Susan E. Spruill and John O. Rawlings
FOUR WAYS TO CREATE GABRIEL'S BIPLOTS FOR GRAPHIC
REPRESENTATION OF PRINCIPAL COMPONENT ANALYSES
I
MIMEO SERIES # 2213
February, 1992
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
MH'EO
SE~IES
Spruill, S. & Rawlings,
FOUR WAYS TO CREATE
ff2213GABRIEL'S BIPLOTS FOR GRAPHIC
REPRESENTATION OF PRINCIPAL COMPNT.
NAME
DATE
I
I
f'_
1
Four Ways to Create Gabriel's Biplots for Graphic
Representation of Principal Component Analyses
by
Susan E. Spruill and John O. Rawlings
-----------/
~~ Institute of Statistics Mimeograph Series No. 2213
February, 1992
(
1
February 14, 1992
Four Ways to Create Gabriel's Biplots for Graphic
Representation of Principal Component Analyses
Principal component analysis is a useful tool for diagnosing
the correlational structure of a dataset, but the results of this
analysis are sometimes difficult to visualize and interpret.
Gabriel's G-H biplots (Gabriel, 1971) are informative plots that
utilize the principal component analysis to show the relationship
among the independent variables (correlation), similarities of
individual datapoints (clustering), and relative importance of
the observations to each independent variable.
The purpose of this exercise was to find ways to utilize SAS
and S-PLUS procedures to create Gabriel's G-H biplots. The
following examples are four programs that produce Gabriel's G-H
biplots. The first program is Gabriel's BIPLOTS program (Gabriel,
1975). Two of the programs are SAS methods (IML and PRINCOMP) and
the fourth utilizes S-PLUS (Statistical Sciences, Inc). The data
being used in the examples are from Dan Richter's Duke Phase II
soils data (1990).
Gabriel, K.R., 1971. The biplot graphic display of matrices with
application to principal component analysis. Biometrika.
58:453-467.
Gabriel, K.R., 1975. Appendix: Computer subroutine BIPLOT. in The
Joy of Statistics. (unpublished).
SAS Institute Inc. SAS/IML User's Guide for Personal Computers.
Version 6 Edition. Cary, NC: SAS Institute Inc.1985. 243 pp.
SAS Institute Inc. The PRINCOMP procedure. SAS User's Guide:
Statistics, Version 5 Edition. Cary, NC: SAS Institute Inc.
1985. pp 621-637.
Statistical Sciences, Inc., 1700 Westlake Ave. N., Suite 500,
Seattle, WA 98109.
.,
')
BIPLOTS PROGRAM
3
1
BIPLOTS is located on the NCSUVM system under "NSBD.LOAD"
,,0
(see JCL statements). Input is in Fortran IV format as follows
(the column positions of these options are important):
col umn #
12345678901234567890123456789012345678901234567 89012345 67890
JCL
IISOILS JOB NCS.ES.C5780,SPRUILL
II EXEC PGM=BIPLOT
IISTEPLIB DD DSN=NSBD.LOAD,DISP=SHR
IIFT03F001 DD SYSOUT=A,DCB=(RECFM=FA,BLKSIZE=133)
IIFT01F001 DD *
DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
TITLE
NROW NCOL ISYM ITRA NDRO NDCO MTRO MTCO IMTRO IMTCO
line 2
54
5
1
BOUND
line 3
line 4
1
1
1
COMP IFMT RHSQl
1
1
RHOSQ2
0
0
RHOSQ3
5.000
4
1 0.0
0.0
0.0
(F9.6,4F10.6)
-0.073317 -0.127631 0.066638 -0.011778 -0.057094
-0.060112 0.052142 -0.058041 -0.045873 -0.239499
-0.099725 -0.095511 -0.087708 -0.146846 -0.106032
-0.071116 0.223842 0.152030 0.049855 0.098617
DATA
-0.077718
-0.053510
-0.097525
-0.060112
column #
-0.152015
-0.139739
-0.081048
-0.116532
-0.321833 -0.212413 0.107515
-0.171496 -0.225526 -0.052645
-0.191140 -0.111440 0.094169
-0.168289 0.003958 0.352205
1234567890123456789012345678901234567 890123456789012345 67 890
The first line after the JCL statements is a TITLE statement
followed in the second line by these program options:
NROW
NCOL
ISYM
= number
= number
ITRA
=
NDRO
=
NDCO
=
=
of rows in data matrix
of columns in data matrix
1 if read entire X matrix
2 if read only lower triangle of symmetric X
3 if read only upper triangle of symmetric X
1 if X is not to be transposed before
decomposition
2 if X is to be transposed before decomposition
type of row deviations to be made before
decomposing X:
1 - not adjusted
2 - deviated from means
3 - weighted
4, 5, 6 - contingency tables
type of column deviations to be made before
decomposing X: (same as NDRO, but only 1, 2, or
3 can be used)
4
MTRO
<~
= for use in weighting rows:
1 - no weights
2 - same as NDRO #3
3 - use inverse variance
4 - input vector of weights
5 - input matrix of inverses
MTCO = same as MTRO but for columns.
IMTRO = 0 - to suppress output of weight matrix
1 - to print out weight matrix
IMTCO = same as IMTRO but for columns.
The third input line contains the options:
BOUND = percent of original sums of squares of X for
residuals before stopping.
COMP = maximum number of components to be calculated.
IFMT = number of format cards to follow.
RHOSQ1, RHOSQ2, and RHOSQ3 = Roy's maximum
characteristic root distribution for printout
of radii of comparison circles.
The fourth line is the FORTRAN format code for the input of the
,~
data matrix which follows in the succeeding lines. More details
on these options are given in Gabriel, 1975.
For this example, the data matrix contains data that is
already centered and scaled to unit length. The singular value
decomposition of X (n*p) is defined as SVD(X) = V*L*W', where
V=roweigenvectors (n*p) , L=root of eigenvalues (p*p) , and
W=column eigenvectors (p*p). Coordinates for the plots were
computed as G=V, the row eigenvectors, and as H=W*L', the column
eigenvectors times the square root of the eigenvalues. The
resulting eigenvalues and percent variances were:
Order
1
2
3
4
5
Eigenvalue
Percent
1.70468
1.37646
0.94223
0.72548
not computed
34.0937
27.5291
18.8445
14.5096
5
•
.~
IISOILS JOB NCS.ES.C5780,SPRUILL
II EXEC PGM=BIPLOT
IISTEPLIB DD DSN=NSBD.LOAD,DISP=SHR
IIFT03F001 DD SYSOUT=A,DCB=(RECFM=FA,BLKSIZE=133)
IIFT01F001 DD *
DUKE FOREST PHASE I I SOILS ANALYSIS - FROM RICHTER (SCALED)
1
1
0
0
5
1
1
1
1
54
0.0
0.0
5.000
4
1 0.0
(F9.6,4F10.6)
-0.073317 -0.127631 0.066638 -0.011778 -0.057094
~0.060112
0.052142 -0.058041 -0.045873 -0.239499
-0.099725 -0.095511 -0.087708 -0.146846 -0.106032
-0.071116 0.223842 0.152030 0.049855 0.098617
0.063128 -0.179259 0.099913 0.129847 0.080822
-0.108528 0.141439 0.144413 0.082639 -0.048196
-0.029302 -0.057000 0.019332 0.078705 -0.106032
-0.156944 0.258653 0.029756 -0.035382 -0.101583
0.032318 -0.145793 0.137598 0.127224 0.125311
-0.031503 -0.062381 0.003697 0.123290 -0.061543
-0.099725 0.046760 -0.162275 0.007892 0.000741
-0.051309 0.065091 0.147620 -0.142912 -0.239499
-0.112930 0.085776 0.062228 0.047233 -0.110481
-0.073317 ~0.157901 0.092296 0.154763 -0.097134
-0.042507 -0.027234 0.125971 0.213773 -0.074890
-0.123933 0.091325 -0.045613 0.077393 -0.074890
-0.150342 0.380071 0.077463 -0.081279 -0.057094
-0.145941 0.106124 0.045792 0.043299 -0.012605
0.124748 -0.078357 -0.034388 -0.036693 -0.070441
0.078533 -0.007054 0.075859 -0.019646 0.147555
0.100540 -0.073144 0.036972 -0.099638 0.049679
0.069730 0.123950 0.202944 0.216396 -0.195010
0.355824 -0.106946 0.055413 -0.212413 0.022986
0.245788 0.012454 0.384552 0.343596 0.174249
-0.082120 0.153716 -0.193545 -0.144223 -0.012605
-0.090923 0.004214 -0.017150 0.006581 -0.154970
0.276598 -0.105937 0.119156 -0.031448 -0.088237
0.190770 -0.025552 0.196931 -0.086524 -0.048196
0.038920 0.102592 0.154435 -0.003910 0.085271
0.120347 -0.006717 0.043386 0.009204 0.054128
0.406441 -0.116700 -0.011938 -0.149469 -0.097134
-0.121733 0.348119 -0.141428 -0.106194 -0.128277
~0.099725
0.233596 -0.015145 -0.064231 -0.217255
-0.011696 0.127818 0.043386 -0.169139 0.325512
0.305208 -0.197085 0.131183 0.062969 -0.083788
0.230383 -0.016303 0.208156 0.136404 -0.030401
-0.029302 0.028766 -0.057640 -0.045873 0.058577
-0.097525 0.133704 -0.127798 0.173121 0.027435
-0.093123 -0.137217 -0.176708 -0.140289 0.005190
-0.093123 0.015985 0.016125 0.076082 0.049679
0.181967 0.000178 0.118755 0.083950 0.174249
-0.011696 -0.118718 -0.197153 -0.287160 0.027435
-0.099725 -0.060027 -0.029177 0.367200 -0.128277
-0.115130 -0.242659 -0.055636 0.007892 0.361103
0.056526 -0.097024 -0.088911 0.089196 -0.146072
-0.075517 -0.107114 -0.219604 -0.129799 -0.017054
0.043322 -0.050441 -0.170694 0.174433 0.076373
-0.117331 -0.093493 -0.134613 -0.133733 -0.030401
-0.077718 0.152875 -0.121383 -0.026203 0.263227
0.014712 0.093343 0.007305 0.013138 0.125311
-0.077718 -0.152015 -0.321833 -0.212413 0.107515
-0.053510 -0.139739 -0.171496 -0.225526 -0.052645
-0.097525 -0.081048 -0.191140 -0.111440 0.094169
-0.060112 -0.116532 -0.168289 0.003958 0.352205
6
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
NROW NCOL ISYM ITRA NDRO NDCO MTRO MTCO IMTRO IMTCO
54
5
1
1
1
1
1
1
0
0
BOUND COMP IFMT
5.000
4
1
VARIABLE
FORMAT FOR
DATA
RHOSQl
0.0
RHOSQ2
0.0
MATRIX
(F9.6,4Fl0.6)
,"
7
RHOSQ3
0.0
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
DATA MATRIX,
-0.07332
-0.06011
-0.09972
-0.07112
0.06313
-0.10853
-0.02930
-0.15694
0.03232
-0.03150
-0.09972
-0.05131
-0.11293
-0.07332
-0.04251
-0.12393
-0.15034
-0.14594
0.12475
0.07853
0.10054
0.06973
0.35582
0.24579
-0.08212
-0.09092
0.27660
0.19077
0.03892
0.12035
0.40644
-0.12173
-0.09972
-0.01170
0.30521
0.23038
-0.02930
-0.09753
-0.09312
-0.09312
0.18197
-0.01170
-0.09972
-0.11513
0.05653
-0.07552
0.04332
-0.11733
-0.07772
0.01471
-0.07772
-0.05351
-0.09753
-0.06011
-0.12763
0.05214
-0.09551
0.22384
-0.17926
0.14144
-0.05700
0.25865
-0.14579
-0.06238
0.04676
0.06509
0.08578
-0.15790
-0.02723
0.09133
0.38007
0.10612
-0.07836
-0.00705
-0.07314
0.12395
-0.10695
0.01245
0.15372
0.00421
-0.10594
-0.02555
0.10259
-0.00672
-0.11670
0.34812
0.23360
0.12782
-0.19708
-0.01630
0.02877
0.13370
-0.13722
0.01599
0.00018
-0.11872
-0.06003
-0.24266
-0.09702
-0.10711
-0.05044
-0.09349
0.15287
0.09334
-0.15201
-0.13974
-0.08105
-0.11653
-0.01178
-0.04587
-0.14685
0.04986
0.12985
0.08264
0.07870
-0.03538
0.12722
0.12329
0.00789
