Download PDF
Methods Inf Med 1988; 27(04): 161-166
DOI: 10.1055/s-0038-1635540
Original Article
Schattauer GmbH

The Relationship Between the MAXABS and the RMS Error

Die Beziehung zwischen den MAXABS und dem RMS-Fehler
A. van Oosterom
*   Laboratory of Medical Physics and Biophysics, University of Nijmegen, The Netherlands
,
G. J. H. Uijen
**   Department of Cardiology, University of Nijmegen, The Netherlands
› Author Affiliations
Further Information

Publication History

Publication Date:
16 February 2018 (online)

  PDF Download  Permissions and Reprints

Summary

Results from mathematical statistics are applied to the analysis of the ratio R between the element having the highest magnitude MAXABS in a sample of size η taken from a Gaussian distribution and the standard deviation RMS of the same sample. Numerically computed . probability density functions of MAXABS are shown for several values of n. For large values of η these density functions are compared with analytical expressions for their asymptotic equivalents. The expectation and variance of MAXABS and RMS are derived and tabulated for different values of n.

These results are applied to the analysis of the residual error resulting from an orthogonal expansion of recorded body surface potentials.

Ergebnisse der mathematischen Statistik werden auf die Analyse des Verhältnisses R zwischen dem Element, das die größte Ausdehnung MAXABS in einer Stichprobe der Größe η aus einer Gauss’schen Verteilung hat, und der Standardabweichung RMS derselben Stichprobe angewandt. Numerisch berechnete Wahrscheinlichkeitsdichte-Funktionen von MAXABS werden für verschiedene Werte von η gezeigt. Für große Werte von η werden diese Dichtefunktionen mit analytischen Ausdrücken für ihre asymptotischen Äquivalente verglichen. Die Erwartung und Varianz von MAXABS und RMS werden für verschiedene Werte von η abgeleitet und aufgelistet.

Diese Ergebnisse werden auf die Analyse des Restfehlers angewandt, der sich aus einer orthogonalen Ausdehnung der registrierten Körperoberflächenpotentiale ergibt.

Key-Words:

Body Surface Potentials - Statistical Models - Error Analysis

Schlüssel-Wörter:

Körperoberflächenpotentiale - Statistische Modelle - Restfehler-Analyse
 

  • REFERENCES

  • 1 Abramowitz M, Stegun I. A. Handbook of Mathematical Functions. New York: Dover Publications; 1965
    Search in Google Scholar
  • 2 Ahmed N, Rao K. R. Orthogonal Transforms for Digital Signal Processing. Berlin: Springer Verlag; 1975
    Search in Google Scholar
  • 3 Damen A. A, Van der Kam J. The use of the singular value decomposition in electrocardiography. Med. Biol. Eng. Comput 1982; 20: 473-482.
    CrossrefPubMedSearch in Google Scholar
  • 4 Forsythe G. E, Malcolm M. A, Moler C. B. Computer Methods for Mathematical Computation. Englewood Cliffs: Prentice Hall; chapter 9 1977
    Search in Google Scholar
  • 5 Galambos J. The Asymptotic Theory of Extreme Order Statistics. New York: John Wiley & Sons; 1978
    Search in Google Scholar
  • 6 Golub G. H, Reinsch C. Singular value decomposition and least squares solution. Numer. Math 1970; 14: 403-420.
    CrossrefPubMedSearch in Google Scholar
  • 7 Gradshteyn I. S, Ryzhik I. M. Table of Integrals, Series and Products. New York: Academic Press; 1980
    Search in Google Scholar
  • 8 Kornreich F, Rautaharju P. M, Warren J. W, Horacek B. M, Dramaix M. Effective extraction of diagnostic ECG waveform information using orthonormal basis functions derived from body surface potential maps. J. Electrocardiol 1985; 18: 341-350.
    CrossrefPubMedSearch in Google Scholar
  • 9 Lux R. L, Evans A. K, Burgers M. J, Wyatt R. F, Abildskov J. A. Redun- dancy reduction for Improved Display and Analysis of Body Surface Potential Maps. Circ. Res 1981; 49: 186-203.
    CrossrefPubMedSearch in Google Scholar
  • 10 Motchenbacher C. D, Fitchen F. C. Lownoise Electronic Design. New York: Wiley & Sons; 1973: 299
    Search in Google Scholar
  • 11 Papoulis A. Probability, Random Variables and Stochastic Processes. London: McGraw-Hill; 1984
    Search in Google Scholar
  • 12 Uijen G. J. H, Heringa A, Oosterom A. van. Data reduction of body surface potentials by means of orthogonal expansions. Trans. IEEE BME-31 . 1984: 706-714.
    Search in Google Scholar
  • 13 Westlake J. R. Numerical Matrix Inversion and Solution of Linear Equations. New York: John Wiley & Sons; 1968
    Search in Google Scholar
  • 14 Wilkinson J. H. The Algebraic Eigenvalue Problem. Oxford: Clarendon Press; 1965
    Search in Google Scholar