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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
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Assisted Selection of Biomarkers by Linear Discriminant Analysis Effect Size LEfSe in Microbiome Data
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Complexity-reduced scheme for feature extraction with linear discriminant analysis.

Yuxi Hou, Iickho Song, Hwang-Ki Min

    IEEE Transactions on Neural Networks and Learning Systems
    |May 9, 2014
    PubMed
    Summary

    Linear Discriminant Analysis (LDA) struggles with small sample sizes (SSS). Null-space-based LDA (NLDA) improves performance but has high complexity. This study introduces a novel, simpler NLDA scheme for SSS problems.

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    Area of Science:

    • Machine Learning
    • Pattern Recognition
    • Data Science

    Background:

    • Linear Discriminant Analysis (LDA) is ill-posed for small sample size (SSS) problems due to within-class scatter singularity.
    • Null-space-based LDA (NLDA) offers improved discriminant performance for SSS problems.
    • The original feature extractor (FE) scheme for NLDA presents a significant complexity burden.

    Purpose of the Study:

    • To address the computational complexity of Null-space-based LDA (NLDA) in small sample size (SSS) scenarios.
    • To derive a novel, computationally efficient feature extractor (FE) for NLDA.
    • To enhance the practicality of NLDA for SSS problems.

    Main Methods:

    • Transforming the problem of finding the NLDA feature extractor (FE) into a linear equation problem.
    • Developing a novel scheme based on this transformation.
    • Evaluating the complexity reduction of the proposed scheme compared to existing methods.

    Main Results:

    • A novel scheme for the NLDA feature extractor (FE) was successfully derived.
    • The proposed scheme significantly reduces the computational complexity associated with NLDA.
    • The method offers a more efficient approach to feature extraction for SSS problems.

    Conclusions:

    • The novel NLDA scheme provides a computationally efficient solution for small sample size (SSS) problems.
    • This approach simplifies the feature extraction process, making NLDA more accessible.
    • Further research can explore the discriminant performance of this reduced-complexity NLDA.