-0.14291
0.04723
0.15476
0.21377
0.07739
-0.08128
0.04330
-0.03669
-0.01965
-0.09964
0.21640
-0.21241
0.34360
-0.14422
0.00658
-0.03145
-0.08652
-0.00391
0.00920
-0.14947
-0.10619
-0.06423
-0.16914
0.06297
0.13640
-0.04587
0.17312
-0.14029
0.07608
0.08395
-0.28716
0.36720
0.00789
0.08920
-0.12980
0.17443
-0.13373
-0.02620
0.01314
-0.21241
-0.22553
-0.11144
0.00396
0.06664
-0.05804
-0.08771
0.15203
0.09991
0.14441
0.01933
0.02976
0.13760
0.00370
-0.16228
0.14762
0.06223
0.09230
0.12597
-0.04561
0.07746
0.04579
-0.03439
0.07586
0.03697
0.20294
0.05541
0.38455
-0.19354
-0.01715
0.11916
0.19693
0.15444
0.04339
-0.01194
-0.14143
-0.01514
0.04339
0.13118
0.20816
-0.05764
-0.12780
-0.17671
0.01612
0.11875
-0.19715
-0.02918
-0.05564
-0.08891
-0.21960
-0.17069
-0.13461
-0.12138
0.00730
-0.32183
-0.17150
-0.19114
-0.16829
8
-0.05709
-0.23950
-0.10603
0.09862
0.08082
-0.04820
-0.10603
-0.10158
0.12531
-0.06154
0.00074
-0.23950
-0.11048
-0.09713
-0.07489
-0.07489
-0.05709
-0.01260
-0.07044
0.14756
0.04968
-0.19501
0.02299
0.17425
-0.01260
-0.15497
-0.08824
-0.04820
0.08527
0.05413
-0.09713
-0.12828
-0.21726
0.32551
-0.08379
-0.03040
0.05858
0.02744
0.00519
0.04968
0.17425
0.02744
-0.12828
0.36110
-0.14607
-0.01705
0.07637
-0.03040
0.26323
0.12531
0.10751
-0.05264
0.09417
0.35221
~,
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
PRODUCT MATRIX·
1 O.99999830D+OO -O.35611405D+OO O.45217659D+OO O.51606429D-Ol O.48697931D-Ol
2 -O.35611405D+OO O.99999938D+OO O.14355015D+OO O.25729240D-Ol -O.16764408D+OO
3 O.45217659D+OO O.14355015D+OO O.l0000003D+Ol O.49558986D+OO -O.84879438D-Ol
4 O.51606429D-Ol O.25729240D-Ol O.49558986D+OO O.l0000009D+Ol -O.19551719D-Ol
5 O.48697931D-Ol -O.16764408D+OO -O.84879438D-Ol -O.19551719D-Ol O.99999897D+OO
EUCLIDEAN NORM
O.499999786D+Ol
9
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
1TH COMPONENT CORRESPONDING TO ROOT
0.329531323D+01
0.130564D+01 (SQUARE OF ROOT
0.170468D+01 )
70 ITERATIONS. REMAINDER
34.093707 PERCENT
65.9063 PERCENT
W(J)
V(I)
0.0119
-0.0690
-0.1326
0.0509
0.1387
0.0552
0.0365
-0.0770
0.1413
0.0441
-0.1242
0.0022
0.0028
0.0939
0.1364
-0.0471
-0.0778
-0.0247
0.0251
0.0592
0.0240
0.2142
0.0940
0.4259
-0.2005
-0.0383
0.1696
0.1472
0.0840
0.0728
0.1079
-0.1854
-0.0837
-0.0671
0.2308
0.2550
-0.0633
-0.0478
-0.1736
0.51827827D+00 -0.96441436D-01
0.0080
-0.0467
-0.0897
0.0344
0.0939
0.0373
0.0247
-0.0521
0.0956
0.0298
-0.0840
0.0015
0.0019
0.0635
0.0923
-0.0319
-0.0527
-0.0167
0.0170
0.0400
0.0163
0.1449
0.0636
0.2882
-0.1357
-0.0259
0.1148
0.0996
0.0569
0.0493
0.0730
-0.1255
-0.0566
-0.0454
0.1562
0.1725
-0.0428
-0.0323
-0.1175
-0.0015
0.0087
0.0167
-0.0064
-0.0175
-0.0069
-0.0046
0.0097
-0.0178
-0.0056
0.0156
-0.0003
-0.0003
-0.0118
-0.0172
0.0059
0.0098
0.0031
-0.0032
-0.0074
-0.0030
-0.0270
-0.0118
-0.0536
0.0253
0.0048
-0.0214
-0.0185
-0.0106
-0.0092
-0.0136
0.0233
0.0105
0.0085
-0.0291
-0.0321
0.0080
0.0060
0.0219
0.67732351D+00
0.0105
-0.0610
-0.1173
0.0450
0.1227
0.0488
0.0322
-0.0681
0.1250
0.0390
-0.1098
0.0019
0.0024
0.0830
0.1206
-0.0417
-0.0688
-0.0218
0.0222
0.0523
0.0212
0.1894
0.0831
0.3766
-0.1773
-0.0339
0.1500
0.1302
0.0743
0.0644
0.0954
-0.1640
-0.0740
-0.0593
0.2041
0.2255
-0.0560
-0.0423
-0.1536
10
0.51180928D+00 -0.37024567D-01
0.0079
-0.0461
-0.0886
0.0340
0.0927
0.0369
0.0244
-0.0514
0.0944
0.0295
-0.0830
0.0015
0.0018
0.0627
0.0912
-0.0315
-0.0520
-0.0165
0.0168
0.0395
0.0161
0.1431
0.0628
0.2846
-0.1340
-0.0256
0.1133
0.0984
0.0562
0.0487
0.0721
-0.1239
-0.0559
-0.0448
0.1542
0.1704
-0.0423
-0.0319
-0.1160
-0.0006
0.0033
0.0064
-0.0025
-0.0067
-0.0027
-0.0018
0.0037
-0.0068
-0.0021
0.0060
-0.0001
-0.0001
-0.0045
-0.0066
0.0023
0.0038
0.0012
-0.0012
-0.0029
-0.0012
-0.0104
-0.0045
-0.0206
0.0097
0.0019
-0.0082
-0.0071
-0.0041
-0.0035
-0.0052
0.0090
0.0040
0.0032
-0.0112
-0.0123
0.0031
0.0023
0.0084
"
-0.0014
0.1618
-0.2115
0.0973
-0.0638
0.0226
-0.1864
-0.0014
-0.1611
-0.1228
0.0043
-0.2729
-0.1868
-0.1782
-0.1110
-0.0009
0.1095
-0.1431
0.0658
-0.0432
0.0153
-0.1261
-0.0010
-0.1090
-0.0831
0.0029
-0.1847
-0.1264
-0.1206
-0.0751
0.0002
-0.0204
0.0266
-0.0123
0.0080
-0.0028
0.0235
0.0002
0.0203
0.0155
-0.0005
0.0344
0.0235
0.0224
0.0140
-0.0012
0.1431
-0.1870
0.0860
-0.0564
0.0200
-0.1648
-0.0013
-0.1424
-0.1086
0.0038
-0.2413
-0.1652
-0.1576
-0.0982
11
-0.0009
0.1081
-0.1413
0.0650
-0.0426
0.0151
-0.1246
-0.0009
-0.1076
-0.0821
0.0029
-0.1824
-0.1248
-0.1191
-0.0742
0.0001
-0.0078
0.0102
-0.0047
0.0031
-0.0011
0.0090
0.0001
0.0078
0.0059
-0.0002
0.0132
0.0090
0.0086
0.0054
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
1TH RESIDUAL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
-0.0814
-0.0134
-0.0100
-0.1055
-0.0308
-0.1458
-0.0540
-0.1049
-0.0633
-0.0613
-0.0157
-0.0528
-0.1148
-0.1368
-0.1348
-0.0920
-0.0977
-0.1292
0.1078
0.0385
0.0843
-0.0752
0.2922
-0.0424
0.0536
-0.0650
0.1618
0.0911
-0.0179
0.0711
0.3334
0.0037
-0.0431
0.0337
0.1490
0.0578
0.0135
-0.0652
0.0244
-0.0922
0.0725
0.1314
-0.1656
-0.1261
0.0435
-0.1122
0.2302
-0.1618
0.1484
-0.0524
0.2490
-0.1280
-0.0568
0.0311
0.0654
0.0861
-0.1461
-0.0101
0.0854
0.3703
0.1030
-0.0752
0.0004
-0.0701
0.1509
-0.0951
0.0661
0.1285
-0.0006
-0.0846
-0.0070
0.1132
0.0025
-0.1031
0.3248
0.2231
0.1194
-0.1680
0.0158
0.0208
0.1277
-0.1591
0.0158
0.0206
-0.1453
-0.0478
0.0561
0.0030
0.0295
0.1071
-0.0228
0.0956
-0.0129
0.0978
0.0126
-0.0353
-0.0525
0.1457
0.0598
0.0093
0.0053
-0.0039
0.1463
0.0676
-0.0566
0.0235
0.0157
0.0136
-0.0277
0.0079
-0.0162
0.0167
-0.0308
0.0667
0.0801
-0.0210
-0.1074
0.0225
0.0589
0.1027
-0.0729
-0.0173
-0.0017
-0.0855
-0.0232
0.0173
-0.0243
-0.0101
-0.1152
12
-0.0197
0.0002
-0.0582
0.0159
0.0371
0.0458
0.0543
0.0160
0.0328
0.0938
0.0909
-0.1444
0.0454
0.0920
0.1226
0.1089
-0.0293
0.0598
-0.0535
-0.0592
-0.1157
0.0733
-0.2752
0.0590
-0.0102
0.0322
-0.1448
-0.1849
-0.0601
-0.0395
-0.2216
0.0177
-0.0083
-0.1243
-0.0913
-0.0340
-0.0036
0.2051
-0.0243
0.0770
-0.0242
-0.1458
0.3022
-0.0565
-0.2428
-0.1124
0.1011
0.0875
-0.0455
-0.1043
-0.1053
0.1321
-0.0594
-0.0053
-0.2394
-0.1103
-0.0926
-0.0683
-0.0772
-0.0609
-0.0138
-0.0692
0.1504
0.0508
-0.1847
0.0275
0.1948
-0.0223
-0.1568
-0.0800
-0.0411
0.0893
0.0576
-0.0919
-0.1372
-0.2213
0.3223
-0.0726
-0.0181
0.0555
0.0251
-0.0032
0.0496
0.1821
0.0172
-0.1236
,.
44
45
46
47
48
49
50
51
52
53
54
-0.0720
0.0412
0.0506
0.0443
-0.0083
0.0054
0.0118
0.1069
0.0729
0.0231
0.0150
-0.2507
-0.0942
-0.1306
-0.0506
-0.1138
0.1374
0.0939
-0.1864
-0.1633
-0.1035
-0.1305
0.0008
-0.1089
-0.0548
-0.1694
0.0078
-0.0127
0.0035
-0.0805
-0.0063
-0.0335
-0.0701
0.0505
0.0741
-0.0052
0.1754
-0.0261
0.0559
0.0102
-0.0301
-0.1007
0.0077
0.0781
CANONICAL WEIGHTS
FOR COLS
0.518278
-0.096441
0.677324
0.511809
-0.037025
FOR ROWS
0.011896
-0.069014
-0.132588
0.050851
0.138740
0.055150
FOR ROWS
0.036467
-0.076958
0.141298
0.044095
-0.124151
0.002175
FOR ROWS
0.002766
0.093862
0.136411
-0.047142
-0.077810
-0.024685
FOR ROWS
0.025082
0.059163
0.024026
0.214162
0.093974
0.425889
FOR ROWS
-0.200535
-0.038326
0.169611
0.147225
0.084037
0.072849
FOR ROWS
0.107928
-0.185395
-0.083716
-0.067110
0.230825
0.254973
FOR ROWS
-0.063302
-0.047801
-0.173641
-0.001366
0.161793
-0.211495
FOR ROWS
0.097292
-0.063786
0.022588
-0.186386
-0.001416
-0.161064
FOR ROWS
-0.122849
0.004331
-0.272894
-0.186799
-0.178239
-0.110993
13
0.3580
-0.1450
-0.0261
0.0763
-0.0382
0.2573
0.1255
0.0943
-0.0617
0.0856
0.3468
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
PRODUCT MATRIX
1 O.54209893D+OO -0.27090794D+00 -0.14623940D+00 -0.40057758D+00 0.81409169D-Ol
2 -O.27090794D+00 O.98414419D+OO 0.25490365D+00 O.10987183D+OO -0.17373100D+00
3 -0.14623940D+00 O.25490365D+00 0.21794707D+00 -O.95356886D-Ol -0.42130027D-Ol
4 -0.40057758D+00 0.10987183D+00 -0.95356886D-Ol 0.55346089D+OO 0.12751227D-Ol
5 0.81409169D-Ol -0.17373100D+OO -0.42130027D-Ol O.12751227D-Ol 0.99766216D+00
EUCLIDEAN NORM
0.329531323D+Ol
14
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
2TH COMPONENT CORRESPONDING TO ROOT
0.191885761D+Ol
0.117322D+Ol (SQUARE OF ROOT
0.137646D+Ol )
53 ITERATIONS. REMAINDER
27.529124 PERCENT
38.3772 PERCENT
W(J)
V(I)
-0.0195
0.1212
-0.0311
0.1635
-0.1133
0.1885
0.0378
0.2490
-0.0920
0.0269
0.0391
0.1384
0.1569
0.0211
0.0999
0.1397
0.3006
0.1417
-0.0828
-0.0796
-0.1174
0.2070
-0.2475
-0.0006
0.0602
0.0924
-0.1221
-0.0557
0.0417
-0.0595
-0.2249
0.2529
0.2403
-0.0712
-0.1648
-0.0167
-0.0142
0.1260
-0.1136
-0.44116641D+00
0.0101
-0.0628
0.0161
-0.0846
0.0586
-0.0976
-0.0196
-0.1289
0.0476
-0.0139
-0.0202
-0.0716
-0.0812
-0.0109
-0.0517
-0.0723
-0.1556
-0.0733
0.0429
0.0412
0.0608
-0.1071
0.1281
0.0003
-0.0312
-0.0478
0.0632
0.0288
-0.0216
0.0308
0.1164
-0.1309
-0.1244
0.0368
0.0853
0.0086
0.0073
-0.0652
0.0588
0.70660841D+00
-0.0162
0.1005
-0.0258
0.1355
-0.0939
0.1563
0.0313
0.2064
-0.0763
0.0223
0.0324
0.1148
0.1300
0.0175
0.0828
0.1158
0.2492
0.1175
-0.0687
-0.0660
-0.0973
0.1716
-0.2052
-0.0005
0.0499
0.0766
-0.1012
-0.0462
0.0345
-0.0493
-0.1865
0.2096
0.1992
-0.0590
-0.1367
-0.0138
-0.0117
0.1045
-0.0942
0.20393611D+00
-0.0047
0.0290
-0.0074
0.0391
-0.0271
0.0451
0.0090
0.0596
-0.0220
0.0064
0.0093
0.0331
0.0375
0.0050
0.0239
0.0334
0.0719
0.0339
-0.0198
-0.0190
-0.0281
0.0495
-0.0592
-0.0002
0.0144
0.0221
-0.0292
-0.0133
0.0100
-0.0142
-0.0538
0.0605
0.0575
-0.0170
-0.0394
-0.0040
-0.0034
0.0301
-0.0272
15
0.27873814D+00 -0.43219424D+00
-0.0064
0.0397
-0.0102
0.0535
-0.0370
0.0616
0.0124
0.0814
-0.0301
0.0088
0.0128
0.0453
0.0513
0.0069
0.0327
0.0457
0.0983
0.0463
-0.0271
-0.0260
-0.0384
0.0677
-0.0809
-0.0002
0.0197
0.0302
-0.0399
-0.0182
0.0136
-0.0195
-0.0736
0.0827
0.0786
-0.0233
-0.0539
-0.0054
-0.0046
0.0412
-0.0371
0.0099
-0.0615
0.0158
-0.0829
