主成份(principal component) 分析是因子分析的一個特例，在此特例中，前面幾個主成份即為選定之共同因子。
(A)
定義： eigenvector of eigenvalue
則
，
(B)
在(A)中即為主成份分析：尋找使得= 有最大的變異
且
在(B)中,共同因子，尋找使得第一個共同因子貢獻最大變異，此變異(Communality)應為。但在PC中變異已標準化至1，故將在PC中係數改變至。
而定義共同因子 f1 即為Y1，而其係數為
取前面m個PC當作共同因子,則
， where
共同因子模式
即
共同因子，
特有因子，
：loading of ith observation on the jth common factor
則X之Variance-Covariance matrix，
一般假設
(1)共同因子在分解成為：
主要分解的變異部份為：
共同因子所能解釋的部份：communality。
(2)若T為一 orthogonal matrix
， where，
且，，
在因子分析中可旋轉至任一軸，使得共同因子容易解釋，經旋轉後之共同解釋變異部份不變。
一般步驟為先固定communalities為SMC(squared multiple correlation)
即
所得之作為communality(diagonal 部份之值)
此時Var-Cov matrix 成為
(Reduced Correlation Matrix)
依照PC做法並旋轉使得係數(loading)之 variance 為最大(此種旋轉稱為Varimax)，但此時矩陣不一定為正定，故eigenvalue可能出現負值。
資料分析講義：因子分析Factor Analysis
options nodate nonotes ps=60;
data factor1;
input pop school employ service house;
cards;
5700 12.8 2500 270 25000
1000 10.9 600 10 10000
3400 8.8 1000 10 9000
3800 13.6 1700 140 25000
4000 12.8 1600 140 25000
8200 8.3 2600 60 12000
1200 11.4 400 10 16000
9100 11.5 3300 60 14000
9900 12.5 3400 180 18000
9600 13.7 3600 390 25000
9600 9.6 3300 80 12000
9400 11.4 4000 100 13000
;
proc factor data=factor1;
run;
proc factor prior=smc data=factor1 preplot
rotate=varimax reorder plot;
run;
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 5 Average = 1
Eigenvalue Difference Proportion Cumulative
1 2.87331359 1.07665350 0.5747 0.5747
2 1.79666009 1.58182321 0.3593 0.9340
3 0.214836 0.11490283 0.0430 0.9770
4 0.09993405 0.08467868 0.0200 0.9969
5 0.01525537 0.0031 1.0000
2 factors will be retained by the MINEIGEN criterion.
Factor Pattern
FACTOR1 FACTOR2
POP 0.58096 0.802
SCHOOL 0.76704 -0.54476
EMPLOY 0.67243 0.72605
SERVICE 0.93239 -0.10431
HOUSE 0.79116 -0.55818
Variance explained by each factor
FACTOR1 FACTOR2
2.873314 1.796660
Final Communality Estimates: Total = 4.669974
POP SCHOOL EMPLOY SERVICE HOUSE
0.987826 0.885106 0.979306 0.880236 0.937500
Initial Factor Method: Principal Factors
Prior Communality Estimates: SMC
POP SCHOOL EMPLOY SERVICE HOUSE
0.968592 0.822285 0.969181 0.785724 0.847019
Eigenvalues of the Reduced Correlation Matrix: Total = 4.39280116 Average = 0.87856023
Eigenvalue Difference Proportion Cumulative
1 2.73430084 1.01823217 0.6225 0.6225
2 1.71606867 1.67650586 0.3907 1.0131
3 0.03956281 0.008626 0.0090 1.0221
4 -.02452345 0.04808427 -0.0056 1.0165
5 -.07260772 -0.0165 1.0000
2 factors will be retained by the PROPORTION criterion.
Factor Pattern
FACTOR1 FACTOR2
SERVICE 0.879 -0.15847
HOUSE 0.74215 -0.57806
EMPLOY 0.71447 0.67936
SCHOOL 0.71370 -0.55515
POP 0.62533 0.76621
Variance explained by each factor
FACTOR1 FACTOR2
2.734301 1.716069
Final Communality Estimates: Total = 4.450370
POP SCHOOL EMPLOY SERVICE HOUSE
0.978113 0.8175 0.971999 0.797743 0.884950
Initial Factor Method: Principal Factors
Plot of Factor Pattern for FACTOR1 and FACTOR2
FACTOR1
1
D .9
.8
E
B .7 C
A
.6
.5
.4
.3
.2
F
.1 A
C
-1 -.9-.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0T
O
-.1 R
2
-.2
-.3
-.4
-.5
-.6
-.7
-.8
-.9
-1
POP=A SCHOOL=B EMPLOY=C SERVICE=D HOUSE=E
Rotation Method: Varimax
Orthogonal Transformation Matrix
1 2
1 0.785 0.61446
2 -0.61446 0.785
Rotated Factor Pattern
FACTOR1 FACTOR2
HOUSE 0.94072 -0.00004
SCHOOL 0.90419 0.00055
SERVICE 0.79085 0.41509
POP 0.02255 0.98874
EMPLOY 0.14625 0.97499
Variance explained by each factor
FACTOR1 FACTOR2
2.349857 2.100513
Final Communality Estimates: Total = 4.450370
POP SCHOOL EMPLOY SERVICE HOUSE
0.978113 0.8175 0.971999 0.797743 0.884950
Rotation Method: Varimax
Plot of Factor Pattern for FACTOR1 and FACTOR2
FACTOR1
1
E
.B
.8 D
.7
.6
.5
.4
.3
.2
C F
.1 A
C
-1 -.9-.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 A.0T
O
-.1 R
2
-.2
-.3
-.4
-.5
-.6
-.7
-.8
-.9
-1
POP=A SCHOOL=B EMPLOY=C SERVICE=D HOUSE=E
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