Time Series Components Separation Based on Singular Spectral Analysis Visualization: an HJ-biplot Method Application
AbstractThe extraction of essential features of any real-valued time series is crucial for exploring, modeling and producing, for example, forecasts. Taking advantage of the representation of a time series data by its trajectory matrix of Hankel constructed using Singular Spectrum Analysis, as well as of its decomposition through Principal Component Analysis via Partial Least Squares, we implement a graphical display employing the biplot methodology. A diversity of types of biplots can be constructed depending on the two matrices considered in the factorization of the trajectory matrix. In this work, we discuss the called HJ-biplot which yields a simultaneous representation of both rows and columns of the matrix with maximum quality. Interpretation of this type of biplot on Hankel related trajectory matrices is discussed from a real-world data set.
T. Alexandrov, A method of trend extraction using Singular Spectrum Analysis, REVSTAT, Statistical Journal, vol. 7, n. 1, pp. 1-22, 2009.
K. Gabriel, The biplot graphic display of matrices with application to principal component analysis, Biometrika, vol. 58, n. 3, pp. 453-467, 1971
M.P. Galindo, An alternative of simultaneous representation: HJ-biplot, Questii, vol. 10, 1, pp. 13-23, 1986.
P. Geladi and B.R. Kowalsky, Partial Least Squares regression: a tutorial, Analytica Chimica Acta, vol. 185, pp. 1-17, 1986.
N. Golyandina, V. Korobeynikov and A. Zhigljavsky, Singular Spectrum Analysis with R, 1st ed., Springer, Berlin, 2018.
N. Golyandina, V. Nekrutkin and A. Zhigljavsky, Analysis of Time Series Structure: SSA and Related Techniques, 1st ed. Chapman & Hall/CRC, Boca Raton, Florida, 2001.
N. Golyandina and A. Shlemov, Variations of Singular Spectrum Analysis for separability improvement: non-orthogonal decompositions of time series, Statistics and its Interface, vol. 8, n. 3, pp. 277-294, 2015.
M. Greenacre, Biplots in Practice, FBBVA, Bilbao, Biscay, 2010.
A.B. Nieto, M.P. Galindo, V. Leiva and P.V. Galindo, A methodology for biplots based on bootstrapping with R, Colombian Journal of Statistics, vol. 37, n. 2, pp. 367-397, 2014.
S. Moritz, T. Bartz-Beielstein, imputeTS: Time Series Missing Value Imputation in R, The R Journal, vol. 9, n. 1, pp. 207-218, 2017. https://doi.org/10.32614/RJ-2017-009.
NOAA Homepage, https://www.esrl.noaa.gov/gmd/ccgg/trends/, last accessed 24/05/2019.
R Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2019 https://www.R-project.org/.
V.E. Vinzi and G. Russolillo, Partial Least Squares algorithms and methods, WIREs Comput Stat, vol. 5, pp. 1-19, 2013.
H. Wold, Estimation of principal components and related models by iterative least squares, in Multivariate Analysis, edited by P.R. Krishnaiah, Academic Press, New York, pp. 391–420, 1966.
S. Wold, C. Albano, W.J. Dunn, K. Esbensen, S. Hellberg, E. Johansson and M. Sjostrom, Pattern recognition: finding and using regularities in multivariate data, in Food Research and Data Analysis, edited by H. Martens and H. Russwurm, Applied Science Publishers, London, pp. 147-189, 1983.
K. Wright, nipals: Principal Components Analysis using NIPALS or Weighted EMPCA, with Gram-Schmidt Orthogonalization. R package version 0.7, 2020. https://CRAN.R-project.org/package=nipals
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