PACS numbers: 05.10.Gg,05.40.-a,05.45.-a
The interplay of randomness **with** nonlinearity gives rise to a variety of phenomena both of practical and theoretical significance that have been the subject of many studies since the pioneering work of Stratonovich [1, 2, 3, 4] For example, electronic and Josephson junctions subject to thermal **noise** [5, 6], the Faraday instability of surface waves [7] and even human rythmic movements [8] can be modelized by nonlinear oscillators subject to various sources of **noise** (for a **general** reference on this subject, see e.g., [9]). In the regime where the dissipation of the effective nonlinear **oscillator** can be neglected, the **random** perturbation injects energy into the system and physical observables such as the oscillator’s mechanical energy, root-mean-square position and velocity, grow algebraically **with** time [10, 11, 12]. This instability saturates when dissipative effects (that scale typically as the square-root of the energy) become large enough to overcome the injection rate and the system reaches an non-equilibrium steady state. In series of recent papers [13, 14, 15], we have derived analytical results for quasi-Hamiltonian nonlinear **random** oscillators subject to external or internal **random** perturbation that is either a **Gaussian** white **noise** or an Ornstein-Uhlenbeck process. In particular, we have calculated the scaling exponents characterize the algebraic growth of the energy, the position and the velocity of the **oscillator**.

Show more
We remark that the class of **Gaussian** **random** fields satisfying Condition (C) is large. It not only includes fractional Brownian sheets of index H = (H 1 , . . . , H N ),
the operator-scaling **Gaussian** fields **with** stationary increments in Xiao (2009b) and solutions to the stochastic heat equation [in all these cases, φ(r) = r], but also the following subclass that can be constructed from **general** subordinators. For the definition of a completely monotone function and its connection to the Laplace exponent of a subordinator, see Berg and Forst (1975), Bertoin (1996) or Sato (1999).

Show more
22 Read more

Department of Electronics, Optronics and Signal 10 Avenue Edouard Belin, 31055 Toulouse France
Abstract—Adaptive detection of a Swerling I-II type target in **Gaussian** **noise** **with** unknown covariance matrix is addressed in this paper. The most celebrated approach to this problem is Kelly’s generalized likelihood ratio test (GLRT), derived under the hypothesis of deterministic target amplitudes. While this conditional model is ubiquitous, we investigate here the equivalent GLR approach for an unconditional model where the target amplitudes are treated as **Gaussian** **random** variables at the design of the detector. The GLRT is derived which is shown to be the product of Kelly’s GLRT and a corrective, data dependent, term. Numerical simulations are provided to compare the two approaches.

Show more
in recent works from various research communities. Finding their roots in geostatistics and spatial statistics **with** optimal linear prediction and Kriging [1, 2], **random** field models for prediction have become a main stream topic in machine learning (under the **Gaussian** Process Regression terminology, see, e.g., [3]), **with** a spectrum ranging from metamodeling and adaptive design approaches for time-consuming simulations in science and engineering [4, 5, 6, 7]) to theoretical Bayesian statistics in function spaces (See [8, 9, 10] and references therein). Often, a **Gaussian** **random** field model is assumed for the function of interest, and so all prior assumptions on this function are incorporated through the corresponding mean function and covariance kernel. Here we focus on **random** field models for which the covariance kernel exists, and we discuss some mathematical properties of associated realisations (or paths) depending on the kernel, both in the **Gaussian** case and in a more **general** second-order framework.

Show more
27 Read more

3) Dynamic PET Image Simulation : The generation process
that takes realistic count rates properties into consideration is detailed in [35]. To summarize, activity concentration images are first computed from the input phantom and TACs, apply- ing the decay of the positron emitter **with** respect to the provided time frames. To mimic the partial volume effect, a stationary 4mm FWHM isotropic 3D **Gaussian** point spread function (PSF) is applied, followed by a down-sampling to a 128 × 128 × 64 image matrix of 2.2 × 2.2 × 2.8 mm 3 voxels. Data is then projected **with** respect to real crystal positions of the Siemens Biograph TruePoint TrueV scanner, taking attenuation into account. A scatter distribution is computed from a radial convolution of this signal. A **random** distribution is computed from a fan-sum of the true-plus-scatter signal. Realistic scatter and **random** fractions are then used to scale all distributions and compute the prompt sinograms. Finally, Poisson **noise** is applied based on a realistic total number of counts for the complete acquisition. Data were reconstructed using the standard ordered-subset expectation maximization (OSEM) algorithm (16 subsets) including a 4mm FWHM 3D **Gaussian** PSF modeling as in the simulation. Two images, referred to as 6it and 50it, are considered for the analysis: the 6th iteration without post-smoothing, and the 50th iteration post-smoothed **with** a 4mm FWHM 3D **Gaussian** kernel [41]. A set of 64 independent samples of each phantom were generated to assess consistent statistical performance.

