# Nonlinear multivariate and time series analysis by neural by William W. Hsieh By William W. Hsieh

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The number of hidden neurons used here is l2 = m2 = 3; cruder approximations to X result from using smaller l2 and m2 . 30 (b) 3 2 2 1 1 y3 y2 (a) 3 0 0 −1 −1 −2 −2 −3 −4 −2 0 y1 2 −3 −4 4 −2 (c) 0 y1 2 4 (d) 2 1 1 0 y3 y3 3 0 −1 −1 −2 2 −2 2 0 −3 −4 −2 0 y2 2 y2 4 0 −2 −2 y1 Figure 11: The NLCCA mode 1 in y-space shown as a string of overlapping small circles. The thin solid curve is the theoretical mode Y , and the thin dashed line, the CCA mode. 31 (a) SLPA 10 PC3 5 0 −5 −10 20 10 0 20 −10 PC2 0 −20 −20 PC1 (b) SSTA 40 PC3 20 0 −20 40 20 50 0 PC2 0 −20 −50 PC1 Figure 12: The NLCCA mode 1 between the tropical Pacific (a) SLPA and (b) SSTA, plotted as (overlapping) squares in the PC1 -PC2 -PC3 3-D space.

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In a separate panel beneath each contour plot, the PC of each SSA mode is also plotted as a time series, (where each tick mark on the abscissa indicates the start of a year). The time of the PC is synchronized to the lag time of 0 month in the space-time eigenvector. 39 100 x3 50 0 −50 −100 100 50 100 x2 0 50 0 −50 −50 −100 −100 x1 Figure 20: The NLSSA mode 1 for the tropical Pacific SLPA. The PCs of SSA modes 1 to 8 were used as inputs x1 , . . cir network, with the resulting NLSSA mode 1 shown as (densely overlapping) crosses in the x1 -x2 -x3 3-D PC space.

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