# Multidimensional Discrimination Techniques—Theory and Application

## Abstract

The problem of discriminating between earthquakes and underground nuclear explosions is formulated as a problem in pattern recognition. As such it may be separated into two stages, feature extraction and classification. Various ways of doing feature extraction will be discussed. Among the techniques mentioned will be univariate and multivariate autoregressive representation, Karhunen-Loève expansions, geophysical parameters and spectral parameters. The ordinary multivariate Gaussian classification algorithm will be reviewed, but some more recent methods will also be mentioned and practical problems in designing a classifier will be discussed. The theory described will be illustrated on a relatively large data base of Eurasian earthquakes and explosions.

## Keywords

Feature Vector Feature Extraction Rayleigh Wave Principal Component Method Underground Nuclear Explosion## Preview

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## References

- 1.Kanal, L. “Patterns in pattern recognition: 1968–1974”, IEEE Trans, on Information Theory, IT-20, pp. 698–722, 1974.Google Scholar
- 2.Fu, K.S. “Syntactic Methods in Pattern Recognition,” Academic Press 1974.Google Scholar
- 3.Kelly, E.J. “A study of two short period discriminants”, M.I.T., Lincoln Lab. Tech. Note 1968–8–1968.Google Scholar
- 4.Weichert, D.H. “Short period sprectral discriminant for earthquake and explosion differentiation”, Z. Geophys,, 37, pp. 147–152, 1971.Google Scholar
- 5.Dahlman, O. and H. Israelson, “Monitoring Underground Nuclear Explosions”, Elsevier, Amsterdam, 1977.Google Scholar
- 6.Douglas, A. “Seismic source identification: “A review of past and present research efforts”, Proceedings of NATO advanced study institute on Identification of Seismic Sources-Earthquake and Underground Explosions, E.S. Husebye, editor pp. 1–48, 1981.Google Scholar
- 7.Dargahi-Noubary, G.R. and P.J. Laycock, “Spectral ratio discriminants and information theory,” Tech. Report No. 116, Dept. of Mathematics, Univ. of Manchester, 1979.Google Scholar
- 8.Chen, C.H. “Seismic pattern recognition”, Geoexploration, 16, pp. 133–146, 1978.CrossRefGoogle Scholar
- 9.Brolley, J.E. “Preprocessing of seismic signals for pattern recognition”, Proceeding of NATO advanced study instute on Pattern Recognition and Signal Processing, C.H. Chen, editor, pp. 367–386, Sijthoff & Noordhoff 1978.CrossRefGoogle Scholar
- 10.Christofferson, A. and E.S. Husebye, “Least squares qignal estimation techniques in analysis of seismic array recorded P-waves”, Geophys. J.R. Astron, Soc., 38, pp. 525–552, 1974.Google Scholar
- 11.Tjøstheim, D. “Improved seismic discrimination using pattern recognition”, Phys. Earth. Planet. Inter., 16, pp. 85–108, 1978.CrossRefGoogle Scholar
- 12.Sandvin, O. and D. Tjøstheim, “Multivariate autoregressive representation of seismic P-wave signals with application to short-period discrimination,” Bull. Seism. Soc. Am., 68, pp. 735–756, 1978.Google Scholar
- 13.Tjøstheim, D. and O. Sandvin, “Multivariate autoregressive feature extraction and the recognition of multichannel waveforms”, IEEE Trans, on Pattern Analysis and Machine Intelligence, PAMI-1, pp. 80–86, 1979.Google Scholar
- 14.Anderson, T.W., “Introduction to Multivariate Statistical Analysis”, Wiley, New York, 1958.Google Scholar
- 15.Rao, C.R. “Linear Statistical Inference and Its Applications”, Wiley, New York, 2nd. ed., 1973.CrossRefGoogle Scholar
- 16.Young, T.Y. and T.W. Calvert, “Classification, Estimation and Pattern Recognition”, American Elsevier, New York, 1974.Google Scholar
- 17.Gnanadesikan, R. “Methods for Statistical Data Analysis of Multivariate Observations,” Wiley, New York, 1977.Google Scholar
- 18.Fukunaga, K. and W.L.G. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering”, IEEE Trans, on Computers, C-19, pp. 311–318, 1970.Google Scholar
- Foley, D.H. “Orthonormal expansion study for waveform processing system,” Rome Air Develop. Center, AF Systems Command, Griffiss AFB, New York, Tech. Rep. RADC-TR-73–168, 1973.Google Scholar
- 20.Foley, D.H. and J.W. Sammon Jr., “An optimal set of discriminant vectors”, IEEE Trans, on Computer, C-24, pp. 281–289, 1975.Google Scholar
- 21.Atal, B.S. and S.L. Hanauer, “Speech analysis and synthesis by linear prediction of the speech wave,” J. Acoust. Soc. Amer., 50, pp. 637–655, 1971.CrossRefGoogle Scholar
- 22.Itakura, F. and S. Saito, “A statistical method for estimation of speech spectral density and formant frequencies,” Electron Commun. Japan, 53-A, pp. 36–43, 1970.Google Scholar
- 23.Rosenberg, A.E„ and M.R. Sambur, “New techniques for automatic speaker verification,” IEEE Trans. Acoust. Speech Signal Processing, ASSP-23, pp. 169–176, 1975.CrossRefGoogle Scholar
