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Under The Hood
Theory
Principal Component Analysis is essentially a change of basis for your dataset. Say your dataset has N variables and P samples. Each sample of your dataset has a natural representation as a point in a space of N dimensions, with its values being the coordinates in the basis {(1, 0...0), (0, 1, 0...0), ..., (0...0, 1)}.
PCA finds a new basis using linear combinations of this basis. This new basis is made of the principal components. There are thus N principal components.
PCA uses two steps to find these linear combinations:
1. Calculate the covariance matrix K for the variables. For each pair of variables, K stores their joint variability over the samples or covariance:
Samples
Sample
Variable A
Variable B
Variable C
s1
1
2
3
s2
4
5
6
s3
7
8
9
s4
10
11
12
There are no rows in this table
Covariance matrix
Name
A
B
C
A
cov(A,A)
cov(A,B)
cov(A,C)
B
cov(B,A)
cov(B,B)
cov(B,C)
C
cov(C,A)
cov(C,B)
cov(C,C)
There are no rows in this table
cov(A, B) = [ (4-1)x (5-2) + (7-1)x (8-2) + (10-1)x(11-2)
+ (1-4)x (2-5) + (7-4)x (8-5) + (10-4)x(11-5)
+ (1-7)x (2-8) + (4-7)x (5-8) + (10-7)x(11-8)
+ (1-10)x(2-11) + (4-10)x(5-11) + (7-10)x(8-11) ]/32
Thus the covariance matrix K is a NxN matrix.
2. Calculate the eigenvectors (V1, V2, ... VN) and eigenvalues (a1, a2,... aN) of the covariance matrix by solving:
K x V = a V
The eigenvectors are the principal components and form the new basis (once ordered by decreasing eigenvalue). These vectors are orthogonal to each others.
The coordinates of the eigenvectors in the old basis are what the Loadings sync table returns. They form the change-of-basis matrix.
The coordinates of the samples in the new basis are what the Principal Components sync table returns.
The last row of the Loadings sync table is the percentage explained, defined as
{a1/(a1+...+aN), (a1+a2)/(a1+...+aN), ...., 1}
Practice
Below are the sync tables we used for our examples. Coda allows only one instance of each table per doc, so both Principal Components and Loadings contain all our datasets. They are separated using the Group column, see .
Principal Components
Group
Label
Pc1
Pc2
Drinking
France
-1.395
-1.619
Drinking
Italy
-1.760
-0.808
Drinking
Switzerland
-1.102
-0.372
Drinking
Austria
-0.332
1.120
Drinking
UK
0.162
0.931
Drinking
USA
0.445
0.405
Drinking
Russia
3.409
-2.056
Drinking
Czech Republic
1.403
2.076
Drinking
Japan
-0.722
-0.126
Drinking
Mexico
-0.108
0.448
MovieReviews
Thor: Love and Thunder
1.340
-1.202
MovieReviews
The Northman
1.794
0.252
MovieReviews
Top Gun: Maverick
-0.002
1.339
MovieReviews
Minions: The Rise of Gru
-2.815
-0.558
MovieReviews
Lightyear
0.014
-1.020
MovieReviews
The Batman
-0.332
1.188
HeightWeight
1
-1.632
-0.051
HeightWeight
2
1.849
-0.753
HeightWeight
3
2.054
0.997
HeightWeight
4
0.992
0.795
HeightWeight
5
0.951
1.068
HeightWeight
6
0.042
-0.505
HeightWeight
7
1.518
0.169
HeightWeight
8
1.297
-0.205
HeightWeight
9
-0.896
-0.860
HeightWeight
10
-0.814
0.039
HeightWeight
11
-0.518
0.545
HeightWeight
12
-0.894
-0.653
HeightWeight
13
0.032
-0.223
HeightWeight
14
-0.584
0.021
HeightWeight
15
-0.538
-0.778
HeightWeight
16
1.900
-0.389
HeightWeight
17
-0.408
0.678
HeightWeight
18
1.186
0.676
HeightWeight
19
1.827
-0.564
HeightWeight
20
-0.487
0.111
HeightWeight
21
0.787
0.875
HeightWeight
22
1.304
0.626
HeightWeight
23
-3.362
-0.105
