Ecological Archives E085-004-A1

Jean Thioulouse, Monique Simier, and Daniel Chessel. 2004. Simultaneous analysis of a sequence of paired ecological tables. Ecology 85:272–283.

Appendix C. STATICO computations.

All the computations described in this appendix were made with ADE-4 software. We are not aware of any other software capable of doing a STATICO analysis. This method could be programmed in any language, but the computation steps might be different from those described here. The tables given in this appendix are intended to allow all the computations to be redone more easily.

FIRST STEP

The first step is a normed PCA of the environmental variable tables and a centered PCA of the species tables. In ADE-4, the KtabUtil module allows sequences of tables to be handled automatically, starting from a sequence of tables stored in the same file. Many operations like transposition, centering, normalization, etc. are possible. All the separate analyses (which need not to be performed here ) can be computed with the “Separate analyses” option of the KTA-MFA module.

Table C1: Table of normed environmental variables (variables in columns, sites in rows, dates one below the other). In ADE-4, this table is obtained with the KTabUtil module (“InitKTab” and “NormalizeKTab” options).

-0.7213 -0.9277 0.0429 0.1230 0.2602 -0.0929 -0.0942 -0.0988 -0.0306 -0.1564
-0.1803 -0.3110 -0.4721 0.4183 -1.6093 0.2673 0.5023 0.3470 -0.1591 0.3128
-0.1803 -0.1002 0.0429 0.0492 0.3180 -0.0249 -0.0628 -0.0531 0.0795 -0.1598
0.3607 0.3321 0.3004 0.0492 0.5685 -0.0113 -0.0628 -0.0449 0.0795 0.0529
0.9017 0.5535 0.0429 -0.0246 0.3951 -0.0045 -0.0314 -0.0400 0.1896 0.0900
-0.1803 0.4533 0.0429 -0.6152 0.0675 -0.1337 -0.2511 -0.1102 -0.1591 -0.1395
-0.6312 -0.4217 0.6437 -0.3199 0.5139 -0.529 -0.2773 -0.6221 -0.4772 -0.7477
-0.6312 -0.3268 -1.1587 0.4921 -0.9315 0.7419 0.9470 0.9601 -0.8809 0.4405
0.4508 -0.2214 -0.6437 0.566 -0.4304 0.3341 -0.0576 0.6499 0.3854 0.7274
0.9918 -0.0105 -0.1287 0.1969 0.1478 0.1303 -0.0262 0.1600 0.5139 0.3730
0.4508 0.0949 0.9012 -0.0246 0.4369 -0.5697 -0.2459 -0.6041 0.8075 -0.2042
-0.6312 0.8855 0.3862 -0.9105 0.2634 -0.1076 -0.3401 -0.5437 -0.3487 -0.5890
-0.8115 -0.3119 0.1717 -0.2584 0.0996 -0.4701 -0.5337 -0.4210 -0.2618 -0.7499
0.2705 -0.1116 -0.8583 1.3657 -0.9219 1.8678 1.8208 1.6086 -1.0326 1.4340
-0.2705 -0.0272 -0.6008 0.2584 -0.0353 -0.0963 -0.0942 0.0084 0.9861 0.4889
0.2705 0.0044 -0.0858 -0.0369 0.3887 -0.4429 -0.2511 -0.3638 0.7659 -0.2200
-0.2705 -0.0641 0.6866 -0.4060 0.0996 -0.4701 -0.4081 -0.4161 0.3071 -0.3381
0.8115 0.5104 0.6866 -0.9228 0.3694 -0.3885 -0.5337 -0.4161 -0.7647 -0.6149
0.0902 -1.1816 -0.5579 -0.1353 0.0000 -0.2039 -0.1727 -0.1663 -0.3671 -0.2841
0.0902 -0.4753 0.2146 0.3814 0.0000 0.3330 0.3610 0.2174 0.1468 0.1918
0.0902 -0.1432 0.2146 0.5291 0.0000 0.2786 0.3296 0.2632 0.1835 0.6138
0.0902 0.8214 0.2146 -0.0615 0.0000 0.0136 -0.0471 -0.0242 0.1101 -0.0714
-0.4508 0.2521 -0.0429 -0.0615 0.0000 -0.1971 -0.1727 -0.1026 0.2202 -0.1457
0.0902 0.7265 -0.0429 -0.6521 0.0000 -0.2243 -0.2982 -0.1875 -0.2936 -0.3044

 

Table C2: Table of centered species abundances (species in columns, sites in rows, dates one below the other). In ADE-4, this table is obtained with the “InitKTab” and “CentringKTab” options of the KTabUtil module.

