TY - JOUR

T1 - Mixing two sets of noisy measurements changes the N-dependence of resolution to a fourth-root power law

AU - Mahani, Alireza S.

AU - Carlsson, A. E.

AU - Wessel, R.

PY - 2004/5/7

Y1 - 2004/5/7

N2 - If noise is uncorrelated during repeated measurements of the same physical variable, averaging these measurements improves the accuracy of estimating the variable. When two values of a variable are measured separately, the smallest separation of these two values that can be discriminated with a certain reliability (resolution) is inversely proportional to the square root of the number of measurements employed. However, if measurements for these two values are mixed together, they need to be clustered before being averaged. Distinguishing mixed clusters with small separations can. be thought of as a problem of deciding the number of components in a finite mixture model. Using the likelihood ratio, the second-moment estimator, and the k-means clustering methods, we will show that a similarly defined resolution for the mixed scenario is, approximately, inversely proportional to the fourth-root of the number of measurements. The observed fourth-root law is explained in terms of some more intuitive properties of the problem. We also conclude that, assuming that the fourth-root law is universal, the methods reported here are near-optimal.

AB - If noise is uncorrelated during repeated measurements of the same physical variable, averaging these measurements improves the accuracy of estimating the variable. When two values of a variable are measured separately, the smallest separation of these two values that can be discriminated with a certain reliability (resolution) is inversely proportional to the square root of the number of measurements employed. However, if measurements for these two values are mixed together, they need to be clustered before being averaged. Distinguishing mixed clusters with small separations can. be thought of as a problem of deciding the number of components in a finite mixture model. Using the likelihood ratio, the second-moment estimator, and the k-means clustering methods, we will show that a similarly defined resolution for the mixed scenario is, approximately, inversely proportional to the fourth-root of the number of measurements. The observed fourth-root law is explained in terms of some more intuitive properties of the problem. We also conclude that, assuming that the fourth-root law is universal, the methods reported here are near-optimal.

UR - http://www.scopus.com/inward/record.url?scp=2442696431&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/37/18/002

DO - 10.1088/0305-4470/37/18/002

M3 - Article

AN - SCOPUS:2442696431

SN - 0305-4470

VL - 37

SP - 4913

EP - 4937

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 18

ER -