Fig. 1
Dimensionality reduction by random projection. Original data X is projected onto a random matrix R to have a lower dimensional subspace P.
Fig. 2
Uncertainties of random projections. Mean and 95% confidence limits of the variance explained by the PCs (a) 1–30 and (b) 2–30 calculated from 100 realisations of projections of RP10% and RP1%. The explained variance of the first eigenvalue is excluded from subfigure (b) to show more details. In RP, the spatial dimension of the original data matrix is reduced.
Fig. 3
Explained variance of the 30 first PCs with their 95% confidence limits. (a–b) Original and RP10%, (c–d) Original and RP1%. The explained variance of the first eigenvalue is excluded from subfigures (b) and (d) to show more details. In RP, the spatial (1) and temporal (2) dimensions are reduced. The confidence limits are obtained by re-sampling the original and projected data sets 100 times, and the PCA of each sample is calculated.
Fig. 4
Spatial patterns of PC1–PC8 loadings. Comparison of the original, RP10% and RP1% data sets. In RP, the temporal dimension is reduced.
Fig. 5
Spatial patterns of PC9–PC12 loadings. Comparison of the original, RP10% and RP1% data sets. In RP, the temporal dimension is reduced.
Fig. 6
Correlation of the original and projected (RP10% and RP1%) PC loadings. In RP, the temporal dimension is reduced.
Fig. 7
Spectra of PC1–PC12 scores. (a) The original data set, (b) RP10% and (c) RP1%. In RP, the spatial dimension is reduced.
Fig. 8
Correlation of the original and projected (RP10% and RP1%) PC scores. In RP, the spatial dimension is reduced.
Fig. 9
Explained variance (%) of PCs with their 95% confidence limits estimated by bootstrapping. PCs (a) 1–30 and (b) 2–30 of the three-dimensional atmospheric temperature data set (the spatial dimension is reduced by RP) are shown. The explained variance of the first eigenvalue is excluded from subfigure (b) to show more details.
Fig. 10
Spectra of PC1–PC12 scores of the three-dimensional atmospheric temperature data set (the spatial dimension is reduced by RP).
Fig. 11
The spatial patterns of the PC5 loadings of the atmospheric temperature data set (the spatial dimension is reduced by RP) between 1000 and 400 hPa. The spatial patterns are approximated using the method explained in the Appendix.
Fig. 12
The spatial patterns of the PC5 loadings of the atmospheric temperature data set (the spatial dimension is reduced by RP) between 300 and 30 hPa. The spatial patterns are approximated using the method explained in the Appendix.