Project: Forecasting Sales

Step 1: Plan Your Analysis

Answer the following questions to help you plan out your analysis:

1. Does the dataset meet the criteria of a time series dataset? Make sure to explore all four key characteristics of a time series data.

The four key characteristics of a time series data set are:

• “The series is over a continuous time interval
• Sequential measurements across that interval
• There is equal spacing between every two-consecutive measurement
• Each time unit within the time interval has at most one data point”

Hence, yes, the dataset meets the above criteria of a time series dataset as follows:
the dataset holds sixty-nine records which represent the time from first entry in January
2008 till September 2013. Moreover, by transforming the ‘month’ column into ‘year’
column and ‘month’ column, the other characteristics of a time series data can be
verified.

2. Which records should be used as the holdout sample? The records 66-69 should be used as the holdout sample, which satisfied the business requirement of four-month forecast.

Step 2: Determine Trend, Seasonal, and Error components

Answer this question:

1. What are the trend, seasonality, and error of the time series? Show how you were able to determine the components using time series plots. Include the graphs. The trend, seasonality, and error (i.e. remainder) are components of the time series, they can be decomposed via the TS Plot tool as shown in the graphs below:

This graph shows the time series before using the TS Plot tool.

This graph shows the annual seasonal pattern where the annual highest sales occur
in Novembers while the lowest are in Mays, with continuous increment and decrement
over the years (e.g. November 2012 is higher than November 2011).

This graph shows the steady rise of sales over the years except between July 2009

and January 2010 where sales were low.

This graph shows the error of the time series. The occurrence of constant variance in

the middle is more often the rest of the graph.

Step 3: Build your Models

Answer these questions:

1. What are the model terms for ETS? Explain why you chose those terms. a. Describe the in-sample errors. Use at least RMSE and MASE when examining results The model terms for ETS are (M, A, M) for the following reasons: - First, the occurrence of constant variance of error is more often and smaller in the middle than the rest of the graph, the error type should be applied as multiplicatively (M), consequently. - Also, because of the linear upward trend, the trend type should be applied as additively (A). - Finally, seasonality type should be applied as multiplicatively (M) due the (slight) growth of sales.

Nevertheless, testing the ETS (M, A, M) as a damper model and undamped model
(separately) on a selected period of time for both model, the result of damped ETS
(M, A, M) is better in term of lower AIC and MASE.

The RMSE assists in quantitatively measure the closeness between the
forecasted variable and the actual data. It measures by the same unit as the
original data.

In the graph below, the MASE is less than one, which reflects that the prediction
model is good.

2. What are the model terms for ARIMA? Explain why you chose those terms. Graph the Auto-Correlation Function (ACF) and Partial Autocorrelation Function Plots (PACF) for the time series and seasonal component and use these graphs to justify choosing your model terms. a. Describe the in-sample errors. Use at least RMSE and MASE when examining results The model is built as ARIMA (p, d, q)(P, d, Q)m since the given time series has seasonality, and thereby the model terms are seven. m=12 as the seasonal period equals 12 months. Some ACF and PACF graphs are explained below:

In this graph, serial correlation appears at the 12 and 24 lags (0 reduction in-

between)

trend/pattern returns after the series was differenced to be stationary.

The final stationary time series after differencing the seasonal difference.

The ARIMA model is configured as ARIMA(0,1,1)(0,1,0)12 for the following reasons:

• Since ACF lag-1 term is negative and the a sharp is obvious, then p=0 and q=
• Since ACF lag-1 is negative, then no other SAR terms is needed (i.e. P=0).
• Since all seasonal legs present no spike, then no SMA term is needed (i.e. Q=0).
• Since both seasonal and non-seasonal differencing are used, then d=1 and D=

The model is good since MASE (of ARIMA) is less than 1.

b. Regraph ACF and PACF for both the Time Series and Seasonal Difference and
include these graphs in your answer.

Step 4 : Forecast

Answer these questions.

1. Which model did you choose? Justify your answer by showing: in-sample error measurements and forecast error measurements against the holdout sample. From step 3, the in-sample errors of ETS are shown in the graph below:

While the in-sample errors of ARIMA are:

However, the ARIMA model is chosen (over the ETS model) since it predicts values that
are closer to the original points of the forecasted period. Also, ARIMA model has lower
error measurements for both of RMSE and MASE (see the graphs above). Nevertheless,

a comparison of time series models is shown below:

2. What is the forecast for the next four periods? Graph the results using 95% and 80% confidence intervals.

The following graphs show the desired results: