Figure 1.
Systematic error in the spectrum of the sample covariance matrix for different ratios of sample size to dimensionality.
Figure 2.
Two dimensional example of a -factor model.
The arrows show the direction of the single factor and the orthogonal complement. The covariance matrices of the factor model (dashed) and the uncorrelated noise
(dotted) are shown as ellipsoids of constant likelihood. The peanut-shaped solid line shows the directional variances (
) of the factor model along all directions
.
Figure 3.
Average ratio between Factor Analysis and true variances in the factor subspace and the orthogonal complement.
Ratios of sample size to dimensionality ,
and
.
. Average over 150 datasets.
Figure 4.
DVA algorithm.
Figure 5.
Illustration of the DVA algorithm.
Depicted are directional variances for the estimated (red/dashed) and true Factor model covariance matrix (blue/solid). The blue squares indicate true variances along the estimated factor direction and the direction of the orthogonal complement. The DVA method (green/dash-dotted) aims at stretching and compressing the estimated covariance peanut such that the variances in these directions correspond to the true ones.
Figure 6.
Comparison of the systematic error in standard Factor Analysis and DVA Factor Analysis.
Left: systematic error. Right: normalized standard deviation of the error. Simulations for different ratios of sample size to dimensionality (,
and
).
. Correction factors estimated on
generated data sets. Mean over 150 simulations.
Figure 7.
Comparison of the mean absolute relative error for standard Factor Analysis and the DVA Factor Analysis for different ratios of sample size to dimensionality (,
and
).
. Correction factors estimated on
generated data sets. Mean over 150 simulations.
Figure 8.
Comparison of the mean absolute relative error for standard Factor Analysis and the DVA Factor Analysis for different ratios of sample size to dimensionality (,
and
).
Note that the y-axis has different scaling for the factor subspace and the orthogonal complement. . Correction factors estimated on
generated data sets. Mean over 150 simulations.
Table 1.
Portfolio risk.
Figure 9.
Regularization dependency of the realized portfolio risk in the US market.
Left: mean absolute deviation. Right: variance.
Figure 10.
Regularization dependency of the realized portfolio risk in the EU market.
Left: mean absolute deviation. Right: variance.
Figure 11.
Regularization dependency of the realized portfolio risk in the HK market.
Left: mean absolute deviation. Right: variance.
Table 2.
Portfolio risk under regularization.