Cette réponse consiste à montrer des similitudes et des différences de calcul concrètes entre l'ACP et l'analyse factorielle. Pour les différences théoriques générales entre eux, voir les questions / réponses 1 , 2 , 3 , 4 , 5 .
Ci-dessous, je ferai, étape par étape, l' analyse en composantes principales (ACP) des données sur l'iris (espèces "setosa" uniquement), puis je ferai une analyse factorielle des mêmes données. L'analyse factorielle (FA) sera effectuée par la méthode de l'axe principal itératif ( PAF ) qui est basée sur l'approche PCA et permet ainsi de comparer PCA et FA étape par étape.
Données sur l'iris (setosa uniquement):
id SLength SWidth PLength PWidth species
1 5.1 3.5 1.4 .2 setosa
2 4.9 3.0 1.4 .2 setosa
3 4.7 3.2 1.3 .2 setosa
4 4.6 3.1 1.5 .2 setosa
5 5.0 3.6 1.4 .2 setosa
6 5.4 3.9 1.7 .4 setosa
7 4.6 3.4 1.4 .3 setosa
8 5.0 3.4 1.5 .2 setosa
9 4.4 2.9 1.4 .2 setosa
10 4.9 3.1 1.5 .1 setosa
11 5.4 3.7 1.5 .2 setosa
12 4.8 3.4 1.6 .2 setosa
13 4.8 3.0 1.4 .1 setosa
14 4.3 3.0 1.1 .1 setosa
15 5.8 4.0 1.2 .2 setosa
16 5.7 4.4 1.5 .4 setosa
17 5.4 3.9 1.3 .4 setosa
18 5.1 3.5 1.4 .3 setosa
19 5.7 3.8 1.7 .3 setosa
20 5.1 3.8 1.5 .3 setosa
21 5.4 3.4 1.7 .2 setosa
22 5.1 3.7 1.5 .4 setosa
23 4.6 3.6 1.0 .2 setosa
24 5.1 3.3 1.7 .5 setosa
25 4.8 3.4 1.9 .2 setosa
26 5.0 3.0 1.6 .2 setosa
27 5.0 3.4 1.6 .4 setosa
28 5.2 3.5 1.5 .2 setosa
29 5.2 3.4 1.4 .2 setosa
30 4.7 3.2 1.6 .2 setosa
31 4.8 3.1 1.6 .2 setosa
32 5.4 3.4 1.5 .4 setosa
33 5.2 4.1 1.5 .1 setosa
34 5.5 4.2 1.4 .2 setosa
35 4.9 3.1 1.5 .2 setosa
36 5.0 3.2 1.2 .2 setosa
37 5.5 3.5 1.3 .2 setosa
38 4.9 3.6 1.4 .1 setosa
39 4.4 3.0 1.3 .2 setosa
40 5.1 3.4 1.5 .2 setosa
41 5.0 3.5 1.3 .3 setosa
42 4.5 2.3 1.3 .3 setosa
43 4.4 3.2 1.3 .2 setosa
44 5.0 3.5 1.6 .6 setosa
45 5.1 3.8 1.9 .4 setosa
46 4.8 3.0 1.4 .3 setosa
47 5.1 3.8 1.6 .2 setosa
48 4.6 3.2 1.4 .2 setosa
49 5.3 3.7 1.5 .2 setosa
50 5.0 3.3 1.4 .2 setosa
Nous avons 4 variables numériques à inclure dans nos analyses: SLength SWidth PLength PWidth , et les analyses seront basées sur des covariances , ce qui revient à dire que nous analysons des variables centrées . (Si nous avons choisi d'analyser les corrélations qui analyseraient des variables standardisées. Une analyse basée sur des corrélations produit des résultats différents de l'analyse basée sur des covariances.) Je n'afficherai pas les données centrées. Appelons ces matrices de données X.
Étapes de l' ACP :
Step 0. Compute centered variables X and covariance matrix S.
