the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

multiantenna glr detection of rank one signals with known power spectrum in white noise with unknown spatial correlation multiple antenna detection of a gaussian signal with spatial rank one in temporally white gaussian noise with arbitrary and unknown spatial covariance is considered this is motivated by spectrum sensing problems in the context of dynamic spectrum access in which several secondary networks coexist but do not cooperate creating a background of spatially correlated broadband interference when the temporal correlation of the signal of interest is assumed known up to a scale factor the corresponding generalized likelihood ratio test is shown to yield a scalar optimization problem closed form expressions of the test are obtained for the general signal spectrum case in the low signal to noise ratio snr regime as well as for signals with binary valued power spectrum in arbitrary snr the two resulting detectors turn out to be equivalent an asymptotic approximation to the test distribution for the low snr regime is derived closely matching empirical results from spectrum sensing simulation experiments

adaptive steganalysis of least significant bit replacement in grayscale natural images this paper deals with the detection of hidden bits in the least significant bit lsb plane of a natural image the mean level and the covariance matrix of the image considered as a quantized gaussian random matrix are unknown an adaptive statistical test is designed such that its probability distribution is always independent of the unknown image parameters while ensuring a high probability of hidden bits detection this test is based on the likelihood ratio test except that the unknown parameters are replaced by estimates based on a local linear regression model it is shown that this test maximizes the probability of detection as the image size becomes arbitrarily large and the quantization step vanishes this provides an asymptotic upper bound for the detection of hidden bits based on the lsb replacement mechanism numerical results on real natural images show the relevance of the method and the sharpness of the asymptotic expression for the probability of detection

cfar detection strategies for distributed targets under conic constraints in this paper we deal with the problem of adaptive detection of mismatched mainlobe targets and or sidelobe interfering signals that are distributed in range to this end we investigate the impact of modeling the actual useful signal as a vector belonging to a proper cone with axis the nominal steering vector as a means to improve the robustness of the decision rule in presence of mainlobe targets similarly in order to improve the rejection capabilities of the decision rule in presence of sidelobe interferers we study the effects of replacing the usual noise only hypothesis with a noise plus interferers hypothesis where interferers belong to the complement of a cone with axis the nominal steering vector at the design stage we resort to the two step glrt based design procedure to this end we assume that a set of training data is available namely data free of signal components but sharing the same gaussian distribution of the noise in the cells under test remarkably proposed detectors possess the cfar property under the noise only hypothesis the performance assessment conducted by monte carlo simulation is aimed at assessing the effectiveness of proposed solutions also in comparison to existing ones

area and hausdorff dimension of julia sets of entire functions we show the julia set of a sin z has positive area and the action of a sin z on its julia set is not ergodic the julia set of a exp z has hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero

derivation of the bias of the normalized sample covariance matrix in a heterogeneous noise with application to low rank stap filter in a previous work we have developed a low rank lr spatio temporal adaptive processing stap filter when the disturbance is modeled as the sum of a low rank spherically invariant random vector sirv clutter and a zero mean white gaussian noise this lr stap filter is built from the normalized sample covariance matrix nscm and exhibits good robustness properties to secondary data contamination by target components in this correspondence we derive the bias of the nscm with this noise model we show that the eigenvectors estimated from the nscm are unbiased the new expressions of the expectation of nscm eigenvalues are also given from these results we also show that the estimate of the clutter subspace projector based on the nscm used in our lr stap is a consistent estimate of the true one results on numerical data validates the theoretical approach

clutter subspace estimation in low rank heterogeneous noise context this paper addresses the problem of the clutter subspace projector csp estimation in the context of a disturbance composed of a low rank lr heterogeneous clutter modeled here by a spherically invariant random vector sirv plus a white gaussian noise wgn in such context the corresponding lr adaptive filters and detectors require less training vectors than classical methods to reach equivalent performance unlike classical adaptive processes which are based on an estimate of the noise covariance matrix cm the lr processes are based on a csp estimate this csp estimate is usually derived from a singular value decomposition svd of the cm estimate however no maximum likelihood estimator mle of the cm has been derived for the considered disturbance model in this paper we introduce the fixed point equation that mle of the csp satisfies for a disturbance composed of a lr sirv clutter plus a zero mean wgn a recursive algorithm is proposed to compute this solution numerical simulations validate the introduced estimator and illustrate its interest compared to the current state of art

