6# 马远航
Stata里面的 pca 命令，就是做主成分分析的（我用的Stata S/E 11）
帮助文件抄在下面


help pca                                           dialogs:  pca  pcamat      
                                                  also see:  pca postestimation
-------------------------------------------------------------------------------

Title

    [MV] pca -- Principal component analysis


Syntax

    Principal component analysis of data

        pca varlist [if] [in] [weight] [, options]


    Principal component analysis of a correlation or covariance matrix

        pcamat matname , n(#) [options pcamat_options]



    options              description
    -------------------------------------------------------------------------
    Model 2
      components(#)      retain maximum of # principal components; factors()
                           is a synonym
      mineigen(#)        retain eigenvalues larger than #; default is 1e-5
      correlation        perform PCA of the correlation matrix; the default
      covariance         perform PCA of the covariance matrix
      vce(none)          do not compute VCE of the eigenvalues and vectors;
                           the default
      vce(normal)        compute VCE of the eigenvalues and vectors assuming
                           multivariate normality

    Reporting
      level(#)           set confidence level; default is level(95)
      blanks(#)          display loadings as blank when |loadings| < #
      novce              suppress display of SEs even though calculated
    # means              display summary statistics of variables

    Advanced
      tol(#)             advanced option; see Options for details
      ignore             advanced option; see Options for details

    + norotated          display unrotated results, even if rotated results
                           are available (replay only)
    -------------------------------------------------------------------------
    # means is not allowed with pcamat.
    + norotated is not available in the dialog box.

    pcamat_options       description
    -------------------------------------------------------------------------
    Model
      shape(full)        matname is a square symmetric matrix; the default
      shape(lower)       matname is a vector with the rowwise lower triangle
                           (with diagonal)
      shape(upper)       matname is a vector with the rowwise upper triangle
                           (with diagonal)
      names(namelist)    variable names; required if matname is triangular
      forcepsd           modifies matname to be positive semidefinite
    * n(#)               number of observations
      sds(matname2)      vector with standard deviations of variables
      means(matname3)    vector with means of variables
    -------------------------------------------------------------------------
    * n() is required for pcamat.

    bootstrap, by, jackknife, rolling, statsby, and xi are allowed with pca;
      see prefix.  However, bootstrap and jackknife results should be
      interpreted with caution; identification of the pca parameters involves
      data-dependent restrictions, possibly leading to badly biased and
      overdispersed estimates.
    Weights are not allowed with the bootstrap prefix.
    aweights are not allowed with the jackknife prefix.
    aweights and fweights are allowed with pca; see weight.
    See [MV] pca postestimation for features available after estimation.


Menu

    pca

        Statistics > Multivariate analysis > Factor and principal component
            analysis > Principal component analysis (PCA)

    pcamat

        Statistics > Multivariate analysis > Factor and principal component
            analysis > PCA of a correlation or covariance matrix


Description

    Principal component analysis (PCA) is a statistical technique used for
    data reduction.  The leading eigenvectors from the eigen decomposition of
    the correlation or covariance matrix of the variables describe a series
    of uncorrelated linear combinations of the variables that contain most of
    the variance.  In addition to data reduction, the eigenvectors from a PCA
    are often inspected to learn more about the underlying structure of the
    data.

    pca and pcamat display the eigenvalues and eigenvectors from the PCA
    eigen decomposition.  The eigenvectors are returned in orthonormal form,
    i.e., orthogonal (uncorrelated) and normalized (with unit length, L'L =
    I).  pcamat provides the correlation or covariance matrix directly.  For
    pca the correlation or covariance matrix is computed from the variables
    in varlist.

    pcamat allows the correlation or covariance matrix to be specified as a k
    x k symmetric matrix with row and column names set to the variable names
    or as a k(k+1)/2 long row or column vector containing the lower or upper
    triangle C along with the names() option providing the variable names.
    See the shape() option for details.