3.00 2.50 2.67 6.83 0.00 0.00 -0.83 2.50 3.33 0.00 3.00 0.00 -2.33
-1.00 -4.50 0.67 -2.17 0.00 0.00 -0.83 -2.50 -1.67 0.00 -1.00 0.00 -2.33
-1.00 0.50 -2.33 -2.17 0.00 0.00 -0.83 -0.50 -0.67 0.00 -1.00 0.00 -2.33
-1.00 -1.50 -1.33 -2.17 0.00 0.00 -0.83 0.50 0.33 0.00 -1.00 0.00 -2.33
-1.00 0.50 -1.33 -2.17 0.00 0.00 4.17 -2.50 -1.67 0.00 -1.00 0.00 1.67
1.00 2.50 1.67 1.83 0.00 0.00 -0.83 2.50 0.33 0.00 1.00 0.00 7.67
5.00 1.33 0.67 -0.83 7.17 -0.67 -1.00 1.00 5.00 0.00 0.00 -1.17 -1.67
-1.00 -5.67 -0.33 -0.83 -2.83 -0.67 -1.00 -1.00 -2.00 0.00 0.00 -1.17 -3.67
-1.00 0.33 -1.33 -0.83 -2.83 1.33 -1.00 -1.00 -2.00 0.00 0.00 -1.17 -3.67
-1.00 1.33 1.67 -0.83 -2.83 1.33 -1.00 -1.00 0.00 0.00 0.00 3.83 1.33
-1.00 0.33 -0.33 1.17 0.17 -0.67 3.00 -1.00 -2.00 0.00 0.00 0.83 3.33
-1.00 2.33 -0.33 2.17 1.17 -0.67 1.00 3.00 1.00 0.00 0.00 -1.17 4.33
3.00 -2.67 -1.50 0.00 5.50 4.00 -2.33 1.17 4.17 -0.50 3.83 -0.50 -2.50
-1.00 -7.67 -8.50 0.00 -3.50 -2.00 -2.33 -3.83 -4.83 -0.50 -3.17 -0.50 -2.50
-1.00 1.33 0.50 0.00 -3.50 -2.00 -2.33 -3.83 -0.83 -0.50 -0.17 -0.50 -2.50
-1.00 2.33 3.50 0.00 -3.50 1.00 -2.33 1.17 0.17 0.50 0.83 1.50 1.50
1.00 2.33 2.50 0.00 0.50 -2.00 5.67 0.17 -0.83 1.50 1.83 0.50 3.50
-1.00 4.33 3.50 0.00 4.50 1.00 3.67 5.17 2.17 -0.50 -3.17 -0.50 2.50
2.33 0.50 -1.17 0.00 3.33 2.33 -2.33 0.17 3.50 -0.33 1.83 0.00 0.00
-0.67 -2.50 -2.17 0.00 -2.67 0.33 -2.33 0.17 -1.50 -0.33 -1.17 0.00 0.00
-0.67 -5.50 -5.17 0.00 -2.67 -3.67 -2.33 -2.83 -4.50 -0.33 -2.17 0.00 0.00
-0.67 0.50 1.83 0.00 -2.67 0.33 -1.33 -0.83 0.50 -0.33 -0.17 0.00 0.00
0.33 3.50 2.83 0.00 0.33 1.33 5.67 -0.83 0.50 1.67 2.83 0.00 0.00
-0.67 3.50 3.83 0.00 4.33 -0.67 2.67 4.17 1.50 -0.33 -1.17 0.00 0.00

 

SECOND STEP

Co-inertia analyses of the pairs of tables. The results of the separate co-inertia analyses (which need not be computed) are not given here. Only the series of cross-tables is needed to continue to PTA. The table below gives the sequence of cross-tables, with species in columns, environmental variables in rows, and dates one below the other, as in the preceding tables.

