# Forecasting Multivariate Time Series
## 3 Experimental Results and Discussion
### Experiment 1: Univariate Series
Along with other particulars, the report included details of the missingness type of the data. The framework summarized the data as missing completely at random (MCAR), normally distributed, and stationary, having no season and trend. Figure 5a,b show the trend and season split of the time series and the distribution of the data, respectively. The summary of the performance of all models applied to the data was computed and stored on a sheet. The summary sheet showed there were about 1270 models applied. These 1270 models were generated as a result of different combinations formed by preprocessing and forecasting techniques. For instance, first, a preprocessing strategy like forward fill was applied to impute the missing data, then all other forecasting techniques were applied for univariate forecasting, including statistical and machine learning techniques, and then each model was evaluated using four different metrics of performance evaluation. The experiment involved trying multiple preprocessing techniques and models, including interpolation, forward and backward filling, moving averages, and deep learning techniques like GAN for imputation. Each model underwent evaluation on the test dataset using various metrics. The results indicated that ARIMA (p, d, q) with order (7, 1, 2) performed best when missing data were imputed using CMA. The Ljung-Box test confirmed the absence of autocorrelation in the white noise and model. This ARIMA model yielded an MAE of 0.56997 and RMSE of 0.6099. The top five models predominantly belonged to ARIMA, utilizing different imputation techniques such as spline order 3, polynomial order 5, backward filling, and padding. After ARIMA, the exponential moving average performed well with multiple imputation techniques including linear, sliner, piecewise polynomial, and derivative techniques. The first 85 models, which performed well, were all statistical models. Conversely, the CMA-based model processed by a polynomial with degree 9 performed the worst, with an MAE of 28.4643 and RMSE of 36.5159. Figure 6 illustrates the performance of the best and worst models in the experiment, with Figure 6a displaying the values predicted by the ARIMA (7,1,2) model on the dataset imputed by CMA compared with the original values of the test set, while Figure 6b showcases the performance of the CMA model. Support vector regression with a polynomial kernel model imputed using time interpolation emerged as the best in the machine learning category, yielding an MAE of 3.6430 and RMSE of 4.7306. The errors computed for the mentioned models are tabulated in Table 2.  
Figure 5: (**a**) Decomposition graph of closing price to split the trend and seasonality component. The graph shows a slight upward trend and no seasonality. (**b**) The graph shows normal data distribution of closing price data. The third part of the experiment focused on analyzing cryptocurrency data, particularly Bitcoin (BTC-USD) from the Coinbase exchange. Cryptocurrency operates independently of traditional banking systems, utilizing blockchain technology for transaction authentication and record keeping. The dataset, spanning approximately two years from Aug 2020 to Aug 2022, consisted of 747 records of Bitcoin prices in USD. To evaluate the framework's performance in handling missing data, 30% of the dataset was randomly removed. Preptimize processed the data, identifying them as stationary with no discernible seasonality or trend. Despite the dataset's numeric nature and absence of skewness, it exhibited considerable variance, indicating significant price volatility over the period. Figure 7a,b show the trend and season split of the time series and distribution of the data, respectively. Various imputation techniques, including deep learning methods, interpolation, and moving averages, were employed to address the missing values, with the dataset subsequently split into training and testing sets for forecasting.