0.0574
-0.0956
-0.0192
-0.1263
0.0466
-0.0136
-0.0198
-0.0702
-0.0795
-0.0107
-0.0506
-0.0708
-0.1524
-0.0718
0.0420
0.0404
0.0595
-0.1049
0.1255
0.0003
-0.0305
-0.0469
0.0619
0.0282
-0.0211
0.0302
0.1140
-0.1282
-0.1219
0.0361
0.0836
0.0084
0.0072
-0.0639
0.0576
0.0472
-0.0919
-0.1797
0.1308
-0.2437
-0.0201
-0.0988
-0.0630
-0.0562
-0.0030
0.0089
-0.2083
-0.1280
-0.1065
-0.2056
-0.0244
0.0476
0.0930
-0.0677
0.1261
0.0104
0.0512
0.0326
0.0291
0.0016
-0.0046
0.1078
0.0663
0.0551
0.1064
0.0391
-0.0762
-0.1490
0.1084
-0.2020
-0.0167
-0.0819
-0.0523
-0.0466
-0.0025
0.0074
-0.1727
-0.1061
-0.0883
-0.1705
0.0113
-0.0220
-0.0430
0.0313
-0.0583
-0.0048
-0.0236
-0.0151
-0.0134
-0.0007
0.0021
-0.0498
-0.0306
-0.0255
-0.0492
16
0.0154
-0.0301
-0.0588
0.0428
-0.0797
-0.0066
-0.0323
-0.0206
-0.0184
-0.0010
0.0029
-0.0681
-0.0419
-0.0348
-0.0672
-0.0239
0.0466
0.0911
-0.0663
0.1236
0.0102
0.0501
0.0320
0.0285
0.0015
-0.0045
0.1056
0.0649
0.0540
0.1043
;>
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
2TH RESIDUAL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
-0.0915
0.0493
-0.0261
-0.0209
-0.0894
-0.0483
-0.0344
0.0240
-0.1109
-0.0474
0.0045
0.0189
-0.0336
-0.1259
-0.0831
-0.0198
0.0579
-0.0559
0.0649
-0.0027
0.0235
0.0319
0.1641
-0.0427
0.0847
-0.0172
0.0986
0.0623
0.0036
0.0402
0.2170
0.1346
0.0813
-0.0031
0.0637
0.0492
0.0062
0.0000
-0.0344
-0.0678
0.0249
0.0384
-0.0979
-0.1100
-0.0571
-0.0864
0.0947
-0.0679
-0.0079
-0.0838
0.0426
-0.0517
-0.0791
-0.0013
-0.0494
-0.0439
-0.1635
-0.0928
-0.0304
0.1210
-0.0144
-0.0065
0.0664
0.0272
-0.0207
0.1101
0.0666
0.0786
-0.0772
0.0166
0.0392
0.0786
0.0518
0.0834
0.1151
0.0238
0.1784
-0.0314
0.0296
0.0325
0.0232
-0.0649
-0.0233
0.0968
0.0036
-0.1562
0.0608
-0.0260
0.0370
0.0679
0.0043
0.0505
-0.0220
0.0382
0.0347
-0.0417
-0.0618
0.1126
0.0223
0.0043
-0.0186
-0.0373
0.0743
0.0337
-0.0367
0.0426
0.0438
-0.0360
0.0315
0.0081
-0.0306
-0.0054
-0.0016
0.0801
0.0702
-0.0068
-0.0536
-0.0380
0.0014
0.1198
-0.0335
-0.0133
0.0017
-0.1157
0.0040
0.0060
-0.0023
0.0329
-0.1465
17
-0.0134
-0.0394
-0.0481
-0.0376
0.0742
-0.0159
0.0420
-0.0654
0.0629
0.0850
0.0781
-0.1896
-0.0059
0.0852
0.0900
0.0632
-0.1276
0.0135
-0.0264
-0.0331
-0.0773
0.0056
-0.1943
0.0592
-0.0299
0.0020
-0.1049
-0.1667
-0.0737
-0.0200
-0.1480
-0.0650
-0.0869
-0.1010
-0.0374
-0.0285
0.0011
0.1639
0.0129
0.0616
0.0059
-0.0871
0.2594
-0.0664
-0.1814
-0.1282
0.1840
0.0301
0.0500
-0.0851
0.0209
0.0855
-0.0458
0.0145
-0.1692
-0.0308
-0.0819
-0.0177
-0.0064
0.0916
0.0580
-0.1112
0.1100
-0.0087
-0.0797
-0.0980
0.1945
0.0082
-0.1100
-0.1419
-0.0693
0.1105
0.0275
-0.2060
-0.0090
-0.0994
0.2862
-0.1562
-0.0265
0.0483
0.0890
-0.0608
0.0736
0.1355
-0.0739
-0.0573
44
45
46
47
48
49
50
51
52
53
54
-0.1981
0.0308
-0.0006
0.0117
-0.0374
0.0039
0.0164
-0.0009
0.0066
-0.0321
-0.0914
-0.0487
-0.0775
-0.0486
0.0016
-0.0672
0.1399
0.0865
-0.0137
-0.0571
-0.0152
0.0400
0.0591
-0.1041
-0.0311
-0.1544
0.0213
-0.0120
0.0013
-0.0307
0.0243
-0.0080
-0.0209
0.1302
0.0807
0.0271
0.1960
-0.0077
0.0569
0.0073
0.0381
-0.0588
0.0425
0.1454
CANONICAL WEIGHTS
FOR COLS
-0.441166
0.706608
0.203936
0.278738
-0.432194
FOR ROWS
-0.019483
0.121247
-0.031099
0.163500
-0.113258
0.188486
FOR ROWS
0.037808
0.248984
-0.091979
0.026881
0.039057
0.138430
FOR ROWS
0.156864
0.021064
0.099855
0.139652
0.300629
0.141685
FOR ROWS
-0.082848
-0.079617
-0.117405
0.206959
-0.247513
-0.000635
FOR ROWS
0.060195
0.092398
-0.122067
-0.055695
0.041657
-0.059511
FOR ROWS
-0.224924
0.252881
0.240329
-0.071175
-0.164838
-0.016660
FOR ROWS
-0.014153
0.126009
-0.113584
0.047222
-0.091920
-0.179704
FOR ROWS
0.130770
-0.243676
-0.020144
-0.098844
-0.063033
-0.056162
FOR ROWS
-0.002995
0.008915
-0.208346
-0.128039
-0.106533
-0.205639
18
0.2345
-0.1552
-0.0762
0.0443
-0.0667
0.2558
0.1300
-0.0113
-0.1266
0.0315
0.2426
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
PRODUCT MATRIX
1 0.27420240D+00 o .15817718D+00 -0.22400014D-Ol -0.23131496D+00 -0.18103905D+00
2 0.15817718D+00 0.29688617D+00 0.56552307D-Ol -0.16123309D+00 0.24662765D+00
3 -0.22400014D-Ol 0.56552307D-Ol 0.16070036D+00 -0.17360119D+00 0.79190792D-Ol
4 -0.23131496D+00 -0.16123309D+00 -0.17360119D+00 0.44651724D+00 0.17857148D+00
5 -0.18103905D+00 0.24662765D+00 0.79190792D-Ol 0.17857148D+00 0.74055144D+00
EUCLIDEAN NORM
0.191885761D+Ol
19
10UKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALEO)
3TH COMPONENT CORRESPONOING TO ROOT
0.9766314960+00
0.9706830+00 (SQUARE OF ROOT
0.9422260+00 )
62 ITERATIONS. REMAINOER
18.844530 PERCENT
19.5326 PERCENT
W(J) -0.336857710+00
V(I)
-0.0457
0.0149
-0.1984
0.0649
-0.1333
0.0436
0.1642
-0.0537
0.0788
-0.0258
0.0528
-0.0173
-0.0557
0.0182
-0.0114
0.0037
0.1319
-0.0431
0.0012
-0.0004
0.0429
-0.0140
-0.2378
0.0778
-0.0232
0.0076
-0.0138
0.0045
0.0383
-0.0125
0.0233
-0.0076
0.0230
-0.0075
0.0735
-0.0240
-0.1309
0.0428
0.0917
-0.0300
-0.0439
0.0143
-0.0808
0.0264
-0.2076
0.0679
0.2169
-0.0709
-0.0244
0.0080
0.0323
-0.0989
-0.1985
0.0649
-0.1452
0.0475
0.0748
-0.0245
-0.0028
0.0084
-0.3044
0.0995
-0.0663
0.0217
-0.1472
0.0481
0.2317
-0.0758
-0.1774
0.0580
-0.0480
0.0157
0.0445
-0.0146
0.1475
-0.0482
-0.0441
0.0144
0.13 597 6500+00
-0.0060
-0.0262
-0.0176
0.0217
0.0104
0.0070
-0.0074
-0.0015
0.0174
0.0002
0.0057
-0.0314
-0.0031
-0.0018
0.0051
0.0031
0.0030
0.0097
-0.0173
0.0121
-0.0058
-0.0107
-0.0274
0.0286
-0.0032
-0.0131
-0.0262
-0.0192
0.0099
0.0011
-0.0402
-0.0088
-0.0194
0.0306
-0.0234
-0.0063
0.0059
0.0195
-0.0058
0.132622130-01
-0.0006
-0.0026
-0.0017
0.0021
0.0010
0.0007
-0.0007
-0.0001
0.0017
0.0000
0.0006
-0.0031
-0.0003
-0.0002
0.0005
0.0003
0.0003
0.0009
-0.0017
0.0012
-0.0006
-0.0010
-0.0027
0.0028
-0.0003
-0.0013
-0.0026
-0.0019
0.0010
0.0001
-0.0039
-0.0009
-0.0019
0.0030
-0.0023
-0.0006
0.0006
0.0019
-0.0006
20
0.409711060+00
-0.0182
-0.0789
-0.0530
0.0653
0.0313
0.0210
-0.0222
-0.0045
0.0525
0.0005
0.0171
-0.0946
-0.0092
-0.0055
0.0152
0.0093
0.0091
0.0293
-0.0521
0.0365
-0.0174
-0.0321
-0.0825
0.0863
-0.0097
-0.0393
-0.0790
-0.0577
0.0298
0.0034
-0.1210
-0.0264
-0.0586
0.0922
-0.0705
-0.0191
0.0177
0.0587
-0.0175
0.836658970+00
-0.0371
-0.1611
-0.1083
0.1333
0.0640
0.0429
-0.0453
-0.0092
0.1072
0.0010
0.0348
-0.1931
-0.0189
-0.0112
0.0311
0.0189
0.0186
0.0597
-0.1063
0.0745
-0.0356
-0.0656
-0.1686
0.1762
-0.0198
-0.0803
-0.1612
-0.1179
0.0608
0.0068
-0.2472
-0.0539
-0.1196
0.1882
-0.1441
-0.0390
0.0362
0.1198
-0.0358
0.1097
0.1241
-0.1128
0.0702
0.3198
-0.1227
-0.0613
0.1150
-0.0569
0.2626
0.1216
0.0043
-0.1439
0.0540
0.3075
-0.0359
-0.0406
0.0369
-0.0230
-0.1046
0.0401
0.0200
-0.0376
0.0186
-0.0858
-0.0398
-0.0014
0.0471
-0.0177
-0.1005
0.0145
0.0164
-0.0149
0.0093
0.0422
-0.0162
-0.0081
0.0152
-0.0075
0.0347
0.0161
0.0006
-0.0190
0.0071
0.0406
0.0014
0.0016
-0.0015
0.0009
0.0041
-0.0016
-0.0008
0.0015
-0.0007
0.0034
0.0016
0.0001
-0.0019
0.0007
0.0040
21
0.0436
0.0494
-0.0449
0.0279
0.1272
-0.0488
-0.0244
0.0457
-0.0226
0.1044
0.0484
0.0017
-0.0572
0.0215
0.1223
0.0891
0.1008
-0.0916
0.0570
0.2597
-0.0996
-0.0498
0.0934
-0.0462
0.2132
0.0988
0.0035
-0.1169
0.0439
0.2497
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
3TH RESIDUAL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
-0.1064
-0.0155
-0.0697
0.0328
-0.0636
-0.0310
-0.0526
0.0203
-0.0678
-0.0470
0.0185
-0.0589
-0.0412
-0.1304
-0.0706
-0.0121
0.0654
-0.0319
0.0221
0.0273
0.0092
0.0055
0.0963
0.0282
0.0768
-0.0495
0.0337
0.0149
0.0281
0.0430
0.1175
0.1129
0.0332
0.0726
0.0057
0.0335
0.0208
0.0483
-0.0488
-0.0319
0.0655
0.0015
-0.0749
-0.1039
-0.0309
-0.0688
0.0730
-0.0783
-0.0148
-0.0764
0.0441
-0.0692
-0.0793
-0.0069
-0.0180
-0.0409
-0.1617
-0.0979
-0.0335
0.1180
-0.0242
0.0108
0.0543
0.0330
-0.0100
0.1375
0.0380
0.0818
-0.0642
0.0428
0.0583
0.0688
0.0507
0.1235
0.1239
0.0433
0.1478
-0.0080
0.0360
0.0266
0.0037
-0.0591
-0.0378
0.0804
0.0185
-0.1655
0.0614
-0.0235
0.0387
0.0658
0.0033
0.0499
-0.0212
0.0384
0.0330
-0.0417
-0.0624
0.1156
0.0225
0.0044
-0.0190
-0.0376
0.0740
0.0328
-0.0351
0.0414
0.0444
-0.0349
0.0342
0.0053
-0.0303
-0.0041
0.0009
0.0819
0.0692
-0.0069
-0.0496
-0.0371
0.0033
0.1168
-0.0312
-0.0127
0.0012
-0.1176
0.0046
0.0046
-0.0039
0.0343
-0.1474
22
0.0048
0.0395
0.0050
-0.1029
0.0428
-0.0368
0.0641
-0.0609
0.0104
0.0845
0.0610
-0.0951
0.0033
0.0906
0.0747
0.0540
-0.1367
-0.0158
0.0257
-0.0696
-0.0599
0.0377
-0.1117
-0.0271
-0.0202
0.0413
-0.0259
-0.1090
-0.1034
-0.0234
-0.0270
-0.0386
-0.0283
-0.1932
0.0332
-0.0094
-0.0167
0.1052
0.0304
0.0179
-0.0435
-0.0422
0.2315
-0.0293
-0.0202
-0.0199
0.0507
-0.0339
0.0072
-0.0398
0.0302
-0.0217
-0.0468
-0.0203
0.0239
-0.0119
-0.0707
-0.0488
-0.0253
0.0729
-0.0017
-0.0049
0.0356
0.0269
-0.0141
0.0706
0.0183
0.0280
-0.0297
0.0193
0.0486
0.0497
0.0206
0.0412
0.0448
0.0201
0.0980
-0.0122
0.0125
0.0122
-0.0308
-0.0250
-0.0155
0.0347
0.0177
-0.1143
44
45
46
47
48
49
50
51
52
53
54
-0.0935
-0.0093
-0.0206
0.0493
-0.0560
0.0897
0.0562
0.0005
-0.0404
-0.0144
0.0091
-0.0909
-0.0613
-0.0406
-0.0135
-0.0597
0.1052
0.0704
-0.0142
-0.0381
-0.0223
-0.0006
0.0550
-0.1025
-0.0303
-0.1558
0.0220
-0.0154
-0.0002
-0.0307
0.0262
-0.0087
-0.0249
0.0030
0.1295
0.0514
0.1502
0.0149
-0.0475
-0.0410
0.0364
-0.0016
0.0210
0.0231
CANONICAL WEIGHTS
FOR COLS
-0.336858
0.135976
0.013262
0.409711
0.836659
FOR ROWS
-0.045707
-0.198421
-0.133344
0.164157
0.078816
0.052788
FOR ROWS
-0.055724
-0.011387
0.131950
0.001238
0.042911
-0.237811
FOR ROWS
-0.023234
-0.013814
0.038338
0.023295
0.022956
0.073549
FOR ROWS
-0.130940
0.091684
-0.043868
-0.080809
-0.207551
0.216919
FOR ROWS
-0.024352
-0.098886
-0.198528
-0.145154
0.074822
0.008427
FOR ROWS
-0.304370
-0.066310
-0.147245
0.231734
-0.177374
-0.048019
FOR ROWS
0.044538
0.147547
-0.044060
