(a) Plot the data and comment on whether trend, seasonality or other patterns can be significant. (b) Look for extreme outliers in the data. You might want to replace such extreme outliers with plausible alternatives (i.e. an average of the number of calls on the same day). Of course, in reality it is important to understand why such outliers showed up (public holidays, unique events etc.). (c) Separate the data into a training set and a test set. The recommendation is to have 70 to 80% of the data in the train set. Make sure your separation does not divide a week since there seems to be weekly seasonality. (d) In practice call center managers forecast the demand for day t by the observed demand from day t 7 (i.e.) Ft = Dt7. Use this as a benchmark and find the RMSE and the MAPE on the train data and on the test data (separately). (For this homework, we only consider one-period ahead forecasts (i.e. to forecast the number of calls for day t, we can use observed demand from all past days up to and including day t 1).)

(e) You can try improving on the practice benchmark by exponentially smoothing over the same day of the week: Ft = (Dt7) + (1 )Ft7. Implement this seasonal smoothing forecast on the train data and find the value of that minimizes the RMSE (you should at least experiment with a few values of ). (f) For the optimal value of from the above, implement the seasonal smoothing forecast on the test data. (g) Develop an ARIMA model for forecasting the number of calls on the training data. Show all steps (autocorrelation and partial autocorrelation functions, differencing etc.) to justify your model. (h) Report the RMSE and the MAPE of the model on the training data. (i) Using the model from above forecast the number of calls for the test data. Report the RMSE and the MAPE. (j) Using the model from part f., forecast the demand for the test data. Report the RMSE and MAPE. Comment on the quality of the forecast.

(k) Prepare a short report that includes your comments from the analysis and results and a summary table with the following information (model specification and RMSE and MAPE for the model on the train and tests sets. Also include your forecasting recommendations to the manager of this call center. Method Spec. RMSE (train) RMSE(test) MAPE(train) MAPE(test) Benchmark - S. Smoothing ARIMA Note that the model specification for seasonal exponential smoothing is: = ... and for an ARIMA is something like Dt = 10+0.5Dt1+t .

