## 4 Brief Literature Survey on ANNs Applied to Time Series Modeling
### Double SOM for Long-Term Time Series Prediction  
In Section 3.5.2, we outlined how to use the basic self-organizing map (SOM) to forecast observations from univariate time series and how to combine an SOM model with a radial basis function network (RBFN) in order to improve forecast accuracy. Additionally, in Section 3.5.4, we outlined a methodology that combines SOM models with local autoregressive models, once again seeking to improve SOM's output accuracy. [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. [64] concentrate mainly on forecasting long-term but not-so-accurate time series trends rather than dealing with the more traditional problem of finding accurate short-term time series predictions. [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. [64] describe a technique based on a double application of the SOM model and sketch a proof of its stability. They work in the context of global NAR-like models, i.e., nonlinear autoregressive prediction models, without moving average terms (they mention the possibility of adding exogenous variables to their model). Their goal is to build a global model to be used in long-term predictions, with the view of obtaining future trends and their means and confidence intervals. In some forecasting problems, it is interesting to predict several values of the series in one bloc, rather than a single \(\hat{y}_{t+1}\) scalar value. In such a case, the prediction problem, from a nonlinear autoregressive approach, has the form  
\[\langle\hat{y}_{t+k},\hat{y}_{t+k-1},\ldots,\hat{y}_{t+1}\rangle=f(y_{t},y_{t-1},\ldots,y_{t-p+1}) \tag{25}\]  
Size \(p\) of the regressor vector is not necessarily equal to the forecasting horizon \(k\). However, in many cases, \(p\) will be a multiple of \(k\). A key concept, called _series of deformations_, is defined by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. [64] as  
\[d_{t}=y_{t+k}-y_{t}. \tag{26}\]  
[PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. [64] also define a regressor vector in the deformation space as  
\[D_{t}=\langle d_{t},d_{t-1},\ldots,d_{t-p+1}\rangle. \tag{27}\]  
Regressors \(Y_{t}=\langle y_{t},y_{t-1},\ldots,y_{t-p+1}\rangle\) are arranged into classes, using a one-dimensional Kohonen map; this map performs local averages, which helps to reduce over-fitting. A one-dimensional Kohonen map with \(n_{t}\) centroids (or codevectors) \(A_{i}\) is thus organized in the space of regressors; each regressor \(Y_{t}\) is associated with a centroid \(A_{i(t)}\) according to the nearest neighbor rule. A Kohonen map in the deformation space is also formed: a one-dimensional Kohonen map with \(n_{d}\) centroids \(B_{j}\) is thus organized in the space of deformations; each deformation \(D_{t}\) is associated with a centroid \(B_{j(t)}\) according to the nearest neighbor rule. After both Kohonen (SOM) maps for the regressor and deformation spaces are formed, [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. [64] proceed next to the construction of a _transition table_, whose entries are defined by  
\[T_{i,j}=P(B_{j}|A_{i}). \tag{28}\]  
\(T_{i,j}\) is the empirical probability that deformation \(D_{t}\) is associated with centroid \(B_{j}\) when the corresponding regressor \(Y_{t}\) is associated with centroid \(A_{i}\). All terms on each row \(i\) of the table sum to \(1\); row \(i\), regarded as a vector \(\mu_{i}\), represents the empirical law of deformations conditional to class \(i\). The modeling of past behavior of the time series is derived from the organization of one-dimensional SOMs in regressor space and warp space constituted by the evaluation of the transition table.