Published on March 1, 2014
Airline Industry- Sales Forecasting Time Series Analysis Presented By: Sudhanshu Ranjan
TABLE OF CONTENTS SI NO. TITLE OF THE SLIDES SLIDE NO 1 INTRODUCTION 3 2 DATA PREPARTION 4 3 MODEL PREPARATION & ESTIMATION 8 4 FORECASTING 11 5 APPENDIX 13
INTRODUCTION OBJECTIVE To predict the ticket sales of the international airline industry for the projected year 1961 Data Description: Dataset contains 10 years sales of airline industry from Jan 1949 to Dec 1960 on monthly basis. Dataset : Sashelp.AIR Variables: DATE & AIR(international airline travel (thousands)) Tools Used: SAS 9.3, MS Excel 2013 Dataset
DATA PREPARATION Volatility Check The plot of the data with time on horizontal axis and time series on vertical axis provides an indication for volatility. A fan shaped or an inverted fan shaped plot define high volatility. To reduce volatility: 1. Fan shaped plot: Use ‘log’ or ‘square root’ transformation 2. For inverted fan shaped plot: Use ‘exponential’ or ‘square’ transformation Fan shaped plot with volatility
DATA PREPARATION Volatility Check After transformation of AIR variable into logarithmic variable the volatility got reduced, hence constant variance within the observation. Code: PROC GPLOT DATA=a; PLOT log_air*date; RUN; Fan shaped plot without volatility
DATA PREPARATION Non-Stationarity Check If the data is completely random with no fixed pattern, it is called non-stationary data and cannot be used for future forecasting. This is checked by ‘Augmented Dickey-Fuller Unit Root Test’ (ADF). After 1st differencing of the logarithmic variable, which removed the nonstationarity in data.
DATA PREPARATION Seasonality Check The Auto Correlation function (ACF) gives the correlation between yt & yt-s. where ‘s’ is the period of lag. If the ACF gives high values at fixed interval, that interval can be considered as the period of seasonality. A differencing of same order will de-seasonalize the data. In our dataset, It was found that ACF gave high values at fixed intervals of 12 (Hence, S=12)
MODEL PREPARATION & ESTIMATION Creation of Development & Validation Sample Depending upon the number of future time points to be forecast, we set aside few of the most recent time points data as the validation sample. The rest of the data which is the development sample, is used to generate forecasts for different models. Development Sample: From year 1949 to 1959 Validation Sample: year 1960 MINIC (Minimum Information Criteria) option under PROC ARIMA generates the minimum BIC (Bayesian Information Criteria) Model after exploring all the possible combinations of ‘p’ (Auto Regressive) and ‘q’ (Moving Average) lags from 0 to 5 (default).
MODEL PREPARATION & ESTIMATION Selecting p & q values During analysis BIC(3,0) gives the minimum values (-6.3503). Total possible combination of p & q: 4*4=16 but (0,0) combination is for αt (Alpha t) and εt (Epsilon t), hence considerable combination,16-1=15. We consider all the models in the neighbourhood of this model and for each of them generate AIC (Akaike Information Criteria) and SBC (Schwartz Bayesian Criteria) and calculate the average of them. We select the top 6-7 models based on relatively lower value of the average and for each of them generated the forecasts value. Avg. of AIC & SBC
MODEL PREPARATION & ESTIMATION MAPE Calculation The forecasts generated from each combination selected from AIC & SBC are separately compared with the actual values of the same time point stored in the validation dataset (V) and ‘MAPE’ (Mean Absolute Percentage Error) is calculated. Then select the combination of p & q which has the minimum MAPE value and that combination is applied on entire data to generate the final forecast for the year 1961. Once the forecast value is generated through final model, we convert the forecasted value to original format by taking exponential for convenience in comparison. MAPE
FORECASTING Forecasted values Date Sales(In thousand) Jan-61 Feb-61 Mar-61 Apr-61 May-61 Jun-61 Jul-61 Aug-61 Sep-61 Oct-61 Nov-61 Dec-61 444.82 420.47 453.76 499.38 511.44 579.87 674.35 657.19 551.07 500.22 423.30 469.02
Date Forecasted value for the year 1961 has similar pattern as Actual value Sep-61 May-61 Jan-61 Sep-60 May-60 Jan-60 Sep-59 May-59 Jan-59 Sep-58 May-58 Jan-58 Sep-57 May-57 Jan-57 Sep-56 May-56 Jan-56 Sep-55 AIR May-55 Jan-55 Sep-54 May-54 Jan-54 Sep-53 May-53 Jan-53 Sep-52 May-52 Jan-52 Sep-51 May-51 Jan-51 Sep-50 May-50 Jan-50 Sep-49 May-49 Jan-49 International Airline Travel (Thousands) FORECASTING Graphical Representation Actual Vs Forecast Forcast_value 700 600 500 400 300 200 100 0
Non-Stationarity Check ADF Test Before & After Taking 1st Difference Before After Non- Stationary data got converted into stationary after taking 1st difference
Seasonality Check Constant Pattern check for seasonality On Every 12 Lag Autocorrelation function(ACF) gives high value at every twelve interval, Hence 12 Years is the period of seasonality.
Seasonality Check After Removal of Seasonality: After removing the seasonality by 12th order differencing
Selection of possible combination of P & Q P & Q value By observation of MIC table, it makes clear that the minimum value of the matrix is (-6.3503) corresponding to AR3 and MA 0. Therefore p=3 & q=0.
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