Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0, 1) while under the alternative, there is an unknown path along which random variables have distribution N(μ, 1), μ> 0, and distribution N(0, 1) away from it. For which values of the mean shift μ can one reliably detect and for which values is this impossible?
Consider, for example, the usual regular lattice with vertices of the form
{(i, j) : 0≤i, −i≤j≤i and j has the parity of i}
and oriented edges (i, j)→(i+1, j+s), where s=±1. We show that for paths of length m starting at the origin, the hypotheses become distinguishable (in a minimax sense) if
, while they are not if μm≪1/log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there, the asymptotic threshold is μm≈m−1/4.
We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian and for other graphs. The concept of the predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
Alternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
References
[1] Ahuja, R. K., Magnanti, T. L. and Orlin, J. B. (1993). Network Flows. Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs, NJ.
[2] Arias-Castro, E., Donoho, D. and Huo, X. (2005). Near-optimal detection of geometric objects by fast multiscale methods. IEEE Trans. Inform. Theory 51 2402–2425.
[3] Arias-Castro, E., Donoho, D. and Huo, X. (2006). Adaptive multiscale detection of filamentary structures in a background of uniform random points. Ann. Statist. 34 326–349.
[4] Bahadur, R. and Ranga Rao, R. (1960). On deviations of the sample mean. Ann. Math. Statis. 31 1015–1027.
Mathematical Reviews (MathSciNet):
MR116413
[5] Baik, J. and Rains, E. M. (2001a). The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 205–281.
[6] Baik, J. and Rains, E. M. (2001b). Symmetrized random permutations. In Random Matrix Models and Their Applications (P. Bleher and A. Its, eds.) 1–19. Cambridge Univ. Press.
[7] Benjamini, I., Pemantle, R. and Peres, Y. (1998). Unpredictable paths and percolation. Ann. Probab. 26 1198–1211.
[8] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
Mathematical Reviews (MathSciNet):
MR433619
[9] Buffet, E., Patrick, A. and Pulé, J. V. (1993). Directed polymers on trees: A martingale approach. J. Phys. A 26 1823–1834.
[10] Candes, E. J., Charlton, P. R. and Helgason, H. (2006). Detecting highly oscillatory signals by chirplet path pursuit. Technical report, California Institute of Technology.
[11] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in random environments: A review. In Stochastic Analysis on Large Scale Interacting Systems 115–142. Math. Soc. Japan, Tokyo.
[12] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51 817–840.
Mathematical Reviews (MathSciNet):
MR971033
[13] Donoho, D. L. and Huo, X. (2002). Beamlets and multiscale image analysis. In Multiscale and Multiresolution Methods. Lecture Notes in Comput. Sci. Eng. 20 149–196. Springer, Berlin.
[14] Feller, W. (1957). The numbers of zeros and of changes of sign in a symmetric random walk. Enseignement Math. (2) 3 229–235.
Mathematical Reviews (MathSciNet):
MR97864
[15] Feller, W. (1968). An Introduction to Probability Theory and Its Applications I, 3rd ed. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR228020
[16] Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.
[17] Häggström, O. and Mossel, E. (1998). Nearest-neighbor walks with low predictability profile and percolation in 2+ε dimensions. Ann. Probab. 26 1212–1231.
[18] Hoffman, C. (1998). Unpredictable nearest neighbor processes. Ann. Probab. 26 1781–1787.
[19] Ingster, Y. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-fit Testing under Gaussian Models. Springer, New York.
[20] Kulldorff, M. (1997). A spatial scan statistic. Comm. Statist. Theory Methods 26 1481–1496.
[21] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (K. B. Athreya and P. Jagers, eds.) 217–221. Springer, New York.
[22] Patil, G. P., Balbus, J., Biging, G., Jaja, J., Myers, W. L. and Taillie, C. (2004). Multiscale advanced raster map analysis system: Definition, design and development. Environ. Ecol. Stat. 11 113–138.
[23] Pemantle, R. (1995). Tree-indexed processes. Statist. Sci. 10 200–213.
[24] Zhong, Y., Jain, A. and Dubuisson-Jolly, M.-P. (2000). Object tracking using deformable templates. IEEE Trans. Pattern Anal. Mach. Intell. 2 544–549.