    The vce(normal) option of pca and pcamat provides standard errors of the
    eigenvalues and eigenvectors and aids in interpreting the eigenvectors.
    See Remarks for a discussion of the underlying assumptions.

    Scores, residuals, rotations, scree plots, score plots, loading plots,
    and more are available after pca and pcamat; see [MV] pca postestimation.

7# houquan
Options

        +---------+
    ----+ Model 2 +----------------------------------------------------------

    components(#) and mineigen(#) specify the maximum number of components
        (eigenvectors or factors) to be retained.  components() specifies the
        number directly, and mineigen() specifies it indirectly, keeping all
        components with eigenvalues greater than the indicated value.  The
        options can be specified individually, together, or not at all.
        factors() is a synonym for components().

        components(#) sets the maximum number of components (factors) to be
        retained.  pca and pcamat always display the full set of eigenvalues
        but display eigenvectors only for retained components.  Specifying a
        number larger than the number of variables in varlist is equivalent
        to specifying the number of variables in varlist, and is the default.

        mineigen(#) sets the minimum value of eigenvalues to be retained.
        The default is 1e-5 or the value of tol() if specified.

        Specifying components() and mineigen() affects only the number of
        components to be displayed and stored in e(); it does not enforce the
        assumption that the other eigenvalues are 0.  In particular, the
        standard errors reported when vce(normal) is specified do not depend
        on the number of retained components.

    correlation and covariance specify that principal components be
        calculated for the correlation matrix and covariance matrix,
        respectively.  The default is correlation.  Unlike factor analysis,
        PCA is not scale invariant; the eigenvalues and eigenvectors of a
        covariance matrix differ from those of the associated correlation
        matrix.  Usually, a PCA of a covariance matrix is meaningful only if
        the variables are expressed in the same units.

        For pcamat, do not confuse the type of the matrix to be analyzed with
        the type of matname.  Obviously, if matname is a correlation matrix
        and the option sds() is not specified, it is not possible to perform
        a PCA of the covariance matrix.

    vce(none|normal) specifies whether standard errors are to be computed for
        the eigenvalues, the eigenvectors, and the (cumulative) percentage of
        explained variance (confirmatory PCA). These standard errors are
        obtained assuming multivariate normality of the data and are valid
        only for a PCA of a covariance matrix.  Be cautious if applying these
        to correlation matrices.

        +-----------+
    ----+ Reporting +--------------------------------------------------------

    level(#) specifies the confidence level, as a percentage, for confidence
        intervals.  The default is level(95) or as set by set level.  level()
        is allowed only with vce(normal).

    blanks(#) shows blanks for loadings with absolute value smaller than #.
        This option is ignored when specified with vce(normal).

    novce suppresses the display of standard errors, even though they are
        computed, and displays the PCA results in a matrix/table style.  You
        can specify novce during estimation in combination with vce(normal).
        More likely, you will want to use novce during replay.

    means displays summary statistics of the variables over the estimation
        sample.  This option is not available with pcamat.

        +----------+
    ----+ Advanced +---------------------------------------------------------

    tol(#) is an advanced, rarely used option and is available only with
        vce(normal).  An eigenvalue, ev_i, is classified as being close to
        zero if ev_i < tol * max(ev).  Two eigenvalues, ev_1 and ev_2, are
        "close" if abs(ev_1-ev_2) < tol*max(ev).  The default is tol(1e-5).
        See option ignore and Remarks below.

    ignore is an advanced, rarely used option and is available only with
        vce(normal).  It continues the computation of standard errors and
        tests, even if some eigenvalues are suspiciously close to zero or
        suspiciously close to other eigenvalues, violating crucial
        assumptions of the asymptotic theory used to estimate standard errors
        and tests.  See Remarks below.

    The following option is available with pca and pcamat but is not shown in
    the dialog box:

    norotated displays the unrotated principal components, even if rotated
        components are available.  This option may be specified only when
        replaying results.