0.109709
0.124123
-0.112824
FOR ROWS
0.070224
0.319777
-0.122678
-0.061284
0.115021
-0.056869
FOR ROWS
0.262550
0.121624
0.004292
-0.143916
0.054009
0.307483
23
-0.0252
-0.0556
-0.0264
-0.0491
-0.0205
0.0425
0.0313
-0.0148
-0.0097
-0.0123
-0.0071
lDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
PRODUCT MATRIX
1 O.16728506D+OO O.20133560D+OO -O.18190640D-Ol -O.10127426D+OO O.84513269D-Ol
2 O.20133560D+OO O.27946478D+OO O.54853145D-Ol -O.21372552D+OO O.13943440D+OO
3 -O.18190640D-Ol O.54853145D-Ol O.16053464D+OO -O.17872094D+OO O.68735900D-Ol
4 -O.10127426D+OO -O.21372552D+OO -O.17872094D+OO O.28835219D+OO -O.14441274D+OO
5 O.84513269D-Ol O.13943440D+OO O.68735900D-Ol -O.14441274D+OO O.80994829D-Ol
EUCLIDEAN NORM
O.976631496D+OO
24
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
4TH COMPONENT CORRESPONDING TO ROOT
0.251154285D+00
0.851750D+00 (SQUARE OF ROOT
0.725477D+00 )
18 ITERATIONS. REMAINDER
14.509550 PERCENT
5.0231 PERCENT
W(J)
V (I)
-0.1110
-0.0704
-0.0753
0.1758
-0.1221
0.0210
-0.1411
0.1048
-0.0810
-0.1641
-0.0670
0.0750
-0.0450
-0.2544
-0.1735
-0.0872
0.2550
-0.0092
-0.0143
0.1239
0.0925
-0.0473
0.2503
0.0654
0.1035
-0.1062
0.0692
0.1667
0.1719
0.0748
0.1528
0.1646
0.0721
0.3409
-0.0407
0.0460
0.0436
-0.0995
-0.0903
0.35854040D+00
-0.0339
-0.0215
-0.0230
0.0537
-0.0373
0.0064
-0.0431
0.0320
-0.0247
-0.0501
-0.0205
0.0229
-0.0138
-0.0777
-0.0530
-0.0266
0.0779
-0.0028
-0.0044
0.0378
0.0283
-0.0144
0.0765
0.0200
0.0316
-0.0324
0.0211
0.0509
0.0525
0.0228
0.0467
0.0503
0.0220
0.1041
-0.0124
0.0141
0.0133
-0.0304
-0.0276
0.58204681D+00
-0.0550
-0.0349
-0.0373
0.0871
-0.0606
0.0104
-0.0700
0.0520
-0.0401
-0.0814
-0.0332
0.0372
-0.0223
-0.1261
-0.0860
-0.0432
0.1264
-0.0046
-0.0071
0.0614
0.0459
-0.0234
0.1241
0.0324
0.0513
-0.0526
0.0343
0.0826
0.0852
0.0371
0.0758
0.0816
0.0357
0.1690
-0.0202
0.0228
0.0216
-0.0493
-0.0448
0.27199628D+00 -0.58919296D+00
-0.0257
-0.0163
-0.0174
0.0407
-0.0283
0.0049
-0.0327
0.0243
-0.0188
-0.0380
-0.0155
0.0174
-0.0104
-0.0589
-0.0402
-0.0202
0.0591
-0.0021
-0.0033
0.0287
0.0214
-0.0110
0.0580
0.0152
0.0240
-0.0246
0.0160
0.0386
0.0398
0.0173
0.0354
0.0381
0.0167
0.0790
-0.0094
0.0107
0.0101
-0.0231
-0.0209
25
0.0557
0.0353
0.0378
-0.0882
0.0613
-0.0105
0.0708
-0.0526
0.0406
0.0824
0.0336
-0.0376
0.0226
0.1277
0.0871
0.0438
-0.1280
0.0046
0.0072
-0.0622
-0.0464
0.0237
-0.1256
-0.0328
-0.0519
0.0533
-0.0348
-0.0836
-0.0863
-0.0375
-0.0767
-0.0826
-0.0362
-0.1711
0.0204
-0.0231
-0.0219
0.0499
0.0453
0.33397600D+00
-0.0316
-0.0200
-0:0214
0.0500
-0.0347
0.0060
-0.0401
0.0298
-0.0230
-0.0467
-0.0191
0.0213
-0.0128
-0.0724
-0.0494
-0.0248
0.0725
-0.0026
-0.0041
0.0352
0.0263
-0.0135
0.0712
0.0186
0.0294
-0.0302
0.0197
0.0474
0.0489
0.0213
0.0435
0.0468
0.0205
0.0970
-0.0116
0.0131
0.0124
-0.0283
-0.0257
-0.0563
0.1249
0.0604
-0.3966
-0.0959
-0.1899
..:0.0920
-0.1615
-0.0757
0.1543
0.1124
-0.0503
-0.0374
-0.0435
-0.0233
-0.0172
0.0381
0.0184
-0.1211
-0.0293
-0.0580
-0.0281
-0.0493
-0.0231
0.0471
0.0343
-0.0154
-0.0114
-0.0133
-0.0071
-0.0279
0.0619
0.0299
-0.1966
-0.0476
-0.0941
-0.0456
-0.0800
-0.0375
0.0765
0.0557
-0.0249
-0.0185
-0.0215
-0.0116
-0.0130
0.0289
0.0140
-0.0919
-0.0222
-0.0440
-0.0213
-0.0374
-0.0175
0.0358
0.0260
-0.0116
-0.0087
-0.0101
-0.0054
26
0.0282
-0.0627
-0.0303
0.1990
0.0481
0.0953
0.0462
0.0810
0.0380
-0.0774
-0.0564
0.0252
0.0188
0.0218
0.0117
-0.0160
0.0355
0.0172
-0.1128
-0.0273
-0.0540
-0.0262
-0.0459
-0.0215
0.0439
0.0320
-0.0143
-0.0106
-0.0124
-0.0066
IDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
4TH RESIDUAL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
-0.0725
0.0060
-0.0467
-0.0209
-0.0263
-0.0374
-0.0095
-0.0117
-0.0430
0.0031
0.0390
-0.0818
-0.0275
-0.0527
-0.0176
0.0145
-0.0125
-0.0290
0.0264
-0.0106
-0.0191
0.0199
0.0198
0.0082
0.0452
-0.0171
0.0126
-0.0360
-0.0244
0.0202
0.0708
0.0626
0.0112
-0.0315
0.0181
0.0195
0.0075
0.0787
-0.0212
-0.0147
0.0274
-0.0169
0.0462
-0.0489
0.0040
-0.0315
-0.0141
-0.0177
-0.0253
-0.0064
-0.0079
-0.0290
0.0021
0.0263
-0.0552
-0.0185
-0.0356
-0.0119
0.0098
-0.0084
-0.0196
0.0178
-0.0071
-0.0129
0.0135
0.0134
0.0055
0.0305
-0.0115
0.0085
-0.0243
-0.0165
0.0136
0.0478
0.0423
0.0075
-0.0212
0.0122
0.0131
0.0050
0.0531
-0.0143
-0.0099
0.0185
-0.0114
0.0312
0.0871
-0.0072
0.0561
0.0251
0.0316
0.0450
0.0115
0.0141
0.0517
-0.0037
-0.0469
0.0983
0.0330
0.0634
0.0212
-0.0174
0.0150
0.0349
-0.0318
0.0127
0.0229
-0.0240
-0.0238
-0.0099
-0.0543
0.0205
-0.0151
0.0433
0.0294
-0.0242
-0.0851
-0.0753
-0.0134
0.0378
-0.0218
-0.0234
-0.0090
-0.0945
0.0255
0.0177
-0.0329
0.0203
-0.0555
27
-0.0509
0.0042
-0.0328
-0.0147
-0.0185
-0.0263
-0.0067
-0.0082
-0.0302
0.0022
0.0274
-0.0574
-0.0193
-0.0370
-0.0124
0.0102
-0.0087
-0.0204
0.0186
-0.0074
-0.0134
0.0140
0.0139
0.0058
0.0317
-0.0120
0.0088
-0.0253
-0.0172
0.0142
0.0497
0.0440
0.0078
-0.0221
0.0127
0.0137
0.0052
0.0552
-0.0149
-0.0103
0.0192
-0.0119
0.0325
0.0023
-0.0002
0.0015
0.0007
0.0008
0.0012
0.0003
0.0004
0.0014
-0.0001
-0.0012
0.0026
0.0009
0.0017
0.0006
-0.0005
0.0004
0.0009
-0.0008
0.0003
0.0006
-0.0006
-0.0006
-0.0003
-0.0014
0.0005
-0.0004
0.0011
0.0008
-0.0006
-0.0023
-0.0020
-0.0004
0.0010
-0.0006
-0.0006
-0.0002
-0.0025
0.0007
0.0005
-0.0009
0.0005
-0.0015
44
45
46
47
48
49
50
51
52
53
54
-0.0642
0.0487
0.0075
0.0986
-0.0329
0.0426
0.0219
0.0159
-0.0290
-0.0011
0.0162
-0.0433
0.0328
0.0051
0.0665
-0.0222
0.0287
0.0147
0.0107
-0.0196
-0.0008
0.0109
0.0772
-0.0585
-0.0090
-0.1184
0.0395
-0.0512
-0.0263
-0.0191
0.0349
0.0013
-0.0195
-0.0451
0.0342
0.0053
0.0692
-0.0231
0.0299
0.0153
0.0111
-0.0204
-0.0008
0.0114
CANONICAL WEIGHTS
FOR COLS
0.358540
0.582047
0.271996
-0.589193
0.333976
FOR ROWS
-0.111039
-0.070384
-0.075251
0.175758
-0.122148
0.021022
FOR ROWS
-0.141132
0.104833
-0.080955
-0.164125
-0.067014
0.074972
FOR ROWS
-0.045044
-0.254434
-0.173517
-0.087229
0.255012
-0.009184
FOR ROWS
-0.014253
0.123910
0.092549
-0.047293
0.250345
0.065420
FOR ROWS
0.103491
-0.106188
0.069247
0.166685
0.171947
0.074781
FOR ROWS
0.152837
0.164643
0.072058
0.340913
-0.040723
0.046033
FOR ROWS
0.043617
-0.099494
-0.090318
-0.056277
0.124895
0.060390
FOR ROWS
-0.396623
-0.095921
-0.189877
-0.092013
-0.161459
-0.075677
FOR ROWS
0.154329
0.112359
-0.050276
-0.037417
-0.043463
-0.023314
28
0.0020
-0.0016
-0.0002
-0.0031
0.0010
-0.0014
-0.0007
-0.0005
0.0009
0.0000
-0.0005
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
4
CO-ORDINATES FOR BIPLOT (FOR AT MOST FIRST 6 COMPONENTS)
G -VECTOR FOR ROWS (G=V)
--------
(AA)
(BB)
(LL)
(WW)
(AH)
(AT)
(BF)
(MM)
(XX)
(AJ)
(AU)
(BG)
(CC)
(NN)
(YY)
(AK)
(AV)
(BH)
.(AW)
(BJ)
1
0.0119
-0.1242
0.0240
0.1079
0.1618
-0.2729
-0.0690
0.0022
0.2142
-0.1854
-0.2115
-0.1868
-0.1326
0.0028
0.0940
-0.0837
0.0973
-0.1782
0.0509
0.0939
0.4259
-0.0671
-0.0638
-0.1110
2
-0.0195
0.0391
-0.1174
-0.2249
-0.0919
-0.2083
0.1212
0.1384
0.2070
0.2529
-0.1797
-0.1280
-0.0311
0.1569
-0.2475
0.2403
0.1308
-0.1065
3
-0.0457
0.0429
-0.0439
-0.3044
0.1241
0.0043
-0.1984
-0.2378
-0.0808
-0.0663
-0.1128
-0.1439
4
-0.1110
-0.0670
0.0925
0.1528
0.1249
-0.0503
-0.0704
0.0750
-0.0473
0.1646
0.0604
-0.0374
(EE)
(QQ)
(FF)
(RR)
(AC)
(AE)
(AY)
(GG)
(SS)
(AD)
(AP)
(AZ)
(AQ)
(BC)
(JJ)
(00)
(AF)
(AR)
(BD)
0.1387
0.1364
-0.2005
0.2308
0.0226
0.0552
-0.0471
-0.0383
0.2550
-0.1864
0.0365
-0.0778
0.1696
-0.0633
-0.0014
-0.0770
-0.0247
0.1472
-0.0478
-0.1611
0.1413
0.0251
0.0840
-0.1736
-0.1228
0.0441
0.0592
0.0728
-0.0014
0.0043
0.1635
0.0211
-0.0006
-0.0712
-0.2437
-0.2056
-0.1133
0.0999
0.0602
-0.1648
-0.0201
0.1885
0.1397
0.0924
-0.0167
-0.0988
0.0378
0.3006·
-0.1221
-0.0142
-0.0630
0.2490
0.1417
-0.0557
0.1260
-0.0562
-0.0920
-0.0828
0.0417
-0.1136
-0.0030
0.0269
-0.0796
-0.0595
0.0472
0.0089
-0.1333
-0.0232
-0.2076
-0.1472
0.0702
0.0540
0.1642
-0.0138
0.2169
0.2317
0.3198
0.3075
0.0788
0.0383
-0.0244
-0.1774
-0.1227
0.0528
0.0233
-0.0989
-0.0480
-0.0613
-0.0557
0.0230
-0.1985
0.0445
0.1150
-0.0114
0.0735
-0.1452
0.1475
-0.0569
0.1319
-0.1309
0.0748
-0.0441
0.2626
0.0012
0.0917
0.0084
0.1097
0.1216
-0.0753
-0.0450
0.2503
0.0721
-0.3966
-0.0435
0.1758
-0.2544
0.0654
0.3409
-0.0959
-0.0233
-0.1221
-0.1735
0.1035
-0.0407
-0.1899
0.0210
-0.0872
-0.1062
0.0460
-0.0920
-0.1411
0.2550
0.0692
0.0436
-0.1615
0.1048
-0.0092
0.1667
-0.0995
-0.0757
-0.0810
-0.0143
0.1719
-0.0903
0.1543
-0.1641
0.1239
0.0748
-0.0563
0.1124
(DD)
(PP)
(ZZ)
(AL)
(AB)
(AM)
(AX)
(AN)
(HH)
(TT)
(KK)
(W)
(AG)
(AS)
(BE)
(
29
H -VECTOR FOR COLUMNS (H=LAMDA*W)
(00)
(01)
(02)
(03)
(04)
1
0.6767
-0.1259
0.8843
0.6682
-0.0483
2
-0.5176
0.8290
0.2393
0.3270
-0.5071
3
-0.3270
0.1320
0.0129
0.3977
0.8121
4
ODIAG(X'RR'X)
0.3054
0.4958
0.2317
-0.5018
0.2845
0.999998D+00
0.999999D+00
0.100000D+01
0.100000D+01
0.999999D+00
OTHE FOLLOWING ARE GOODNESS OF FIT COEFFICIENTS FOR COLUMNS
oRANK
oRANK
oRANK
oRANK
oRANK
oRANK
oRANK
oRANK
1
1
2
2
3
3
4
4
0.4579
0.5421
0.7258
0.2742
0.8327
0.1673
0.9260
0.0740
0.0159
0.9841
0.7031
0.2969
0.7205
0.2795
0.9663
0.0337
0.7821
0.2179
0.8393
0.1607
0.8395
0.1605
0.8931
0.1069
0.4465
0.5535
0.5535
0.4465
0.7116
0.2884
0.9635
0.0365
INTERLEAVED WITH RESIDUAL SUM OF SQUARES
0.0023
0.9977
0.2594
0.7406
0.9190
0.0810
0.9999
0.0001
30
IDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER' (SCALED)
G AND H VECTOR PLOT
-0.45
-0.27
-0.09
0.09
0.27
11-------------------1-------------------1-------------------1-------------------1-------------------1
o ---
0.88
o .81
0.75
0.69
0.63
0.56
0.50
0.44
0.38
0.31
0.25
0.19
0.13
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
01
SS
HH
AJ
AI<
XX
FF
I
NN
I
MM
0.06
I
I
I
LL
AN
-0.31
-0.38
-0.44
I
-0.50
I
I
I
-0.56
-0.63
-0.69
-0.75
-0.81
-0.88
-0.94
0.17
0.14
0.11
0.05
0.08
I
I
0.02
ZZI
-0.02
I
WW
??
