Before using the formulas of the factorial experiment to analyze the decisions and the results it is imperative to set the results that will be considered in comparison. The experimental procedure employed uses several different representations of the same data with several different methods. In order to use the factorial design, in the most clear and simple version, it is necessary to narrow the results to one by decision. In this section the results will be commented and a model will be selected for the factorial experiment comparison. The results obtained can be found in AppendixB, which are plenty and cumbersome to interpret. The parameter ranges defined in the regression task section remain unchanged in these results. In the lines of the decisions presented in the experimental definition, the following a priori conclusions can be obtained:
• The best result of the MKL method is surpassed by the best results of the RBF and MDARC0 kernels by a very slim margin. (0.02505 vs 0.02448 and 0.02497)
• All kernel methods except MKL perform better with the inclusion of exogenous variables, but with different degree of improvement.
SPX DP EP DY SVAR BM 0 0:4 1 0:4 0 2 0:4 0 3 0:4 0 4 0:4 0 5 0:4 0 0 0 0 6 0:4 0, 3 7 0:4 0, 3 8 0:4 0, 3 9 0:4 0, 3 10 0:4 0, 3 0, 3 0, 3 0, 3 11 0:4 0:4 12 0:4 0:4 13 0:4 0:4 14 0:4 0:4 15 0:4 0:4 0:4 0:4 0:4 16 0 0:4 17 0 0:4 18 0 0:4 19 0 0:4 20 0 0:4 0:4 0:4 0:4 21 0:4 0:4 22 0 0:4 23 0:4 0 24 0:4 0 0 0 0 0 25 0:4 0, 3 26 0:4 0, 3 0, 3 0, 3 0, 3 0, 3 27 0:4 0:4 0:4 0:4 0:4 0:4 28 0 0:4 0:4 0:4 0:4 0:4
In terms of the decision of using MKL or not the results of the RBF kernel are surprising, as it surpasses the much more complex MKL algorithm. MDARC0 also gets an improvement by using the exogenous variables and it becomes the single best performing, time-dependent kernel function. The main difference between MDARC0 and MDARC1 relates again to the over-fitting, as can be observed in the minuscule training error of MDARC1 for its best result. The ratio between the test error and the train error can be a useful tool to determine the degree of over-fitting. This ratio will be defined simply as RatioofOverfitting = TestError /TrainingError and quantifies how much the model fits to the training data. The results of this comparison can be found at figure 8. As the figure shows the RBF and MDARC0 kernels suffer less than the half of over-fitting than MKL. It is possible that this fact means that the MKL model produces worst results as it falls on this problem. The notable exception is VAR which has the lowest ratio, however it outputs noticeably worst results, perhaps a result of bad fitting.
Figure 8: Ratio of over-fitting for all kernel methods
As shown in the regression task, the inclusion of exogenous variables in the MKL model decreases the predictive capacity. However the results reported in AppendixB show that every other kernel method increases their predictive capacity by including this additional data. The most notable case, again, is the fact that RBF improves the most compared with other methods shifting its results from 0.0293 using only endogenous variables to 0.0245. Other improvements near the 15% performance increase can be appreciated in the VAR kernel. MDARC0 also improves in about 10% its test MSE. The rest of kernel methods only increase their results by 5% or less. Another sources of interesting information are the data sets with which the models obtain said increases in performance. The data sets are 1, 2, 17, 18, and 22. In the case of 1 and 2 they are selected by TGA and MDARC1, the models that tend to over-fitting and not improve much with the exogenous variables. The rest of data sets have common facts, which are the lack of lags of the SPX index (i.e. the endogenous variable) and that all of them include a single exogenous variable with four lags (Dividend Price, Dividend Yield and Book-to-market ratio). This contrasts with
Factor Levels - + 1 Kernel Method(KM) RBF MKL 2 Variables(Var) Endogenous Exogenous 3 Validation Method(VM) OOS CV
Table 9: The different factor levels for each decision of the Factorial Experiment.
the clear tendency of MKL to favor the SPX index with maximum lags. It is safe to conclude that, in the methods with increased performance by the introduction of exogenous variables, the endogenous variable has a reduced impact contrary to the methods that saw little or no impact in the use of exogenous variables. Looking at the results it is possible that the models like MKL and MDARC1 over-fit towards the SPX index as it is the most similar to the output variable, but the models like RBF and MDARC0 unravel correlations between the output and the exogenous variables that the other methods apparently do not do.
The decision between using or not the Cross-Validation technique is also looked upon. As introduced in [3] Cross-Validation is not demonstrated to work with non-independent signals, non-stationary distributions of data and/or non-auto-regressive models. The results of this experimental approach indicate positive results using this technique. All of the kernel methods get increases in performance when this technique is employed, and most of them also had their best results using it. The only exception is RBF which, in general, improves with CV however the best result is found using OOS by a slim margin. Although the improvement of this technique is empirically true, the results are not conclusive on the effects that this technique can have in similar data sets or methodologies. The effects are slim however, less than a 10% improvement in all the cases.