# Bootstrap

Least-squares fitting of compartmental model
(or function) to measured data provides the model (function) parameters, and the sum-of-squares
(SS) as an estimate of the goodness of the fit, but not the errors on the estimated model
(function) parameters. With the means of classical statistics we could estimate the
*confidence intervals* (CI), if the measurement errors of the data were normally
distributed, with a known variance. Usually, this not the case, and most certainly not regarding
the PET data.
However, *Monte Carlo simulation* based *bootstrap method* allows estimation of
the errors on the estimated model (function) parameters without knowing the distribution and
variance of the errors in the data.

In the bootstrap method the model is fitted to the data as usual, but the residuals for the individual data points are stored, and randomly picked to generate synthetic datasets. The synthetic datasets are then fitted using the same method, each dataset fit providing us a set of bootstrapped parameter estimates. From these we can estimate confidence intervals for the model parameters.

Bootstrap method can be combined with AIC in model selection (Burnham & Anderson, 2002). Spectral analysis can be used in combination with bootstrapping (Turkheimer et al., 1998).

## PET image

Bootstrap approach has been validated for estimation of the statistical properties of PET and SPECT reconstructed images, including variance and noise correlation (Buvat, 2002; Dahlbom, 2002).

Patient movement during the PET scan can be detected using bootstrap method, if list-mode data is saved (Huang et al., 2011).

## See also:

## References

Burnham KP and Anderson DR. *Model Selection and Multimodel Inference: A Practical
Information-Theoretical Approach*, 2nd ed., 2002, Springer. ISBN 0-387-95364-7.

Efron B. Bootstrap methods: another look at the Jackknife. *Ann Statist.* 1979;
7(1): 1-26. doi: 10.1214/aos/1176344552.

Efron B, Tibshirani RJ: *An Introduction to the Bootstrap*. Chapman and Hall, 1993.

Hughes IG, Hase TPA: *Measurements and their Uncertainties - A Practical Guide to Modern Error
Analysis*. Oxford University Press, 2010. ISBN 978-0-19-956632-7.

Ikoma Y, Ito H, Yamaya T, Kitamura K, Takano A, Toyama H, Suhara T.
Evaluation of error on parameter estimates in the quantitative analysis of receptor studies with
positron emission tomography. *IEEE Nuc Sci Sym Con. 2005*: 2683-2685.
doi: 10.1109/NSSMIC.2005.1596889.

Ikoma Y, Shidahara M, Ito H, Seki C, Suhara T, Kanno I. Evaluation of optimal scan time by
bootstrap approach for quantitative analysis in PET receptor study.
positron emission tomography. *IEEE Nuc Sci Sym Con. 2006*.
doi: 10.1109/NSSMIC.2006.354335.

Ikoma Y, Ito H, Arakawa R, Okumura M, Seki C, Shidahara M, Takahashi H, Kimura Y, Kanno I,
Suhara T. Error analysis for PET measurement of dopamine D_{2} receptor occupancy
by antipsychotics with [^{11}C]raclopride and [^{11}C]FLB 457.
*Neuroimage* 2008; 42(4): 1285-1294.

Wasserman L. *All of Nonparametric Statistics.* Springer, 2006.
ISBN-13: 978-0387-25145-5

Tags: Fitting, Bootstrap, Variance, Noise, Spectral analysis

Updated at: 2019-01-19

Created at: 2014-10-08

Written by: Vesa Oikonen