Table C3: Co-inertia analysis cross-table. In ADE-4, this table is obtained with the “MatchTwoKTabs” option of the KTabUtil module.

-0.54 -0.27 -0.60 -1.20 0.00 0.00 0.75 -0.63 -0.57 0.00 -0.54 0.00 0.30
-0.47 -0.01 -0.48 -1.09 0.00 0.00 0.46 -0.26 -0.53 0.00 -0.47 0.00 1.12
0.04 0.32 -0.11 0.09 0.00 0.00 0.04 0.24 0.16 0.00 0.04 0.00 0.10
-0.12 -0.53 -0.09 -0.23 0.00 0.00 -0.02 -0.37 -0.08 0.00 -0.12 0.00 -1.04
0.20 1.26 -0.38 0.44 0.00 0.00 0.33 0.66 0.48 0.00 0.20 0.00 0.38
-0.11 -0.29 -0.04 -0.23 0.00 0.00 0.00 -0.20 -0.13 0.00 -0.11 0.00 -0.23
-0.15 -0.51 -0.01 -0.31 0.00 0.00 -0.03 -0.34 -0.19 0.00 -0.15 0.00 -0.44
-0.10 -0.34 0.00 -0.22 0.00 0.00 -0.03 -0.21 -0.14 0.00 -0.10 0.00 -0.21
-0.07 0.04 -0.17 -0.15 0.00 0.00 0.16 -0.09 -0.04 0.00 -0.07 0.00 -0.14
-0.15 -0.38 -0.04 -0.33 0.00 0.00 0.08 -0.27 -0.19 0.00 -0.15 0.00 -0.17
-0.63 0.48 0.15 -0.17 -1.25 0.48 0.09 -0.63 -0.72 0.00 0.00 0.98 0.30
-0.42 0.55 -0.04 0.47 -0.07 -0.08 0.36 0.45 -0.05 0.00 0.00 0.02 1.14
0.64 1.37 0.17 0.49 1.78 -0.26 0.73 0.47 0.90 0.00 0.00 0.19 1.67
-0.32 -0.82 -0.08 -0.46 -1.15 0.25 -0.32 -0.71 -0.76 0.00 0.00 0.16 -1.19
0.51 1.13 0.21 0.28 1.25 -0.09 0.38 0.35 0.78 0.00 0.00 0.27 1.16
-0.53 -0.84 -0.10 -0.24 -1.24 0.15 -0.42 -0.25 -0.63 0.00 0.00 -0.08 -0.88
-0.28 -1.11 -0.05 -0.25 -0.81 -0.03 -0.28 -0.32 -0.50 0.00 0.00 -0.10 -0.85
-0.62 -1.22 -0.16 -0.47 -1.70 0.27 -0.58 -0.57 -0.94 0.00 0.00 -0.07 -1.50
-0.48 0.77 0.03 0.09 -0.62 0.30 0.42 -0.39 -0.56 0.00 0.00 0.70 0.75