asymptotic properties of robust complex covariance matrix estimates in many statistical signal processing applications the estimation of nuisance parameters and parameters of interest is strongly linked to the resulting performance generally these applications deal with complex data this paper focuses on covariance matrix estimation problems in non gaussian environments and particularly the m estimators in the context of elliptical distributions first this paper extends to the complex case the results of tyler in d tyler robustness and efficiency properties of scatter matrices biometrika vol 70 no 2 p 411 1983 more precisely the asymptotic distribution of these estimators as well as the asymptotic distribution of any homogeneous function of degree 0 of the m estimates are derived on the other hand we show the improvement of such results on two applications directions of arrival doa estimation using the multiple signal classification music algorithm and adaptive radar detection based on the adaptive normalized matched filter anmf test

adaptive beamformer orthogonal rejection test research in the area of signal detection in the presence of unknown interference has resulted in a number of adaptive detection algorithms examples of such algorithms include the adaptive matched filter amf the generalized likelihood ratio test glrt and the adaptive coherence estimator ace each of these algorithms results in a tradeoff between detection performance for matched signals and rejection performance for mismatch signals this paper introduces a new detection algorithm we call the adaptive beamformer orthogonal rejection test abort our test decides if an observation contains a multidimensional signal belonging to one subspace or if it contains a multidimensional signal belonging to an orthogonal subspace when unknown complex gaussian noise is present in our analysis we use a statistical hypothesis testing framework to develop a generalized likelihood ratio decision rule we evaluate the performance of this decision rule in both the matched and mismatched signal cases our results show that for constant power complex gaussian noise if the signal is matched to the steering vector abort glrt and amf give approximately equivalent probability of detection higher than that of ace which trades detection probability for an extra invariance to scale mismatch between training and test data of these four tests ace is most selective and therefore least tolerant of mismatch whereas amf is most tolerant of mismatch and therefore least selective abort and glrt offer compromises between these extremes with abort more like ace and with glrt more like amf

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

adaptive detection of a subspace signal in signal dependent interference this paper deals with the problem of adaptive detection of subspace signals embedded in thermal noise and clutter that depends on the transmitted signal to this end at the design stage we assume that the signal dependent sd clutter shares the same subspace as the target signals as customary a set of secondary data free of signal components is also assumed available two adaptive detectors referred to as the sd rao and sd wald are proposed by resorting to the rao test and wald test design criteria unlike the classical rao and wald tests which are derived by dividing the complex parameter into the real and imaginary parts the proposed detectors treat the complex parameter as a single quantity to reduce the computational burden moreover we derive the theoretical false alarm probabilities and detection probabilities and show that both the two proposed detectors exhibit the constant false alarm rate property numerical results show that the proposed detectors achieve a detection performance improvement over the conventional multidimensional detectors

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

performance analysis of covariance matrix estimates in impulsive noise this paper deals with covariance matrix estimates in impulsive noise environments physical models based on compound noise modeling spherically invariant random vectors sirv compound gaussian processes allow to correctly describe reality e g range power variations or clutter transitions areas in radar problems however these models depend on several unknown parameters covariance matrix statistical distribution of the texture disturbance parameters that have to be estimated based on these noise models this paper presents a complete analysis of the main covariance matrix estimates used in the literature four estimates are studied the well known sample covariance matrix mscm and a normalized version mn the fixed point fp estimate mfp and a theoretical benchmark mtfp among these estimates the only one of practical interest in impulsive noise is the fp the three others which could be used in a gaussian context are in this paper only of academic interest i e for comparison with the fp a statistical study of these estimates is performed through bias analysis consistency and asymptotic distribution this study allows to compare the performance of the estimates and to establish simple relationships between them finally theoretical results are emphasized by several simulations corresponding to real situations