HeightWeight
24
0.306
-0.037
HeightWeight
25
0.748
0.981
HeightWeight
26
-0.122
0.417
HeightWeight
27
1.952
-0.155
HeightWeight
28
0.088
0.423
HeightWeight
29
-1.634
-0.599
HeightWeight
30
-1.703
0.126
HeightWeight
31
-0.842
-1.986
HeightWeight
32
-1.162
0.397
HeightWeight
33
-0.132
-0.037
HeightWeight
34
1.498
-0.434
HeightWeight
35
2.164
-0.642
HeightWeight
36
0.550
-0.369
HeightWeight
37
0.443
1.281
HeightWeight
38
-0.460
-0.249
HeightWeight
39
0.193
0.295
HeightWeight
40
-2.634
0.209
HeightWeight
41
0.057
-0.395
HeightWeight
42
-1.149
0.870
HeightWeight
43
1.359
0.113
HeightWeight
44
0.607
0.593
HeightWeight
45
-1.946
-0.510
HeightWeight
46
0.353
-0.172
HeightWeight
47
0.701
1.481
HeightWeight
48
-0.706
-0.524
HeightWeight
49
1.651
0.288
HeightWeight
50
0.871
0.040
HeightWeight
51
1.885
0.456
HeightWeight
52
-0.279
0.173
HeightWeight
53
0.743
-0.949
HeightWeight
54
-0.261
-1.100
HeightWeight
55
-1.157
0.716
HeightWeight
56
2.035
0.409
HeightWeight
57
2.592
0.799
HeightWeight
58
-0.464
0.564
HeightWeight
59
-1.044
0.115
HeightWeight
60
0.240
0.539
HeightWeight
61
-0.439
0.617
HeightWeight
62
0.993
0.228
HeightWeight
63
0.278
0.022
HeightWeight
64
-0.248
0.437
HeightWeight
65
1.521
-0.564
HeightWeight
66
-0.943
-1.140
HeightWeight
67
1.306
0.497
HeightWeight
68
0.247
0.407
HeightWeight
69
-1.667
-1.135
HeightWeight
70
-0.359
0.064
HeightWeight
71
-0.853
1.100
HeightWeight
72
1.455
-0.651
HeightWeight
73
1.497
0.061
HeightWeight
74
-2.775
-0.108
HeightWeight
75
0.186
-0.033
HeightWeight
76
-0.989
0.170
HeightWeight
77
0.822
0.523
HeightWeight
78
-0.561
1.239
HeightWeight
79
-0.041
-1.331
HeightWeight
80
-0.358
-0.198
HeightWeight
81
0.689
0.186
HeightWeight
82
-0.741
0.112
HeightWeight
83
2.430
0.899
HeightWeight
84
-0.507
0.710
HeightWeight
85
0.567
-0.345
HeightWeight
86
1.154
0.294
HeightWeight
87
0.593
1.023
HeightWeight
88
1.038
-0.522
HeightWeight
89
0.452
-1.173
HeightWeight
90
0.356
0.151
HeightWeight
91
0.617
0.486
HeightWeight
92
1.678
0.009
HeightWeight
93
2.082
-0.499
HeightWeight
94
-0.436
-1.347
HeightWeight
95
1.138
-0.400
HeightWeight
96
1.222
-0.681
HeightWeight
97
-1.030
0.180
HeightWeight
98
-1.891
1.403
HeightWeight
99
-0.376
0.484
HeightWeight
100
-0.352
-1.037
HeightWeight
101
-2.608
-0.363
HeightWeight
102
-0.128
0.498
HeightWeight
103
0.557
0.266
HeightWeight
104
-2.542
-0.837
HeightWeight
105
-0.818
-0.679
HeightWeight
106
-0.195
-0.246
HeightWeight
107
-0.352
-0.141
HeightWeight
108
-0.641
0.554
HeightWeight
109
0.614
-0.472
HeightWeight
110
-1.500
1.496
HeightWeight
111
-0.389
-0.221
HeightWeight
112
0.788
-0.742
HeightWeight
113
1.078
-0.555
HeightWeight
114
-1.635
-0.177
HeightWeight
115
0.444
-1.057
HeightWeight
116
0.319
-0.162
HeightWeight
117
0.636
0.009
HeightWeight
118
1.182
-0.137
HeightWeight
119
-1.574
0.240
HeightWeight
120
0.893
0.251
HeightWeight
121
-1.217
0.349
HeightWeight
122
-1.346
-0.807
HeightWeight
123
0.598
-0.474
HeightWeight
124
0.838
0.116
HeightWeight
125
-1.207
-1.200
HeightWeight
126
-1.078
0.613
HeightWeight
127
-1.575
-0.519
HeightWeight
128
-0.475
-0.461
HeightWeight
129
1.112
0.427
HeightWeight
130
1.234
-0.435
HeightWeight
131
0.433
0.200
HeightWeight
132
1.401
-0.990
HeightWeight
133
-0.270
-1.109
HeightWeight
134
-1.497
1.091
HeightWeight
135
1.202
-1.094
HeightWeight
136
0.615
0.418
HeightWeight
137
-1.366
0.114
HeightWeight
138
-0.210
0.162
HeightWeight
139
3.597