Options unique to pcamat

        +-------+
    ----+ Model +------------------------------------------------------------

    shape(shape_arg) specifies the shape (storage mode) for the covariance or
        correlation matrix matname.  The following shapes are supported:

        full specifies that the correlation or covariance structure of k
            variables is stored as a symmetric k x k matrix.  Specifying
            shape(full) is optional in this case.

        lower specifies that the correlation or covariance structure of k
            variables is stored as a vector with k(k+1)/2 elements in rowwise
            lower-triangular order:

                C(11) C(21) C(22) C(31) C(32) C(33) ... C(k1) C(k2) ... C(kk)

        upper specifies that the correlation or covariance structure of k
            variables is stored as a vector with k(k+1)/2 elements in rowwise
            upper-triangular order:

                C(11) C(12) C(13) ... C(1k) C(22) C(23) ... C(2k) ...  C(k-1
                    k-1) C(k-1 k) C(kk)

    names(namelist) specifies a list of k different names, which are used to
        document output and to label estimation results and are used as
        variable names by predict.  By default, pcamat verifies that the row
        and column names of matname and the column or row names of matname2
        and matname3 from the sds() and means() options are in agreement.
        Using the names() option turns off this check.

    forcepsd modifies the matrix matname to be positive semidefinite (psd)
        and so to be a proper covariance matrix.  If matname is not positive
        semidefinite, it will have negative eigenvalues.  By setting negative
        eigenvalues to 0 and reconstructing, we obtain the least-squares
        positive-semidefinite approximation to matname.  This approximation
        is a singular covariance matrix.

    n(#) is required and specifies the number of observations.

    sds(matname2) specifies a k x 1 or 1 x k matrix with the standard
        deviations of the variables.  The row or column names should match
        the variable names, unless the names() option is specified.  sds()
        may be specified only if matname is a correlation matrix.

    means(matname3) specifies a k x 1 or 1 x k matrix with the means of the
        variables.  The row or column names should match the variable names,
        unless the names() option is specified.  Specify means() if you have
        variables in your dataset and want to use predict after pcamat.


Remarks

    Technical note:

    pca and pcamat with the vce(normal) option assume that

        (A1) the variables are multivariate normal distributed and

        (A2) the variance-covariance matrix of the observations has all
             distinct and strictly positive eigenvalues.

    Under assumptions A1 and A2, the eigenvalues and eigenvectors of the
    sample covariance matrix can be seen as maximum likelihood estimates for
    the population analogues and they are asymptotically (multivariate)
    normal distributed.  Be cautious in interpreting because the asymptotic
    variances are rather sensitive to violations of assumption A1 (and A2).
    Wald tests of hypotheses that are in conflict with assumption A2 (e.g.,
    testing that the first and second eigenvalue are the same) produce
    incorrect p-values.

    Because the statistical theory for a PCA of a correlation matrix is much
    more complicated, pca and pcamat compute standard errors and tests of a
    correlation matrix as if it were a covariance matrix.  This will usually
    lead to some underestimation of standard errors, but we believe that this
    problem is smaller than the consequences of deviations from normality.

    We suggest that you conduct tests for marginal normality of the variables
    (see [R] sktest and [R] swilk), but recall that marginal normality does
    not imply multivariate normality.


