 

Distance for Ordinal Variables Number usually has order. When we have sequence of number 1, 2, 3, we can say that 3 is higher than 2 and 1, while 2 is higher than 1. When we discuss about nominal scale, we neglect this characteristic of number. When we have categorical data and we assign each set of category to nonarbitrary numbers in an orderly manner, we call this measurement as ordinal scale. Ordinal scale play very important role in behavioral survey because it is relatively easy to design, easy to answer by respondent. Here are examples of ordinal scale
A note should be given to distinguish ordering, rank and nominal variable. Both ordering and rank are ordinal variables, though the labels are category. Nominal variable is best represented as existence of the choice, without order . Ordinal variable emphasize the sequence, or order of the choice .
Example: We have set of fruits: {Grape, Mangoes, Banana, Apple and Orange } and here is my rank of preference and the ordering of my preference
Given the vector [Grape, Mangoes, Banana, Apple and Orange ], my rank vector is [ 5, 1, 3, 4, 2] or 51342 for short while my ordering vector is [Mangoes, Orange , Banana, Apple, Grape] or MOBAG for short.
To compute dissimilarity or distance between two rank or two ordering or two rating vectors, the most common methods are Normalized Rank Transformation
Some nice relationship between ordinal distances are given by Marden, 1995 that
If is the total number of ranks (that we rank 1 as the best and as the worst), then
Except the first methods (i.e. Normalized Rank Transformation) where we assume rank as quantitative variable, the other methods are utilized special for ordinal variable. Distance for ordinal variables is a measure of spatial disorder between two rank / ordering vectors. We shall name the two rank/ordering vectors as pattern vector and disorder vector . Patternvector has order or sequences that disordervector want to achieve. Patternvector serves as example, guide or goal that the disordervector will reach after a number of transformations or operations. Distance for ordinal variables measures the minimum number of operation steps to make disordervector into patternvector. The different between several distances of ordinal variables are based on the type of operations. Example: We asked three persons name Alex, Brian and Cherry their ranking preference over three choices of public transport mode to go to school: Bus, Train and Van. The results are tabulated as follow:
Distance of preference between Alex and Cherry is zero because they have the same ordering preference. How about distance between Alex and Brian? Arbitrarily, we can set A = [Bus, Van, Train] as patternvector and B=[Van, Bus, Train] as disordervector, or we can also set B=[Van, Bus, Train] as patternvector and A=[Bus, Van, Train] as disordervector. Either ways will give the same result because distance is symmetry: d(A,B) = d(B, A).
Preferable reference for this tutorial is Teknomo, Kardi. Similarity Measurement. http:\\people.revoledu.com\kardi\ tutorial\Similarity\




© 2006 Kardi Teknomo. All Rights Reserved. Designed by CNV Media 