Permap: What is the Objective Function?

Most objective functions are defined to be the sum over all object pairs of the pair's weight factor times the square of the pair's badness measure Bij. This is called a weighted sum-squares form. Mathematically, the sum-squares objective function equation is:

Sum-Squares form: Objective Function = [SUM(all i, j < i) Wij Bij ^2 ] / NC


Focusing only on the form of the above equation, one can see how the individual badness terms are combined to form the quantity to be minimized. This separation between the definition of Bij and how the Bij are accumulated represents a level of rigor that is not always adhered to. Some authors implicitly assume a form for the combination of the badness terms and then use the badness name for the name of the objective function. This causes some confusion. For example, if one says that Stress is the criteria, does that specify how badness is measured or how the entire objective function is defined? Both usages can be found in the literature. Here, we takes Stress, Stress1, SStress, Multiscale, and Fractional to define the badness measure, but it is just as common to find authors that use these terms to specify the overall objective function. The sum-squares form of the objective function is the most commonly used form. This may be because it is very simple while still being an analytical function (i.e., all derivatives exist) and an even function (i.e., f(x) = f(-x)), or perhaps it is because of the analogy with common statistical estimators that are based on maximum likelihood arguments. For whatever reason, it was used by the originators of the badness terms Stress, SStress, Multiscale, and Fractional. However, you will find other objective functions that use a "root-sum-squares" form. For instance, Kruskal introduced Stress1 and Stress2 (Kruskal's Stress2 never gained wide acceptability and is not discussed further here) that use the root-sum-squares form.

Mathematically, the root-sum-squares objective function equation is:

Root-Sum-Squares form: 

Objective Function = SQRT{[SUM(all i, j < i) Wij Bij ^2 ] / NC}



Even though the root-sum-squares form was used originally to define Stress1, you will find some early papers that redefine Stress1 using the sum-squares way of combining the badness terms. Similarly, even though the sum-squares form was used by the originators of SStress, some papers use a root-sum-squares formula for it. Clearly, the situation is confusing. It would be better had the early workers in the field given more thought to maintaining consistency in their definitions, but it didn't happen. The situation is tolerable only because it makes no difference to the configuration of the map whether the sum-squares or root-sum-squares form is used. The only difference lies in the value of the objective function that is reported for a particular map. The root-sum-squares form makes Stress1 values considerably larger than Stress and all the other objective functions which use the sum-squares form. In fact, making the values larger was Kruskal's stated reason for using the square root in his definition. Permap uses the definition which conforms to the originator's definition. Thus, it is to be understood that the Stress1 badness always uses the root-sum-squares form of the objective function and that all the rest use the sum-squares form. This means that if you use the BADNESS FN shortcut button to cycle through the various badness functions the objective function value will increase significantly when Stress1 is activated whereas the other functions will have values that are of comparable magnitude. Because MDS maps found using Stress and Stress1 are exactly the same (because the badness function is exactly the same), one might want to eliminate one or the other from the BADNESS FN shortcut rotation list. This can be done using the option box at the lower left side of the Analysis Parameters / Objective Functions screen.



Normalization Constants:

 Normalization of a quantity is achieved by dividing it by a constant, NC that is chosen to achieve three purposes. First, you want the units of measurement to cancel out. That is, you would like to get the same results regardless of whether the original data were expressed in centimeters or inches. Second, you want large problems to produce the same intensive results as small problems. This means the NC must serve as a scale factor. Third, you want values to fall in the zero-one range so that the results based on them will be easily interpreted and do not require scale-related weights. This last goal is the overarching goal of normalization, but it cannot always be achieved such as when the quantity being normalized is not bounded. Fortunately, this is not a problem for MDS objective functions. Normalization of the objective function does not influence the resulting MDS map at all. It only changes the numerical value of the objective function value. Still, to facilitate comparisons across studies it is best to normalize the objective function. Therefore, Permap always displays normalized objective function values.


Permap uses the following NC factors:


Stress: NC = Sum (all i, j < i) Wij Dij^2

Stress1: NC = Sum (all i, j < i) Wij dij^2

SStress: NC = Sum (all i, j < i) Wij Dij^4

Multiscale: NC = Sum (all i, j < i) Wij

Fractional: NC = Sum (all i, j < i) Wij


By convention, if Wij is not used then they are set equal to one. In this case the sum over the lower left corner of the weights matrix reduces to N (N - 1) / 2 where N is the number of objects in the analysis. Finally, if you carefully examine the preceding formulas you notice that there is an inconsistency in the Stress1 NC definition. The normalizing constant for Stress1 is not a constant at all. So, in this case, the term "normalizing constant" should be replaced with "normalizing factor." The imprecision is tolerated to gain the pedagogical value of the taxonomy and to avoid arguments with Stress1 advocates.