Examples


    Standard PCA for descriptive use
        . sysuse auto
        . pca trunk weight length headroom
        . pca trunk weight length headroom, comp(2) covariance

    PCA assuming multivariate normality of the data
        . webuse bg2
        . pca bg2cost*, vce(normal)

    PCA of a covariance or correlation matrix
        . matrix S = ( 10.167, 22.690,  2.040  \ ///
                       22.690, 56.949,  3.788  \ ///
                        2.040,  3.788,  0.688  )
        . matrix rownames S = visual hearing taste
        . matrix colnames S = visual hearing taste
        . pcamat S, n(979) comp(2)

    Same as above
        . matrix S = ( 10.167, 22.690, 2.040, ///
                               56.949, 3.788, ///
                                       0.688 )
        . pcamat S, n(979) shape(upper) comp(2) names(visual hearing taste)

8# houquan
Saved results

    pca and pcamat without the vce(normal) option save the following in e():

    Scalars        
      e(N)                number of observations
      e(f)                number of retained components
      e(rho)              fraction of explained variance
      e(trace)            trace of e(C)
      e(lndet)            ln of the determinant of e(C)
      e(cond)             condition number of e(C)

    Macros         
      e(cmd)              pca (even in the case of pcamat)
      e(cmdline)          command as typed
      e(Ctype)            correlation or covariance
      e(wtype)            weight type
      e(wexp)             weight expression
      e(title)            title in output
      e(properties)       nov noV eigen
      e(rotate_cmd)       program used to implement rotate
      e(estat_cmd)        program used to implement estat
      e(predict)          program used to implement predict
      e(marginsnotok)     predictions disallowed by margins

    Matrices      
      e(C)                p x p correlation or covariance matrix
      e(means)            1 x p matrix of means
      e(sds)              1 x p matrix of standard deviations
      e(Ev)               1 x p matrix of eigenvalues (sorted)
      e(L)                p x f matrix of eigenvectors = components
      e(Psi)              1 x p matrix of unexplained variance

    Functions      
      e(sample)           marks estimation sample

    pca and pcamat with the vce(normal) option save the above, as well as the
    following:

    Scalars        
      e(v_rho)            variance of e(rho)
      e(chi2_i)           chi-squared statistic for test of independence
      e(df_i)             degrees of freedom for test of independence
      e(p_i)              significance of test of independence
      e(chi2_s)           chi-squared statistic for test of sphericity
      e(df_s)             degrees of freedom for test of sphericity
      e(p_s)              significance of test of sphericity
      e(rank)             rank of e(V)

    Macros         
      e(vce)              multivariate normality
      e(properties)       b V

    Matrices      
      e(b)                1 x p+fp coefficient vector (all eigenvalues and
                            retained eigenvectors)
      e(Ev_bias)          1 x p matrix: bias of eigenvalues
      e(Ev_stats)         p x 5 matrix with statistics on explained variance
      e(V)                variance-covariance matrix of the estimates e(b)


Also see

    Manual:  [MV] pca

      Help:  [MV] pca postestimation;
             [R] tetrachoric, [MV] biplot, [MV] canon, [MV] factor, [D]
             corr2data, [R] alpha


我试了一下：
        . sysuse auto
        . pca trunk weight length headroom


输出如下：
Principal components/correlation                  Number of obs    =        74
Number of comp.  =         4
Trace            =         4
Rotation: (unrotated = principal)             Rho              =    1.0000

--------------------------------------------------------------------------
Component    Eigenvalue   Difference         Proportion   Cumulative
-------------+------------------------------------------------------------
Comp1       3.02027      2.36822             0.7551       0.7551
Comp2       .652053       .37494             0.1630       0.9181
Comp3       .277113      .226551             0.0693       0.9874
Comp4      .0505616            .             0.0126       1.0000
--------------------------------------------------------------------------

Principal components (eigenvectors)

--------------------------------------------------------------------
Variable     Comp1     Comp2     Comp3     Comp4  Unexplained
-------------+----------------------------------------+-------------
trunk    0.5068    0.2327   -0.8249    0.0921            0
weight    0.5221   -0.4536    0.2677    0.6708            0
length    0.5361   -0.3903    0.1370   -0.7358            0
headroom    0.4280    0.7667    0.4786   -0.0057            0
--------------------------------------------------------------------



Comp1 Comp2 ... 就是主成分了

非常感谢，祝你新年好运！

predict p1 p2 p3 ... pn, score