Langkah-langkah yang dilakukan dalam analisis faktor dibandingkan dengan langkah-langkah yang dilakukan dalam PCA

Saya tahu cara melakukan PCA (analisis komponen utama), tetapi saya ingin tahu langkah-langkah yang harus digunakan untuk analisis faktor.

Untuk melakukan PCA, mari kita pertimbangkan beberapa matriks , misalnya:$A$

         3     1    -1
         2     4     0
         4    -2    -5
        11    22    20

Saya telah menghitung matriks korelasinya B = corr(A):

        1.0000    0.9087    0.9250
        0.9087    1.0000    0.9970
        0.9250    0.9970    1.0000

Kemudian saya telah melakukan dekomposisi nilai eigen [V,D] = eig(B), menghasilkan vektor eigen:

        0.5662    0.8209   -0.0740
        0.5812   -0.4613   -0.6703
        0.5844   -0.3366    0.7383

dan nilai eigen:

        2.8877         0         0
             0    0.1101         0
             0         0    0.0022

Gagasan umum di balik PCA adalah memilih komponen yang signifikan, membentuk matriks baru yang memiliki vektor vektor kolom, maka kita perlu memproyeksikan matriks asli (dalam PCA itu nol berpusat). Tetapi dalam analisis faktor, misalnya kita harus memilih komponen yang memiliki nilai singular yang lebih tinggi , dan kemudian kita menggunakan rotasi faktor, tolong katakan padaku bagaimana ini dilakukan? Misalnya dalam hal ini.$1$

Tolong bantu saya memahami langkah-langkah analisis faktor, dibandingkan dengan langkah-langkah PCA.

Jawaban ini adalah untuk menunjukkan persamaan dan perbedaan komputasi yang konkret antara PCA dan analisis Faktor. Untuk perbedaan teoretis umum di antara mereka, lihat pertanyaan / jawaban 1 , 2 , 3 , 4 , 5 .

Di bawah ini saya akan lakukan, langkah demi langkah, analisis Principal Component (PCA) dari data yang iris ( "Setosa" spesies saja) dan kemudian akan melakukan analisis faktor dari data yang sama. Analisis faktor (FA) akan dilakukan dengan metode Iterative principal axis ( PAF ) yang didasarkan pada pendekatan PCA dan dengan demikian membuat seseorang dapat membandingkan PCA dan FA langkah demi langkah.

Data iris (hanya setosa):

  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 

Kami memiliki 4 variabel numerik untuk dimasukkan dalam analisis kami: SLength SWidth PLength PWidth , dan analisis akan didasarkan pada kovariansi , yang sama dengan mengatakan bahwa kami menganalisis variabel terpusat . (Jika kami memilih untuk menganalisis korelasi yang akan menganalisis variabel standar. Analisis berdasarkan korelasi menghasilkan hasil yang berbeda dari analisis berdasarkan kovariansi.) Saya tidak akan menampilkan data yang terpusat. Sebut saja matriks data ini X.

Langkah-langkah PCA :

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 
  ...

FA (langkah-langkah metode ekstraksi sumbu utama):

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).

Setelah ekstraksi (ditunjukkan di atas), rotasi opsional dapat terjadi. Rotasi sering dilakukan di FA. Kadang-kadang dilakukan di PCA dengan cara yang persis sama. Rotasi memutar matriks pemuatan A ke dalam beberapa bentuk "struktur sederhana" yang memfasilitasi interpretasi banyak faktor (kemudian skor yang dirotasi dapat dihitung ulang). Karena rotasi bukan yang membedakan FA dari PCA secara matematis dan karena itu adalah topik besar yang terpisah, saya tidak akan menyentuhnya.