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Nonnegative Matrix Factorizations for Intelligent Data Analysis

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Non-negative Matrix Factorization Techniques

Part of the book series: Signals and Communication Technology ((SCT))

Abstract

We discuss nonnegative matrix factorization (NMF) techniques from the point of view of intelligent data analysis (IDA), i.e., the intelligent application of human expertize and computational models for advanced data analysis. As IDA requires human involvement in the analysis process, the understandability of the results coming from computational models has a prominent importance. We therefore review the latest developments of NMF that try to fulfill the understandability requirement in several ways. We also describe a novel method to decompose data into user-defined—hence understandable—parts by means of a mask on the feature matrix, and show the method’s effectiveness through some numerical examples.

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Notes

  1. 1.

    Source: Cisco® Visual Networking Index (VNI) Forecast (2010–2015).

  2. 2.

    Since the values of the variance of data depends on the scale of the variables, usually the original data contained in X are subject to a standardization process so that each variable has mean zero and standard deviation one.

  3. 3.

    Henceforth a matrix is denoted with an uppercase letter, e.g., X, its elements with the corresponding lowercase letter, e.g., \(x_{\textit{ij}}\), a column vector in lowercase boldface, e.g., \(\mathbf {x}_i\).

  4. 4.

    The use of NMF in clustering applications will be detailed in Sect. 2.4.2.

  5. 5.

    In this chapter, we mainly consider NMF based on the error function described in (2.8), but other divergence measures could be used (e.g., generalized Kullback–Leibler divergence, \(\alpha \)-divergence). Anyway, technical details apart, the general ideas described in the section still hold.

  6. 6.

    The function in (2.10) yields values in the interval [0,1], where 0 indicates the minimum degree of sparseness obtained when all the elements \(x_i\) have the same absolute value, while 1 indicates the maximum degree of sparseness, which is reached when only one component of the vector x is different from zero.

  7. 7.

    The symbol \(W_{:i}\) denotes the ith column of W. The same applies for H.

  8. 8.

    The dataset is available at http://alumni.cs.ucr.edu/~titus/ (accessed: March 25th 2015).

  9. 9.

    http://sat.collegeboard.org.

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Authors and Affiliations

  1. Department of Informatics University of Bari, University Campus “Ernesto Quagliariello”, Via E. Orabona, 4, 70125, Bari, Italy

    G. Casalino & C. Mencar

  2. Department of Mathematics University of Bari, University Campus “Ernesto Quagliariello”, Via E. Orabona, 4, 70125, Bari, Italy

    N. Del Buono

Authors
  1. G. Casalino
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  2. N. Del Buono
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  3. C. Mencar
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Correspondence to G. Casalino .

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  1. University of Technology Sydney, Sydney, New South Wales, Australia

    Ganesh R. Naik

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Casalino, G., Del Buono, N., Mencar, C. (2016). Nonnegative Matrix Factorizations for Intelligent Data Analysis. In: Naik, G. (eds) Non-negative Matrix Factorization Techniques. Signals and Communication Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48331-2_2

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  • DOI: https://doi.org/10.1007/978-3-662-48331-2_2

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