a unifying framework for adaptive radar detection in homogeneous plus structured interference part i on the maximal invariant statistic this paper deals with the problem of adaptive multidimensional multichannel signal detection in homogeneous gaussian disturbance with unknown covariance matrix and structured deterministic interference the aforementioned problem corresponds to a generalization of the well known generalized multivariate analysis of variance gmanova in this part i of the paper we formulate the considered problem in canonical form and after identifying a desirable group of transformations for the considered hypothesis testing we derive a maximal invariant statistic mis for the problem at hand furthermore we provide the mis distribution in the form of a stochastic representation finally strong connections to the mis obtained in the open literature in simpler scenarios are underlined

a cfar adaptive subspace detector for second order gaussian signals we study the problem of detecting subspace signals described by the second order gaussian sog model in the presence of noise whose covariance structure and level are both unknown such a detection problem is often called gauss gauss problem in that both the signal and the noise are assumed to have gaussian distributions we propose adaptive detectors for the sog model signals based on a single observation and multiple observations with a single observation the detector can be derived in a manner similar to that of the generalized likelihood ratio test glrt but the unknown covariance structure is replaced by sample covariance matrix based on training data the proposed detectors are constant false alarm rate cfar detectors as a comparison we also derive adaptive detectors for the first order gaussian fog model based on multiple observations under the same noise condition as for the sog model with a single observation the seemingly ad hoc cfar detector for the sog model is a true glrt in that it has the same form as the glrt cfar detector for the fog model we give an approximate closed form of the probability of detection and false alarm in this case furthermore we study the proposed cfar detectors and compute the performance curves

performance of the adaptive sidelobe blanker detection algorithm in homogeneous environments the adaptive sidelobe blanker asb algorithm is a two stage detector consisting of a first stage adaptive matched filter amf detector followed by a second stage detector called the adaptive coherence or cosine estimator ace only those data test cells that survive both detection thresholdings are declared signal target bearing we provide exact novel closed form expressions for the resulting probability of detection pd and false alarm pfa for the asb algorithm and demonstrate that under homogeneous data conditions with no signal array response mismatch that i the asb is a constant false alarm rate cfar algorithm ii the asb has a higher or commensurate pd for a given pfa than both the amf and the ace and iii the asb has an overall performance that is commensurate with kelly s 1986 benchmark generalized likelihood ratio test glrt a compact statistical summary is derived providing distributions and dependencies among the glrt amf and the ace decision statistics

adaptive detection and estimation in the presence of useful signal and interference mismatches this paper considers adaptive detection and estimation in the presence of useful signal and interference mismatches we assume a homogeneous environment where the random disturbance components from the primary and secondary data share the same covariance matrix moreover the data under test contains a deterministic interference vector in addition to the possible useful signal we focus on the situation where an energy fraction of both the useful signal and the deterministic interference may lie outside their nominal subspaces conical uncertainty model under these conditions we devise a procedure for the computation of the joint maximum likelihood ml estimators of the useful signal and interference vectors resorting to a suitable rank one decomposition of a semidefinite program sdp problem optimal solution hence we use the aforementioned estimators for the synthesis of adaptive receivers based on different generalized likelihood ratio test glrt criteria at the analysis stage we assess the performance of the new detectors in comparison with some decision rules available in open literature

absence of wandering domains for some real entire functions with bounded singular sets let f be a real entire function whose set s f of singular values is real and bounded we show that if f satisfies a certain function theoretic condition the sector condition then f has no wandering domains our result includes all maps of the form z mapsto lambda frac sinh z z a with lambda 0 and a in mathbb r we also show the absence of wandering domains for certain non real entire functions for which s f is bounded and f n s f rightarrow infty uniformly as a special case of our theorem we give a short elementary and non technical proof that the julia set of the exponential map f z e z is the entire complex plane furthermore we apply similar methods to extend a result of bergweiler concerning baker domains of entire functions and their relation to the postsingular set to the case of meromorphic functions

