In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with longrange-dependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.

Abstract: We present a unified statistical theory for assessing the significance of apparent signal observed in noisy difference images. The results are usable in a wide range of applications, including astrophysics, but are discussed with particular reference to images which represent changes in cerebral blood flow elicited by a specific cognitive or sensorimotor task. Our main result is an estimate of the p-value for local maxima of Gaussian, t, χ 2 and F fields over search regions of any shape or size in any number of dimensions. This unifies the p-values for large search areas in 2-D (Friston et al. 1991), large search regions in 3-D (Worsley et al. 1992), and the usual uncorrected p-value at a single pixel or voxel.

The maximum of a Gaussian random field was used by Worsley et al. (...

Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to in nite (countably or continuous) index sets. GPs have been applied in a large number of elds to a diverse range of ends, and very many deep theoretical analyses of various properties are available. This paper gives an introduction to Gaussian processes on a fairly elementary level with special emphasis on characteristics relevant in machine learning. It draws explicit connections to branches such as spline smoothing models and support vector machines in which similar ideas have been investigated.

this paper are concerned with approximate evaluation of the significance level of the test defined by (1.5), i.e., the probability when = 0 that X max exceeds a constant threshold, say b. First order approximations for this can easily be derived from the results going back to Belyaev and Pitaberg (1972) (see Adler, 1981, Theorem 6.9.1, p. 160) who give the the following. Suppose Y (r) is a zero mean, unit variance, stationary random field defined on an interval S ae IR

Most existing inductive learning algorithms work under the assumption that their training examples are already tagged. There are domains, however, where the tagging procedure requires significant computation resources or manual labor. In such cases, it may be beneficial for the learner to be active, intelligently selecting the examples for labeling with the goal of reducing the labeling cost. In this paper we present LSS---a lookahead algorithm for selective sampling of examples for nearest neighbor classifiers. The algorithm is looking for the example with the highest utility, taking its effect on the resulting classifier into account. Computing the expected utility of an example requires estimating the probability of its possible labels. We propose to use the random field model for this estimation. The LSS algorithm was evaluated empirically on seven real and artificial data sets, and its performance was compared to other selective sampling algorithms. The experiments show that the proposed algorithm outperforms other methods in terms of average error rate and stability.

Introduction Fractal analysis of computer traffic has received considerable attention since the seminal work of Leland and al. [11] who provided experimental evidence that some traces of data traffic exhibit long range dependence (LRD). This is a typical fractal feature which is not found with the classical Poisson models. An important issue since then has been to propose "physical" models that lead to such fractal behavior. A popular model [27] is based on the superposition of simple i.i.d ON/OFF sources which ON and/or OFF periods follow a heavy tailed law (P r(X ? ) ¸ c \Gammaff ; 1 ! ff ! 2). When properly normalized, the resulting traffic is a fractional Brownian motion (fBm) of LRD exponent H = (3 \Gamma ff)=2. Several practical implications of LRD traffic have consequently been investigated, e.g. the queuing behavior [15] (see

We generalize the definition of the fractional Brownian motion of exponent H to the case where H is no longer a constant, but a function of the time index of the process. This allows us to model non stationary continuous processes, and we show that H(t) and 2 \Gamma H(t) are indeed respectively the local Holder exponent and the local box and Hausdorff dimension at point t. Finally, we propose a simulation method and an estimation procedure for H(t) for our model.

In this paper we present an approach to making inferences about generic activations in groups of subjects using fMRI. In particular we suggest that activations common to all subjects reflect aspects of functional anatomy that may be ‘‘typical’ ’ of the population from which that group was sampled. These commonalities can be identified by a conjunction analysis of the activation effects in which the contrasts, testing for an activation, are specified separately for each subject. A conjunction is the joint refutation of multiple null hypotheses, in this instance, of no activation in any subject. The motivation behind this use of conjunctions is that fixed-effect analyses are generally more ‘‘sensitive’ ’ than equivalent random-effect analyses. This is because fixed-effect analyses can harness the large degrees of freedom and small scan-to-scan variability (relative to the variability in responses from subject to subject) when assessing the significance of an estimated response. The price one pays for the apparent sensitivity of fixed-effect analyses is that the ensuing inferences pertain to, and only to, the subjects studied. However, a conjunction analysis, using a fixedeffect model, allows one to infer: (i) that every subject studied activated and (ii) that at least a certain proportion of the population would have shown this effect. The second inference depends upon a meta-analytic formulation in terms of a confidence region for this proportion. This approach retains the sensitivity of fixed-effect analyses when the inference that only a substantial proportion of the population activates is sufficient.

This paper studies the Hadwiger characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Hadwiger characteristic, like the Euler characteristic, counts the number of connected components in the excursion set minus the number of `holes'. For high thresholds the Hadwiger characteristic is a measure of the number peaks. The geometry of excursion sets has been studied by Adler (1981) who defined the IG (integral geometry) characteristic of excursion sets as a multidimensional analogue of the number of `upcrossings' of threshold by a unidimensional process. The IG characteristic equals the Euler characteristic of an excursion set provided that the set does not touch the boundary of the volume, and Adler (1981) found the expected IG charactersitic for a stationary random field inside a fixed volume. Worsley et al. (1992) used the IG characteristic as an estimator of the number of regions of activation of positron emission tomography (PET) images of blood flow in the brain, and Worsley et al. (1993) derived the exact bias of this estimator. Unfortunately the IG characteristic is only defined on intervals, it is not invariant under rotations and it only partially counts connected regions that touch the boundary. This is important since activation often occurs in the cortical regions near the boundary of the brain. In this paper we study the Hadwiger characteristic, which is defined on arbitrary sets, is invariant under rotations and does count connected regions whether they touch the boundary or not. Our main result is a simple expression for the expected Hadwiger characteristic for an isotropic stationary random field in two and three dimensions, and on a smooth surface embedded in three di...