Covariances S (= X'*X/(n-1) matrix: see /stats//a/22520/3277)
.12424898 .09921633 .01635510 .01033061
.09921633 .14368980 .01169796 .00929796
.01635510 .01169796 .03015918 .00606939
.01033061 .00929796 .00606939 .01110612
Step 1.1. Decompose data X or matrix S to get eigenvalues and right eigenvectors.
You may use svd or eigen decomposition (see /stats//q/79043/3277)
Eigenvalues L (component variances) and the proportion of overall variance explained
L Prop
PC1 .2364556901 .7647237023
PC2 .0369187324 .1193992401
PC3 .0267963986 .0866624997
PC4 .0090332606 .0292145579
Eigenvectors V (cosines of rotation of variables into components)
PC1 PC2 PC3 PC4
SLength .6690784044 .5978840102 -.4399627716 -.0360771206
SWidth .7341478283 -.6206734170 .2746074698 -.0195502716
PLength .0965438987 .4900555922 .8324494972 -.2399012853
PWidth .0635635941 .1309379098 .1950675055 .9699296890
Step 1.2. Decide on the number M of first PCs you want to retain.
You may decide it now or later on - no difference, because in PCA values of components do not depend on M.
Let's M=2. So, leave only 2 first eigenvalues and 2 first eigenvector columns.
Step 2. Compute loadings A. May skip if you don't need to interpret PCs anyhow.
Loadings are eigenvectors normalized to respective eigenvalues: A value = V value * sqrt(L value)
Loadings are the covariances between variables and components.
Loadings A
PC1 PC2
SLength .32535081 .11487892
SWidth .35699193 -.11925773
PLength .04694612 .09416050
PWidth .03090888 .02515873
Sums of squares in columns of A are components' variances, the eigenvalues
Standardized (rescaled) loadings.
St. loading is Loading / sqrt(Variable's variance);
these loadings are computed if you analyse covariances, and are suitable for interpretation of PCs
(if you analyse correlations, A are already standardized).
PC1 PC2
SLength .92300804 .32590717
SWidth .94177127 -.31461076
PLength .27032731 .54219930
PWidth .29329327 .23873031
Step 3. Compute component scores (values of PCs).
Regression coefficients B to compute Standardized component scores are: B = A*diag(1/L) = inv(S)*A
B
PC1 PC2
SLength 1.375948338 3.111670112
SWidth 1.509762499 -3.230276923
PLength .198540883 2.550480216
PWidth .130717448 .681462580
Standardized component scores (having variances 1) = X*B
PC1 PC2
.219719506 -.129560000
-.810351411 .863244439
-.803442667 -.660192989
-1.052305574 -.138236265
.233100923 -.763754703
1.322114762 .413266845
-.606159168 -1.294221106
-.048997489 .137348703
...
Raw component scores (having variances = eigenvalues) can of course be computed from standardized ones.
In PCA, they are also computed directly as X*V
PC1 PC2
.106842367 -.024893980
-.394047228 .165865927
-.390687734 -.126851118
-.511701577 -.026561059
.113349309 -.146749722
.642900908 .079406116
-.294755259 -.248674852
-.023825867 .026390520
...
Étapes FA (méthode d'extraction d'axe principal itérative):
Step 0.1. Compute centered variables X and covariance matrix S.
Step 0.2. Decide on the number of factors M to extract.
(There exist several well-known methods in help to decide, let's omit mentioning them. Most of them require that you do PCA first.)
Note that you have to select M before you proceed further because, unlike in PCA, in FA loadings and factor values depend on M.
Let's M=2.
Step 0.3. Set initial communalities on the diagonal of S.
Most often quantities called "images" are used as initial communalities (see /stats//a/43224/3277).
Images are diagonal elements of matrix S-D, where D is diagonal matrix with diagonal = 1 / diagonal of inv(S).
(If S is correlation matrix, images are the squared multiple correlation coefficients.)
With covariance matrix, image is the squared multiple correlation multiplied by the variable variance.
S with images as initial communalities on the diagonal
.07146025 .09921633 .01635510 .01033061
.09921633 .07946595 .01169796 .00929796
.01635510 .01169796 .00437017 .00606939
.01033061 .00929796 .00606939 .00167624
Step 1. Decompose that modified S to get eigenvalues and right eigenvectors.