-0.05
-0.08
-0.11
I
BG
I
I
I
I
I
AU
BF
0.20
I
AY
I
I
I
I
0.23
I
I
I
I
I
UU
I
0.26
I
Z
I
I
0.29
I
AE
AL
I
0.32
021
I
BC
I
-0.25
AP
CC
I
0.35
I
I
I
0.41
0.38
I
AB
BD
0.44
I
AV
I
-0.19
3
I
I
-0.00
-0.13
DD
I
I
I
I
I
-0.06
\,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
-0.14
-0.17
-0.20
I
BJ
AH
AW
04
YY
0
I
I
I
I
I
I
I
I
I
I
I
I
-0.23
I
-0.26
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
11-------------------1-------------------1-------------------1-------------------1-------------------1
-0.94
-0.56
-0.19
0.19
0.56
31
-0.29
-0.32
-0.35
-0.38
-0.41
-0.44
1DUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
J -VECTOR FOR ROWS
(J=LAMDA*V)
-------------------------(JJ)
(00)
(KK)
(W)
(AQ)
(BC)
(AF)
(AR)
(BD)
(AS)
0.0476
-0.1016
0.2215
-0.0826
-0.0018
-0.1005
-0.0322
0.1922
-0.0624
-0.2103
0.1845
0.0327
0.1097
-0.2267
-0.1604
0.0576
0.0772
0.0951
-0.0018
0.0057
0.2211
0.1638
0.1084
-0.0195
-0.1160
0.0444
0.3527
-0.1432
-0.0166
-0.0740
0.2921
0.1662
-0.0653
0.1478
-0.0659
-0.1079
-0.0972
0.0489
-0.1333
-0.0035
0.0315
-0.0934
-0.0698
0.0554
0.0105
0.0765
0.0372
-0.0236
-0.1722
-0.1191
0.0512
0.0226
-0.0960
-0.0466
-0.0595
-0.0541
0.0223
-0.1927
0.0432
0.1116
-0.0111
0.0714
-0.1409
0.1432
-0.0552
0.1281
-0.1271
0.0726
-0.0428
0.2549
0.0012
0.0890
0.0082
0.1065
0.1181
0.1497
-0.2167
0.0557
0.2904
-0.0817
-0.0199
-0.1040
-0.1478
0.0881
-0.0347
-0.1617
0.0179
-0.0743
-0.0904
0.0392
-0.0784
-0.1202
0.2172
0.0590
0.0372
-0.1375
0.0893
-0.0078
0.1420
-0.0847
-0.0645
-0.0690
-0.0121
0.1465
-0.0769
0.1314
-0.1398
0.1055
0.0637
-0.0479
0.0957
(03)
(04)
(AA)
(BB)
(CC)
(LL)
(MM)
(NN)
(WW)
(XX)
(AH)
(AT)
(BF)
(AJ)
(BG)
(YY)
(AK)
(AV)
(BH)
(DD)
(PP)
(ZZ)
(AL)
(AW)
(BJ)
1
0.0155
-0.1621
0.0314
0.1409
0.2112
-0.3563
-0.0901
0.0028
0.2796
-0.2421
-0.2761
-0.2439
-0.1731
o .0036
0.1227
-0.1093
0.1270
-0.2327
0.0664
0.1225
0.5561
-0.0876
-0.0833
-0.1449
2
-0.0229
0.0458
-0.1377
-0.2639
-0.1078
-0.2444
0.1423
0.1624
0.2428
0.2967
-0.2108
-0.1502
-0.0365
0.1840
-0.2904
0.2820
0.1534
-0.1250
3
-0.0444
0.0417
-0.0426
-0.2954
0.1205
0.0042
-0.1926
-0.2308
-0.0784
-0.0644
-0.1095
-0.1397
4
-0.0946
-0.0571
0.0788
0.1302
0.1064
-0.0428
-0.0599
0.0639
-0.0403
0.1402
0.0514
-0.0319
K -VECTOR
(AU)
FOR COLUMNS
(EE)
(QQ)
(AB)
(AM)
(AX)
(AC)
(AN)
(AY)
(FF)
(GG)
(SS)
(AD)
(AP)
(AZ)
0.1811
0.1781
-0.2618
0.3014
0.0295
0.0720
-0.0616
-0.0500
0.3329
-0.2434
0.1918
0.0247
-0.0007
-0.0835
-0.2859
-0.2413
-0.1329
0.1172
0.0706
-0.1934
-0.0236
-0.1294
-0.0226
-0.2015
-0.1429
0.0682
0.0524
0.1593
-0.0134
0.2106
0.2249
0.3104
0.2985
-0.0641
-0.0384
0.2132
0.0614
-0.3378
-0.0370
(RR)
(HH)
(TT)
(AE)
(K=W)
(01)
(02)
0.5183
-0.0964
0.6773
0.5118
-0.0370
2
-0.4412
0.7066
0.2039
0.2787
-0.4322
3
-0.3369
0.1360
0.0133
0.4097
0.8367
0.3585
0.5820
0.2720
-0.5892
0.3340
1
4
ODIAG(XC'CX' )
0.295041D-01
0.219186D-01
0.384200D-01
0.291927D-01
0.887911D-01
0.210732D+00
0.105441D-01
0.165690D+00
0.114120D+00
0.691651D-01
0.103989D+00
0.722133D-01
0.341302D-01
0.326379D-01
0.183740D+00
0.744438D-01
0.205691D+00
o.248582D-01
0.595667D-01
0.731221D-01
0.697246D-01
0.292210D-01
o. 110702D+00
o. 116067D+00
0.784341D-01
0.498070D-01
0.189404D+00
0.904867D-01
0.238855D-01
0.373784D-01
0.146268D+00
0.856372D-01
0.152923D+00
0.174439D-01
0.825408D-01
0.105435D+00
OTHE FOLLOWING ARE GOODNESS OF FIT COEFFICIENTS FOR ROWS
(BE)
(
----------------------(00)
(AG)
0.694940D-01
0.385276D-01
0.182923D+00
0.186766D+00
0.431765D-01
0.160189D+00
0.846251D-01
0.698172D-01
0.739011D-01
0.617905D-01
0.106445D+00
0.366917D-01
0.356868D+00
0.194254D-01
0.116201D+00
o. 136314D+00
0.594369D-01
0.169578D+00
INTERLEAVED WITH RESIDUAL SUM OF SQUARES
32
ORANK
ORANK
ORANK
ORANK
ORANK
ORANK
ORANK
1
1
2
2
3
3
4
0.0082
0.6820
0.1174
0.0001
0.5031
0.0003
0.0487
0.2080
0.4722
0.4549
0.0839
0.1014
0.1034
0.0564
0.0971
0.0283
0.4654
0.0367
0.1388
0.1748
0.4315
0.2788
0.4657
0.0337
0.5345
0.0806
0.8664
0.7721
0.0767
0.4430
0.0942
0.3189
0.1029
0.0502
0.5670
0.9537
0.6478
0.0523
0.6553
0.0002
0.5273
0.5594
0.0974
0.0337
0.0175
0.7175
0.0000
0.7440
0.2254
0.0013
0.6703
0.0293
0.0123
0.5642
0.0610
0.1064
0.7328
0.0296
0.0384
0.1238
0.0861
0.0572
0.0367
0.0380
0.0566
0.0336
0.0197
0.1726
0.0939
0.0357
0.0391
0.0281
0.0206
0.0282
0.0282
0.0681
0.1717
0.0477
0.0202
0.0301
0.0617
0.0487
0.0311
0.0104
0.1909
0.1251
0.1041
0.1452
0.0694
0.0054
0.0037
0.0705
0.0270
0.0174
0.0400
0.0601
0.1496
0.1988
0.0489
0.0233
0.0698
0.0152
0.0884
0.0248
0.0625
0.0259
0.7365
0.0460
0.4100
0.2479
0.0197
0.5254
0.8819
0.1486
0.4554
0.2164
0.7262
0.6518
0.8753
0.8195
0.1932
0.7365
0.9177
0.7814
0.6247
0.3604
0.1804
0.4305
0.6830
0.9376
0.5321
0.8664
0.8282
0.4368
0.6283
0.4813
0.3341
0.7167
0.4247
0.7980
0.7879
0.0958
0.8005
0.9570
0.6740
0.3459
0.8817
0.1761
0.6647
0.8855
0.2394
0.4311
0.0287
0.8804
0.0784
0.8171
0.2255
0.0057
0.9857
0.0287
0.0102
0.7782
0.0408
0.0801
0.9442
0.0283
0.0045
0.4671
0.0493
0.0566
0.0190
0.0243
0.0077
0.0067
0.0177
0.0482
0.0086
0.0080
0.0274
0.0187
0.0196
0.0194
0.0093
0.0091
0.0874
0.0477
0.0153
0.0184
0.0412
0.0444
0.0287
0.0055
0.1212
0.0371
0.0246
0.1383
0.0320
0.0050
0.0034
0.0487
0.0093
0.0144
0.0284
0.0156
0.1260
0.1170
0.0484
0.0099
0.0643
0.0109
0.0884
0.0247
0.0027
0.0926
0.7815
0.0234
0.9463
0.7485
0.0041
0.8067
0.8951
0.0904
0.7360
0.2189
0.8105
0.6716
0.9178
0.8332
0.3267
0.7392
0.9188
0.9203
0.8490
0.9138
0.1805
0.6625
0.7450
0.9797
0.7494
0.9907
0.8345
0.7191
0.9637
0.7131
0.4563
0.7201
0.8389
0.8205
0.9639
0.4267
0.9855
0.9757
0.8512
0.6215
0.9050
0.8263
0.8363
0.9735
0.2675
0.8995
0.3134
0.9233
0.2569
0.8683
0.7947
0.5664
0.9858
0.0268
0.0084
0.9633
0.0037
0.0268
0.9814
0.0115
0.0040
0.9924
0.0239
0.0564
0.0132
0.0229
0.0051
0.0062
0.0148
0.0477
0.0084
0.0029
0.0110
0.0025
0.0196
0.0115
0.0075
0.0030
0.0468
0.0033
0.0147
0.0092
0.0040
0.0246
0.0235
0.0054
0.0340
o .0330
0.0042
0.0877
0.0023
0.0028
0.0016
0.0282
0.0074
0.0030
0.0139
0.0036
0.1214
0.0207
0.0342
0.0063
0.0519
0.0078
0.0234
0.0108
0.0027
0.3958
0.8661
0.0039
0.9983
0.7868
0.0014
0.8757
0.9334
0.0013
0.9836
0.8693
0.9662
0.9849
0.9230
0.9809
0.9859
0.9971
0.9955
0.9220
0.9141
0.9188
0.9986
0.9889
0.9577
0.9908
0.9929
0.9994
0.9220
0.9697
0.9951
0.9485
0.9531
0.9290
33
ORANK
4
0.9193
0.9275
0.9964
0.9780
0.9930
0.9889
0.9821
0.7179
0.9805
0.9580
0.9700
0.9929
0.9563
0.9319
0.8385
0.9977
0.5278
0.9382
0.9461
0.9348
0.9955
0.0178
0.0052
0.9729
0.0001
0.0227
0.9999
0.0074
0.0026
0.9947
0.0015
0.0094
0.0023
0.0011
0.0048
0.0007
0.0003
0.0005
0.0005
0.0029
0.0063
0.0024
0.0000
0.0004
0.0012
0.0014
0.0013
0.0002
0.0069
0.0010
0.0005
0.0044
0.0020
0.0014
0.0170
0.0133
0.0004
0.0034
0.0011
0.0013
0.0002
0.0210
0.0015
0.0007
0.0025
0.0010
0.0072
0.0140
0.0080
0.0002
0.0330
0.0037
0.0062
0.0016
0.0009
0.0029
0.0000
0.0009
34
IDUKE FOREST PHASE II SOILS ANALYSIS - FROM RICHTER (SCALED)
J AND K VECTOR PLOT
-0.59
-0.35
-0.12
o ---
0.12
0.35
11-------------------1-------------------1-------------------1-------------------1-------------------1
I
I
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
01
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
SS
AJ
??
XX
NN
MM
DD
I
0.15
0.10
-0.00
I
I
I
AC
LL
I
BD
I
I
-0.10
I
I
I
I
-0.15
-0.20
I
-0.25
I
I
I
-0.35
I
-0.50
-0.55
-0.60
-0.65
-0.70
-0.75
Z
AL
AS
UU
AY
JJAT
EE
WW
??
BG
AD
0.57
0.53
0.49
0.45
0.41
0.37
0.33
0.30
0.26
0.22
0.18
0.14
I
I
I
0.10
I
0.06
I
I
I
0.02
ZZI
-0.02
I
I
-0.06
I
I
I
I
-0.14
-0.10
I
AM
AU
I
I
-0.18
I
-0.22
I
BF
I
-0.45
AN
BC
I
-0.40
AP
CC
I
-0.30
I
AB
I
I
I
-0.05
02
AV
I
0.05
03
FF
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
BJ
AW
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
AH
I
YY
I
04
00
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
11-------------------1-------------------1-------------------1-------------------1-------------------1
-0.75
-0.45
-0.15
0.15
35
0.45
-0.26
-0.30
-0.33
-0.37
-0.41
-0.45
-0.49
-0.53
-0.57
SAS/IML
The second analysis is labelled as SAS/IML. This program
uses matrix algebra to center and scale the data matrix, compute
the singular value decomposition of this matrix and create the G
and H coordinates. The SVD subroutine was called to compute the
biplots. See page 6 of SAS/IML output. The vectors were hand
drawn from the (0,0) point to the H coordinate labels (T, C, D,
M, and S). Notice that the entire plot is scaled on both axes to
range between -1 and 1. This helps to standardize the plots.
37
**********************************************************************
** DATA: /spruill/sasuser/richter-soils/working.biplots
**
**
**
**
** PGM:
/spruill/sasuser/richter-soils/soils_biplot.pgm
**
**
** analysis of Duke phase I I growth data using soil measurements
**
** made by Dan Richter ( using IML for SVD of X matrix)
**
*********************************************************************.,
options Is=80 PS=60i
data workingiinfile 'working.biplots'i
input @1 block 0 r m t s c di
runi
proc imlireset nologi
use workingi
read all into ai
id=a ( 11: 54,1: 3 I ) i
x=a ( 11: 54,4: 8 I ) i
xmean=x ( 1+, I ) /nrow (x) i
xphi=shape(l,nrow(x),l)i
xcent=x-xphi*xmeani
ss=xcent'*xcenti
std=sqrt(inv(diag(ss)))i
xscaled=xcent*stdi n_l=53i
call svd(v,l,w,xscaled)i eigenvec=wi
Isq=(I##2)ieigenval=lsqi
prop=lsqllsq ( 1+, I ) i
h=w*diag(l')i g=Vi
prin=xscaled*w*sqrt(n_l)i
vscaled=v*2i
r = { l l m ll
Ilt ll
IISII
Ile ll
"dll};
rl= {""ecl" "vec2" "vec3" "vec4" "vec5"
print
print
print
print
print
print
xscaled(lcolname=rl)i
eigenval I prOPi
eigenvec(lrowname=r colname=rll)i
g ( Icolname=rll ) i
prin(lcolname=rll)i
h ( Irowname=r I) i
*********************.,
* setting up the
* ,.
* scaled X matrix
* ,.
* from the data set *i
*********************.,
* n_l =(# of obs.-l)*i
*********************.,
* computing the
* ,.
* singular value
* ,.
* decomposition of * ,.
* XSCALED, and
* ,.
* relabelling the
* ,.
* vectors to match * ,.
* the names that
* ,.
* Gabriel uses.
*i
*********************.,
* concatenating matrices *i
idv=idl Ivscaledi
*********************.,
* creating SAS
*i
* datasets for use *i
* in making the
*i
* biplots.
*i
*********************.,
create pcs from h (Irowname=rl)i
append from h (Irowname=rl)i
create ids from idvi
append from idvi
quiti
*********************************************************************.,
* The end of IML. Next steps are to plot the eigenvectors of the
* ,.
* variables and the principal component scores of the data in the
* ,.
* G-H format that Gabriel uses.
