Advances in multidimensional unfolding Busing, F.M.T.A.

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Advances in multidimensional unfolding

Busing, F.M.T.A.

Citation

Busing, F. M. T. A. (2010, April 21). Advances in multidimensional unfolding. Retrieved from https://hdl.handle.net/1887/15279

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glossary of solutions

This glossary contains terms directly related with degenerate solutions, which, in general, are zero stress solutions that fail to preserve the structure that is contained in the data. Other terms can be found through the subject index.

Trivial solution — A perfect but meaningless solution having zero stress.

Often, this solution can be avoided by using a proper normalization of the stress function. The most renowned trivial solutions are the two-point or object-sphere solution for unconditional unfolding using raw stress, the two-plus-two-point solution using stress-

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, and the object-point solution for row-conditional unfolding using stress-

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with an improper summation for the normalization factor.

Absolutely or completely degenerate solution — A flawless but purposeless zero stress solution without variation in the transformed preferences or the distances that cannot be avoided by using a proper normalization of the stress function. An absolutely degenerate solution resembles a trivial solution, such as the object-sphere solution, but differs in computational respect to that solution.

Partially degenerate solution — An impeccable but ineffectual row-conditional zero stress solution without variation in the rows of the transformed prefer- ences or the distances that cannot be avoided by using a proper normalization of the stress function. A partially degenerate solution resembles a trivial solution, for example an object-point solution, but deviates in view of computational aspects.

Extremely, near(ly), quasi, strongly, or approximately degenerate solution — A solution that tends to a absolutely or partially degenerated solution, but deviates from this solution due to non-convergence, local minima, or other anomalies. Such a solution is difficult to identify, because only part of the solution is absolutely or partially degenerate. Commonly, this solution shows conspicuously low variation in either the transformed preferences or the distances and has a strikingly low stress value.

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glossary of solutions

For the following solutions, subjects (rows) are represented by a small dot (one subject) or a large dot (multiple subject) and objects (columns) are represented by a plus sign (one object) or a cross (multiple objects).

Two-point solution — This zero stress solution holds only one distance value for all subject-object distances.

This equal distance solution is the result of minimizing raw stress without a normalization factor. The transformation of the preferences results in one value for all transformed preferences, i.e., γ = c, and with distances identical to this value, i.e., d = c, r-stress kγ − dk²is equal to zero.

Two-plus-two-point solution — This zero stress solution contains one distance value for all subject- object distances, except for one subject-object distance, which has a different value. This solution is the result of minimizing stress-

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with the sum-of- squares of the distances as normalization factor. Due to the one differing distance value, and correspond- ing transformed preferences, the normalization factor kdk² remains unequal to zero, while stress-

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kγ − dk²/kdk²is equal to zero.

Object-sphere solution — This zero stress solution forms an extension of the two-point solution as there only exist one value for the distances. Whether the subjects are on the edge of the circle with the object in the center or the objects are on the fringe with the subjects in the middle, is of subordinate significance, since only subject-object distances matter for the loss function.

Object-point solution — This zero stress solution con- cerns a specific row-conditional solution with an improper normalization factor. Each subject has one value for the subject-object distances, but different values may exist for different subjects. A normalization factor, even a variance-based normalization factor as used in stress-

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or s-stress-

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, that considers all distances simultaneously (i.e., improper summation), remains unequal to zero, while the raw stress part of the loss function is equal to zero.

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