-0.740
HeightWeight
140
0.928
-0.552
HeightWeight
141
1.104
-0.041
HeightWeight
142
-1.826
0.192
HeightWeight
143
-0.090
-0.134
HeightWeight
144
0.163
-0.121
HeightWeight
145
-2.195
-0.177
HeightWeight
146
-0.523
-0.749
HeightWeight
147
0.617
0.165
HeightWeight
148
-2.104
0.410
HeightWeight
149
0.486
-0.068
HeightWeight
150
0.966
0.280
HeightWeight
151
0.472
-0.687
HeightWeight
152
0.465
0.865
HeightWeight
153
-0.914
0.157
HeightWeight
154
0.057
1.471
HeightWeight
155
2.199
-1.074
HeightWeight
156
-2.640
0.144
HeightWeight
157
2.981
0.772
HeightWeight
158
-1.254
-0.926
HeightWeight
159
2.167
-0.800
HeightWeight
160
-1.631
0.293
HeightWeight
161
0.810
1.444
HeightWeight
162
-2.322
0.272
HeightWeight
163
1.217
0.182
HeightWeight
164
-0.276
-0.590
HeightWeight
165
-0.898
1.026
HeightWeight
166
-0.294
0.296
HeightWeight
167
-0.549
-0.869
HeightWeight
168
-0.748
0.462
HeightWeight
169
-0.293
0.261
HeightWeight
170
-1.657
-0.229
HeightWeight
171
0.267
-0.812
HeightWeight
172
-0.171
-0.185
HeightWeight
173
-0.221
-0.083
HeightWeight
174
-1.428
0.518
HeightWeight
175
2.857
-1.429
HeightWeight
176
-1.739
-0.908
HeightWeight
177
-0.633
0.139
HeightWeight
178
-1.298
0.313
HeightWeight
179
-1.057
0.379
HeightWeight
180
-0.970
0.028
HeightWeight
181
0.155
0.767
HeightWeight
182
-1.408
0.108
HeightWeight
183
-1.493
-0.641
HeightWeight
184
0.110
-0.465
HeightWeight
185
0.197
1.451
HeightWeight
186
-0.611
0.839
HeightWeight
187
0.268
-0.191
HeightWeight
188
-0.868
0.123
HeightWeight
189
-0.332
0.382
HeightWeight
190
-1.475
-1.570
HeightWeight
191
1.867
0.234
HeightWeight
192
-1.847
-0.076
HeightWeight
193
0.155
0.512
HeightWeight
194
0.934
1.327
HeightWeight
195
1.709
0.463
HeightWeight
196
-1.161
0.406
HeightWeight
197
-1.347
-0.006
HeightWeight
198
0.169
-0.042
HeightWeight
199
0.040
-0.011
HeightWeight
200
1.293
-1.215
There are no rows in this table
Loadings
Group
Variable Name
Principal Component1
Principal Component2
Principal Component3
Principal Component4
Principal Component5
Principal Component6
Drinking
Spirits
0.35
-0.57
-0.214
-0.635
-0.329
0
Drinking
Wine
-0.45
-0.38
-0.618
0.448
-0.276
0
Drinking
Beer
0.07
0.72
-0.425
-0.207
-0.497
0
Drinking
Life Expectancy
-0.58
0.09
-0.269
-0.567
0.506
0
Drinking
Heart Disease Rate
0.58
0.04
-0.565
0.177
0.559
0
Drinking
Drinking Percentage Explained
0.46
0.78
0.898
0.983
1.000
1
MovieReviews
Variety
0.58
-0.20
0.650
0.449
0.000
0
MovieReviews
The New York Times
0.57
0.22
-0.698
0.369
0.000
0
MovieReviews
Vanity Fair
-0.44
0.63
0.162
0.617
0.000
0
MovieReviews
RogerEbert.com
-0.37
-0.72
-0.253
0.531
0.000
0
MovieReviews
MovieReviews Percentage Explained
0.65
0.96
1.000
1.000
1.000
1
HeightWeight
Height
0.71
-0.71
0.000
0.000
0.000
0
HeightWeight
Weight
0.71
0.71
0.000
0.000
0.000
0
HeightWeight
HeightWeight Percentage Explained
0.78
1.00
1.000
1.000
1.000
1
There are no rows in this table
Variable Names Drinking
Name
Description
Spirits
in litres, per year
Wine
in litres, per year
Beer
in litres, per year
Life Expectancy
in years
Heart Disease Rate
per 100.000 persons
There are no rows in this table
Variable Names Movie Reviews
Name
Description
Variety
The New York Times
Vanity Fair
RogerEbert.com
There are no rows in this table
Variable Names Height vs Weight
Name
Description
Height
in inches
Weight
in pounds
There are no rows in this table
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