# Walking Back the Data Quantity Assumption to Improve Time Series Prediction in Deep Learning
## 3 Methodology
### Walking Back  
The proposed methodology is based on the hypothesis that a larger number of observations increases both the computational cost and the training time of the models, without necessarily guaranteeing an improvement in predictive performance. This phenomenon is because, although more data can provide more information, they can also increase the complexity of the model, which can lead to overfitting, where the model learns non-generalizable patterns from the training data.  
There is a gap in the literature regarding the optimal number of observations needed to efficiently train neural networks. This ideal amount is not universal and depends largely on the characteristics inherent to the time series, such as its seasonality, trends, periodicity, and noise level. Previous studies have shown that the optimal amount of training data is not always linearly proportional to the accuracy of the models, as performance can degrade when irrelevant or excessively noisy historical data are introduced.  
The choice to split the time series in half (50%) as a starting point is based on the search for a balance between the amount of data used for training and the amount reserved for subsequent iterations. Fifty percent represents a compromise between a sample large enough to capture general patterns and an appropriate starting point for progressive evaluation, avoiding computational overhead from the start. Various studies in incremental learning suggest that splitting the data into large initial segments can help identify the point at which the model starts to overfit historical data rather than improve performance.  
By splitting only half of the observations, the model has enough information to learn general relationships in the time series without being overly dependent on noisy or irrelevant patterns in the early data. In fact, in many cases of time series prediction, more recent data tend to have more relevance in future prediction than distant historical data, making a progressive and selective approach to data input appropriate. This is also aligned with the practice of using moving or dynamic data windows, widely used in time series forecasting, which suggests that using recent data may be more beneficial than including the entire series.  
The progressive inclusion of the additional 10% of data in each iteration allows for a flexible and adaptive approach to model evaluation, as one can monitor at each step how the model's performance changes as more historical data are introduced. This approach is derived from the methodology of cross-validation and learning curve analysis, where the idea is to find the point at which the model reaches its optimal performance without the addition of additional data generating substantial improvements. In some cases, adding more data can even degrade performance due to the phenomenon known as diminishing returns [58], which occurs when additional data brings redundant information or noise that reduces the model's ability to generalize. The sequence of action of the model is reflected in Figure 3.  
Figure 3: Walking back.  
Formally, let \(x=\{x_{1},\ldots,x_{N}\}\in\mathrm{R}^{\mathrm{N}}\) by the full time series where \(N\) is the total number of time steps. Each iteration \(k\in\{1,\;2,\;3,\;4,\;5,\;6\}\) defines a subset \(x^{(k)}\) of the full time series as:  
\[x^{(k)}=\left\{x\bigg{(}\bigg{[}\frac{N}{2}+1-(k-1)\frac{N}{10}\bigg{]}\bigg{)},\ldots,x(N)\right\} \tag{8}\]