Total order ranking method

Applying a total order ranking method, like desirability or utility functions, to the experimental attributes yi,. .., y . an experimental ranking, rexp, is calculated. According to the experimental ranking, a specific experimental rank is associated to each i-th element ... 

In the next step, the total order ranking method is applied to the model attributes xh. .., xp, defining a model ranking, rmod and according to that, a model rank is associated to each i-th element rrd =f(xu,xi2,...,xip) rank ... 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

DART (Decision Analysis by Ranking Techniques) is a free available sofiware implementing both partial-order ranking and several total ranking methods [DART- Milano Chemometrics, 2007]. 

In the first phase, elements are ranked according to the experimental attributes describing them. Thus, a partial or total ranking method is selected and applied to the experimental attributes providing a diagram of partially ordered elements or a totally ordered element sequence, respectively. In the second phase the same ranking method previously applied to the experimental attributes, is now applied to a selected subset of model attributes, and the elements are ranked according to the selected model attributes. 

The following simple example illustrates how ONR and SNR is calculated a scenario ranked nr 577 (concerning outage compensation or Styrel) among a total number of 1000 rank levels will be given the ONR/SNR (1000-577)71000 = 0.423. The ONR and SNR will always vary within the interval [0,1]. For the subset of scenarios without affected top priority customers (priority 0 or 1) the correlation between the two ranking methods are analysed. In order to do this we use Spearmans rank correlation coefficient. 

Given an array X of size I x J x K, two slices in, say, the third mode are needed in order to be able to use the generalized rank annihilation method. These may be formed as weighted averages of all the slices. A sensible way to define two such samples is to determine two slices that preserve the total variation in X maximally in a least squares sense. Additionally these two slices must be within an I -dimensional subspace (R is the number of components in the PARAFAC model) in the first and second mode in order to maximize directly the appropriateness of the span of the data matrices. Thus, two slices Gi and G2 of size R x R are sought such that these are representative of the variation in X. This may be accomplished in a least squares sense by fitting a Tucker3 model with dimensionality R x R x 2,... 

---