Abstract

Evaluation schemes, e.g., least-squares fitting, are not generally applicable to any types of experiments. If the evaluation schemes were not derived from a measurement model that properly described the experiment to be evaluated, poorer precision or accuracy than attainable from the measured data could result. We outline ways in which statistical data evaluation schemes should be derived for all types of experiment, and we demonstrate them for laser-spectroscopic experiments, in which pulse-to-pulse fluctuations of the laser power cause correlated variations of laser intensity and generated signal intensity. The method of maximum likelihood is demonstrated in the derivation of an appropriate fitting scheme for this type of experiment. Statistical data evaluation contains the following steps. First, one has to provide a measurement model that considers statistical variation of all enclosed variables. Second, an evaluation scheme applicable to this particular model has to be derived or provided. Third, the scheme has to be characterized in terms of accuracy and precision. A criterion for accepting an evaluation scheme is that it have accuracy and precision as close as possible to the theoretical limit. The fitting scheme derived for experiments with pulsed lasers is compared to well-established schemes in terms of fitting power and rational functions. The precision is found to be as much as three times better than for simple least-squares fitting. Our scheme also suppresses the bias on the estimated model parameters that other methods may exhibit if they are applied in an uncritical fashion. We focus on experiments in nonlinear spectroscopy, but the fitting scheme derived is applicable in many scientific disciplines.

© 2003 Optical Society of America

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References

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  1. P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).
  2. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Griffin, London, 1973), Vol. 2.
  3. J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.
  4. R. A. Fisher, “On an absolute criterion for fitting frequency curves,” Mess. Math. 41, 155–160 (1912).
  5. P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000).
    [CrossRef]
  6. S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970).
  7. H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973).
    [CrossRef]
  8. A. Celmiņš, “Least squares adjustment with finite residuals for non-linear constraints and partially correlated data,” in Nineteenth Conference of Army Mathematicians (U.S. Army Research Office, Washington, D.C., 1973), AROD-73-3-PT-2, pp. 809–858.
  9. W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
    [CrossRef]
  10. W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
    [CrossRef]
  11. MATLAB Optimization Toolbox, V. 2.2. Mathworks: http://www.mathworks.com/access/helpdesk/help/toolbox/optim/optim.shtml .
  12. D. Taupin, Probabilities, Data Reduction and Error Analysis in the Physical Sciences (Éditions de Physique, Les Ulis, France, 1988).
  13. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971), Vol. II.
  14. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, Mass., 1992).
  15. International Organization for Standardization, Guide to Expression of Uncertainty in Measurement, 2nd ed. (ISO, Geneva, 1995).
  16. W. A. Fuller, Measurement Error Models (Wiley, New York, 1987).
    [CrossRef]
  17. R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).
  18. J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
    [CrossRef]
  19. B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Rep. (National Institute of Standards and Technology, Gaithersburg, Md., 1994); http://physics.nist.gov/pubs/guidelines/contents.html .
  20. W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1996).
    [CrossRef]
  21. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Overseas Publishers, Amsterdam, 1996).
  22. E. L. Dereniak, D. G. Growe, Optical Radiation Detectors (Wiley, New York, 1984).
  23. W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991).
    [CrossRef]
  24. J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977), pp. 39–41, 44.

2001

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

2000

P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000).
[CrossRef]

1991

W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991).
[CrossRef]

1981

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

1980

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

1973

H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973).
[CrossRef]

1912

R. A. Fisher, “On an absolute criterion for fitting frequency curves,” Mess. Math. 41, 155–160 (1912).

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, Mass., 1992).

Brandt, S.

S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970).

Britt, H. I.

H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973).
[CrossRef]

Caroll, R. J.

R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).

Celminš, A.

A. Celmiņš, “Least squares adjustment with finite residuals for non-linear constraints and partially correlated data,” in Nineteenth Conference of Army Mathematicians (U.S. Army Research Office, Washington, D.C., 1973), AROD-73-3-PT-2, pp. 809–858.

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1996).
[CrossRef]

Dereniak, E. L.

E. L. Dereniak, D. G. Growe, Optical Radiation Detectors (Wiley, New York, 1984).

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Overseas Publishers, Amsterdam, 1996).

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971), Vol. II.

Fisher, R. A.

R. A. Fisher, “On an absolute criterion for fitting frequency curves,” Mess. Math. 41, 155–160 (1912).

Fuller, W. A.

W. A. Fuller, Measurement Error Models (Wiley, New York, 1987).
[CrossRef]

Garthwaite, P. H.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).

