A free group of piecewise linear transformations

2011 ◽  
Vol 125 (2) ◽  
pp. 141-146
Author(s):  
Grzegorz Tomkowicz
Keyword(s):  
Free Group ◽  
Piecewise Linear ◽  
Linear Transformations

scholarly journals Interconversion of Plasma Free Thyroxine Values from Assay Platforms with Different Reference Intervals Using Linear Transformation Methods

Biology ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 45
Author(s):  
Fanwen Meng ◽  
Jacqueline Jonklaas ◽  
Melvin Khee-Shing Leow
Keyword(s):  
Linear Transformation ◽  
Piecewise Linear ◽  
Equilibrium Dialysis ◽  
Accurate Method ◽  
Reference Intervals ◽  
Thyroid Stimulating Hormone ◽  
Free Thyroxine ◽  
Thyroid Function Tests ◽  
Linear Transformations ◽  
Transformation Methods

Clinicians often encounter thyroid function tests (TFT) comprising serum/plasma free thyroxine (FT4) and thyroid stimulating hormone (TSH) measured using different assay platforms during the course of follow-up evaluations which complicates reliable comparison and interpretation of TFT changes. Although interconversion between concentration units is straightforward, the validity of interconversion of FT4/TSH values from one assay platform to another with different reference intervals remains questionable. This study aims to establish an accurate and reliable methodology of interconverting FT4 by any laboratory to an equivalent FT4 value scaled to a reference range of interest via linear transformation methods. As a proof-of-concept, FT4 was simultaneously assayed by direct analog immunoassay, tandem mass spectrometry and equilibrium dialysis. Both linear and piecewise linear transformations proved relatively accurate for FT4 inter-scale conversion. Linear transformation performs better when FT4 are converted from a more accurate to a less accurate assay platform. The converse is true, whereby piecewise linear transformation is superior to linear transformation when converting values from a less accurate method to a more robust assay platform. Such transformations can potentially apply to other biochemical analytes scale conversions, including TSH. This aids interpretation of TFT trends while monitoring the treatment of patients with thyroid disorders.

Induction of Similarity Measures and Medical Diagnosis Support Rules through Separable, Linear Data Transformations

2006 ◽  
Vol 45 (02) ◽  
pp. 200-203 ◽  
Author(s):  
L. Bobrowski
Keyword(s):  
Medical Diagnosis ◽  
Euclidean Distance ◽  
Piecewise Linear ◽  
Similarity Measures ◽  
Case Based Reasoning ◽  
Data Sets ◽  
Linear Transformations ◽  
Data Transformations ◽  
Criterion Functions ◽  
Learning Data

Summary Objectives: To improve the medical diagnosis support rules based on comparisons of diagnosed patients with similar cases (precedents) archived in a clinical database. The case-based reasoning (CBR) or the nearest neighbors (K-NN) classifications, which operate on referencing (learning) data sets, belong to this scheme. Methods: Inducing similarity measure through special linear transformations of the referencing sets aimed at the best separation of these sets. Designing separable transformations can be based on dipolar models and minimization of the convex and piecewise linear (CPL) criterion functions in accordance with the basis exchange algorithm. Results: Separable linear transformations allow for some data sets to decrease the error rate of the K-NNclassification rule based on the Euclidean distance. Such results can be seen on the example of data sets taken from the Heparsystem of diagnosis support. Conclusions: Medical diagnosis support based on the CBRor the K-NNrules can be improved through separable transformations of the referencing sets.

scholarly journals Standard deviation of recurrence times for piecewise linear transformations

2017 ◽  
Vol 10 (01) ◽  
pp. 1750009
Author(s):  
Mimoon Ismael ◽  
Rodney Nillsen ◽  
Graham Williams
Keyword(s):  
Standard Deviation ◽  
Dynamical Systems ◽  
Piecewise Linear ◽  
Recurrence Formula ◽  
Recurrence Time ◽  
Linear Transformations ◽  
Recurrence Times ◽  
Borel Set ◽  
Bounded Interval ◽  
The Mean

This paper is concerned with dynamical systems of the form [Formula: see text], where [Formula: see text] is a bounded interval and [Formula: see text] comes from a class of measure-preserving, piecewise linear transformations on [Formula: see text]. If [Formula: see text] is a Borel set and [Formula: see text], the Poincaré recurrence time of [Formula: see text] relative to [Formula: see text] is defined to be the minimum of [Formula: see text], if the minimum exists, and [Formula: see text] otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.

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