Abstract
We introduce the concept of generalization for models of functional neuroactivation, and show how it is affected by the number, N, of neuroimaging scans available. By plotting generalization as a function of N (i.e. a “learning curve”) we demonstrate that while simple, linear models may generalize better for small N's, more flexible, low-biased nonlinear models, based on artificial neural networks (ANN's), generalize better for larger N's. We demonstrate that for sets of scans of two simple motor tasks—one set acquired with [O15]water using PET, and the other using fMRI—practical N's exist for which “generalization crossover” occurs. This observation supports the application of highly flexible, ANN models to sufficiently large functional activation datasets.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
H. Akaike. Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics, 21:243–247, 1969.
C. M. Bishop. Neural Networks for Pattern Recognition. Clarendon Press, Oxford, 1995.
J. S. Bridle. Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters. Advances in Neural Information Processing Systems, 2:211–217, 1990.
R. O. Duda and P. E. Hart. Pattern Classification and Scene Analysis. John Wiley & Sons, 1973.
B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability. Chapman & Hall, 1993.
J. H. Friedman. On bias, variance, 0/1-loss, and the curse-of-dimensionality. Journal of Knowledge Discovery and Data Mining, 1996. In press.
K. J. Friston, J.-P. Poline, A. P. Holmes, C. D. Frith, and R. S. J. Frackowiak. A multivariate analysis of PET activation studies. Human Brain Mapping, 4:140–151, 1996.
B. Hassibi and D. G. Stork. Optimal brain surgeon. Advances in Neural Information Processing Systems, 5:164–174, 1992.
J. Hertz, A. Krogh, and R. G. Palmer. Introduction to the Theory of Neural Computation. Addison-Wesley, 1994.
M. Hintz-Madsen, M. W. Pederson, L. K. Hansen, and J. Larsen. Design and evaluation of neural skin classifiers. In Y. Tohkura, S. Katagiri, and E. Wilson, editors, Proceedings of 1996 IEEE Workshop on Neural Networks for Signal Processing, pages 223–230, 1996.
J. E. Jackson. A User's Guide to Principal Components. Wiley Series on Probability and Statistics, John Wiley and Sons, 1991.
B. Lautrup, L. K. Hansen, I. Law, N. Mørch, C. Svarer, and S. C. Strother. Massive weight sharing: A cure for extremely ill-posed problems. In H. J. Hermann, D. E. Wolf, and E. P. Pöppel, editors, Proceedings of Workshop on Supercomputing in Brain Research: From Tomography to Neural Networks, HLRZ, KFA Jülich, Germany, pages 137–148, 1994.
Le Cun, Y., J. S. Denker, and S. Solla. Optimal brain damage. Advances in Neural Information Processing Systems, 2:598–605, 1990.
K. V. Mardia, J. T. Kent, and J. M. Bibby. Multivariate Analysis. Academic Press, 1979.
J. R. Moeller and S. C. Strother. A regional covariance approach to the analysis of functional patterns in positron emission tomographic data. Journal of Cerebral Blood Flow and Metabolism, 11:A121–A135, 1991.
J. R. Moeller, S. C. Strother, J. J. Sidtis, and D. A. Rottenberg. Scaled subprofile model: A statistical approach to the analysis of functional patterns in positron emission tomographic data. Journal of Cerebral Blood Flow and Metabolism, 7:649–658, 1987.
J. Moody. Prediction risk and architecture selection for neural networks. In V. Cherkassky, J. H. F. H., and H. Wechsler, editors, From Statistics to Neural Networks, Theory and Pattern Recognition Applications, pages 147–165. Springer Verlag, 1992.
N. Mørch, L. K. Hansen, I. Law, S. C. Strother, C. Svarer, B. Lautrup, U. Kjems, N. Lange, and O. B. Paulson. Generalization and the bias-variance trade-off in models of functional activation. IEEE Transactions on Medical Imaging, 1996. Submitted.
N. Mørch, U. Kjems, L. K. Hansen, C. Svarer, I. Law, B. Lautrup, S. Strother, and K. Rehm. Visualization of neural networks using saliency maps. In Proceedings of 1995 IEEE International Conference on Neural Networks, volume 4, pages 2085–2090, 1995.
N. Murata, S. Yoshizawa, and S.-I. Amari. Network information criterion—determining the number of hidden units for an artificial neural network model. IEEE Transactions on Neural Networks, 5:865–872, 1994.
M. I. Posner and M. E. Raichle. Images of Mind. W. H. Freeman, 1994.
S. C. Strother, J. R. Anderson, K. A. Schaper, J. J. Sidtis, J. S. Liow, R. P. Woods, and D. A. Rottenberg. Principal component analysis and the scaled subprofile model compared to intersubject averaging and statistical parametric mapping: I. “Functional connectivity” of the human motor system studied with [15O]water PET. Journal of Cerebral Blood Flow and Metabolism, 15:738–753, 1995.
S. C. Strother, J. R. Anderson, K. A. Schaper, J. J. Sidtis, and D. A. Rottenberg. Linear models of orthogonal subspaces & networks from functional activation PET studies of the human brain. In Y. Bizais, C Barillot, and R. D. Paola, editors, Proceedings of the 14th International Conference on Information Processing in Medical Imaging, pages 299–310. Kluwer Academic Publishers, 1995.
C. Svarer, L. K. Hansen, and J. Larsen. On design and evaluation of tapped-delay neural network architectures. In H. R. Berenji et al., editors, Proceedings of 1993 IEEE International Conference on Neural Networks, pages 45–51, 1993.
J. Talairach and P. Tournoux. Co-planar stereotaxic atlas of the human brain. Thieme Medical Publishers Inc., New York, 1988.
A. W. Toga and J. C. Mazziotta. Brain Mapping. Academic Press, 1996.
R. P. Woods, S. R. Cherry, and J. C. Mazziotta. A rapid automated algorithm for accurately aligning and reslicing positron emission tomography images. Journal of Computer Assisted Tomography, 16:620–633, 1992.
R. P. Woods, J. C. Mazziotta, and S. R. Cherry. Automated image registration. In K. Uemura et al., editors, Quantification of Brain Function. Tracer Kinetics and Image Analysis in Brain PET, pages 391–400. Elsevier Science Publishers B. V., 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mørch, N. et al. (1997). Nonlinear versus linear models in functional neuroimaging: Learning curves and generalization crossover. In: Duncan, J., Gindi, G. (eds) Information Processing in Medical Imaging. IPMI 1997. Lecture Notes in Computer Science, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63046-5_20
Download citation
DOI: https://doi.org/10.1007/3-540-63046-5_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63046-3
Online ISBN: 978-3-540-69070-2
eBook Packages: Springer Book Archive