*i
*********************************************************************.,
38
data pc;set pcs;
pcl=coll; pc2=co12;
drop coll-co15;
pc3=co13;
pc4=co14;
pc5=co15;
data idandviset ids;
block=colli o3=co12; ph=co13;
el=co14; e2=co15i e3=co16; e4=co17; e5=co18;
drop coll-co18;
data all;set pc idandv;
OX=Oi oy=Oi
proc print data=all(obs=lO)i
proc plot;
plot pc2*pcl=r oy*ox='.' e2*el=block/overlay
haxis =-1 to 1 by .25 vaxis=-l to 1 by .25;
title 'plot of first and second principle component'i
runi
39
16:10 Friday, November 2, 1990
SAS
XSCALED
m
-0.073317
-0.060112
-0.099725
-0.071116
0.0631282
~0.108528
-0.029302
-0.156944
0.032318
-0.031503
-0.099725
-0.051309
-0.11293
-0.073317
-0.042507
-0.123933
-0.150342
-0.145941
0.1247484
0.0785332
0.1005405
0.0697303
0.3558245
0.2457883
-0.08212
-0.090923
0.2765984
0.1907702
0.0389202
0.120347
0.4064411
-0.121733
-0.099725
-0.011696
0.3052078
0.2303832
-0.029302
-0.097525
-0.093123
-0.093123
0.1819673
-0.011696
-0.099725
-0.11513
0.056526
-0.075517
0.0433217
-0.117331
-0.077718
0.0147122
-0.077718
-0.05351
-0.097525
-0.060112
t
-0.127631
0.0521417
-0.095511
0.2238423
-0.179259
0.1414395
-0.057
0.2586533
-0.145793
-0.062381
0.0467603
0.0650907
0.0857755
-0.157901
-0.027234
0.0913251
0.3800713
0.106124
-0.078357
-0.007054
-0.073144
0.1239499
-0.106946
0.0124538
0.1537158
0.0042136
-0.105937
-0.025552
0.1025924
-0.006717
-0.1167
0.3481192
0.2335961
0.1278178
-0.197085
-0.016303
0.0287662
0.1337037
-0.137217
0.0159854
0.0001775
-0.118718
-0.060027
-0.242659
-0.097024
-0.107114
-0.050441
-0.093493
0.152875
0.0933431
-0.152015
-0.139739
-0.081048
-0.116532
s
0.0666384
-0.058041
-0.087708
0.1520299
0.099913
0.1444128
0.0193323
0.0297556
0.1375975
0.0036972
-0.162275
0.14762
0.0622285
0.0922959
0.1259715
-0.045613
0.0774627
0.0457916
-0.034388
0.0758591
0.0369718
0.2029441
0.0554132
0.3845515
-0.193545
-0.01715
0.1191562
0.1969306
0.1544353
0.0433862
-0.011938
-0.141428
-0.015145
0.0433862
0.1311832
0.2081558
-0.05764
-0.127798
-0.176708
0.0161251
0.1187553
-0.197153
-0.029177
-0.055636
-0.088911
-0.219604
-0.170694
-0.134613
-0.121383
0.0073053
-0.321833
-0.171496
-0.19114
-0.168289
40
c
-0.011778
-0.045873
-0.146846
0.0498553
0.1298471
0.0826388
0.0787048
-0.035382
0.1272244
0.1232904
0.0078923
-0.142912
0.0472326
0.1547626
0.213773
0.0773934
-0.081279
0.0432986
-0.036693
-0.019646
-0.099638
0.2163956
-0.212413
0.3435958
-0.144223
0.006581
-0.031448
-0.086524
-0.00391
0.0092037
-0.149469
-0.106194
-0.064231
-0.169139
0.0629687
0.1364038
-0.045873
0.1731214
-0.140289
0.0760821
0.0839501
-0.28716
0.3671999
0.0078923
0.0891955
-0.129799
0.1744327
-0.133733
-0.026203
0.0131377
-0.212413
-0.225526
-0.11144
0.0039583
d
-0.057094
-0.239499
-0.106032
0.0986174
0.0808218
-0.048196
-0.106032
-0.101583
0.1253109
-0.061543
0.0007415
-0.239499
-0.110481
-0.097134
-0.07489
-0.07489
-0.057094
-0.012605
-0.070441
0.1475554
0.0496795
-0.19501
0.022986
0.1742488
-0.012605
-0.15497
-0.088237
-0.048196
0.0852707
0.0541284
-0.097134
-0.128277
-0.217255
0.3255117
-0.083788
-0.030401
0.0585773
0.0274349
0.0051904
0.0496795
0.1742488
0.0274349
-0.128277
0.3611029
-0.146072
-0.017054
0.0763729
-0.030401
0.263227
0.1253109
0.1075152
-0.052645
0.0941685
0.3522051
1
SAS
16:10 Friday, November 2, 1990
EIGENVAL
L
PROP
1.7046854 1.305636 0.3409371
1.3764558 1.1732245 0.2752912
0.9422275 0.970684 0.1884455
0.7254771 0.8517494 0.1450954
0.2511542 0.5011529 0.0502308
EIGENVEC
m
t
s
e
d
vee1
0.5182809
-0.096445
0.6773226
0.5118073
-0.037023
vee2
-0.441164
0.706608
0.2039395
0.2787401
-0.432195
vee3
-0.336857
0.1359768
0.0132626
0.4097121
0.8366586
41
vee4
0.358541
0.5820473
0.2719949
-0.589193
0.3339764
vee5
0.542895
0.3662442
-0.652293
0.3812418
-0.017296
2
16:10 Friday, November 2, 1990
SAS
G
vee1
0.0118963
-0.069014
-0.132588
0.0508503
0.1387404
0.0551489
0.0364666
-0.076959
0.1412979
0.0440954
-0.124151
0.0021749
0.0027658
0.0938616
0.1364107
-0.047143
-0.077811
-0.024686
0.025082
0.0591633
0.0240265
0.2141615
0.093976
0.4258884
-0.200536
-0.038326
0.1696119
0.1472257
0.0840369
0.0728491
0.1079298
-0.185397
-0.083717
-0.06711
0.2308257
0.2549732
-0.063302
-0.047802
-0.173641
-0.001366
0.1617937
-0.211494
0.0972908
-0.063785
0.022588
-0.186386
-0.001416
-0.161063
-0.122849
0.0043314
-0.272893
-0.186799
-0.178238
-0.110992
vee2
-0.019482
0.1212471
-0.031099
0.1635
-0.113258
0.188487
0.0378085
0.2489842
-0.091978
0.026881
0.0390561
0.1384306
0.1568634
0.0210639
0.0998558
0.1396521
0.3006285
0.1416843
-0.082848
-0.079617
-0.117406
0.2069599
-0.247512
-0.000633
0.0601936
0.0923977
-0.122066
-0.055694
0.0416583
-0.059511
-0.224923
0.2528799
0.2403291
-0.071176
-0.164837
-0.016659
-0.014153
0.1260084
-0.113585
0.0472224
-0.091919
-0.179705
0.1307707
-0.243676
-0.020144
-0.098845
-0.063033
-0.056162
-0.002996
0.0089153
-0.208348
-0.128039
-0.106534
-0.20564
vee3
-0.045708
-0.198421
-0.133344
0.1641575
0.0788156
0.052788
-0.055724
-0.011388
0.1319499
0.0012378
0.0429113
-0.237811
-0.023234
-0.013815
0.0383382
0.0232957
0.0229556
0.0735487
-0.130941
0.0916846
-0.043867
-0.08081
-0.207551
0.2169194
-0.024353
-0.098886
-0.198527
-0.145154
0.0748219
0.0084271
-0.30437
-0.06631
-0.147245
0.2317331
-0.177373
-0.048019
0.044538
0.1475466
-0.04406
0.1097095
0.1241231
-0.112824
0.0702251
0.3197767
-0.122678
-0.061284
0.1150213
-0.056868
0.26255
0.1216241
0.0042922
-0.143917
0.0540082
0.3074831
42
vec4
-0.111039
-0.070385
-0.075251
0.1757579
-0.122148
0.021022
-0.141132
0.1048327
-0.080956
-0.164126
-0.067014
0.0749712
-0.045044
-0.254435
-0.173517
-0.087229
0.2550115
-0.009184
-0.014253
0.12391
0.0925486
-0.047293
0.2503448
0.0654194
0.1034914
-0.106188
0.0692472
0.1666844
0.171947
0.0747816
0.1528373
0.1646439
0.0720583
0.3409121
-0.040723
0.0460332
0.0436168
-0.099494
-0.090317
-0.056277
0.1248948
0.0603899
-0.396622
-0.095921
-0.189876
-0.092013
-0.161459
-0.075677
0.1543289
0.1123597
-0.050276
-0.037417
-0.043463
-0.023314
vee5
-0.266421
0.0219008
-0.171723
-0.076812
-0.096673
-0.13764
-0.035029
-0.043131
-0.158173
0.011387
0.143334
-0.300606
-0.100902
-0.193864
-0.064704
0.0533145
-0.045792
-0.106769
0.0971518
-0.038855
-0.070173
0.0733213
0.0727981
0.0302053
0.1660121
-0.06274
0.0462478
-0.132494
-0.089791
0.0741244
0.2601951
0.2302576
0.041029
-0.115635
0.066647
0.0715413
0.0273851
0.2891547
-0.07806
-0.054022
0.100533
-0.062216
0.1698431
-0.236099
0.1789481
0.0275928
0.3623002
-0.120904
0.1565031
0.0803142
0.0583084
-0.10662
-0.004119
0.0596164
3
SAS
PRIN
vec1
0.113076
-0.655993
-1.260274
0.4833411
1.3187513
0.5241998
0.346621
-0.731509
1.3430615
0.4191349
-1.180079
0.0206731
0.0262898
0.8921705
1. 2966078
-0.448105
-0.739611
-0.234641
0.2384092
0.5623577
0.2283759
2.0356427
0.8932582
4.0481434
-1. 906126
-0.364296
1. 6121909
1.3994056
0.7987854
0.6924431
1. 0258915
-1. 762226
-0.795742
-0.63789
2.1940387
2.4235646
-0.601694
-0.454362
-1.650484
-0.012986
1. 5378775
-2.010292
0.9247661
-0.606288
0.2147026
-1. 771627
-0.013461
-1.530934
-1.167698
0.0411706
-2.593893
-1.775555
-1.694183
-1.055003
vec2
-0.166403
1. 0355965
-0.26562
1. 3964865
-0.967356
1.6099054
0.32293
2.1266247
-0.785602
0.2295959
0.3335862
1.1823641
1.3398026
0.1799107
0.8528886
1.192797
2.5677293
1.210154
-0.707621
-0.680023
-1.002786
1. 7676867
-2.114052
-0.00541
0.5141258
0.7891879
-1.042591
-0.475697
0.3558121
-0.508295
-1.92111
2.159899
2.0526997
-0.607925
-1.407903
-0.14229
-0.120886
1.0762629
-0.970154
0.403336
-0.785103
-1. 534898
1.1169387
-2.081284
-0.172055
-0.844256
-0.538379
-0.479692
-0.025588
0.0761473
-1.779542
-1.093611
-0.909924
-1.75641
16:10 Friday, November 2, 1990
vec3
-0.323002
-1. 40218
-0.942298
1.1600493
0.5569653
0.373036
-0.393782
-0.080472
0.9324484
0.0087473
0.3032407
-1.680536
-0.16419
-0.097625
0.2709237
0.1646232
0.1622203
0.5197455
-0.925317
0.6479065
-0.309997
-0.571056
-1.466695
1.532901
-0.172093
-0.698799
-1.402931
-1. 025761
0.5287431
0.0595517
-2.150886
-0.468592
-1.040536
1. 6375851
-1.253441
-0.339335
0.3147362
1.0426658
-0.311358
0.7752825
0.8771391
-0.797294
0.4962586
2.2597618
-0.866928
-0.433076
0.8128193
-0.401871
1. 8553587
0.8594792
0.0303316
-1.017014
0.3816591
2.1728865
43
vec4
-0.688536
-0.436443
-0.46662
1.0898449
-0.75742
0.1303541
-0.875135
0.6500497
-0.501994
-1. 017717
-0.415544
0.4648839
-0.279308
-1.577707
-1.07595
-0.540893
1.5812827
-0.056948
-0.08838
0.7683446
0.573878
-0.293257
1. 5523453
0.4056542
0.6417324
-0.658453
0.42939
1. 0335816
1. 066214
0.4637078
0.947718
1.0209291
0.4468211
2.1139378
-0.252516
0.2854439
0.2704606
-0.616948
-0.560042
-0.348961
0.7744514
0.3744672
-2.459388
-0.59479
-1.177391
-0.570558
-1. 001177
-0.469263
0.9569674
0.6967233
-0.311753
-0.232018
-0.269509
-0.144564
vec5
-0.972024
0.0799039
-0.626521
-0.280243
-0.352706
-0.502171
-0.1278
-0.157363
-0.577084
0.041545
0.5229465
-1.096746
-0.368137
-0.707304
-0.23607
0.1945152
-0.167069
-0.38954
0.3544533
-0.141761
-0.256023
0.2675091
0.2656002
0.1102025
0.6056866
-0.228903
0.1687326
-0.483397
-0.327597
0.2704392
0.9493082
0.8400828
0.149692
-0.421888
0.243158
0.2610146
0.0999131
1. 0549658
-0.284796
-0.197097
0.3667895
-0.226992
0.6196635
-0.861397
0.652883
0.1006709
1.3218335
-0.44111
0.5709935
0.293022
0.2127353
-0.388999
-0.015027
0.2175074
4
SAS
H
m
t
s
c
d
Exiting IML.
OBS
R
1
2
3
4
5
6
7
8
9
10
OBS
1
2
3
4
5
6
7
8
9
10
m
t
s
c
d
0.6766863
-0.125922
0.8843367
0.6682341
-0.048339
PCl
0.67669
-0.12592
0.88434
0.66823
-0.04834
-0.517584
0.8290098
0.2392668
0.3270247
-0.507062
PC2
-0.51758
0.82901
0.23927
0.32702
-0.50706
16:10 Friday, November 2, 1990
-0.326982
0.1319905
0.0128738
0.397701
0.8121311
PC3
-0.32698
0.13199
0.01287
0.39770
0.81213
0.3053871
0.4957584
0.2316715
-0.501844
0.2844642
PC4
0.30539
0.49576
0.23167
-0.50184
0.28446
5
0.2720734
0.1835444
-0.326898
0.1910604
-0.008668
PC5
BLOCK
03
0.27207
0.18354
-0.32690
0.19106
-0.00867
3
3
3
3
3
PH
3
4
5
3
4
El
0.02379
-0.13803
-0.26518
0.10170
0.27748
E2
-0.03896
0.24249
-0.06220
0.32700
-0.22652
E3
-0.09142
-0.39684
-0.26669
0.32831
0.15763
44
E4
-0.22208
-0.14077
-0.15050
0.35152
-0.24430
E5
-0.53284
0.04380
-0.34345
-0.15362
-0.19335
3
3
3
4
4
OX
OY
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
plot of first and second principle component
6
16:10 Friday, November 2, 1990
Symbol is value of R.
Symbol used is '.'.
Symbol is value of BLOCK.
Plot of PC2*PC1.
Plot of OY*OX.
Plot of E2*E1.
PC2
1. 00 +
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1. 00
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
I
I
I
+
I
3
4
3
s
4
3
5
3
5
5
5
5
5
5
-+--------+-------~+--------+--------+--------+--------+--------+--------+
-1.00
-0.75
-0.50
-0.25
0.00
1. 00
PC1
NOTE: 54 obs had missing values.
63 obs hidden.