Show more
12 Read more

Abstract— This paper shows that the capacity achieving input distribution for a fading **Gaussian** broadcast channel is not **Gaussian** in **general**. The construction of non-**Gaussian** dis- tributions that strictly outperform **Gaussian** ones, for certain characterized fading distributions, is provided. The ability of analyzing non-**Gaussian** input distributions **with** closed form expressions is made possible in a local setting. It is shown that there exists a specific coordinate system, based on Hermite polynomials, which parametrizes **Gaussian** neighborhoods and which is particularly suitable to study the entropic operators encountered **with** **Gaussian** **noise**.

Show more
by the null space of some linear operator T , i.e. for which
T (Z) = 0 (a.s.). (1)
As we first develop in **general** second-order settings, an impressive diversity of path properties including invariances under group actions or sparse ANOVA decompositions of multivariate paths can be encapsulated in the framework of Eq. 1. Furthermore, in the particular case of **Gaussian** **random** fields, a more **general** class of path properties (notably some degeneracy properties involving differential operators) can be covered through the link between op- erators on the paths and operators on the reproducing kernel Hilbert space [22] associated **with** the **random** field, and also through an additional repre- sentation of Z in terms of **Gaussian** measures on Banach spaces.

Show more
24 Read more

Surprisingly enough, considering α t p as a **random** variable has received little attention, and the quasi totality of recent studies followed the lead of [2] and considered α t p as deter- ministic parameters. To the best of the authors knowledge, no references have addressed detection of a **Gaussian** signal in col- ored **noise** **with** unknown covariance matrix (while the case of white **noise** has been examined thoroughly). In [8], detection of an arbitrary **Gaussian** signal is addressed but this signal is not aligned on a known signature. However, a stochastic assump- tion for α t p makes sense to take into account the unpredictable fluctuation of the radar cross-section. Indeed, the widely ac- cepted Swerling I-II target model [9], [10] corresponds to as- suming that α t p are independent and drawn from a complex **Gaussian** distribution **with** zero mean and unknown variance P , i.e., α t p ∼ CN (0, P ). This is the approach we take in this paper. In fact, when T p = 1, we have a constant but **random** amplitude along the CPI which corresponds to a Swerling I target, while for T p > 1, the target amplitude varies randomly from CPI to CPI **with** no intra-pulse fluctuation, which leads to a Swerling II target. Compared to the classical “conditional” model where α t p are treated as deterministic unknowns, it may be felt that the “unconditional” model suffers some drawbacks. Firstly, a statistical assumption is made about α t p while this is not the case in the conditional model. Secondly, derivations in the conditional model involve a simple linear least-squares problem **with** respect to α t p while the derivations in the uncon- ditional model are more complicated, see below. On the other hand, a drawback of the conditional model is that the number of unknowns grows **with** T p while it is constant in the uncon- ditional model. Therefore, an unconditional model is worthy of investigation and thus we address the equivalent of (1) in a stochastic framework, i.e., we consider the problem

Show more
12 Read more

in a simple manner to the case of a bounded domain D ⊂ R d .
A **general** construction of wavelet expansions of the form (2) for **Gaussian** processes was recently proposed in [22], in the **general** framework of Dirichlet spaces. Here the needed assumptions are that the covariance operator commutes **with** the operator that defines the Dirichlet structure. The latter does not have a simple explicit form in the case of Mat´ ern covariances on a domain D, which makes this approach difficult to analyze and implement in our setting. Let us also mention the construction in [18], where an orthonormal basis for the RKHS is built by a direct Gram-Schmidt process, however, **with** generally no size and localization bounds on the resulting basis functions.

Show more
25 Read more

In the last two columns there is no further surprise. They emphasize, though, that the results differ **with** respect to the DoE.
Combined the results suggest that the (solely) ortho-additive kernel is not suitable for predictions if the data is generated by any other kernel. Of course the experiment treats only a small number of kernels but it is plausible that a kernel which does not make allowances for additive data cannot depict the complexity of some (more **general**) dataset. The same applies to the sparse additive and the full additive kernel (which cannot gather the non-additive part). But, interestingly, we can see it only in the experiment **with** the LHS design. This shows us that the DoE can have a critical impact.

