Summary
“Instability” of a statistical operation, Φ, means that small changes in the data, D, can lead to relatively large changes in the data description, Φ(D). A “singularity” is a data set at which Φ is infinitely unstable to changes in the data, essentially a discontinuity of Φ. Near the set, S, of its singularities Φ will be unstable. A general topological theory gives conditions under which singularity will arise and gives lower bounds on the dimension, dim S, of S. dim S is related to the probability of getting data near S. We describe this theory by showing how it manifests itself in several common forms of data analysis and state some of the mathematical results.
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Ellis, S.P. (2003). Instability in Nonlinear Estimation and Classification: Examples of a General Pattern. In: Denison, D.D., Hansen, M.H., Holmes, C.C., Mallick, B., Yu, B. (eds) Nonlinear Estimation and Classification. Lecture Notes in Statistics, vol 171. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21579-2_27
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DOI: https://doi.org/10.1007/978-0-387-21579-2_27
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