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Circulation typing with fuzzy rotated T-mode principal component analysis: methodological considerations

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Abstract

Circulation typing using rotated T-mode decomposition (i.e., variable is time series and observation is grid points) principal component analysis (PCA) requires up to four important decisions to optimize the actual classification output (i.e., the time series and associated maps): (i) the determination of the most appropriate number of rotated PCs to retain, (ii) the selection of a simple structure rotation algorithm, (iii) the consideration of constructing a hyperplane width threshold used to separate noise from the signal by establishing a confidence interval for a zero PC loading, and (iv) deciding on the various classification types, e.g., hard/overlapping, hard/non-overlapping, fuzzy/overlapping. This study examines the sensitivity of the validity of the classification outputs to each of these decisions. Rather than adhere to the traditional eigenvalue methods that make this decision prior to rotation, we develop a method to select the optimal number of rotated PCs to retain, using the congruence coefficient to assess the validity of each PC after rotation, by documenting the distance between the PCs to the patterns portrayed in the similarity matrix from which those PCs were derived (e.g., correlation, covariance). After an optimal number of components is determined, the PC loadings are rotated using a number of algorithms to establish the sensitivity of the validity of the solution to various orthogonal (Varimax) and oblique rotations (Direct Oblimin, Promax, Simplimax). In the present study, the Direct Oblimin rotation method is found to perform most accurately in the congruence coefficient matching. Next, the accuracy of the solution to the choice of the hyperplane width threshold is examined and the threshold is found to be critical in affecting the shape of the maps and the time series. For the classification types tested, we find for the two overlapping classifications, the congruence matching to the similarity matrix patterns yields excellent matches for hyperplane width threshold magnitude values within 0.2 to 0.3; thus, that range is found optimal for separating noise from the signal in this study. As the hyperplane width was increased beyond 0.3, the congruence match with the similarity matrix patterns decreased. Overlapping classifications that allow for the assignment of observation to more than one circulation type are found to have higher congruence matches with the similarity matrix patterns compared to non-overlapping classifications. Further, fuzzy/overlapping classifications, that provide non-binary weights for every observation, are found to have the largest congruence matches with the correlation patterns. Thus, despite the trade-off of less simplicity, fuzzy/overlapping classifications can be considered as an alternative to hard classifications when the goal is to interpret patterns that are well supported by the similarity matrix.

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ERA5 data are available at https://cds.climate.copernicus.eu/cdsapp#!/dataset.

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Authors and Affiliations

  1. Department of Geography, Kent State University, Kent, OH, USA

    Chibuike Chiedozie Ibebuchi

  2. ClimRISE Lab, Kent State University, Kent, OH, USA

    Chibuike Chiedozie Ibebuchi

  3. School of Meteorology, University of Oklahoma, Norman, OK, USA

    Michael B. Richman

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  1. Chibuike Chiedozie Ibebuchi
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Appendix

Appendix

Tables 10 and 11

Table 10 Congruence matches between the rotated PC loadings to the correlation patterns for oblique solutions (m = 2, 3, 4) under the Promax method
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Table 11 Congruence coefficient between the anomaly SLP composite maps of the CTs from the same principal component. The CTs are fuzzy overlapping CTs at T = 0.2
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Figures 9, 1011, 12, 13 and 14

Fig. 9
figure 9

Idealized locations of synoptic rain-bearing systems in the study region during their active periods. CW: cross-equatorial northeast trade wind; MCT: Mozambique Channel Trough; SIOHP: western branch of the South Indian Ocean high-pressure AGC: Agulhas warm current; CF: cold fronts; SAOHP: South Atlantic Ocean high-pressure; AC: Angola warm current; BC: Benguela cold current; ITCZ: Inter-tropical convergence Zone during its southward track in austral summer; KL: Kalahari low; AL: Angola low; SICZ: South Indian Ocean Convergence Zone

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Fig. 10
figure 10

Standardized DJF annual mean loadings of PC3 and annual mean SAM index from 1979 to 2020

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Fig. 11
figure 11

Standardized SLP anomaly map on 1984–09-04

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Fig. 12
figure 12

Distribution of the PC loadings from all 4 retained PCs. The horizontal dashed line indicates the 0.92 PC loading magnitude

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Fig. 13
figure 13

Histogram of the congruence coefficients (in absolute value) between the PC scores and the 15,341 daily SLP observations, for the non-overlapping (a), and overlapping (b) classification. The vertical line indicates the 0.92 congruence coefficient. For the overlapping classification, the weighted mean map of the 4 PC scores (i.e., from the four retained PCs and using the PC loadings as the weights) is matched to the SLP anomaly map of the day. For the non-overlapping, (only one) PC score that has the largest loading magnitude is matched to the SLP anomaly of the day

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Fig. 14
figure 14

Effect of sampling variability on the PC loadings. The X-axis contains the original PC loadings and the Y-axis contains the count of PC loadings after resampling the data that changed to within the range of \(-\) 0.05 to + 0.05. The red, yellow, brown, blue, and green vertical lines demarcate PC loadings values of 0.1, 0.2, 0.3, 0.4, and 0.5 respectively

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Ibebuchi, C.C., Richman, M.B. Circulation typing with fuzzy rotated T-mode principal component analysis: methodological considerations. Theor Appl Climatol 153, 495–523 (2023). https://doi.org/10.1007/s00704-023-04474-5

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