Use eigen decomposition, not svd. (Usually some last eigenvalues will be negative.)
Eigenvalues L
F1 .1782099114
F2 .0062074477
-.0030958623
-.0243488794
Eigenvectors V
F1 F2
SLength .6875564132 .0145988554 .0466389510 .7244845480
SWidth .7122191394 .1808121121 -.0560070806 -.6759542030
PLength .1154657746 -.7640573143 .6203992617 -.1341224497
PWidth .0817173855 -.6191205651 -.7808922917 -.0148062006
Leave the first M=2 values in L and columns in V.
Step 2.1. Compute loadings A.
Loadings are eigenvectors normalized to respective eigenvalues: A value = V value * sqrt(L value)
F1 F2
SLength .2902513607 .0011502052
SWidth .3006627098 .0142457085
PLength .0487437795 -.0601980567
PWidth .0344969255 -.0487788732
Step 2.2. Compute row sums of squared loadings. These are updated communalities.
Reset the diagonal of S to them
S with updated communalities on the diagonal
.08424718 .09921633 .01635510 .01033061
.09921633 .09060101 .01169796 .00929796
.01635510 .01169796 .00599976 .00606939
.01033061 .00929796 .00606939 .00356942
REPEAT Steps 1-2 many times (iterations, say, 25)
Extraction of factors is done.
Final loadings A and communalities (row sums of squares in A).
Loadings are the covariances between variables and factors.
Communality is the degree to what the factors load a variable, it is the "common variance" in the variable.
F1 F2 Comm
SLength .3125767362 .0128306509 .0978688416
SWidth .3187577564 -.0323523347 .1026531808
PLength .0476237419 .1034495601 .0129698323
PWidth .0324478281 .0423861795 .0028494498
Sums of squares in columns of A are factors' variances.
Standardized (rescaled) loadings and communalities.
St. loading is Loading / sqrt(Variable's variance);
these loadings are computed if you analyse covariances, and are suitable for interpretation of Fs
(if you analyse correlations, A are already standardized).
F1 F2 Comm
SLength .8867684574 .0364000747 .7876832626
SWidth .8409066701 -.0853478652 .7144082859
PLength .2742292179 .5956880078 .4300458666
PWidth .3078962532 .4022009053 .2565656710
Step 3. Compute factor scores (values of Fs).
Unlike component scores in PCA, factor scores are not exact, they are reasonable approximations.
Several methods of computation exist (/stats//q/126885/3277).
Here is regressional method which is the same as the one used in PCA.
Regression coefficients B to compute Standardized factor scores are: B = inv(S)*A (original S is used)
B
F1 F2
SLength 1.597852081 -.023604439
SWidth 1.070410719 -.637149341
PLength .212220217 3.157497050
PWidth .423222047 2.646300951
Standardized factor scores = X*B
These "Standardized factor scores" have variance not 1; the variance of a factor is SSregression of the factor by variables / (n-1).
F1 F2
.194641800 -.365588231
-.660133976 -.042292672
-.786844270 -.480751358
-1.011226507 .216823430
.141897664 -.426942721
1.250472186 .848980006
-.669003108 -.025440982
-.050962459 .016236852
...
Factors are extracted as orthogonal. And they are.
However, regressionally computed factor scores are not fully uncorrelated.
Covariance matrix between computed factor scores.
F1 F2
F1 .864 .026
F2 .026 .459
Factor variances are their squared loadings.
You can easily recompute the above "standardized" factor scores to "raw" factor scores having those variances:
raw score = st. score * sqrt(factor variance / st. scores variance).
Après l'extraction (illustrée ci-dessus), une rotation facultative peut avoir lieu. La rotation est souvent effectuée dans FA. Parfois, cela se fait dans PCA exactement de la même manière. La rotation fait pivoter la matrice de chargement A en une certaine forme de "structure simple" qui facilite considérablement l'interprétation des facteurs (les scores pivotés peuvent alors être recalculés). Étant donné que la rotation n'est pas ce qui différencie mathématiquement FA de PCA et parce que c'est un grand sujet distinct, je ne le toucherai pas.