-0.75 -0.70 -0.12 -0.36 -1.74 0.37 -0.33 -0.64 -1.04 0.00 0.00 0.24 -0.96
-0.63 0.54 0.32 0.00 -0.32 -0.27 0.45 0.59 -0.41 -0.05 -1.13 0.05 0.59
-0.23 0.62 0.51 0.00 0.17 -0.05 0.42 0.47 0.07 -0.02 -0.43 -0.01 0.36
0.34 1.62 1.76 0.00 1.63 0.47 1.60 1.56 1.04 0.21 0.41 0.09 1.20
-0.31 -2.41 -2.58 0.00 -1.89 -0.74 -1.46 -1.90 -1.59 -0.14 -0.54 -0.08 -1.20
0.10 1.58 1.76 0.00 0.71 0.48 0.50 1.03 0.95 0.10 0.44 0.15 0.67
-0.47 -2.83 -3.22 0.00 -1.54 -0.89 -1.02 -1.66 -1.91 -0.23 -1.28 -0.23 -1.09
-0.49 -2.75 -3.08 0.00 -1.78 -0.93 -1.08 -1.73 -1.97 -0.18 -1.18 -0.15 -1.02
-0.42 -2.47 -2.80 0.00 -1.46 -0.81 -0.97 -1.56 -1.69 -0.20 -1.08 -0.19 -1.01
-0.07 1.52 1.74 0.00 -1.21 -0.26 -0.36 -0.52 0.22 0.23 0.95 0.31 0.18
-0.61 -2.05 -2.43 0.00 -2.17 -1.17 -1.07 -1.96 -1.93 -0.15 -1.06 -0.13 -1.00
-0.03 -0.32 -0.26 0.00 -0.03 -0.12 -0.51 0.08 -0.05 -0.15 -0.26 0.00 0.00
-0.55 0.87 1.36 0.00 -0.21 -0.38 1.08 0.38 -0.19 0.08 -0.26 0.00 0.00
-0.29 -0.36 -0.14 0.00 -0.63 -0.33 -0.06 -0.16 -0.54 -0.01 -0.31 0.00 0.00
-0.08 -1.08 -1.03 0.00 -0.93 -0.30 -0.64 -0.68 -0.74 -0.02 -0.21 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
-0.13 -0.66 -0.55 0.00 -0.56 -0.25 -0.45 -0.26 -0.48 -0.07 -0.28 0.00 0.00
-0.12 -0.75 -0.67 0.00 -0.61 -0.26 -0.49 -0.33 -0.53 -0.06 -0.26 0.00 0.00
-0.10 -0.52 -0.45 0.00 -0.44 -0.22 -0.30 -0.24 -0.41 -0.03 -0.20 0.00 0.00
-0.15 -0.29 -0.19 0.00 -0.60 -0.16 0.07 -0.34 -0.43 0.07 -0.05 0.00 0.00
-0.17 -0.93 -0.83 0.00 -0.71 -0.48 -0.46 -0.47 -0.77 -0.05 -0.35 0.00 0.00