true trees are dense we show that any compact connected set k in the plane can be approximated by the critical points of a polynomial with two critical values equivalently k can be approximated in the hausdorff metric by a true tree in the sense of grothendieck s dessins d enfants

adaptive double subspace signal detection in gaussian background part ii partially homogeneous environments in this part of the paper we continue to study the problem of detecting a double subspace signal in gaussian noise precisely we address the detection problem in partially homogeneous environments where the primary and secondary data share the same covariance matrix up to an unknown scaling factor we derive the generalized likelihood ratio test glrt rao test wald test and their two step versions we also introduce three spectral norm tests snts all these detectors possess the constant false alarm rate cfar property moreover various kinds of special cases of these detectors are given at the stage of performance evaluation we consider two cases one is the case of no signal mismatch the other is more general namely the case of signal mismatch including the column space signal mismatch and row space signal mismatch

glrt based direction detectors in homogeneous noise and subspace interference in this paper we derive and assess decision schemes to discriminate resorting to an array of sensors between the h0 hypothesis that data under test contain disturbance only i e noise plus interference and the h1 hypothesis that they also contain signal components along a direction which is a priori unknown but constrained to belong to a given subspace of the observables the disturbance is modeled in terms of complex normal random vectors plus deterministic interference assumed to belong to a known subspace we assume that a set of noise only secondary data is available which possess the same statistical characterization of noise in the cells under test at the design stage we resort to either the plain generalized likelihood ratio test glrt or the two step glrt based design procedure the performance analysis conducted resorting to simulated data shows that the one step glrt performs better than the detector relying on the two step design procedure when the number of secondary data is comparable to the number of sensors moreover it outperforms a one step glrt based subspace detector when the dimension of the signal subspace is sufficiently high

radar detection of distributed targets in homogeneous interference whose inverse covariance structure is defined via unitary invariant functions in this paper we deal with the problem of detecting an extended target embedded in homogeneous gaussian interference with unknown but structured covariance matrix we model the possible target echo from each range bin under test as a deterministic signal with an unknown scaling factor accounting for the target response at the design stage we exploit some a priori knowledge about the operating environment enforcing the inverse interference plus noise covariance matrix to belong to a set described via unitary invariant continuous functions hence we derive the constrained maximum likelihood ml estimates of the unknown parameters under both the h0 and h1 hypotheses and design the generalized likelihood ratio test glrt for the considered decision problem at the analysis stage we assess the performance of the devised glrt for some covariance matrix uncertainty sets of practical relevance both for spatial and doppler processing the results highlight that correct use of the a priori knowledge can lead to a detection performance quite close to the optimum receiver which supposes the perfect knowledge of the interference plus noise covariance matrix

adaptive subspace detectors we use the theory of generalized likelihood ratio tests glrts to adapt the matched subspace detectors msds of scharf 1991 and of scharf and frielander 1994 to unknown noise covariance matrices in so doing we produce adaptive msds that may be applied to signal detection for radar sonar and data communication we call the resulting detectors adaptive subspace detectors asds these include kelly s 1987 glrt and the adaptive cosine estimator ace of kaurt and scharh see ibid vol 47 p 2538 41 1999 and of scharf and mcwhorter see proc 30th asilomar conf signals syst comput pacific grove ca 1996 for scenarios in which the scaling of the test data may deviate from that of the training data we then present a unified analysis of the statistical behavior of the entire class of asds obtaining statistically identical decompositions in which each asd is simply decomposed into the nonadaptive matched filter the nonadaptive cosine or t statistic and three other statistically independent random variables that account for the performance degrading effects of limited training data