Date Index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 January 2004 Thursday 2 January 2004 Friday 3 January 2004 Saturday 4 January 2004 Sunday 5 January 2004 Monday 6 January 2004 Tuesday 7 January 2004 Wednesday 8 January 2004 Thursday 9 January 2004 Friday 10 January 2004 Saturday 11 January 2004 Sunday 12 January 2004 Monday 13 January 2004 Tuesday 14 January 2004 Wednesday 15 January 2004 Thursday 16 January 2004 Friday 17 January 2004 Saturday 18 January 2004 Sunday 19 January 2004 Monday 20 January 2004 Tuesday 21 January 2004 Wednesday 22 January 2004 Thursday 23 January 2004 Friday 24 January 2004 Saturday 25 January 2004 Sunday 26 January 2004 Monday 27 January 2004 Tuesday 28 January 2004 Wednesday 29 January 2004 Thursday 30 January 2004 Friday 31 January 2004 Saturday 1 February 2004 Sunday 2 February 2004 Monday 3 February 2004 Tuesday 4 February 2004 Wednesday 5 February 2004 Thursday 6 February 2004 Friday 7 February 2004 Saturday 8 February 2004 Sunday 9 February 2004 Monday 10 February 2004 Tuesday 19 Daily Total 16569 65464 31497 19224 77684 59906 55797 60059 55839 28543 17584 65796 60592 64398 67930 63068 30314 17282 72992 54926 42214 48590 39122 23878 15942 67318 60156 51558 61653 72903 33842 6710 20 21 22 23 24 25 26 27 28 29 30 31 32 33 7460 8967 34 35 36 37 38 14916 84738 73019 30603 17855 39 40 72939 61653 41 42 43 44 45 46 59497 54517 40448 23622 19244 77754 67098 62671 57651 47 48 49 50 51 59259 52 53 54 55 56 27102 15702 70700 57556 58561 55623 60561 30814 18016 57 58 11 February 2004 Wednesday 12 February 2004 Thursday 13 February 2004 Friday 14 February 2004 Saturday 15 February 2004 Sunday 16 February 2004 Monday 17 February 2004 Tuesday 18 February 2004 Wednesday 19 February 2004 Thursday 20 February 2004 Friday 21 February 2004 Saturday 22 February 2004 Sunday 23 February 2004 Monday 24 February 2004 Tuesday 25 February 2004 Wednesday 26 February 2004 Thursday 27 February 2004 Friday 28 February 2004 Saturday 29 February 2004 Sunday 1 March 2004 Monday 2 March 2004 Tuesday 3 March 2004 Wednesday 4 March 2004 Thursday 5 March 2004 Friday 6 March 2004 Saturday 7 March 2004 Sunday 8 March 2004 Monday 9 March 2004 Tuesday 10 March 2004 Wednesday 11 March 2004 Thursday 12 March 2004 Friday 13 March 2004 Saturday 14 March 2004 Sunday 15 March 2004 Monday 16 March 2004 Tuesday 17 March 2004 Wednesday 18 March 2004 Thursday 59 60 61 72875 62 64845 63 64 65 66 67 68 59088 57285 60640 29611 16739 66571 58911 61044 57316 58155 29180 22012 69 70 71 72 73 74 75 76 74703 64450 62201 56528 77 78 Date Index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 January 2004 Thursday 2 January 2004 Friday 3 January 2004 Saturday 4 January 2004 Sunday 5 January 2004 Monday 6 January 2004 Tuesday 7 January 2004 Wednesday 8 January 2004 Thursday 9 January 2004 Friday 10 January 2004 Saturday 11 January 2004 Sunday 12 January 2004 Monday 13 January 2004 Tuesday 14 January 2004 Wednesday 15 January 2004 Thursday 16 January 2004 Friday 17 January 2004 Saturday 18 January 2004 Sunday 19 January 2004 Monday 20 January 2004 Tuesday 21 January 2004 Wednesday 22 January 2004 Thursday 23 January 2004 Friday 24 January 2004 Saturday 25 January 2004 Sunday 26 January 2004 Monday 27 January 2004 Tuesday 28 January 2004 Wednesday 29 January 2004 Thursday 30 January 2004 Friday 31 January 2004 Saturday 1 February 2004 Sunday 2 February 2004 Monday 3 February 2004 Tuesday 4 February 2004 Wednesday 5 February 2004 Thursday 6 February 2004 Friday 7 February 2004 Saturday 8 February 2004 Sunday 9 February 2004 Monday 10 February 2004 Tuesday 19 Daily Total 16569 65464 31497 19224 77684 59906 55797 60059 55839 28543 17584 65796 60592 64398 67930 63068 30314 17282 72992 54926 42214 48590 39122 23878 15942 67318 60156 51558 61653 72903 33842 6710 20 21 22 23 24 25 26 27 28 29 30 31 32 33 7460 8967 34 35 36 37 38 14916 84738 73019 30603 17855 39 40 72939 61653 41 42 43 44 45 46 59497 54517 40448 23622 19244 77754 67098 62671 57651 47 48 49 50 51 59259 52 53 54 55 56 27102 15702 70700 57556 58561 55623 60561 30814 18016 57 58 11 February 2004 Wednesday 12 February 2004 Thursday 13 February 2004 Friday 14 February 2004 Saturday 15 February 2004 Sunday 16 February 2004 Monday 17 February 2004 Tuesday 18 February 2004 Wednesday 19 February 2004 Thursday 20 February 2004 Friday 21 February 2004 Saturday 22 February 2004 Sunday 23 February 2004 Monday 24 February 2004 Tuesday 25 February 2004 Wednesday 26 February 2004 Thursday 27 February 2004 Friday 28 February 2004 Saturday 29 February 2004 Sunday 1 March 2004 Monday 2 March 2004 Tuesday 3 March 2004 Wednesday 4 March 2004 Thursday 5 March 2004 Friday 6 March 2004 Saturday 7 March 2004 Sunday 8 March 2004 Monday 9 March 2004 Tuesday 10 March 2004 Wednesday 11 March 2004 Thursday 12 March 2004 Friday 13 March 2004 Saturday 14 March 2004 Sunday 15 March 2004 Monday 16 March 2004 Tuesday 17 March 2004 Wednesday 18 March 2004 Thursday 59 60 61 72875 62 64845 63 64 65 66 67 68 59088 57285 60640 29611 16739 66571 58911 61044 57316 58155 29180 22012 69 70 71 72 73 74 75 76 74703 64450 62201 56528 77 78