Statistical Science

The spectral envelope and its applications

David S. Stoffer, David E. Tyler, and David A. Wendt

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The concept of the spectral envelope was recently introduced as a statistical basis for the frequency domain analysis and scaling of qualitative-valued time series. In the process of developing the spectral envelope methodology, many other interesting extensions became evident. In this article we explain the basic concept and give numerous examples of the usefulness of the technology. These examples include analyses of DNA sequences, finding optimal transformations for the analysis of real-valued time series, residual analysis, detecting common signals in many time series,and the analysis of textures.

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Statist. Sci., Volume 15, Number 3 (2000), 224-253.

First available in Project Euclid: 24 December 2001

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Spectral envelope optimal scaling Fourier analysis latent roots and vectors principal components canonical correlation signal detection optimal transformations coherency random fields categorical-valued time series EEG sleep states DNA US GNP growth rate residual analysis long range dependence matching sequences functional magnetic resonance imaging (fMRI) pain perception textures image retrival


Stoffer, David S.; Tyler, David E.; Wendt, David A. The spectral envelope and its applications. Statist. Sci. 15 (2000), no. 3, 224--253. doi:10.1214/ss/1009212816.

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  • Carstens, E. (1998). Isoflurane anesthesia blunts cerebral responses to noxious and innocuous stimuli: A functional MRI study. Life Sciences 61 PL349-PL354.
  • Bina, M. (1994). Periodicity of dinucleotides in nucleosomes derived from siraian virus 40 chromatin. J. Molecular Biology 235 198-208.
  • Bloomfield, P. (1976). Fourier Analysis of Time Series: An Introduction. Wiley, New York.
  • Breiman, L. and Friedman, J. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580-619.
  • Brillinger, D. R. (1975). Time Series: Data Analysis and Theory. (2nd ed., 1981.) Holden-Day, San Francisco.
  • Brillinger, D. R. (1980). Analysis of variance problems under time series models. Handbook of Statistics (P. R. Krishnaiah, ed.) 1 237-278, North Holland, Amsterdam.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Brodatz, P. (1966). Textures: A Photographic Album for Artists and Designers. Dover Publications, New York.
  • Bruce, A. and Gao, H-Y. (1996). Applied Wavelet Analysis with S-PLUS. Springer, New York.
  • Chatfield, C. (1989). The Analysis of Time Series: An Introduction, 4th ed. Chapman and Hall, London.
  • Cooley, J. W. and Tukey, J. W. (1965). An algorithm for the machine calculation of the complex Fourier series. Math. Comp. 19 297-301. Cornette, J. L., Cease, K. B., Margaht, H., Spouge, J. L.,
  • Berzofsky, J. A. and DeLisi, C. (1987). Hydrophobicity scales and computational techniques for detecting amphipathic structures in proteins. J. Molecular Biology 195 659- 685.
  • Ding, Z., Granger, C. W. J. and Engle, R. F. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1 83-106.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
  • Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
  • Eisenberg, D., Weiss, R. M. and Terwillger, T. C. (1984). The hydrophobic moment detects periodicity in protein hydrophobicity. Proc. Nat. Acad. Sci. U.S.A. 81 140-144.
  • Ferryanto, Sg. (1995). On estimation of the Walsh-Fourier spectral density of two dimensional strictly homogeneous random fields. J. Nonparametr. Statist. 5 391-407.
  • Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
  • Fuller, W. A. (1995). Introduction to Statistical Times Series, 2nd ed. Wiley, New York.
  • Greenacre, M. J. (1984). Theory and Applications of Correspondence Analysis. Academic Press, London.
  • Hannan, E. J. (1970). Multiple Time Series. Wiley, New York.
  • Ioshikhes, I., Bolshoy, A. and Trifonov, E. N. (1992). Preferred positions of AA and TT dinucleotides in aligned nucleosomal DNA sequences. J. Biomolecular Structure and Dynamics 9 1111-1117.
