# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 4 Results
### Nonlinearity Detection in Small Simulated TS  
As stated in the introduction, the first step is to generate the two bases for choosing the fusion algorithm. For this, for each basis, we generate 10 systems of each quadratic model of Equations (4a)-(4c), with randomly chosen parameters, under the stability constraint for the bilinear and QARMA models. The normal excitation variance is calculated for NLER equal to 0.25, 0.5, 0.75. For all 90 cases (30 nonlinear models and 3 NLER values), we generate one thousand trials of nonlinear TS and the corresponding linear TS (but also surrogate data as seen in [30]), over which the 12 indices have been estimated (in fact, using different hyperparameter values of some HT, \(q_{5}\) or \(q_{9}\) for instance, we have estimated 23 indices). This process has been repeated for the TS of length \(N=128\), 256, 512, 1024, 2048 and 4096 (under assessing the effect of the TS length on the HT performance), leading to more than 50 million index estimates. We have reproduced \(\tilde{R}\) for each index in Table 3 and for the fusion methods in Table 4.  
* As stated in the Introduction, robustness for a small TS length (\(N=128\), see first line of Table 3) is generally weak. Two HT give fairly good results \(q_{9}\) and \(q_{12}\). Indices \(q_{1}\), \(q_{5}\), \(q_{6}\) and \(q_{10}\) are less robust but not totally inefficient, unlike the remaining HT. As expected, the performance in the HT depends on the nonlinear system. For instance \(q_{9}\), \(q_{10}\) and \(q_{12}\) lead to fairly good decisions on QARMA models. \(q_{1}\) outperforms the others for bilinear/Volterra models. Moreover, the NLER, as previously stated, interferes in the performance. Weakly nonlinear TS led to poor results, in particular, for all TS lengths. Other details, in particular, on the optimal hyperparameters are given in [30]. * Amazingly, the MI under the two hypotheses (see Tables 4 and 5 of [30]), does not make clusters appear inside the 12 HT, all the pairs of HT having the same MI level, even for the closest ones (e.g., \(q_{5}\) and \(q_{6}\)). The only conclusions that we can draw from the MI estimates are: first, the MI increases with the TS length, the HT giving the same \begin{table}
\begin{tabular}{l l c c} \hline \hline
**Method** & **Type** & **Merged Quantity** & **Ref.** \\ \hline Neural networks on probability (NN) & Centralized & Probability & [61] \\ \hline Neural networks on log-probability (NNL) & Centralized & Entropy & [61] \\ \hline Maximum likehood (ML) & Centralized & Probability & [25] \\ \hline Optimal decision (OD) & Parallel distributed & Decision & [59] \\ \hline Decision with security offset (DSO) & Serial distributed & Probability & [25] \\ \hline Maximum of entropy (ME) & Parallel distributed & Decision & [25; 60] \\ \hline Geometric mean (GM) & Centralized & Probability & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tested methods of HT fusion. decision as they become more robust; second, the MI is slightly higher under \(\mathcal{H}_{1}\), the HT giving a similar decision for nonlinear TS unlike the \(\mathcal{H}_{0}\) case. * Obviously, for the selection of our fusion method, we mainly consider the results on the validation basis. In particular, even for NNs, we obtain the best results over the learning basis, while the DSO method gives the best results on the validation basis. The performance remains close for all TS lengths, except for the ML method that improves by passing from \(N=128\) to \(N=4096\). As seen, the input data is a key feature for neural networks, since the NN with entropy as input underperform the NN with probability as input; the reason being that the entropy is not discriminant, as seen in the previous point. As seen, the performance improvement is slight, passing from \(\bar{R}\) from 0.21 for \(q_{9}\) alone to 0.18 for the DSO and as seen in [30] for \(N=128\), the decision is performed by the most robust HT, i.e., \(q_{9}\) at the outset or at the outcome of the DSO process.