a unifying framework for adaptive radar detection in homogeneous plus structured interference part ii detectors design this paper deals with the problem of adaptive multidimensional multichannel signal detection in homogeneous gaussian disturbance with unknown covariance matrix and structured unknown deterministic interference the aforementioned problem extends the well known generalized multivariate analysis of variance gmanova tackled in the open literature in part i of this paper we have obtained the maximal invariant statistic mis for the problem under consideration as an enabling tool for the design of suitable detectors which possess the constant false alarm rate cfar property herein we focus on the development of several theoretically founded detectors for the problem under consideration first all the considered detectors are shown to be function of the mis thus proving their cfarness property second coincidence or statistical equivalence among some of them in such a general signal model is proved third strong connections to well known simpler scenarios analyzed in adaptive detection literature are established finally simulation results are provided for a comparison of the proposed receivers

adaptive array detection of uncertain rank one waveforms adaptive array detection of known within a complex scaling rank one space time waveforms in unknown spatial noise has received considerable attention the two published solutions are the adaptive matched filter and the glrt we expand on this work to consider the case of rank one waveforms that are uncertain i e only partially known more precisely we model the space time steering vector as the kronecker product of two vectors each of which is unknown but is known to lie in a known subspace applications for such a model include detection in the presence of multipath and spectral or polarization diversity in both radar processing and wireless communication using the principle of invariance we construct detectors based on the maximal invariant we show that the snr required to achieve a given detection probability for a given false alarm rate is only weakly impacted by waveform uncertainty thus our detector approaches the performance of earlier detectors which entail known waveforms

a subspace based adaptive sidelobe blanker we propose a modified version of the adaptive sidelobe blanker asb consisting of a generalized likelihood ratio test glrt based subspace detector followed by the adaptive coherence estimator the performance analysis shows that it possesses the constant false alarm rate property with respect to the unknown covariance matrix of the noise in homogeneous environment and that it guarantees a wider range of ldquodirectivityrdquo values with respect to the plain asb the probability of false alarm and the probability of detection the latter for matched signals only have been evaluated in closed form in homogeneous environment and by resorting to monte carlo simulation for the other considered cases

robust multidimensional matched subspace classifiers based on weighted least squares we propose and design two classes of robust subspace classifiers for classification of multidimensional signals our classifiers are based on robust m estimators and the least median of squares principle and we show that they may be unified as iterated reweighted oblique subspace classifiers the performance of the proposed classifiers are demonstrated by two examples noncoherent detection of space time frequency shift keying signals and shape classification of partially occluded two dimensional 2 d objects in both cases the proposed robust subspace classifiers outperform the conventional subspace classifiers

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

false alarm rate of the glrt for subspace signals in subspace interference plus gaussian noise in the presence of interference and gaussian noise with unknown covariance matrix a generalized likelihood ratio test glrt detector has already been developed for detecting a distributed target the target signal and interference are described with subspace models namely the target signals or interference are modeled as linear combinations of the linearly independent columns of a known subspace matrix in this paper we obtain statistical properties of this glrt detector and prove its constant false alarm rate against the noise covariance matrix moreover we derive analytical expressions for the probability of false alarm of the glrt detector specifically exact expressions for the probability of false alarm are obtained for six cases where the target subspace dimension denoted by p is 1 2 and 3 and the number of range cells denoted by h the distributed target occupies is 1 2 and 3 for the general case where p geq 4 and h geq 4 we derive an approximate expression for the probability of false alarm monte carlo simulations show that the exact expressions are valid and the accuracy of the approximate expression is acceptable for moderate or large training data size in practice these expressions can greatly facilitate the threshold setting for the glrt detector for any preassigned probability of false alarm

on the statistical invariance for adaptive radar detection in partially homogeneous disturbance plus structured interference this paper deals with the problem of adaptive vector subspace signal detection in partially homogeneous gaussian disturbance and structured unknown deterministic interference within the framework of invariance theory it is first proved that the maximal invariant statistic mis for the problem at hand is scalar valued and coincides with the well known adaptive normalized matched filter evaluated after data projection in the complementary subspace of the interfering signal second the statistical characterization of the mis under both hypotheses is derived then it is shown the statistical equivalence of two step generalized likelihood ratio test rao and wald tests as well as the more recently considered durbin and gradient test to the above statistic finally simulation results are provided to confirm our findings and analyze the performance trend of the mis with the relevant parameters

