing schemes. The error bound for the rational cubic function is O(h3).

scholarly journals Clustering Single-Cell RNA-Seq Data with Regularized Gaussian Graphical Model

Genes ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 311
Author(s):  
Zhenqiu Liu
Keyword(s):  
Single Cell ◽  
Free Parameter ◽  
Graphical Model ◽  
Expression Patterns ◽  
Information Criterion ◽  
Log P ◽  
Rna Seq ◽  
Clustering Methods ◽  
Wide Range ◽  
Free Parameters

Single-cell RNA-seq (scRNA-seq) is a powerful tool to measure the expression patterns of individual cells and discover heterogeneity and functional diversity among cell populations. Due to variability, it is challenging to analyze such data efficiently. Many clustering methods have been developed using at least one free parameter. Different choices for free parameters may lead to substantially different visualizations and clusters. Tuning free parameters is also time consuming. Thus there is need for a simple, robust, and efficient clustering method. In this paper, we propose a new regularized Gaussian graphical clustering (RGGC) method for scRNA-seq data. RGGC is based on high-order (partial) correlations and subspace learning, and is robust over a wide-range of a regularized parameter λ. Therefore, we can simply set λ=2 or λ=log(p) for AIC (Akaike information criterion) or BIC (Bayesian information criterion) without cross-validation. Cell subpopulations are discovered by the Louvain community detection algorithm that determines the number of clusters automatically. There is no free parameter to be tuned with RGGC. When evaluated with simulated and benchmark scRNA-seq data sets against widely used methods, RGGC is computationally efficient and one of the top performers. It can detect inter-sample cell heterogeneity, when applied to glioblastoma scRNA-seq data.

scholarly journals Local Convexity-PreservingC2Rational Cubic Spline for Convex Data

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd Majid ◽  
Jamaludin Md. Ali
Keyword(s):  
Shape Parameter ◽  
Cubic Spline ◽  
Convex Curve ◽  
Shape Feature ◽  
Local Convexity ◽  
Shape Parameters ◽  
Numerical Examples ◽  
2D Data ◽  
Industrial Demand ◽  
Made In

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data usingC2rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

scholarly journals KAON WEAK DECAYS IN CHIRAL THEORIES

1999 ◽  
Vol 14 (19) ◽  
pp. 1273-1285 ◽  
Author(s):  
M. D. SCADRON
Keyword(s):  
Perturbation Theory ◽  
Free Parameter ◽  
Sigma Model ◽  
Axial Current ◽  
Linear Sigma Model ◽  
Pole Model ◽  
Chiral Perturbation ◽  
Weak Decays ◽  
Partial Conservation ◽  
Free Parameters

The ten nonleptonic weak decays K→2π, K→ 3π, KL→2γ, KS→2γ, KL→π02γ, are predicted for a chiral pole model based on the linear sigma model theory which automatically satisfies the partial conservation of axial current (PCAC) hypothesis. These predictions, agreeing with data to the 5% level and containing no or at most one free parameter, are compared with the results of chiral perturbation theory (ChPT). The latter ChPT approach to one-loop level is known to contain at least four free parameters and then predicts a KL→π0γγ rate which is 60% shy of the experimental value.

Classification and Recursion Operators of Dark Burgers’ Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 175-180 ◽  
Author(s):  
Mei-Dan Chen ◽  
Biao Li
Keyword(s):  
Symbolic Computation ◽  
Free Parameter ◽  
Burgers Equation ◽  
Differential Polynomial ◽  
Higher Order ◽  
Recursion Operators ◽  
Burgers Equations ◽  
Dual Symmetry ◽  
Free Parameters

AbstractWith the help of symbolic computation, two types of complete scalar classification for dark Burgers’ equations are derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark Burgers’ systems; so some special equations including symmetry equation and dual symmetry equation are obtained by selecting the free parameter. Furthermore, two kinds of recursion operators for these dark Burgers’ equations are constructed by two direct assumption methods.

scholarly journals Data Visualization Using Rational Trigonometric Spline

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Uzma Bashir ◽  
Jamaludin Md. Ali
Keyword(s):  
Data Visualization ◽  
The Other ◽  
Shape Features ◽  
Shape Parameters ◽  
Shape Preserving ◽  
Numerical Examples ◽  
Data Points ◽  
Trigonometric Spline ◽  
The Given

This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive, monotone, and constrained curve interpolating schemes, by using aC1piecewise rational cubic trigonometric spline with four shape parameters, are developed. Two of these shape parameters are constrained and the other two are set free to preserve the inherited shape features of the data as well as to control the shape of the curve. Numerical examples are given to illustrate the worth of the work.

scholarly journals Monotone Data Visualization Using Rational Trigonometric Spline Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Farheen Ibraheem ◽  
Maria Hussain ◽  
Malik Zawwar Hussain
Keyword(s):  
Data Visualization ◽  
Spline Interpolation ◽  
Trigonometric Functions ◽  
Rectangular Patch ◽  
Surface Data ◽  
Trigonometric Spline ◽  
Monotone Data ◽  
Cubic Function

Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively. The rational cubic function has four parameters in each subinterval, while the rational bicubic partially blended function has eight parameters in each rectangular patch. The monotonicity of curve and surface data is retained by developing constraints on some of these parameters in description of rational cubic and bicubic trigonometric functions. The remaining parameters are kept free to modify the shape of curve and surface if required. The developed algorithm is verified mathematically and demonstrated graphically.

scholarly journals Shape preserving rational bi-cubic function

2012 ◽  
Vol 13 (3) ◽  
pp. 147-154
Author(s):  
Malik Zawwar Hussain ◽  
Maria Hussain ◽  
Madiha Amjad
Keyword(s):  
Shape Preserving ◽  
Cubic Function
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