# 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) –

$x = p*a + error.$

## 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
{ 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.