robust covariance matrix estimation in heterogeneous low rank context this paper addresses the problem of robust covariance matrix cm estimation in the context of a disturbance composed of a low rank lr heterogeneous clutter plus an additive white gaussian noise the lr clutter is modeled by a spherically invariant random vector with assumed high clutter to noise ratio in such a context adaptive process should require less training samples than classical methods to reach equivalent performance as in a full rank clutter configuration the main issue is that classical robust estimators of the cm cannot be computed in the undersampled configuration to overcome this issue the current approach is based on regularization methods nevertheless most of these solutions are enforcing the estimate to be well conditioned while in our context it should be lr structured this paper therefore addresses this issue and derives an algorithm to compute the maximum likelihood estimator of the cm for the considered disturbance model several relaxations and robust generalizations of the result are discussed performance is finally illustrated on numerical simulations and on a space time adaptive processing for airborne radar application

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on the invariance coincidence and statistical equivalence of the glrt rao test and wald test three common techniques to discriminate between alternatives in a binary hypothesis testing problem are the generalized likelihood ratio test glrt the rao test and the wald test in this paper we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing first of all we focus on the invariance of the rao and wald tests with respect to transformations leaving the testing problem unaltered then we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the glrt statistic finally we present some detection problems usually encountered in practical signal processing applications where the decision variables of the three quoted tests are equivalent namely related by strictly monotonic transformations

information geometry of boltzmann machines a boltzmann machine is a network of stochastic neurons the set of all the boltzmann machines with a fixed topology forms a geometric manifold of high dimension where modifiable synaptic weights of connections play the role of a coordinate system to specify networks a learning trajectory for example is a curve in this manifold it is important to study the geometry of the neural manifold rather than the behavior of a single network in order to know the capabilities and limitations of neural networks of a fixed topology using the new theory of information geometry a natural invariant riemannian metric and a dual pair of affine connections on the boltzmann neural network manifold are established the meaning of geometrical structures is elucidated from the stochastic and the statistical point of view this leads to a natural modification of the boltzmann machine learning rule

modified rao test for multichannel adaptive signal detection the problem of detecting a subspace signal is studied in colored gaussian noise with an unknown covariance matrix in the subspace model the target signal belongs to a known subspace but with unknown coordinates we first present a new derivation of the rao test based on the subspace model and then propose a modified rao test mrt by introducing a tunable parameter the mrt is more general which includes the rao test and the generalized likelihood ratio test as special cases moreover closed form expressions for the probabilities of false alarm and detection of the mrt are derived which show that the mrt bears a constant false alarm rate property against the noise covariance matrix numerical results demonstrate that the mrt can offer the flexibility of being adjustable in the mismatched case where the target signal deviates from the presumed signal subspace in particular the mrt provides better mismatch rejection capacities as the tunable parameter increases

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

adaptive double subspace signal detection in gaussian background part i homogeneous environments in this two part paper we consider the problem of adaptive multidimensional multichannel signal detection in gaussian noise with unknown covariance matrix the test data primary data is assumed as a collection of sample vectors arranged as the columns of a rectangular data array the rows and columns of the signal matrix are both assumed to lie in known subspaces but with unknown coordinates due to this feature of the signal structure we name this kind of signal as the double subspace signal part i of this paper focuses on the adaptive detection in homogeneous environments while part ii deals with the adaptive detection in partially homogeneous environments precisely in this part we derive the generalized likelihood ratio test glrt rao test wald test as well as their two step variations in homogeneous environments three types of spectral norm tests snts are also introduced all these detectors are shown to possess the constant false alarm rate cfar property moreover we discuss the differences between them and show how they work another contribution is that we investigate various special cases of these detectors remarkably some of them are well known existing detectors while some others are still new at the stage of performance evaluation conducted by monte carlo simulations both matched and mismatched signals are dealt with for each case more than one scenario is considered