Growe, D. G.

E. L. Dereniak, D. G. Growe, Optical Radiation Detectors (Wiley, New York, 1984).

Hanna, S. F.

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

Jefferys, W. H.

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

Jolliffe, I. T.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).

Jones, B.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).

Kaminski, C. F.

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

Keeping, E. S.

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Griffin, London, 1973), Vol. 2.

Kenney, J. F.

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

Kuyatt, C. E.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Rep. (National Institute of Standards and Technology, Gaithersburg, Md., 1994); http://physics.nist.gov/pubs/guidelines/contents.html .

Lucht, R. P.

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

Luecke, R. H.

H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973).
[CrossRef]

Macdonald, J. R.

W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991).
[CrossRef]

Mohr, P. J.

P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000).
[CrossRef]

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, Mass., 1992).

Ruppert, D.

R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).

Stefanski, L. A.

R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Griffin, London, 1973), Vol. 2.

Taupin, D.

D. Taupin, Probabilities, Data Reduction and Error Analysis in the Physical Sciences (Éditions de Physique, Les Ulis, France, 1988).

Taylor, B. N.

P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000).
[CrossRef]

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Rep. (National Institute of Standards and Technology, Gaithersburg, Md., 1994); http://physics.nist.gov/pubs/guidelines/contents.html .

Thompson, W. J.

W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991).
[CrossRef]

Tukey, J. W.

J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977), pp. 39–41, 44.

Walewski, J.

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

Am. J. Phys.

W. J. Thompson, J. R. Macdonald, “Correcting parameter bias caused by taking logs of exponential data,” Am. J. Phys. 59, 854–856 (1991).
[CrossRef]

Astron. J.

W. H. Jefferys, “On the method of least squares,” Astron. J. 85, 177–181 (1980).
[CrossRef]

W. H. Jefferys, “On the method of least squares. II,” Astron. J. 86, 149–155 (1981).
[CrossRef]

Mess. Math.

R. A. Fisher, “On an absolute criterion for fitting frequency curves,” Mess. Math. 41, 155–160 (1912).

Phys. Rev. A

J. Walewski, C. F. Kaminski, S. F. Hanna, R. P. Lucht, “Dependence of partially saturated polarization spectroscopy signals on pump intensity and collision rate,” Phys. Rev. A 64, 063816 (2001).
[CrossRef]

Rev. Mod. Phys.

P. J. Mohr, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 1998,” Rev. Mod. Phys. 72, 351–495 (2000).
[CrossRef]

Technometrics

H. I. Britt, R. H. Luecke, “The estimation of parameters in nonlinear implicit models,” Technometrics 15, 233–247 (1973).
[CrossRef]

Other

A. Celmiņš, “Least squares adjustment with finite residuals for non-linear constraints and partially correlated data,” in Nineteenth Conference of Army Mathematicians (U.S. Army Research Office, Washington, D.C., 1973), AROD-73-3-PT-2, pp. 809–858.

J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977), pp. 39–41, 44.

S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, Amsterdam, 1970).

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall, London, 1995).

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Griffin, London, 1973), Vol. 2.

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. Rep. (National Institute of Standards and Technology, Gaithersburg, Md., 1994); http://physics.nist.gov/pubs/guidelines/contents.html .

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1996).
[CrossRef]

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Overseas Publishers, Amsterdam, 1996).

E. L. Dereniak, D. G. Growe, Optical Radiation Detectors (Wiley, New York, 1984).

MATLAB Optimization Toolbox, V. 2.2. Mathworks: http://www.mathworks.com/access/helpdesk/help/toolbox/optim/optim.shtml .

D. Taupin, Probabilities, Data Reduction and Error Analysis in the Physical Sciences (Éditions de Physique, Les Ulis, France, 1988).

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971), Vol. II.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, Mass., 1992).

International Organization for Standardization, Guide to Expression of Uncertainty in Measurement, 2nd ed. (ISO, Geneva, 1995).

W. A. Fuller, Measurement Error Models (Wiley, New York, 1987).
[CrossRef]

R. J. Caroll, D. Ruppert, L. A. Stefanski, Measurement Error in Nonlinear Models, Vol. 63 of Monographs on Statistics and Applied Probability (Chapman Hall, London, 1995).

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Figures (8)

Fig. 1
Fig. 1

Block diagram of the setup for a laser-spectroscopic experiment. The symbols shown in this figure are explained in Subsection 2.B.