45
0.25
0.50
0.75
SAS PRINCOMP
46
SAS PRINCOMP
The third analysis uses SAS PRINCOMP to compute all
necessary portions of the singular value decomposition. The
results are output into
SAS datasets (STATS and PRINX) and
manipulated to produce the graphs. Page one of the output shows
the usual PRINCOMP output. Notice that the eigenvalues
proportions match the
and
previous computations from IML and
BIPLOTS. The output labelled as EIGENVECTORS are the W vectors
and match the results of the SVD of XSCALED and the column
canonical weights of the BIPLOTS output. Page two of the output
is a printout of the STATS dataset as SAS produces it. The
dataset PRINX, listed on page 3, contains what SAS calls the
principal component scores of the observations. They are a
function of the G coordinates and can be computed as
PRINX=G*L*(n-l) or PRINX=XSCALED*W*SQRT(n-l). The latter equation
was used in the SAS/IML example in the matrix called PRIN (see
page 4 of SAS/IML).
Datasets were concatenated, and the center reference
coordinate (0,0) was created in dataset PC. Here the Hand G
coordinates were computed as Hj=Wj*SQRT(EIGVAL) and
Gi =2*(V i /SQRT(EIGVAL*(n-l)). Again, G is arbitrarily rescaled by
a factor of 2 for plotting purposes. G and H were plotted using
the same options as those in the SAS/IML example.
47
**********************************************************************
** DATA: /spruill/sasuser/richter-soils/working.biplots
**
**
**
**
** PGM: /spruill/sasuser/richter-soils/moresoil.pgm
**
**
** analysis of Duke phase II growth data using soil measurements
**
** made by Dan Richter (computing biplots from PROC PRINCOMP)
**
*********************************************************************.,
options ls=80 PS=60i
data workingiinfile 'working.biplots'i
input @I block 0 r m t s c di
*********************************************************************.,
** RUNNING PRINCOMP AND OUTPUTTING THE RESULTS INTO A SAS DATASET **i
** CALLED "STATS".
** i
*********************************************************************.,
proc sortiby block 0 ri
proc princomp out=prinx prefix=w outstat=statsi
var m t s c di
proc print data=statsi
title 'output of dataset STATS'i
proc print data=prinxivar block 0 r wl-w5i
title 'output of dataset PRINX'i
*********************************************************************.,
** PULLING EIGENVALUES FROM PRINCOMP OUTPUT TO BE USED TO COMPUTE **i
** THE "H" COORDINATES FOR GABRIEL'S BIPLOTS. THE EIGENVALUES ARE **i
** RELABELLED AT THIS TIME, BECAUSE THE SAS LABELS ARE MISLEADING. **i
*********************************************************************.,
data eigvaliset statsiif _type_='EIGENVAL'i
eigvall=mi
eigva12=ti
eigva13=si
eigva14=ci
eigva15=di
drop m t s c di
*********************************************************************.,
** PULLING EIGENVECTORS FROM THE PRINCOMP OUTPUT TO USED TO MAKE
**i
** THE "G" COORDINATES FOR GABRIEL'S BIPLOTS. THIS DATASET WILL BE **i
** TRANSPOSED IN ORDER TO HAVE THE PROPER ORIENTATION FOR PLOTTING **i
*********************************************************************.I
data eigveciset statsiif _type_='SCORE'i
proc transpose out=newivar m t s c di
id _name_i
********************************************************************.,
** CONCATENATING THE "EIGVAL" AND TRANSPOSED "EIGVEC" (NEW) DATA- **.,
** SETS WITH THE PRINCIPAL COMPONENT SCORES FOUND IN "PRINX". AT **.,
**.,
** THIS TIME THE ORIGIN IS DEFINED AS "OY" AND "OX", AND THE
**.,
** "H" AND "G" COORDINATES ARE CREATED AS DEFINED BY GABRIEL.
**.,
**
**.,
** THE EIGENVALUES ARE MULTIPLIED BY (N-I). THIS WAS NOT A
**.,
** NECESSARY STEP IN IML BECAUSE THE X MATRIX WAS CENTERED AND
**.,
** SCALED BEFORE THE SINGULAR VALUE DECOMPOSITION.
** THE RESCALING OF THE "G"s BY 2 IS ARBITRARY AND IS USED SIMPLY **.,
**.,
** TO SPREAD THE DATA POINTS OUT ON THE PLOTS. THE FEWER NUMBER
** OF OBSERVATIONS THAT EXIST, THE LESS NECESSARY THIS RESCALING. **.,
**.,
** LIKEWISE, THE LARGER THE NUMBER OF OBSERVATIONS USED, THE
**.,
** LARGER THE SCALER WILL HAVE TO BE IN ORDER TO BE EFFECTIVE.
48
**.,
CAUTIONl 1 1
**.,
** DO NOT USE A SCALER THAT WILL RESULT IN "G"s THAT ARE BEYOND
** RANGE OF -l<=G<=l.
**j
********************************************************************.,
data pCjif _n_=l then set eigvaljset new prinxj
oy=Ojox=Ojdrop m t s C dj
if block=. then dOj
********************.,
h1=w1*sqrt(eigval1)j
*
CREATING "H"
*j
h2=w2*sqrt(eigva12)j
********************.,
h3=w3*sqrt(eigva13)jendj
**
if block ne . then dOj
gl=2*(w1/(sqrt(eigval1*53)))j
g2=2*(w2/(sqrt(eigva12*53)))j
g3=2*(w3/(sqrt(eigva13*53)))jendj
drop _type_ w1-w5;
proc printjtitle 'output of dataset
********************.,
*
CREATING "G"
*;
********************.,
PC, with G"s and H"s';
********************************************************************.,
** PLOTTING THE COORDINATES IN THE G-H FORMAT THAT GABRIEL USES.
**j
,
************~*******************************************************.
proc plotj
plot h2*h1=_name_ oy*ox='.' g2*gl=block/overlay
haxis =-1 to 1 by .25 vaxis=-l to 1 by .25;
title 'plot of FIRST and SECOND principal components'j
runj
49
SAS
12:01 Wednesday, November 7, 1990
Principal Component Analysis
54 Observations
5 Variables
Simple Statistics
M
T
S
C
Mean
26.57870370
0.0538333333
Std
15.60401981
0.0308751289
39.26944444
19.54777778
3.019814815
8.16800694
3.42631089
1.047481239
D
Correlation Matrix
T
S
C
D
-.3561
1.0000
0.1435
0.0257
- .1676
0.4522
0.1435
1.0000
0.4956
-.0849
0.0516
0.0257
0.4956
1.0000
-.0196
0.0487
- .1676
-.0849
-.0196
1.0000
M
M
T
S
C
D
1.0000
-.3561
0.4522
0.0516
0.0487
Eigenvalues of the Correlation Matrix
W1
W2
W3
W4
W5
Eigenvalue
Difference
Proportion
Cumulative
1.70469
1.37646
0.94223
0.72548
0.25115
0.328230
0.434228
0.216750
0.474323
0.340937
0.275291
0.188446
0.145095
0.050231
0.34094
0.61623
0.80467
0.94977
1.00000
Eigenvectors
M
T
S
C
D
W1
W2
W3
W4
W5
0.518281
-.096445
0.677323
0.511807
-.037023
-.441164
0.706608
0.203939
0.278740
-.432195
-.336857
0.135977
0.013263
0.409712
0.836659
0.358541
0.582047
0.271995
-.589193
0.333976
0.542895
0.366244
-.652293
0.381242
-.017296
50
1
OBS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
_TYPE_
output of dataset STATS
2
12:01 Wednesday, November 7, 1990
_NAME_
MEAN
STD
N
CaRR
CaRR
CaRR
CaRR
CaRR
EIGENVAL
SCORE
SCORE
SCORE
SCORE
SCORE
M
T
S
C
D
Wi
W2
W3
W4
W5
M
26.5787
15.6040
54.0000
1.0000
-0.3561
0.4522
0.0516
0.0487
1. 7047
0.5183
-0.4412
-0.3369
0.3585
0.5429
T
S
C
D
39.2694
8.1680
54.0000
-0.3561
1. 0000
0.1435
0.0257
-0.1676
1. 3765
-0.0964
0.7066
0.1360
0.5820
0.3662
19.5478
3.4263
54.0000
0.4522
0.1435
1. 0000
0.4956
-0.0849
0.9422
0.6773
0.2039
0.0133
0.2720
-0.6523
3.0198
1.0475
54.0000
0.0516
0.0257
0.4956
1. 0000
-0.0196
0.7255
0.5118
0.2787
0.4097
-0.5892
0.3812
0.0538
0.0309
54.0000
0.0487
-0.1676
-0.0849
-0.0196
1. 0000
0.2512
-0.0370
-0.4322
0.8367
0.3340
-0.0173
51
3
output of dataset PRINK
12:01 Wednesday, November 7, 1990
OBS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
BLOCK
0
R
W1
W2
W3
W4
W5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
0.02629
0.89217
1.29661
-0.44810
-0.73961
-0.23464
0.11308
-0.65599
-1.26027
0.48334
1.31875
0.52420
0.34662
-0.73151
1.34306
0.41913
-1.18008
0.02067
1.02589
-1.76223
-0.79574
-0.63789
2.19404
2.42356
0.23841
0.56236
0.22838
2.03564
0.89326
4.04814
-1.90613
-0.36430
1.61219
1.39941
0.79879
0.69244
-1.16770
0.04117
-2.59389
-1.77555
-1.69418
-1.05500
-0.60169
-0.45436
-1.65048
-0.01299
1.53788
-2.01029
0.92477
-0.60629
0.21470
-1.77163
-0.01346
-1.53093
1. 33980
0.17991
0.85289
1.19280
2.56773
1.21015
-0.16640
1. 03560
-0.26562
1. 39649
-0.96736
1.60991
0.32293
2.12662
-0.78560
0.22960
0.33359
1.18236
-1. 92111
2.15990
2.05270
-0.60793
-1. 40790
-0.14229
-0.70762
-0.68002
-1. 00279
1. 76769
-2.11405
-0.00541
0.51413
0.78919
-1. 04259
-0.47570
0.35581
-0.50830
-0.02559
0.07615
-1. 77954
-1. 09361
-0.90992
-1. 75641
-0.12089
1.07626
-0.97015
0.40334
-0.78510
-1. 53490
1.11694
-2.08128
-0.17205
-0.84426
-0.53838
-0.47969
-0.16419
-0.09763
0.27092
0.16462
0.16222
0.51975
-0.32300
-1.40218
-0.94230
1.16005
0.55697
0.37304
-0.39378
-0.08047
0.93245
0.00875
0.30324
-1. 68054
-2.15089
-0.46859
-1.04054
1.63759
-1.25344
-0.33934
-0.92532
0.64791
-0.31000
-0.57106
-1.46670
1.53290
-0.17209
-0.69880
-1.40293
-1.02576
0.52874
0.05955
1. 85536
0.85948
0.03033
-1.01701
0.38166
2.17289
0.31474
1. 04267
-0.31136
0.77528
0.87714
-0.27931
-1. 57771
-1. 07595
-0.54089
1.58128
-0.05695
-0.68854
-0.43644
-0.46662
1.08984
-0.75742
0.13035
-0.87513
0.65005
-0.50199
-1.01772
-0.41554
0.46488
0.94772
1. 02093
0.44682
2.11394
-0.25252
0.28544
-0.08838
0.76834
0.57388
-0.29326
1. 55235
0.40565
0.64173
-0.65845
0.42939
1. 03358
1.06621
0.46371
0.95697
0.69672
-0.31175
-0.23202
-0.26951
-0.14456
0.27046
-0.61695
-0.56004
-0.34896
0.77445
0.37447
-2.45939
-0.59479
-1.17739
-0.57056
-1. 00118
-0.46926
-0.36814
-0.70730
-0.23607
0.19452
-0.16707
-0.38954
-0.97202
0.07990
-0.62652
-0.28024
-0.35271
-0.50217
-0.12780
-0.15736
-0.57708
0.04155
0.52295
-1.09675
0.94931
0.84008
0.14969
-0.42189
0.24316
0.26101
0.35445
-0.14176
-0.25602
0.26751
0.26560
0.11020
0.60569
-0.22890
0.16873
-0.48340
-0.32760
0.27044
0.57099
0.29302
0.21274
-0.38900
-0.01503
0.21751
0.09991
1.05497
-0.28480
-0.19710
0.36679
-0.22699
0.61966
-0.86140
0.65288
0.10067
1.32183
-0.44111
52
~0.79729
0.49626
2.25976
-0.86693
-0.43308
0.81282
-0.40187
output of dataset
OBS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
_NAME_
M
T
S
C
D
OBS
OX
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
PC, with G's and H's
12:01 Wednesday, November 7, 1990
EIGVALl
EIGVAL2
EIGVAL3
EIGVAL4
EIGVAL5
1.70469
1. 70469
1.70469
1. 70469
1.70469
1. 70469
1.70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1.70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1.37646
1. 37646
1. 37 646
1. 37646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1.37646
1. 37 646
1. 37 646
1.37646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1. 37 646
1.37646
1.37646
1.37646
1.37646
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
H1
H2
H3
0.67669
-0.12592
0.88434
0.66823
-0.04834
-0.51758
0.82901
0.23927
0.32702
-0.50706
-0.32698
0.13199
0.01287
0.39770
0.81213
53
BLOCK
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
0
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
1
1
1
R
OY
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Gl
G2
G3
0.00553
0.18772
0.27282
-0.09429
-0.15562
-0.04937
0.02379
-0.13803
-0.26518
0.10170
0.27748
0.11030
0.07293
-0.15392
0.28260
0.08819
-0.24830
0.00435
0.21586
-0.37079
-0.16743
0.31373
0.04213
0.19971
0.27930
0.60126
0.28337
-0.03896
0.24249
-0.06220
0.32700
-0.22652
0.37697
0.07562
0.49797
-0.18396
0.05376
0.07811
0.27686
-0.44985
0.50576
0.48066
-0.04647
-0.02763
0.07668
0.04659
0.04591
0.14710
-0.09142
-0.39684
-0.26669
0.32831
0.15763
0.10558
-0.11145
-0.02278
0.26390
0.00248
0.08582
-0.47562
-0.60874
-0.13262
-0.29449
output of dataset
OBS
_NAME_
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
PC I with
GIS and HIs
5
12:01 Wednesday I November 7 1 1990
EIGVALl
EIGVAL2
EIGVAL3
EIGVAL4
EI GVAL 5
BLOCK
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1.70469
1. 70469
1. 70469
1. 70469
1.70469
1.70469
1.70469
1.70469
1.70469
1.70469
1. 70469
1.70469
1.70469
1.70469
1.70469
1. 70469
1.70469
1.70469
1.70469
1.70469
1.70469
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1.37646
1. 37646
1.37646
1. 37646
1. 37646
1. 37646
1. 37646
1. 37646
1. 37646
1. 37646
1.37646
1.37646
1. 37646
1. 37646
1. 37646
1.37646
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
0.25115
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
OBS
a
R
OY
OX
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
1
1
1
2
2
2
3
3
3
4
4
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
5
3
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Hi
H2
H3
54
Gl
G2
G3
-0.13422
0.46165
0.50995
0.05016
0.11833
0.04805
0.42832
0.18795
0.85178
-0.40107
-0.07665
0.33922
0.29445
0.16807
0.14570
-0.24570
0.00866
-0.54579
-0.37360
-0.35648
-0.22198
-0.12660
-0.09560
-0.34728
-0.00273
0.32359
-0.14235
-0.32967
-0.03332
-0.16570
-0.15923
-0.23481
0.41392
-0.49502
-0.00127
0.12039
0.18480
-0.24413
-0.11139
0.08332
-0.11902
-0.00599
0.01783
-0.41670
-0.25608
-0.21307
-0.41128
-0.02831
0.25202
-0.22717
0.09444
-0.18384
0.46347
-0.35475
-0.09604
-0.26188
0.18337
-0.08773
-0.16162
-0.41510
0.43384
-0.04871
-0.19777
-0.39705
-0.29031
0.14964
0.01685
0.52510
0.24325
0.00858
-0.28783
0.10802
0.61497
0.08908
0.29509
-0.08812
0.21942
0.24825
output of dataset
OBS
_NAME_
53
54
55
56
57
58
59
PC, with G's and H's
6
12:01 Wednesday, November 7, 1990
EIGVAL1
EIGVAL2
EIGVAL3
EIGVAL4
EIGVAL5
BLOCK
1.70469
1.70469
1. 70469
1. 70469
1. 70469
1. 70469
1. 70469
1.37646
1.37646
1.37646
1.37646
1. 37646
1.37646
1. 37646
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.94223
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.72548
0.25115
0.25115
0,25115
0.25115
0.25115
0.25115
0.25115
5
5
5
5
5
5
5
OBS
0
R
OY
OX
53
54
55
56
57
58
59
4
5
5
5
6
6
6
5
3
4
5
3
4
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Hi
H2
H3
55
Gl
G2
G3
-0.42299
0.19458
-0.12757
0.04518
-0.37277
-0.00283
-0.32213
-0.35941
0.26154
-0.48735
-0.04029
-0.19769
-0.12607
-0.11232
-0.22565
0.14045
0.63955
-0.24536
-0.12257
0.23004
-0.11374
plot of FIRST and SECOND principal components
7
12:01 Wednesday, November 7, 1990
Symbol is value of JJAME_.