- 24.Whittle, P. “Prediction and Regulation,” Van Nostrand, Princeton, 1963.Google Scholar
- 25.Jones, R.H. “Multivariate autoregression estimation using residuals”, Applied Time Series Analysis, Proceeding from Symposium in Tulsa, Oklahoma, D.F. Findley, Editor, pp. 139–162, Academic Press, 1978.Google Scholar
- 26.Sandvin, O. and D. Tjøstheim, “A numerical comparison of two criteria for determining the order of AR processes”, Journal Matematische Operationsforschung und Statistik, Series Statistics, to appear 1980.Google Scholar
- 27.Kashyap, R. “Optimal feature selection and decision rules in classification problems with time series”, IEEE Trans. Inform. Theory, IT-24, pp. 281–288, 1978.CrossRefGoogle Scholar
- 28.Dargabi-Noubary, G.R., Laycock, P.J. and T. Subba Rao, “Non-linear stochastic models for seismic events with applications in event identification”, Geophys. J.R. Astron. Soc., 55, pp. 655–668, 1978.Google Scholar
- 29.Haskell, N.A. “Total energy and energy spectral density of elastic wave radiation from propagating faults, 2, A Statistical source model,” Bull. Seism. Soc. Amer., 56, pp. 125–140, 1966.Google Scholar
- 30.Haskell, N.A. “Analytic approximation from elastic radiation from a contained underground explosion”. J. Geophys. Res., 72, pp. 2583–2587, 1967.CrossRefGoogle Scholar
- 31.Von Seggern, D. and R. Blandford, “Source time functions and spectra of underground nuclear explosions”, Geophys. J.R. Astron. Soc. 31, pp. 83–97, 1972.Google Scholar
- 32.Aki, K. “Scaling law of seismic spectrum.” J. Geophys. Res., 72, pp. 1217–1232, 1967.CrossRefGoogle Scholar
- 33.Chen, C.H. “A review of statistical pattern recognition”, Proceedings of NATO advanced study institute on Pattern Recognition and Signal Processing, C.H. Chen, editor, pp. 117–132, 1978.Google Scholar
- 34.Cover, T. “The best two independent measurements are not the two best.”, IEEE Trans. Syst,, Man., Cybern. (Corresp.), SMC-4, pp. 116–117, 1974.Google Scholar
- 35.Mucciardi, A.N. and E.E. Gose, “A comparison of seven techniques for choosing subsets of pattern recognition properties”, IEEE Trans. Comput., C-20, pp. 1023–1031, 1971.CrossRefGoogle Scholar
- 36.Fu, K.S. “Sequential Methods in Pattern Recognition and Machine Learning, Academic Press, New York, 1968.Google Scholar
- 37.Murray, G. “A cautionary note on selection of variables in discriminant analysis”, Appl. Statist., 26, pp. 246–250, 1977.CrossRefGoogle Scholar
- 38.Elvers, E. “Seismic event identification by negative evidence”, Bull. Seism. Soc. Am., 64, pp. 1671–1683, 1974.Google Scholar
- 39.Weichert, D.H. and P.W. Basham, “Deterrence and false alarms in seismic discrimination”, Bull Seism. Soc. Am., pp. 1119–1132, 1973.Google Scholar
- 40.Azen, S.P., Breiman, L. and W.S. Meisel, “Modern approaches to Data Analysis”. Course Notes. Technology Service Corporation, Santa Monica, Calif., 1975.Google Scholar
- 41.Efron, B. “The efficiency of logistic regression compared to normal discriminant analysis”, Journal of the American Statistical Association, 70, pp. 892–898, 1975.CrossRefGoogle Scholar
- 42.Press, S.J. and S. Wilson, “Choosing between logistic regression and discriminant analysis”, Journal of the American Statistical Association, 73, pp. 699–705, 1978.CrossRefGoogle Scholar
- 43.Cover, T.M. “A hierarchy of probability density function estimates”, in Frontiers of Pattern Recognition, S. Watanabe, editor, Academic Press, New York, pp. 83–98, 1972.Google Scholar
- 44.Breiman, L., Meisel, W. and E. Purcell, “Variable kernel estimates of multivariate densities and their calibration”, Technology Service Corporation, Santa Monica, Calif., 1975.Google Scholar
- 45.Cover, T.M. and P.F. Hart, “Nearest neighbor pattern classification”, IEEE Trans. Inform, Theory, IT-13, pp. 21–27, 1967.CrossRefGoogle Scholar
- 46.Hills, M. “Allocation rules and their error rates,” J. Roy. Stat. Soc., Ser. B., Vol. 28, pp. 1–31, 1968.Google Scholar
- 47.Lachenbruch, P.A. and R.M. Mickey, “Estimation of error rates in discriminant analysis”, Technometrics, 10, pp. 1–11, 1968.CrossRefGoogle Scholar
- 48.Toussaint, G.T., “Bibliography on estimation and misclassification”, IEEE Trans. Inform. Theory, IT-20, pp. 472–479, 1974.CrossRefGoogle Scholar
- 49.Kanal, L. and Chandrasekaran, “On dimensionality and sample size in statistical pattern classification”, Pattern Recognition, 3, pp. 225–234, 1971.CrossRefGoogle Scholar
- 50.Foley, D.H. “Considerations of sample and feature size”, IEEE Trans. Inform. Theory, IT-18, pp. 618–626, 1972.CrossRefGoogle Scholar
- 51.Tjøstheim, D. and E.S. Husebye, “An improved discriminant for test ban verification using short and long period spectral parameters,” Geophys. Res. Lett., 3, pp. 499–502, 1976.CrossRefGoogle Scholar
- 52.Anderberg, M.R. “Cluster Analysis for Applications”, Academic Press, New York, 1973.Google Scholar