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PDF Submissiоn Only GrаdeScоpe Submissiоn Link (10-minute submission window) Cаnvаs file upload here Question 1 : Data Analysis and Decomposition 1a. Evaluate the stationarity of the time series. In your analysis, include visualizations such as time series plots and autocorrelation function (ACF) plots to examine trends, seasonality, and correlations over time. Provide a thorough explanation of your findings, clearly interpreting the plots and justifying your conclusions about whether the series is stationary. 1b. First, split the time series data into a training set and a test set by using all but the last six points for training and reserving the last six points for testing. Using the training data, fit at least two trend models covered in the course. Evaluate and interpret the model fits with plots, and perform a residual analysis to identify any patterns or anomalies. Based on your results, discuss how well these models capture the trend, and assess their suitability for forecasting the test period. While you don't need to forecast based on the two model, you will need to provide a clear, detailed explanation to support your conclusions. 1c. Using the training set of the time series, fit one seasonal model from the seasonal models discussed in the course. Evaluate the model fit using appropriate plots, and perform a residual analysis to check for patterns or anomalies. Based on your findings, discuss how well the model captures the seasonal patterns and its suitability for forecasting the test period (without necessarily forecasting the test data). Provide a clear explanation to support your conclusions. 1d. Using the training set, fit a non parametric Trend-Seasonal model. Plot the original series along with the fitted values from the model, then compute and examine the residuals and their ACF. Provide an interpretation of the residual analysis, and how this model might or not be suitabile for forecasting, then provide a recommendation on which approach is more appropriate for predicting comparing to the results from 1b and 1c. Note: It may be helpful to prepare the data here to obtain the forecast in the next section. 1e. Compare whether differencing the series yields better results in terms of stationarity, and support your analysis with relevant plots. In addition, provide a detailed and in-depth explanation of the findings. Question 2: ARIMA Modeling. 2a. Using the trend-seasonal model in section 1d, apply the iterative approach for ARMA order selection to determine the ARMA(p,q) model applied to the residuals, considering a maximum of p = 6 and q = 6. Use AICc as the criterion for model selection. 2b. Use now the training original data and iterate to find the optimal ARIMA model, with a maximum of p=6, q=6, and d=1. Evaluate the model using appropriate plots and statistical tests. We recommend setting include.mean = TRUE in the ARIMA function to account for the mean in the model fitting. 2c. Apply a SARIMA(2,0,2)(2,1,0) model with a period of 12 and with drift to the training original data. Use the same tests and plots that were applied in the previous question (2b). Afterward, provide an explanation of the differences and expected outcomes in the predictions when comparing this model to the one used in 2b. Discuss how the inclusion of seasonal components in the SARIMA model may impact the predictions. Question 3: Forecast 3a. Using the models selected in 2a, 2b, and 2c, you will now forecast the test set (the last 6 points). However, it's important to note that the model created in 2a was based on the residuals, not the actual data points. Therefore, to generate forecasts for the actual data, you will need to take additional steps, using also the model from 1c. 3b. Which model would you select for out-of-sample prediction? What makes it the best choice? Support your argument with relevant prediction performance metrics, confidence intervals, or any other appropriate methods you deem necessary to justify your decision.
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