高次元統計解析: 理論と方法論の新しい展開  [in Japanese] High-Dimensional Statistical Analysis: New Developments of Theories and Methodologies  [in Japanese]

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Abstract

<p>本論文は,高次元統計解析の理論と方法論について,最新の展開を紹介する.最近,Aoshima and Yata (2018a) は,強スパイク固有値(Strongly Spiked Eigenvalue: SSE)モデルというノイズモデルを提唱した.高次元データのノイズは巨大かつ非スパースであり,それゆえデータがもつ潜在的な幾何学的構造は破壊され,統計的推測に精度を保証することが困難になる.理論的には,SSEモデルのもとでは,高次元統計解析の根幹を成す高次元漸近正規性が成立しない.Aoshima and Yata (2018a) は,巨大なノイズ構造を精密に解析し,強スパイクするノイズ空間を避けるようなデータ変換法を開発した.この方法を用いれば,データは弱スパイク固有値(Non-SSE: NSSE)モデルに変換され,潜在空間の幾何学的構造が浮き彫りになり,高精度な高次元統計的推測が可能になる.Aoshima and Yata (2018b) は,この方法論を発展させ,高次元判別分析に新たな理論を展開している.本論文は,高次元統計解析の最新の展開について,適宜文献を紹介しながら解説する.</p>

<p>In this paper, we introduce new developments of theories and methodologies in high-dimensional statistical analysis. Recently, Aoshima and Yata (2018a) have provided a noise model called the strongly spiked eigenvalue (SSE) model. Since the noise associated with high dimensional data is huge and non-sparse, the potential geometric structure of the data is destroyed and it is difficult to guarantee the accuracy for statistical inferences. In theory, the high-dimensional asymptotic normality that forms the basis of high-dimensional statistical analysis is not established under the SSE model. Aoshima and Yata (2018a) have developed a data transformation technique that avoids strongly spiked-noise spaces by precisely analyzing the huge noise structure. Using this method, the data is transformed into the non-strongly spiked eigenvalue (NSSE) model, which highlights the geometric structure of the latent space and enables highly accurate high-dimensional statistical inference. Aoshima and Yata (2018b) have applied this methodology to create a new theory for high-dimensional discriminant analysis. In this current paper, we explain the latest developments of high-dimensional statistical analysis while appropriately introducing literature.</p>

Journal

  • Journal of the Japan Statistical Society, Japanese Issue

    Journal of the Japan Statistical Society, Japanese Issue 48(1), 89-111, 2018

    Japan Statistical Society

Codes

  • NII Article ID (NAID)
    130007623071
  • NII NACSIS-CAT ID (NCID)
    AA11989749
  • Text Lang
    JPN
  • Article Type
    journal article
  • ISSN
    0389-5602
  • NDL Article ID
    029303181
  • NDL Call No.
    Z3-1003
  • Data Source
    NDL  IR  J-STAGE 
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