zbMATH — the first resource for mathematics

Estimating dimension from small samples. (English) Zbl 0809.62019
A new algorithm to estimate the dimension of a measure from point samples has been described. It emphasizes that dimension need not be the same at all scales, that is, there need not be a “scaling region”. When applied to sufficiently large samples of Gaussian measures, dimension estimates should be seen to converge to the measures’ true dimensions. The algorithm is verified with this type of measures and is found to be reliable and have less demanding data requirements than conventional estimators. When one has additional knowledge of the convergence rate with cutoff, then extrapolation estimates can provide good estimates from very small samples.
Reviewer: G.Olenev (Tartu)

62F10 Point estimation
37A99 Ergodic theory
Full Text: DOI
[1] Albano, A.M.; Mees, A.I.; deGuzman, G.C.; Rapp, P.E., Data requirements for reliable estimation of correlation dimensions, (), 207-220
[2] Akaike, H., A new look at the statistical identification model, IEEE trans. ann. control, 19, 716-723, (1974) · Zbl 0314.62039
[3] Arnold, B.C., Pareto distributions, (1983), Int. Coop. Publishing House Burtonsville, Maryland · Zbl 1169.62307
[4] Eckmann-P, J.; Ruelle, D., Fundamental limitations for estimating dimensions and Liapunov exponents in dynamical systems, Physica D, 56, 185-187, (1992) · Zbl 0759.58030
[5] Goldie, C.; Smith, R.L., Slow variation with remainder: theory and applications, Q.J. mech. appl. math., 38, 45-71, (1987) · Zbl 0611.26001
[6] Grassberger, P.; Procaccia, I., Characterization of strange attractors, Phys. rev. lett., 50, 346-349, (1983)
[7] Hall, P., On some simple estimates of an exponent of regular variation, J. R. stat. soc. B, 44, 37-42, (1982) · Zbl 0521.62024
[8] Hill, B.M., A simple general approach to inference about the tail of a distribution, Ann. statist., 3, 1163-1174, (1975) · Zbl 0323.62033
[9] Judd, K., An improved estimator of dimension and some comments on providing confidence intervals, Physica D, 56, 216-228, (1992) · Zbl 0761.58027
[10] Nelder, J.A.; Mead, R., A simplex method for function minimization, Comput. J., 7, 308-313, (1965) · Zbl 0229.65053
[11] Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Vetterling, W.T., Numerical recipes: the art of scientific computing, (1986), Cambridge Univ Cambridge · Zbl 0587.65003
[12] Rissanen, J., ()
[13] Smith, L.A., Intrinsic limits on dimension calculations, Phys. lett. A, 133, 283-288, (1988)
[14] Smith, R.L., Estimating dimension in noisy chaotic time series, J. R. stat. soc. B, 54, 229-351, (1992) · Zbl 0775.62246
[15] Schwarz, G., Estimating the dimension of a model, Ann. stat., 6, 461-464, (1978) · Zbl 0379.62005
[16] Mosteller, F.; Tukey, J.W., Data analysis and regression, (1977), Addison-Wesley Reading, MA
[17] Takens, F., On the numerical determination of the dimension of an attractor, (), 99-106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.