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Principal Component Analysis

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Encyclopedia of Mathematical Geosciences

Part of the book series: Encyclopedia of Earth Sciences Series ((EESS))

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Synonyms

Karhunen-Loève decomposition; Principal orthogonal decomposition

Definition

Principal component analysis (PCA) is a statistical technique aimed to explore and reduce the dimensionality of a multivariate dataset. It is based on the key idea of finding a representation of the p-dimensional data through a smaller set of k < p new variables, defined as linear combinations of the original variables. These are obtained as to explain at best the data variability. PCA was first introduced by Pearson (1901); nowadays its formulation has been developed for varied types of Euclidean data, including compositional data and functional data. Extensions to non-Euclidean data are available as well.

PCA as an Optimization Problem

It is given a dataset made of n observations of p variables, organized in a data matrix \( \mathbbm{X}=\left[{x}_{ij}\right] \) in ℝn,p, whose columns are denoted by x·j = (x1j, …, xnj). It is assumed that the variables have null sample mean (\(...

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Bibliography

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Author information

Authors and Affiliations

  1. MOX-Department of Mathematics, Politecnico di Milano, Milano, Italy

    Alessandra Menafoglio

Authors
  1. Alessandra Menafoglio
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Corresponding author

Correspondence to Alessandra Menafoglio .

Editor information

Editors and Affiliations

  1. System Science & Informatics Unit, Indian Statistical Institute- Bangalore Centre, Bangalore, India

    B. S. Daya Sagar

  2. Insititue of Earth Sciences, China University of Geosciences, Bei**g, China

    Qiuming Cheng

  3. School of Natural and Built Environment, Queen's University Belfast, Belfast, UK

    Jennifer McKinley

  4. Canada Geological Survey, Ottawa, ON, Canada

    Frits Agterberg

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Menafoglio, A. (2022). Principal Component Analysis. In: Daya Sagar, B.S., Cheng, Q., McKinley, J., Agterberg, F. (eds) Encyclopedia of Mathematical Geosciences. Encyclopedia of Earth Sciences Series. Springer, Cham. https://doi.org/10.1007/978-3-030-26050-7_256-2

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  • DOI: https://doi.org/10.1007/978-3-030-26050-7_256-2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26050-7

  • Online ISBN: 978-3-030-26050-7

  • eBook Packages: Springer Reference Earth and Environm. ScienceReference Module Physical and Materials ScienceReference Module Earth and Environmental Sciences

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Chapter history

  1. Latest

    Principal Component Analysis
    Published:
    05 August 2022

    DOI: https://doi.org/10.1007/978-3-030-26050-7_256-2

  2. Original

    Principal Component Analysis
    Published:
    11 January 2022

    DOI: https://doi.org/10.1007/978-3-030-26050-7_256-1

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