Symbol used is '.'.
Symbol is value of BLOCK.
Plot of H2*H1.
Plot of OY*OX.
Plot of G2*Gl.
H2
1. 00
+
I
I
I
I
0.75 +
I
I
I
I
0.50 +
I
I
3
4
3
I
I
0.25 +
I
I
I
I
0.00 +
I
S
4
3
5
3
5
I
I
I
5
5
-0.25 +
I
I
I
I
5
5
5
4
-0.50 +
I
I
I
I
-0.75 +
I
I
I
I
-1.00 +
I
5
4
D
M
-+--------+--------+--------+--------+--------+--------+-----~--+--------+
-1.00
-0.75
-0.50
-0.25
0.00
H1
NOTE: 54 obs had missing values.
63 obs hidden.
56
0.25
0.50
0.75
1.00
S-PLUS
The fourth analysis is labelled as $-PLUS in which a
function named BIPLOT has been created. This function also uses
matrix algebra to compute the singular value decomposition of the
data matrix. Like the BIPLOTS program that is located on the
NCSUVM system, the data matrix must already contain data that is
centered and scaled to unit length.
Start with an ASCII file containing the centered and scaled
data (example XSCALED.DAT) :
XSCALED.DAT =>
-0.073317 -0.127631
-0.060112
0.066638 -0.011778 -0.057094
0.052142 -0.058041 -0.045873 -0.239499
-0.099725 -0.095511 -0.087708 -0.146846 -0.106032
0.049855
0.098617
-0.077718 -0.152015 -0.321833 -0.212413
0.107515
-0.071116
0.223842
0.152030
-0.053510 -0.139739 -0.171496 -0.225526 -0.052645
-0.097525 -0.081048 -0.191140 -0.111440
0.094169
-0.060112 -0.116532 -0.168289
0.352205
0.003958
and created an S-PLUS data file using the following commands:
r_list(m=O,t=O,s=O,c=O,d=O}
xscaled_scan ("XSCALED. DAT" ,r) .
Next, convert the new S-PLUS data set, xscaled, into an S-PLUS
matrix called xscaledm, as follows:
xscaledm_cbind(xscaled$m,xscaled$t,xscaled&s,xscaled$c,
xscaled$d} .
The BIPLOT function can be invoked on the matrix xscaledm as
follows:
output_biplot(xscaledm,dataname="Mottling"}
58
where a data set labeled output will be created to contain the
results of the singular value decomposition of xscaledm. If an S~
PLUS graphics window is opened before invoking the BIPLOT
function (e.g. suntools(), Xll()) then a graph of the biplots
will be displayed. The function dataname= will label the graph
with the supplied quote.
59
biplot
funetion(input, a
= 1,
b
= 2,
dataname, rname
= "Row",
ename
= "Column")
(
if(!is.matrix(input»
stop ("biplot () needs a matrix")
if(dataname == "")
stop ("No data name given for heading!")
nr <- nrow(input)
ne <- neol(input)
x.svd <- svd(input)
eolmark <- diag(x.svd$d) %*% t(x.svd$v)
prop <- matrix(x.svd$d A 2/sum(x.svd$d A 2), nrow = ne, neol = 1)
lim <- max (abs (eolmark[a,
]»
lim2 <- max(abs(x.svd$u»
k < - lim/ lim2
oldpar <- par
par (mar = e ( 6 , 6, 8, 6»
plot (eolmark[a,
], eolmark[b, ], xlim = e( - lim, lim), ylim = e(
- lim, lim), xlab = "", ylab = "", type = "n")
points(O, 0, peh = ".")
arrows(e(rep(O, ne», e(rep(O, ne», colmark[a,
], eolmark[b,
],
size = 0.1)
points (k * x. svd$u [, a], k * x. svd$u [, b], peh = "+")
text(k * x.svd$u[, a] - 0.03 * lim, k * x.svd$u[, b], labels = seq(
l:nr), cex
=
0.6)
text (eolmark[a,
+ 0.03 * lim, eolmark[b,
], labels = seq(l:ne),
eex = 0.6)
mtext(paste("Biplots of Dimensions ", a, "and", b, ",", dataname,
"data"), side = 3, line = 6, eex = 1.5)
title(mpg = e(4, 2, 0), xlab = paste ("Dimension " a, ",", ename,
"Information"»
title(mpg = e(3, 1, 0), ylab = paste ("Dimension " b, ",", ename,
"Information") )
axis(3, signif(seq( - k * lim2, k * lim2, length = 5), digits = 2),
labels = e(signif(seq( - lim2, lim2, length = 5), digits = 2»)
mtext(paste("Dimension", a, ",", rname, "Information"), side = 3, line
= 3)
axis(4, signif(seq( - k * lim2, k * lim2, length = 5), digits = 2),
labels = e(signif(seq( - lim2, lim2, length = 5), digits = 2»,
srt = 90)
mtext(paste("Dimension,", b, ",", rname, "Information"), side = 4,
line = 4)
text ( - lim, - lim, label = round (prop [a, 1], digits = 3), eex = 0.6)
arrows ( - lim + 0.1 * lim, - lim, - lim + 0.3 * lim,
- lim, size =
0.05)
text ( - lim, - lim + 0.2 * lim, label = round(prop[b, 1], digits = 3),
eex = 0.6)
arrows ( - lim, - lim + 0.3 * lim, - lim, - lim + 0.5 * lim, size =
0.05)
z <- list (x.svd$u, x.svd$d, eolmark, prop)
names(z) <- e("u", "d", "eolmark", "prop")
par <- oldpar
z
60
Output of S-PLUS BIPLOT FUNCTION
$u:
[1, ]
[2, ]
[3, ]
[4, ]
[5, ]
[6, ]
[7, ]
[8, ]
[9, ]
[10, ]
[11, ]
[12, ]
[13, ]
[14, ]
[15, ]
[16, ]
[17,]
[18, ]
[19, ]
[20, ]
[21, ]
[22, ]
[23, ]
[24, ]
[25, ]
[26, ]
[27, ]
[28, ]
[29, ]
[30, ]
[31, ]
[32, ]
[33, ]
[34, ]
[35, ]
[36, ]
[37, ]
[38, ]
[39, ]
[40, ]
[41, ]
[42, ]
[43, ]
[44, ]
[45, ]
[46, ]
[47, ]
[48, ]
[49, ]
[50, ]
[51, ]
[52, ]
[53, ]
[54, ]
[ , 1]
0.011895803
-0.069013746
-0.132588382
0.050851063
0.138739920
0.055150046
0.036466777
-0.076957816
0.141297763
0.044095321
-0.124150871
0.002175445
0.002766280
0.093861805
0.136410883
-0.047142424
-0.077810029
-0.024684744
0.025081630
0.059162879
0.024025708
0.214162341
0.093974466
0.425888880
-0.200535383
-0.038326163
0.169610994
0.147225495
0.084036785
0.072848818
0.107928341
-0.185395403
-0.083715509
-0.067110008
0.230825107
0.254973230
-0.063301551
-0.047801330
-0.173641136
-0.001365828
0.161793082
-0.211495227
0.097291520
-0.063785814
0.022587782
-0.186386180
-0.001416054
-0.161063502
-0.122848507
0.004331369
-0.272893647
-0.186799307
-0.178238770
-0.110993296
[ , 2]
[ , 3]
[,4]
[ , 5]
0.0194826517 -0.045707425 -0.11103916 -0.26642158
-0.1212472592 -0.198421182 -0.07038373 0.02190028
0.0310985241 -0.133343562 -0.07525109 -0.17172273
-0.1634996842 0.164156958 0.17575785 -0.07681147
0.1132583686 0.078815721 -0.12214807 -0.09667329
-0.1884861821 0.052788184 0.02102217 -0.13763970
-0.0378080935 -0.055723612 -0.14113183 -0.03502841
-0.2489842416 -0.011387406 0.10483300 -0.04313169
0.0919785479 0.131949848 -0.08095516 -0.15817331
-0.0268808202 0.001237775 -0.16412534 0.01138677
-0.0390565804 0.042910675 -0.06701450 0.14333353
-0.1384302661 -0.237810994 0.07497224 -0.30060545
-0.1568637554 -0.023234035 -0.04504363 -0.10090141
-0.0210636780 -0.013814281 -0.25443444 -0.19386484
-0.0998552700 0.038338003 -0.17351734 -0.06470437
-0.1396521836 0.023295233 -0.08722870 0.05331366
-0.3006286176 0.022955793 0.25501168 -0.04579199
-0.1416846614 0.073549240 -0.00918414 -0.10676921
0.0828476068 -0.130940460 -0.01425302 0.09715133
0.0796170394 0.091684340 0.12390964 -0.03885511
0.1174054944 -0.043867678 0.09254865 -0.07017374
-0.2069588799 -0.080809402 -0.04729316 0.07332128
0.2475126055 -0.207550517 0.25034455 0.07279854
0.0006353057 0.216919302 0.06541958 0.03020522
-0.0601949072 -0.024352085 0.10349106 0.16601182
-0.0923982058 -0.098886067 -0.10618768 -0.06274003
0.1220667810 -0.198527892 0.06924706 0.04624768
0.0556949920 -0.145153968 0.16668494 -0.13249374
-0.0416574581 0.074822057 0.17194691 -0.08979065
0.0595109241 0.008426929 0.07478138 0.07412552
0.2249235876 -0.304369821 0.15283744 0.26019490
-0.2528811098 -0.066309888 0.16464322 0.23025708
-0.2403294054 -0.147245392 0.07205813 0.04102950
0.0711754525 0.231733590 0.34091257 -0.11563302
0.1648379534 -0.177373715 -0.04072334 0.06664725
0.0166604138 -0.048019315 0.04603311 0.07154117
0.0141530656 0.044537662 0.04361694 0.02738453
-0.1260088538 0.147546677 -0.09949438 0.28915434
0.1135844718 -0.044060015 -0.09031807 -0.07805915
-0.0472222687 0.109708960 -0.05627685 -0.05402233
0.0919200591 0.124123281 0.12489503 0.10053386
0.1797043799 -0.112824181 0.06039038 -0.06221641
-0.1307701368 0.070224482 -0.39662259 0.16984313
0.2436758242 0.319776947 -0.09592093 -0.23609930
0.0201442300 -0.122677719 -0.18987662 0.17894856
0.0988443214 -0.061284052 -'0.09201262 0.02759313
0.0630329180 0.115021413 -0.16145875 0.36230069
0.056161677 -0.056868623 -0.07567730 -0.120904317
0.002995186 0.262550174 0.15432914 0.156503043
-0.008915284 0.121624463 0.11235912 0.080314776
0.208346203 0.004292493 -0.05027604 0.058308710
0.128038589 -0.143915928 -0.03741740 -0.106619866
0.106532625 0.054008985 -0.04346337 -0.004119236
0.205639286 0.307483182 -0.02331395 0.059617173
61
•
$d:
[1] 1.3056357 1.1732245 0.9706833 0.8517495 0.5011530
$colmark:
[1,
[2,
[3,
[4,
[5,
[ ,4]
[ , 2]
[ , 3]
[ , 5]
[ ,1]
]
0.6766826 -0.1259174 0.88433775 0.6682365 -0.04834057
]
0.5175872 -0.8290103 -0.23926287 -0.3270224 0.50706087
] -0.3269822 0.1319901 0.01287341 0.3976997 0.81213090
]
0.3053866 0.4957581 0.23167270 -0.5018448 0.28446391
]
0.2720737 0.1835448 -0.32689815 0.1910601 -0.00866748
$prop:
[,1]
[1,]
[2,]
[3,]
[4,]
[5,]
•
0.34093707
0.27529124
0.18844530
0.14509550
0.05023088
62
•
•
Biplots of Dimensions 1 and 2 , Mottling data
Dimension 1 , Row Information
-0.43
0.0
-0.21
0.43
0.21
C')
~
o
ci-l
51+
ctS
21 +
E
....
.E
C
34+
48+
E
(\J
C
0
:;::;
~.p-
C
0
ci
52+
E
....
.E
C
0
TO"
(\J
54+
42+
C
0
:;::;
ctS
::l
23+
31+
44+
::0
a:.
3+
0
49+
ci
36+
37+
0
0
24+
(\J
11 +
c:
25+
C
0
'00
26+
C
2 +.38+
16+18
(])
E
.Q
(/)
15+
C
3
Q)
E
43+
Q
4+
Q
6+
~--1
1
32+
I
33s++
TO"
t\!
22+
0
I
17+
0275
J_
0.341
I
•
I
-0.5
I
0.0
Dimension 1 ,Column Information
I
0.5
C')
r~
'(
•
•
.,
l
r
General Comments:
The BIPLOTS program was used as the yard stick on which the
other three options were measured. There were a number of reasons
for wanting to find an alternative to this program:
1)
The program is not widely available or well documented
2)
The options are cryptic and in rigid FORTRAN format
3)
Usually the data matrix has to be manipulated (eg.
centered and/or scaled) before decomposing.
4)
The output is lengthy and cumbersome.
The singular value decomposition and plotting processes can
be easily programmed in matrix language (IML, or S-PLUS), but
this too is cumbersome if one is not familiar with matrix
algebra. Minor details dependent on the dataset format make this
process difficult to generalize for use by nonstatisticians. This
procedure is more general in the BIPLOT function of S-PLUS. The
PRINCOMP procedure on the other hand is available to any
organization using SAS. The process is straight forward and
documented in the SAS manuals, and is generalized to run on any
dataset format or size. The manipulations of the output are
simple SAS data steps that can also be generalized, making this
method more appealing than the IML of BIPLOTS.
Data used in these examples are courtesy of Dan Richter,
1990, from soils analysis of Duke Forest Phase II. This research
was funded by the Southern Commercial Forest Forest Research
Cooperative .
•
63