## 4 Brief Literature Survey on ANNs Applied to Time Series Modeling
### Combining Feedforward Neural Networks and Particle Swarm Optimization for Time Series Forecasting
#### 4.1.1 Particle Swarm Optimization for Artificial Neural Networks
PSO individuals in this combined method are evaluated by a proposed fitness function, which is directly proportional to a metric measuring the degree of synchronization between time series movements in forecasts and corresponding time series movements in validation data. The proposed fitness function is also inversely proportional to the sum of several regression error metrics, among them, mean squared error (MSE) and Theil's U statistic. We now summarize, in the next few lines, the experiments conducted by [PERSON]. [PERSON], [PERSON], [PERSON], and [PERSON] [92], their observations, and their conclusions. All investigated time series were normalized to lie within the interval [0,1] and were divided into three sets: the training set with 50% of the data, the validation set with 25% of the data, and the test set with 25% of the data. Ten particles were used in the PSO algorithm, with 1000 iterations. The standard PSO optimization routine was employed to find the minimum in the parameter space; in this way, an optimal FFNN model is found for each time series. This optimal FFNN was then compared (via the same regression error metrics employed in the proposed fitness function) against a random walk model and a standalone FFNN model trained with the Levenberg-Marquardt algorithm. Benchmarking data used in the experiments consist of two natural phenomena time series (daily starshine measures and yearly sunspot measures) and four financial time series of daily frequency (_Dow Jones Industrial Average Index_ (DJIA), _National Association of Securities Dealers Automated Quotation Index_ (NASDAQ), _Petrobras Stock Values_, and _Dollar-Real Exchange Rate_). From these experiments, it was observed that the proposed model behaved better than the random walk model, the heads or tails experiment, and the standalone FFNN model for the two time series of natural phenomena that were analyzed. Nevertheless, for the four financial time series that were forecast, the proposed model displayed behavior similar to both a random walk model and a heads or tails experiment and behaved slightly better than the standalone FFNN model. It was also observed by the authors that predictions for all analyzed financial series are _dislocated_ one-step-ahead with respect to the original values, noting that this observed behavior is consistent with the work of [PERSON] and [PERSON] [115] and [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] [116], which have shown that the forecast of financial time series denotes a distinctive one-step shift concerning the original data. Finally, [PERSON]. [PERSON], [PERSON], [PERSON], and [PERSON] [92] claim that this behavior can be corrected by a phase prediction adjustment, and conclude that their proposed method is a valid option for predicting financial time series values, obtaining satisfactory forecasting results with an admissible computational cost. Now let us take a look at a similar but more refined approach. Simplified swarm optimization (SSO) [117] is a refinement of PSO, which, of course, can also be employed for adjusting ANN weights. SSO is a swarm intelligence method that also belongs to the evolutionary computation methods. SSO's updating mechanism for particle position is much simpler than that of PSO. In turn, parameter-free improved simplified swarm optimization [93], or ISSO for short, is a refinement of SSO; ISSO treats SSO's tunable meta-parameters as variables in the search space where particles move. The idea here is to reduce human intervention during the optimization process, i.e., minimize the need for manual tuning of meta-parameters. In [PERSON] [93], ISSO is employed for adjusting ANN weights. ISSO uses three different position updating mechanisms: one for updating ANN weights, a second one for updating SSO meta-parameters, and a third one for updating the whole position of a particle if its associated fitness value shows no improvement after several iterations in the process.