Holt-Winter

Holt-Winter methods (Additive and Multiplicative) are applicable to data series which exhibit seasonal variations (e.g. monthly or quarterly). They consists of forecast equation and three smoothing equations. First equation is for Level At, which can be described as typical or “average” value of the data, although it cannot be concerned as statistical average. The second equation calculates Trend Tt, which is predictable increase or decrease in data values over time. The third equation calculates Seasonal component St. Additionally, all three equations have their respective parameter: alpha, beta, gamma. stands for periodicity of the seasonality, e.g. quarterly forecast and seasonality is by quarter, L=4. Than forecast for time-period is as follows:

Screen Shot 2014-05-30 at 12.19.01

The difference between those methods is in the nature of the seasonal component. The additive method is preferred when the variation of seasonal component is almost constant through the series. Seasonal component is expressed in absolute terms and sum up to zero. While multiplicative is used when seasonal variations changes proportionally to the level of the series. It is expressed in relative terms and sum up roughly to L. Double exponential smoothing is simplified version of Additive model. It includes forecast equation:

Screen Shot 2014-05-30 at 12.36.13

(Talluri and Van Ryzin, 2004), (Phumchusri and Mongkolkul, 2012), (Lim et al., 2009)

 

The application of this method on our data set is available here: Holt – Winter

 


 

Explanation

As this method is pretty sophisticated I will try to explain assumptions and calculations I conducted in spreadsheet above.

  1. Firstly, columns D:H calculates Additive approach; I:M – Multiplicative
  2. Secondly; in oder to conduct this method we need two training sets, so the algoritm may “warm up”;
    a) we will forecast 2013
    b) first training set will be the last week of October to calculate Seasonal factor for further calculations as L=7
    c) second training set will be November, December 2012 to “to warm up” – All equations (At, Tt, St)
  3. parameters alpha,beta, gamma are set based on optimization made on historical bookings: 1.01.2010 – 20.10.2012. As a result:
    a) for additive; alpha=0,0701, beta=0,058, gamma=0,01
    b) for multiplicative; alpha=0,01, beta=0,184, gamma=0,07
    c) optimization is described here: Solver
  4. the whole procedure (Holt_Winter method) was undertaken twice; first time for optimization reason and the second one for forecasting. I will only explain it for forecasting and the optimization is basically the same.
  5. Applying Holt-Winter Method:
    a) as we remember, this method consist of three equations, for Level, Trend and Seasonal.
    b) Level:
    – D1036 – was initial value that was set as average of previous bookings
    – D1037 – was the first calculated formula as:  =$P$1*(C1037-F1030)+(1-$P$1)*(D1036+E1036)
    $P$1 – the alpha parameter; C1037 – Zt (number of bookings that day); F1030 – S(t-7) (Seasonal component 7 days before); D1036 – A(t-1)   (Level component a day before); E1036 – T(t-1) (Trend component a day before)
    c) Trend
    – E1036 – was initial value that was set as 1
    – E1037 – was the first calculated formula as:  
    =$R$1*(D1037-D1036)+(1-$R$1)*E1036
    – $R$1 - the beta parameter; D1037 – At (Level component this day);  D1036 – A(t-1) (Level component a day before); E1036 – T(t-1) (Trend component a day before)
    d) Seasonal
    – F1030:F1036 
    - was initial value that was set as number of bookings that day divided into average of bookings of that week
    – F1037 – was the first calculated formula as: =$T$1*(C1037-D1037)+(1-$T$1)*F1030
    – $T$1 - the gamma parameter; C1037 – Zt (number of bookings that day) D1037 – At (Level component this day)F1030 – S(t-7) (Seasonal component 7 days before)
    e) G1037 – forecast equation: 
    =D1036+E1036+F1030;
    D1036 – Level component a day before; E1036 – Trend component a day before; F1036 – component a day before;
    f) Analogically I calculated Multiplicative Holt-Winter
  6. Then I calculated MAPE of the entire forecast

 

 


 

Example

This method is extremely efficient with stable demand as it take additional information about Trend and Seasonal factor into equation. In our case, we achieved MAPE of 7,41% (Additive) and 5,8% (Multiplicative). However, as following example will show, it is basically completely inefficient when the fluctuations of demand are significant. In this example (model 47 ->  alpha = 0,001; beta=0,001 and gamma 0,8779) MAPE accounted to 49,5%

Screen Shot 2014-05-31 at 17.06.17

 

 


 

 

References

 

  1. Lim, C., Chang, C. and Mcaleer, M. (2009) Forecasting h(m)otel guest nights in New Zealand. International Journal of Hospitality Management, vol. 28, no.2, pp. 228-235
  2. Phumchusri, D., Mongkolkul, J. (2012) Hotel Room Demand via Observed Reservation Information. Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference 2012, pp. 1978-1985
  3. Talluri, K. and Van Ryzin, G. (2004) The theory and practice of revenue management. Boston, Mass.: Kluwer Academic Publishers.
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