Etapas realizadas na análise fatorial comparadas às etapas realizadas no PCA

12

Sei como executar o PCA (análise de componentes principais), mas gostaria de saber as etapas que devem ser usadas para a análise fatorial.

Para executar o PCA, vamos considerar uma matriz , por exemplo:UMA

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

Eu calculei sua matriz de correlação B = corr(A):

        1.0000    0.9087    0.9250
        0.9087    1.0000    0.9970
        0.9250    0.9970    1.0000

Então eu fiz a decomposição do autovalor [V,D] = eig(B), resultando em autovetores:

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

e autovalores:

        2.8877         0         0
             0    0.1101         0
             0         0    0.0022

A idéia geral por trás do PCA é escolher componentes significativos, formar uma nova matriz que tenha colunas de autovetores e, em seguida, precisamos projetar a matriz original (no PCA, ele é centrado em zero). Mas na análise fatorial, por exemplo, devemos escolher componentes que tenham valores superiores a 1 valor , e também usar a rotação de fatores, por favor me diga como isso é feito? Por exemplo, neste caso.

Por favor, ajude-me a entender as etapas da análise fatorial, em comparação com as etapas do PCA.

dato datuashvili
fonte

Respostas:

24

Esta resposta é mostrar semelhanças e diferenças computacionais concretas entre a análise PCA e fatorial. Para diferenças teóricas gerais entre eles, consulte as perguntas / respostas 1 , 2 , 3 , 4 , 5 .

Abaixo farei, passo a passo, a análise de componentes principais (PCA) dos dados da íris (apenas espécies "setosa") e, em seguida, farei a análise fatorial dos mesmos dados. A análise fatorial (FA) será feita pelo método do eixo principal iterativo ( PAF ), que é baseado na abordagem PCA e, portanto, permite comparar PCA e FA passo a passo.

Dados da íris (apenas 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 

Temos quatro variáveis ​​numéricas para incluir em nossas análises: SLength SWidth PLength PWidth , e as análises serão baseadas em covariâncias , que é o mesmo que dizer que analisamos variáveis centralizadas . (Se escolhermos analisar correlações que analisariam variáveis ​​padronizadas. A análise baseada em correlações produz resultados diferentes dos que a análise baseada em covariâncias.) Não mostrarei os dados centralizados. Vamos chamar esses dados de matriz X.

Etapas do 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 
  ...

Etapas de FA (método de extração de eixo principal iterativo):

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).

Após a extração (mostrada acima), a rotação opcional pode ocorrer. A rotação é freqüentemente feita na FA. Às vezes, é feito no PCA exatamente da mesma maneira. A rotação gira a matriz de carregamento A em alguma forma de "estrutura simples" que facilita muito a interpretação dos fatores (então as pontuações rotadas podem ser recalculadas). Como a rotação não é o que diferencia a FA do PCA matematicamente e porque é um tópico grande separado, não vou tocá-la.

ttnphns
fonte
Quando você fala sobre "imagens" como comunidades iniciais, fornece um link para outra resposta sua (que discute vários métodos de escolha de comunidades iniciais), mas não menciona "imagens". Parece interessante, você gostaria de expandir essa resposta antiga?
Ameba diz Reinstate Monica
mas a análise fator que parece um pouco estranho para mim, agora eu estou pensando sobre isso e não podia adivinhar
dato datuashvili