Show more
59 Read more

A paradigm of a stochastic dynamical system for a quantitative study of the interplay between randomness and nonlinearity is provided by a particle trapped in a nonlinear confining potential and subject to a **random** **noise** [3, 4]. We showed in [5, 6, 7] that in the limit of a vanishingly small damping rate, such a particle exhibits anomalous diffusion **with** exponents that depend on the form of the confining potential at infinity. For a non-zero damping rate, this anomalous diffusion occurs as intermediate time asymptotics: the particle diffuses in phase space and absorbs energy from the **noise** until the dissipative time scale is reached and the physical observables become stationary. For an additive **Gaussian** white **noise**, it was shown explicitly [6] that, for times larger than the inverse damping rate, the intermediate time P.D.F. matches the canonical distribution. For colored **noise**, we observed from numerical simulations that the anomalous growth exponents are different from those calculated for white **noise**. This behavior strongly suggests that the long time asymptotics in the case of colored **noise** is not identical to the canonical Boltzmann- Gibbs distribution. The physical reason for this change of behavior is the following: the intrinsic period of a nonlinear **oscillator** is a decreasing function of its energy and when the amplitude of the oscillations grows the period eventually becomes shorter than the correlation time of the **noise**. In that case, destructive interference between the fast variable and the **noise** suppresses the energy transfer from the **noise** to the system and the diffusion slows down. In this regime, the correlation time of the **noise** ceases to be the shortest time scale in the system and the **noise** can not be treated perturbatively as white. Therefore, usual perturbative calculations based on small correlation time expansions [8, 9, 10] cannot predict the correct colored **noise** scalings (this was shown explicitely for the pendulum **with** multiplicative **noise**

Show more
20 Read more

3) Dynamic PET Image Simulation : The generation process
that takes realistic count rates properties into consideration is detailed in [35]. To summarize, activity concentration images are first computed from the input phantom and TACs, apply- ing the decay of the positron emitter **with** respect to the provided time frames. To mimic the partial volume effect, a stationary 4mm FWHM isotropic 3D **Gaussian** point spread function (PSF) is applied, followed by a down-sampling to a 128 × 128 × 64 image matrix of 2.2 × 2.2 × 2.8 mm 3 voxels. Data is then projected **with** respect to real crystal positions of the Siemens Biograph TruePoint TrueV scanner, taking attenuation into account. A scatter distribution is computed from a radial convolution of this signal. A **random** distribution is computed from a fan-sum of the true-plus-scatter signal. Realistic scatter and **random** fractions are then used to scale all distributions and compute the prompt sinograms. Finally, Poisson **noise** is applied based on a realistic total number of counts for the complete acquisition. Data were reconstructed using the standard ordered-subset expectation maximization (OSEM) algorithm (16 subsets) including a 4mm FWHM 3D **Gaussian** PSF modeling as in the simulation. Two images, referred to as 6it and 50it, are considered for the analysis: the 6th iteration without post-smoothing, and the 50th iteration post-smoothed **with** a 4mm FWHM 3D **Gaussian** kernel [41]. A set of 64 independent samples of each phantom were generated to assess consistent statistical performance.

Show more
13 Read more

2 **General** algorithm for **random** field generation
If C(h) is a valid autocovariance function , R is a symmetric positive definite Toeplitz (diagonal-constant) matrix.
To generate a realization of the **random** vector Y of normal variables **with** zero mean and autocovariance matrix R, algorithm 1 is used.

17 Read more

To remedy this impossibility of exact computation two different models have been recently proposed in (Abbassi and Pieczynski 2008, Pieczynski 2008). Based on the **general** triplet Markov chains (Pieczynski and Desbouvries 2005), they make the exact computation of optimal Kalman-like filters possible, and the exact calculation of smoothing is also possible, as shown in (Bardel et al. 2009). The **general** idea leading to these models is to consider the independence of the