invariant adaptive detection of range spread targets under structured noise covariance the invariance principle is adopted to develop an exhaustive study for adaptive detection of range spread targets in gaussian noise sharing a block diagonal covariance structure for this problem the usual generalized likelihood ratio principle is intractable in this paper we first determine the largest group of affine transformations that does not alter the decision problem then a maximal invariant identified by this group is derived which can characterize the totality of the invariant detectors and extends the existing results for the point target case a theoretical performance analysis of the maximal invariant is also given finally we propose two classes of invariant detectors which are distinguished by whether a constant false alarm rate cfar is maintained numerical experiments are provided for a comparison of the proposed detectors where a tradeoff between the cfarness and the improved performance has been observed and studied

the cfar adaptive subspace detector is a scale invariant glrt the constant false alarm rate cfar matched subspace detector cfar msd is the uniformly most powerful invariant test and the generalized likelihood ratio test glrt for detecting a target signal in noise whose covariance structure is known but whose level is unknown previously the cfar adaptive subspace detector cfar asd or adaptive coherence estimator ace was proposed for detecting a target signal in noise whose covariance structure and level are both unknown and whose covariance structure is estimated with a sample covariance matrix based on training data we show here that the cfar asd is glrt when the test measurement is not constrained to have the same noise level as the training data as a consequence this glrt is invariant to a more general scaling condition on the test and training data than the well known glrt of kelly 1986

adaptive radar detectors based on the observed fim modified versions of rao wald and durbin tests are considered exploiting an estimator of the fisher information matrix fim in place of the exact one they are asymptotically equivalent under some technical conditions to the standard counterparts and rely on the use of the observed fim ofim which is proportional to the negative hessian of the log likelihood the developed framework is applied to the problem of adaptive radar detection of a point like target in homogeneous or partially homogeneous interference remarkably for both the scenarios it is shown that rao wald and durbin tests with ofim are statistically equivalent to the generalized likelihood ratio test glrt for the specific detection problem namely kelly s detector for the homogeneous environment and the adaptive coherence estimator ace s kraut and l l scharf the cfar adaptive subspace detector is a scale invariant glrt ieee trans signal process vol 47 no 9 pp 2538 2541 sep 1999 also known as adaptive normalized matched filter anmf e conte m lops and g ricci asymptotically optimum radar detection in compound gaussian clutter ieee trans aerosp electron syst vol 31 no 2 pp 617 625 apr 1995 for the partially homogeneous scenario this provides a new interpretation of the mentioned glrts laying the foundations for a better understanding of their theoretical validity

a cfar adaptive subspace detector for first order or second order gaussian signals based on a single observation in this paper we consider the problem of detecting a signal in gaussian noise with unknown covariance matrix in partially homogeneous environments where the test and training data samples share the same noise covariance matrix up to an unknown scaling factor one solution to this problem is the adaptive subspace detector asd with a single or multiple observations however the probabilities of false alarm and detection of this asd have not been obtained yet in this paper these expressions are derived on the basis of a single observation which are confirmed with monte carlo simulations it is shown that the asd has a constant false alarm rate property with respect to both the shared noise covariance matrix structure and the independent scaling of the noise in the test data in addition we prove that for the first order model where the signal of interest is assumed to be a deterministic but unknown vector the asd derived with the generalized likelihood ratio test is consistent with that derived with an ad hoc two step design procedure

mean squared error and threshold snr prediction of maximum likelihood signal parameter estimation with estimated colored noise covariances an interval error based method mie of predicting mean squared error mse performance of maximum likelihood estimators mles is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix an intermediate step central to adaptive array detection and parameter estimation the successful application of mie requires good approximations of two quantities 1 interval error probabilities and 2 asymptotic snrrarrinfin local mse performance of the mle exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein that in conjunction with the union bound provide accurate prediction of the required interval error probabilities the crameacuter rao bound crb often provides adequate prediction of the asymptotic local mse performance of mle the signal parameters however are decoupled from the colored noise parameters in the fisher information matrix for the deterministic signal model rendering the crb incapable of reflecting loss due to colored noise covariance estimation a new modification of the crb involving a complex central beta random variable different from but analogous to the reed mallett and brennan beta loss factor provides a working solution to this problem facilitating mse prediction well into the threshold region with remarkable accuracy

