princomp#

Purpose#

Computes principal components of a data matrix.

Format#

{ p, v, a } = princomp(x, j)#
Parameters:

  • x (NxK data matrix) – \(N > K\), full rank.

  • j (scalar) – number of principal components to be computed (\(j <= K\)).

Returns:

  • p (NxJ matrix) – the first j principal components of x in descending order of amount of variance explained.

  • v (Jx1 vector) – fractions of variance explained.

  • a (JxK matrix) –

    factor loadings such that:

    \[x = p*a + error.\]

Remarks#

Adapted from a program written by Mico Loretan.

The algorithm is based on Theil, Henri “Principles of Econometrics.” Wiley, NY, 1971, 46-56.

Example#

// Create matrix with percent return
// of 4 stocks over 11 time periods
pcnt_return = {  0.0646   1.2326   0.0508  -0.0346,
                -0.1632   0.1806   0.1104   0.1276,
                 1.3477   1.3347   0.1424   0.0159,
                -0.4465  -0.5691  -0.1524  -0.1719,
                 1.6232   1.4690  -0.0192   0.0979,
                 0.3381   0.5307   0.0610   0.0374,
                -0.0383   0.2556   0.0370   0.0518,
                 0.4493   0.3140   0.0177   0.1001,
                 0.5896   0.0542   0.1991   0.2669,
                -0.2218   0.3772   0.1189   0.1234,
                 1.1778  -0.0464  -0.1282   0.2171 };

// Compute: all 4 principal component vectors,
//         percent variance explained
//         matrix of factor loadings
{ p, v, a } = princomp(pcnt_return, 4);

After the code above:

p =   0.2662  -0.6077   0.0965  -0.2951     v = 0.8394     a = 2.4244   2.3264   0.1321   0.2227
      0.0059  -0.1702  -0.3938  -0.1569         0.1436         0.9506  -0.9977  -0.1402   0.1566
      0.5631  -0.0350   0.0953   0.6146         0.0144         0.0317   0.0153  -0.2757  -0.3420
     -0.2170   0.0709   0.4012  -0.0219         0.0026         0.0208  -0.0188   0.1455  -0.1162
      0.6491   0.0491   0.2359  -0.2269
      0.1823  -0.1085  -0.0554   0.0444
      0.0456  -0.1485  -0.1299  -0.1762
      0.1624   0.0654  -0.1032  -0.1584
      0.1445   0.2677  -0.6520   0.2598
      0.0337  -0.3008  -0.3926  -0.2472
      0.2447   0.6267  -0.0115  -0.5214

From the results above, we can see that approximately 83.9% of the variance in the pcnt_return is included in the first principal component vector and another 14.36% is included in the second principal component.

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