This paper presents a new normalization technique based on hyperbolic tangent function (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>), which is continuous and bounded everywhere, uniformly convergent in the whole domain. In this normalization method, the difference of each instance and the mean of a particular feature is divided by the square root of the standard deviation, which shows better classification performance on publicly available datasets. The proposed normalization is an alternative version of existing hyperbolic tangent function-based normalization (hypertan). The comparison with the existing hypertan normalization method is done on the basis of statistical entities, changes in the central behavior of data, entropy analysis, Cohen’s Kappa coefficient analysis and area under the curve using receiver operating characteristic curve. Also, the mathematical proof of the uniform convergence behavior of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> normalization function has been derived using Maclaurin series expansion and validated using convergence tests. Twenty-one publicly available datasets from different domains, feature sizes, instances and classes are selected to validate the performance of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> normalization technique. Shannon entropy has been used to investigate the changes in the feature entropies after hypertan and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> normalized datasets. The unbiased behavior of the proposed normalization method and hypertan normalization along with their significant difference has been investigated using Cohen’s Kappa coefficient and two statistical tests. Area under the receiver operating characteristic curve has been calculated for both the mapping functions. Entropy analysis and area under the receiver operating characteristic curve analysis validate the better performance of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> normalization function. Also, the performance of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$hypertan\_mod$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mi>y</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>_</mi> <mi>m</mi> <mi>o</mi> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> normalization is compared with the existing five normalization methods using classification and statistical analysis.

A continuous, bounded and uniformly converging hypertan-normalization approach for improved classification accuracy and feature entropy

期刊论文
发表日期：

2025年12月9日

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Mannat Mand,
Birmohan Singh,
Vijay Kumar Kukreja

摘要

This paper presents a new normalization technique based on hyperbolic tangent function ( $hypertan\_mod$ $h y p e r t a n_m o d$ ), which is continuous and bounded everywhere, uniformly convergent in the whole domain. In this normalization method, the difference of each instance and the mean of a particular feature is divided by the square root of the standard deviation, which shows better classification performance on publicly available datasets. The proposed normalization is an alternative version of existing hyperbolic tangent function-based normalization (hypertan). The comparison with the existing hypertan normalization method is done on the basis of statistical entities, changes in the central behavior of data, entropy analysis, Cohen’s Kappa coefficient analysis and area under the curve using receiver operating characteristic curve. Also, the mathematical proof of the uniform convergence behavior of the $hypertan\_mod$ $h y p e r t a n_m o d$ normalization function has been derived using Maclaurin series expansion and validated using convergence tests. Twenty-one publicly available datasets from different domains, feature sizes, instances and classes are selected to validate the performance of $hypertan\_mod$ $h y p e r t a n_m o d$ normalization technique. Shannon entropy has been used to investigate the changes in the feature entropies after hypertan and $hypertan\_mod$ $h y p e r t a n_m o d$ normalized datasets. The unbiased behavior of the proposed normalization method and hypertan normalization along with their significant difference has been investigated using Cohen’s Kappa coefficient and two statistical tests. Area under the receiver operating characteristic curve has been calculated for both the mapping functions. Entropy analysis and area under the receiver operating characteristic curve analysis validate the better performance of $hypertan\_mod$ $h y p e r t a n_m o d$ normalization function. Also, the performance of $hypertan\_mod$ $h y p e r t a n_m o d$ normalization is compared with the existing five normalization methods using classification and statistical analysis.