  • Li, C.-S. and Turek, J. (1996). Content-based indexing of earth observing satellite image database with fuzzy attributes. In Symposium on Electronic Imaging: Science and TechnologyStorage and Retrieval for Image and Video Databases IV 2670 438-449. IS&T/SPIE.
  • Liu, F. and Picard, R. W. (1996). Periodicity, directionality, and randomness: Wold features for image modeling and retrieval. IEEE Trans. Pattern Analysis and Machine Intelligence 18 722-733.
  • MacNeill, I. (1977). A test of whether several time series share common periodicities. Biometrika 64 495-508.
  • McDougall, A. J., Stoffer, D. S. and Tyler, D. E. (1997). Optimal transformations and the spectral envelope for realvalued time series. J. Statist. Plann. Inference 57 195-214.
  • McLachlan, A. D. and Stewart, M. (1976). The 14-fold periodicity in alpha-tropomyosin and the interaction with actin. J. Molecular Biology 103 271-298.
  • Michailidis, G. and De Leeuw, J. (1988). The Gifi system of descriptive multivariate analysis. Statist. Sci. 18 307-336. Niblack, W., Barber, R., Equitz, W., Flickner, M., Glasman,
  • E., Petkovic, D., Yanker, P. and Faloutsos, C. (1993). The QBIC project: Querying images by content using color, texture, and shape. IBM RJ 9203 (81511), February 1993.
  • Nishisato, S. (1980). Analysis of Categorical Data: Dual Scaling and its Applications. Univ. Toronto Press.
  • Ogawa, S. and Lee, T. M. (1990). Magnetic resonance imaging of blood vessels at high fields: in vivo and in vitro measurements and image simulation. Magnetic Resonance in Medicine 16 9-18.
  • Ogawa, S., Lee, T. M., Nayak, A. and Glynn, P. (1990). Oxygenation-sensitive contrast in magnetic resonance image of rodent brain at high magnetic fields. Magnetic Resonance in Medicine 14 68-78.
  • Percival, D. B. and Walden, A. T. (1993). Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge Univ. Press.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series 1 and 2. Academic Press, London.
  • Rao, K. R. and Yip, P. (1990). Discrete Cosine Transform. Academic Press, San Diego.
  • Satchwell, S. C., Drew, H. R. and Travers, A. A. (1986). Sequence periodicities in chicken nucleosome core DNA. J. Molecular Biology 191 659-675.
  • Shumway, R. H. and Stoffer, D. S. (2000). Time Series Analysis and Its Applications. Springer, New York.
  • Stoffer, D. S. (1987). Walsh-Fourier analysis of discrete-valued time series. J. Time Ser. Anal. 8 449-467.
  • Stoffer, D. S. (1999). Detecting common signals in multiple time series using the spectral envelope. J. Amer. Statist. Assoc. 94 1341-1356. Stoffer, D. S., Scher, M., Richardson, G., Day, N. and Coble,
  • P. (1988). A Walsh-Fourier analysis of the effects of moderate maternal alcohol consumption on neonatal sleep-state cycling. J. Amer. Statist. Assoc. 83 954-963.
  • Stoffer, D. S. and Tyler, D. E. (1998). Matching sequences: Cross-spectral analysis of categorical time series. Biometrika 85 201-213.
  • Stoffer, D. S., Tyler, D. E. and McDougall, A. J. (1993). Spectral analysis for categorical time series: Scaling and the spectral envelope. Biometrika 80 611-622. Stoffer, D. S., Tyler, D. E., McDougall, A. J. and Schachtel,
  • G. (1993). Spectral analysis of DNA sequences (with discussion). Bulletin of the International Statistical Institute I 345-
  • 361; Discussion: IV 63-69 (1994).
  • Tavar´e, S. and Giddings, B. W. (1989). Some statistical aspects of the primary structure of nucleotide sequences. In Mathematical Methods for DNA Sequences (M. S. Waterman, ed.) 117-131. CRC Press, Boca Raton.
  • Tiao, G. C. and Tsay, R. S. (1994). Some advances in nonlinear and adaptive modeling in time series analysis. J. Forecasting 13 109-131.
  • Tiao, G. C., Tsay, R. S. and Wang, T. (1993). Usefulness of linear transformations in multivariate time series analysis. Empirical Economics 18 567-593.
  • Viari, A., Soldano, H. and Ollivier, E. (1990). A scaleindependent signal processing method for sequence analysis. Computer Applications in the Biosciences 6 71-80.
  • Waterman, M. S. and Vingron, M. (1994). Sequence comparison significance and Poisson approximation. Statist. Sci. 9 367-381.
  • Wendt, D. A. (1999). Analysis of multidimensional categorical data. Ph.D. dissertation, Univ. Pittsburgh.