The optimal scale combination OSC method

The simple multiscale approach can be seen as a subset of the optimal scale combination (OSC) method. In OSC the total number of possible scale combinations is generated. In the simple multiscale approach, scales were increased in a systematic fashion 0, 0 1, 0 1 2, 0 1 2 3, ... In OSC all possible combinations of the J + 1 scales are generated and tested. The combination that gives rise to a regression or classification model with the lowest number of coefficients and the lowest prediction error will be selected. Assume i is the number of scales to be selected from a total of K scales. There 

The idea is illustrated with a simple example. Assume there are three scales 1, 2 and 3, and, f (3) = 7. The seven scale combinations are , 2, 3, 1 2, (1 3), 2 3 and 1 2 3). If the jth scale combination, c j gives rise to a multivariate model then there is a (prediction) error value rj associated with each combination. c j is a set which has members qi, q2, — q where n is the number of scales in the set (e.g. in the sixth combination above qi =2 q2 = 3). Of the doublets (c J ,rj) we are only interested in those models with a low value of rj (prediction error). However, when there are several combinations with the same (or comparable) prediction error, we should follow Occam s razor and choose the smallest one. In this case this is easy to define because it can be based on the total number of wavelet coefficients associated with a particular scale combination c. Function produces the total number of wavelet coefficients 

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