Fig. 2
Fig. 2

Schematics of bias from nonlinearity. (a) When no laser fluctuations occur, the variations of observed laser and signal intensities are caused by detector noise. Here both variations are symmetrical, and a point (, ) obtained from the mean intensities is expected to lie close to the curve that describes the nonlinear dependence. (b) Laser fluctuations contribute essentially to the variation. Although the laser intensities are distributed symmetrically, the signal intensities are not, because of the nonlinear dependence. Hence bias is introduced, as can be seen from point (, ) lying systematically beside (here, above) the curve.

Fig. 3
Fig. 3

Example of a synthetic data set generated with parameters for the laser intensities and the uncertainties that were obtained from a fit to measurement data by Walewski et al.18 There are 11 clusters, with m i = 100 data points each. The clusters are distinguished by different gray levels of the symbols. Thousand of such sets were produced by Monte Carlo simulation and evaluated with fitting schemes described in Subsection 2.C and Appendix B.

Fig. 4
Fig. 4

Estimates of parameter a for a power function [Eq. (26)] for the various fitting schemes (see Appendix B). The results are shown as the relative deviation from initial model parameter a 0. One thousand estimates of the parameter obtained from Monte Carlo simulations were used for each fitting scheme for calculating the distributions. On the ordinate the fitting schemes, log-log, simple LSQ, WLSQ, covariant WLSQ without bias correction (cov WLSQ) and covariant WLSQ with bias correction (bias corr WLSQ) are marked. The interpretation of this plot and the meaning of the crosses (outliers) are explained in Appendix D. Note that the simple LSQ fitting scheme shows a more than three-times-worse precision than the WLSQ scheme.

Fig. 5
Fig. 5

Estimates of parameter b for a power function [Eq. (26)] for the various fitting schemes (see Appendix B). The results are shown as relative deviations from initial model parameter b 0.

Fig. 6
Fig. 6

Estimates of parameter a for a rational function [Eq. (27)] for the various fitting schemes (see Appendix B). The results are shown as relative deviations from initial model parameter a 0.

Fig. 7
Fig. 7

Estimates of parameter l sat for a rational function [Eq. (27)] for various fitting schemes (see Appendix B). The results are shown as relative deviations from initial model parameter l sat,0.

Fig. 8
Fig. 8

Estimates of parameter l sat for a rational function [Eq. (27)] for several fitting schemes (see Appendix B). The results are shown as relative deviations from initial model parameter l sat,0. The σ2i values for this Monte Carlo simulation are ten times smaller than for Fig. 7.

Tables (1)

Tables Icon

Table 1 Fitting Schemesa

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

j: Lijli, σLi2.
Xij=Lij+Eij1,
j: Eij10, σ1i2.
Yij=fiLij+Eij2,
j: Eij20, σ2i2.
=i=1nj=1mi pijxij, yij,
pijindep.xij, yij=12πVXi1/2exp-xij-Xi22VXi×12πVYi1/2exp-yij-Yi22VYi.
pijxij, yij=12πdetσi1/2exp-12δijTσi-1δij,
δij :=xij-Xiyij-Yi.
σi:= VXiCXi, YiCXi, YiVYi.
Xi=li, Yi=fili.
σi=σiσLi, σ1i, σ2i, li, a1, a2,.
=σLi, σ1i, σ2i, li, a1, a2,.
=li, a1, a2,.
χ2 := i=1nj=1miδijTûi-1δij,
X¯i := 1mij=1mi Xij, Y¯i := 1mij=1mi Yij.
=i=1n pix¯i, y¯i.
pix¯i, y¯i12πdetσi/mi1/2exp-12δiTσimi-1δi,
δi:= x¯i-X¯iy¯i-Y¯i.
χ2=i=1nδiTûimi-1δi.
X¯i=Xi=li.
Yijfili+filiLij- li+½filiLij-li2+Eij2.
Y¯i=Yifili+½filiσLi2 fili+1/2filifili CXi, Yi.
CXi, YifiliσLi2,
i=1nCXi, Yi2mi det σi i=1nVXiVYimi det σi.
fl, a, b=alb.
fl, a, lsat=al31+l/lsat2.
χ2=i=1nj=1mixij-li2σxi2+i=1nj=1miyij-fli2σyi2.
χ2=1σy2λ2i=1nj=1mixij-li2+i=1nj=1miyij-fli2,
λ := σyσx.
χ2=i=1nj=1miyij-fxij2σyi2,
χ2=1σy2i=1nj=1miyij-fxij2

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