The outline of this work is as follows. In section 2, we derive **general** results about the stochastic **oscillator** **with** **random** frequency: thanks to dimensional analysis, we reduce the dimension of the parameter space from four to two and show how the Lyapunov exponent can be calculated by using an effective first order Langevin equation; we also recall the exact results for white **noise**. In section 3, we rederive the rigorous functional evolution equation of P.D.F.; although this equation is purely formal and is not closed (it involves a hierarchy of correlation functions), it will be used as a systematic basis for various approximations; we also carry out an exact perturbative expansion of the Lyapunov exponent in the small **noise** limit. In section 4, we consider a mean-field type approximation known as the ‘decoupling Ansatz’ which provides a simple expression for the colored **noise** Lyapunov exponent in terms of the white **noise** Lyapunov exponent. In section 5, we consider two small correlation time approximations that both lead to an effective Markovian evolution : we show that these approximations are fairly accurate in the small **noise** regime. In section 6, we investigate the large correlation time limit by performing an adiabatic elimination : this approximation is quite suitable for the large **noise** regime. The last section is devoted to a synthesis and a discussion of our results.

Show more
18 Read more

4.1 MAP estimation **with** a non-**Gaussian** prior
The use of a quadratic penalization for the regularisation of the least-squares estimation problem (1) suits smooth objects best. However, any discontinuity that might be present in the object, such as edges between distinct regions in an image, will be significantly smoothed as estimated under such a penalization. Alternate penalty functions such as the generalized **Gaussian** Markov **random** field function (Bouman and Sauer, 1993) or the l 2 l 1 function (Char-

20 Read more

is of **general** use whatever the exact properties of the kernel w(r, s).
6. CONCLUSION
In this paper, we consider mainly the problem of restor- ing astronomical images acquired **with** CCD cameras. The nonuniform sensitivity of the detector elements (flat field) is taken into account and the various **noise** eﬀects such as the statistical Poisson e ﬀect during the image formation process and the additive **Gaussian** read-out **noise** are taken into ac- count. We first show that applying the split gradient method (SGM), maximum likelihood algorithms can be obtained in a rigorous way; the relaxed convergent form of such algo- rithm is exhibited and it has been demonstrated that the EM algorithms are nonrelaxed versions of the proposed algo- rithm. The proposed method can be systematically applied in a rigorous way (ensuring convergence and positivity of the solution) to MAP estimation for various convex penalty functions. In this paper three penalty functions have been developed: the quadratic one **with** either constant or smooth prior and the entropic **with** constant prior. Previous attempts of regularization for this model using sieves has been pro- posed by Snyder et al. [ 3 , 4 , 6 ], however the penalty method proposed outperforms “sieves” [ 19 ]. Another approach, pro- posed by Llacer and Nu˜ nez [ 7 , 8 ], uses the penalized ML ap- proach **with** an entropy penalty function but here the con- vergence and the positivity constraint are not always ensured [ 25 ], contrary to our method. The proposed algorithm has been checked on typical astrophysical images **with** antagonis- tic characteristics: diﬀuse or bright points objects. We have shown that a Laplacian operator gives satisfactory results for extended objects and an over-smoothed solution for bright points objects. The reverse seems to occur when the prior is a constant.

Show more
14 Read more

among both novices and experts.
Despite all these attractive features, linear regression is not a silver bullet. Estimated parameters are oversensitive to outliers [1] and OLS may be outperformed by the minimum absolute deviations estimator [2]. Techniques exist to account for non-**Gaussian** distributions, by transforming the data prior to regression [3], by using mixture models [4], or generalized linear models [5]. Common to all these methods, however, is that the regression is designed to assess a central tendency. In contrast, the aim of this paper is to estimate a lower bound.

the likelihood ratio, Eq. ( 2 ), provides a natural unified ranking for the candidate events identified by the search methods. It has a straightforward interpretation—events from each method are ranked according to the ratio of the method’s sensitivity to its background. After forming the joined list of candidate events, calculation of the pos- terior probability distribution or an upper limit on the rate of gravitational-wave emissions can proceed exactly as it would for a single search method. Therefore, there is never a problem consistently accounting for multiple trials in the analysis. If the combined search is interpreted as a count- ing experiment, the procedure for calculating the upper limit from multiple searches is similar to that suggested in [ 5 ]. In that case, classifiers contribute in proportion to their sensitivities. Additionally, our method accounts for infor- mation about the classifier’s background, which is impor- tant when dealing **with** experimental data containing non-**Gaussian** **noise** artifacts.

Show more
12 Read more

The restriction on the location of the zeros of E(s)/F(s) will not be met in practical cases. The penalty for attempting this operation is that the noise power d[r]

22 Read more