# Forecasting Multivariate Time Series
## 3 Experimental Results and Discussion
### Experiment 2: Multivariate Series  
The second experiment focused on evaluating Pretimize's performance on multivariate time series data, with and without missing values. The complete data from the stock market were used in the first part, while the second part introduced missing instances using a dataset on individual household electric power consumption. In the first part, the stock market dataset captured Microsoft's (MSFT) performance based on open, high, low, and close values from August 3, 2020, to August 1, 2022, totaling 503 records. Pretimize began with data profiling, revealing high pairwise correlation among the columns, indicating dependence. However, 'open' and 'high' columns were found to be non-stationary, potentially containing trend and seasonality components, as depicted in Figure 9; the heatmap illustrates the pairwise correlation. Moving to forecasting, as no preprocessing was needed, Pretimize applied various techniques including VAR, VMA, VARMA, MLP, decision tree regressor, random forest, RNN, GRU, GRU-bidirectional, and Stacked LSTM. Transformation was applied to 'open' and 'high' columns to make them stationary before forecasting. Among the models  
Figure 8: (**a**) Comparison of actual and forecasted data generated by Bidirectional LSTM imputed by GAIN. (**b**) Comparison of actual and forecasted series generated by CMA imputed by spline of 5 th-degree polynomial. Figure 9: (**a**) Correlation heatmap showing correlation between pairs in stock market multivariate dataset. (**b**) Correlation heatmap showing correlation between pairs in power consumption multivariate dataset. mentioned in Table 4, GRU with 50 units and 100 epochs emerged as the best performer, with 3.9242 MAE and 5.1312 RMSE. Figure 10a showcases the performance of the GRU model on the multivariate test set. However, statistical models like VAR, VMA, and VARMA showed poor performance, failing to understand the basic pattern in the multivariate series. Figure 10b showcases the performance of the VAR model on the multivariate data. Furthermore, the individual columns' performance analysis indicated that the 'open' and 'high' columns, initially deemed non-stationary, posed challenges for prediction. Despite applying difference transformation, these columns exhibited predicted values diverging significantly from the original data, leading to high RMSE values. The second part of the multivariate series experiment tested Pretimize using the individual household electric power consumption dataset, aiming to understand its behavior with incomplete multivariate time series data [44]. This dataset comprises seven columns, excluding date and time, recording power consumption over four years with a one-minute sampling rate. With around 2,075,259 instances, the dataset exhibited missing values primarily in April 2007. Details are given in Table 5.  
Due to the dataset's size, only one week of data containing missing instances was processed. With 40 preprocessing strategies and 25 forecasting models, a total of 1000 models were evaluated. However, due to computational limitations, the experiment was split into parts, taking several days to generate results. After conducting previous experiments, 18 successful preprocessing techniques were combined with 25 forecasting models for evaluation.