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Principal Component Analysis for Normal-Distribution-Valued Symbolic Data.

Huiwen Wang, Meiling Chen, Xiaojun Shi

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    This study introduces a novel principal component analysis (PCA) for normal-distribution-valued symbolic data, enhancing economic and management analysis by utilizing all variance information. The method accurately constructs observations in PC space, proving effective in simulated tests and explaining stock market risk-return tradeoffs.

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    Area of Science:

    • Statistics
    • Econometrics
    • Data Analysis

    Background:

    • Symbolic data analysis (SDA) extends traditional data analysis to more complex data types.
    • Principal Component Analysis (PCA) is a key technique for dimensionality reduction.
    • Existing PCA methods for symbolic data often do not utilize all available variance information.

    Purpose of the Study:

    • To develop a new analytical Principal Component Analysis (PCA) approach for normal-distribution-valued symbolic data.
    • To derive numerical characteristics and variance-covariance structures for this data type.
    • To improve upon existing representative-type PCA methods in economic and management applications.

    Main Methods:

    • Derivation of numerical characteristics and variance-covariance structure for normal-distribution-valued symbolic data.
    • Development of an analytical PCA approach that incorporates all variance information.
    • Construction of observations in a Principal Component (PC) space using the linear additivity property of normal distributions.

    Main Results:

    • The proposed PCA method utilizes all variance information, unlike traditional representative-type approaches.
    • An accurate method for constructing observations in PC space was developed.
    • Simulated numerical experiments confirmed the effectiveness of the proposed method.

    Conclusions:

    • The new PCA approach offers a more comprehensive analysis of normal-distribution-valued symbolic data.
    • This method has significant potential for applications in economic and management fields.
    • The approach was successfully applied to analyze the risk-return tradeoff in China's stock market.