# Combined methods

With nowadays computing power and storage, hotels use not one method, but several of them and pick the best one or combine them. Apparently, linear combination of the forecasts with appropriate weights can create new, superior method. One way to create
a weight was proposed by Bates and Granger, they define weight (alpha) as:

where MSE is mean squared error of model i, i=1,2 and ro is coefficient of correlation between the errors in the forecasts of two models. Ro is calculated as:

Talluri and Van Ryzin (2004) proposed another method of calculating adaptive weights, which vary over time:

where MSE is mean squared error of model i at time t. When the weight is calculated the combined forecast equation is given by:

where  is forecasted values of model 1, is forecasted values of model 2 and alpha is weight calculated by formulas showed before (Talluri and Van Ryzin, 2004)

The application of this method on our data set is available here: Combined

Explanation

First of all, I decided to use linear regression and Additive Pick-up method as both of them gave the best result

Secondly, I decided to use Bates and Granger formula of calculating weight.

As a result I achieved MAPE lower (by 0,1%) than MAPE of individual methods. Even if the result is small, this proves that combination of methods gives better predictor.

Example

Also in example with very unstable demand, combination of those techniques gave superior result. However, here I combined Seasonal Autoregression Moving Average SARMA (1,1)(1,1) with Linear Regression (3 independent variables Y(t-1), Y(t-2), Y(t-3)).

References:

1. Bates, J. , Granger, C. (1969) The combination of forecasts. OR, pp. 451-468.
2. Talluri, K. and Van Ryzin, G. (2004) The theory and practice of revenue management. Boston, Mass.: Kluwer Academic Publishers.

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