 

THIRD STEP

Partial triadic analysis: a PTA (interstructure, compromise, trajectories) is computed on the series of cross-tables obtained in the second step.

Interstructure

Table C4: Matrix of scalar products between tables (Covv). In ADE-4, this table and all the following ones are obtained with the STATIS module (“Table averaging” option).

0.1088 0.0953 0.1382 0.0242
0.0953 0.4125 0.4978 0.0893
0.1382 0.4978 1.4021 0.2994
0.0242 0.0893 0.2994 0.1545

 

Table C5: Components of the first eigenvector (column 1) and list of eigenvalues (column 2) of the interstructure analysis (i.e., of the matrix in Table C4).

0.1042 1.6901
0.3725 0.2238
0.9005 0.0826
0.1989 0.0813

 

Compromise

Table C6: Compromise table (species in rows and environmental variables in columns).

-0.8658 -0.5214 0.4965 -0.4245 0.3015 -0.6582 -0.5843 -0.6400 -0.2796 -0.8790
0.5753 0.9357 1.9281 -2.7447 1.9771 -3.0281 -3.0937 -2.8168 1.6019 -2.3330
0.2267 0.6635 1.6093 -2.5665 1.6236 -3.0481 -2.9256 -2.6703 1.5213 -2.4036
-0.1868 0.0632 0.1935 -0.1961 0.1487 -0.1146 -0.1260 -0.1994 0.0195 -0.1692
-0.7548 0.0875 2.0066 -2.3142 1.1036 -1.9570 -2.0297 -2.0385 -1.4391 -2.7443
-0.0884 -0.1529 0.2637 -0.6296 0.3959 -0.7895 -0.8953 -0.6728 -0.1552 -1.0095
0.4162 0.7787 1.7068 -1.5661 0.6277 -1.1584 -1.1733 -1.1543 -0.1328 -1.1671
0.2418 0.6353 1.5717 -2.1510 1.1234 -1.6569 -1.7741 -1.6822 -0.6938 -2.1231
-0.7025 -0.0494 1.1857 -1.8750 1.1937 -2.0586 -2.0843 -1.9709 -0.1048 -2.2939
-0.0705 -0.0019 0.1904 -0.1315 0.0882 -0.2206 -0.1716 -0.1863 0.2217 -0.1442
-1.1221 -0.4862 0.3169 -0.5359 0.4172 -1.2220 -1.1279 -1.0195 0.8418 -1.0395
0.4045 0.0002 0.1492 -0.0140 0.2317 -0.2338 -0.1753 -0.1970 0.5358 -0.0263
0.6710 0.8705 1.7159 -1.6302 1.0697 -1.3305 -1.2828 -1.4874 0.4261 -1.2743

 

Table C7: List of eigenvalues of the PCA of the compromise.

1.5480
0.0838
0.0505
0.0054
0.0013
0.0008
0.0002
0.0001
0.0000
0.0000

 

Trajectories

Table C8: Coordinates of the projections of the columns of the series of tables on the first two axes of the compromise. The first two columns give the coordinates of the 10 environmental variables at each date (10 variables × 4 dates = 40 rows). The third and fourth columns give the coordinates of the 13 species at each date (13 species × 4 dates = 52 rows).

0.2047 0.0390 -0.0870 -0.1251
0.0679 0.0898 -0.3964 -0.0169
-0.0788 0.0037 0.0336 -0.1655
0.2135 -0.0825 -0.1806 -0.2778
-0.2946 0.0423 0.0000 0.0000
0.1018 -0.0196 0.0000 0.0000
0.1674 -0.0373 -0.0454 0.1859
0.1062 -0.0197 -0.2473 -0.1264
0.0342 -0.0038 -0.1325 -0.1197
0.1208 -0.0168 0.0000 0.0000
0.1373 0.4817 -0.087 -0.1251
-0.1951 0.1299 0.0000 0.0000
-0.7077 -0.1765 -0.3373 0.0751
0.4890 0.1654 -0.4034 -0.2855
-0.5413 -0.1043 -0.8561 0.2698
0.4424 0.1542 -0.0973 0.0171
0.4108 0.0336 -0.3156 0.0040
0.6655 0.2035 -1.1126 -0.4558
-0.0403 0.3644 0.1595 0.1710
0.5508 0.3351 -0.3668 0.1201
-0.0910 0.2196 -0.3988 -0.2073
-0.2431 0.0939 -0.6357 -0.3171
-1.0950 -0.0664 0.0000 0.0000
1.4228 -0.0077 0.0000 0.0000
-0.8216 0.1375 -0.0345 0.3383
1.5418 -0.1760 -0.9909 0.239
1.5463 -0.0886 -0.3314 -0.1569
1.3787 -0.1213 -1.9446 0.4386
-0.3052 0.6449 -2.1791 0.4429
1.4273 0.1644 0.0000 0.0000
0.1270 -0.0548 -1.3369 -0.5052
-0.3179 0.3146 -0.6616 -0.1723
0.2356 0.1271 -0.9213 -0.1023
0.5712 0.0352 -1.3692 -0.1576
0.0000 0.0000 -1.3553 -0.0849
0.3427 0.0286 -0.1485 0.0442
0.3865 0.0201 -0.7460 0.0389
0.2722 0.0235 -0.1259 0.0841
0.2061 0.1159 -0.8674 0.0793
0.4910 0.0336 -0.0307 -0.0794
    -0.4909 -0.1058
    -0.4768 -0.0398
    0.0000 0.0000
    -0.3404 -0.1975
    -0.1479 -0.1060
    -0.3273 0.0038
    -0.2422 -0.0846
    -0.3124 -0.1662
    -0.0320 0.0048
    -0.1303 -0.0732
    0.0000 0.0000
    0.0000 0.0000


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