robust subspace learning and detection in laplacian noise and interference we address the problem of maximum likelihood subspace learning and detection in the presence of laplacian noise and interference whose subspace may be known or unknown for subspace learning the laplacian problem reduces to a maxmin convex mathematical program with polyhedral cost the minimization involves projection of the measurements onto a subspace orthogonal to the signal and interference spaces and has linear constraints while the maximization produces the subspace and has polyhedral constraints the laplacian noise model for subspace detection and estimation motivated by applications in functional magnetic resonance imaging and applicable in other areas yields maximum likelihood detectors and learned subspaces with unique structure due to the presence of corners in the polyhedrally convex optimization for instance the optimal learned subspace can consist of vectors whose elements take values of 1 or 1 only emergence of such a quantization attests to the robustness property of laplacian learning meaning that the solution is insensitive to perturbation in the data set the resulting detectors are similarly robust to false alarms and have computationally attractive properties

adaptive threshold estimation via extreme value theory determining a detection threshold to automatically maintain a low false alarm rate is a challenging problem in a number of different applications the underlying parametric assumptions of most automatic target detection algorithms are invalid therefore thresholds derived using these incorrect distribution assumptions do not produce desirable results when applied to real sensor data monte carlo methods for threshold determination work well but tend to perform poorly when targets are present in order to mitigate these effects we propose an algorithm using extreme value theory through the use of the generalized pareto distribution gpd and a kolmogorov smirnov statistical test unlike previous work based on gpd estimates this algorithm incorporates a way to adaptively maintain low false alarm rates in the presence of targets both synthetic and real world detection results demonstrate the usefulness of this algorithm

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

statistical analysis of the nonhomogeneity detector for non gaussian interference backgrounds we derive the nonhomogeneity detector nhd for non gaussian interference scenarios and present a statistical analysis of the method the non gaussian interference scenario is assumed to be modeled by a spherically invariant random process sirp we present a method for selecting representative homogeneous training data based on our statistical analysis of the nhd for finite sample support used in covariance estimation in particular an exact theoretical expression for the nhd test statistic probability density function pdf is derived performance analysis of the nhd is presented using both simulated data and measured data from the multichannel airborne radar measurement mcarm program a performance comparison with existing nhd approaches is also included

adaptive detection of range distributed targets a modified generalized likelihood ratio test mglrt for the adaptive detection of a target or targets that are distributed in range is derived the unknown parameters associated with the hypothesis test are the complex amplitudes in range of the desired target and the unknown covariance matrix of the additive interference which is assumed to be characterized as complex zero mean correlated gaussian random variables the target s or targets complex amplitudes are assumed to be distributed across the entire input data block sensor spl times range results on probabilities of false alarm and detection are derived and a bounded constant false alarm rate cfar detector is developed simulation results are presented it is shown that the derived mglrt of range distributed targets is much more effective in detecting targets distributed in range than an m out of k detector which is cascaded with a single point target kelly 1986 detector

subspace partitioning for target detection and identification detection of a given target or set of targets from observed data is a problem countered in many applications regardless of the algorithm selected detection performance can be severely degraded when the subspace defined by the target data set is singular or ill conditioned high correlations between target components and their linear combinations lead to false positives and misidentifications especially for subspace based detectors in this paper we propose a subspace partitioning scheme that allows for detection to be performed in a number of better conditioned subspaces instead of the original subspace the proposed technique is applied to raman spectroscopic data analysis through both simulation and experimental results we demonstrate the improvement in the overall detection performance when using the proposed subspace partitioning scheme in conjunction with several subspace detection methods that are commonly used in practice

from image deblurring to optimal investments maximum likelihood solutions for positive linear inverse problems the problem of recovering an input signal from a blurred output in an input output system with linear distortion is ubiquitous in science and technology when the blurred output is not degraded by statistical noise the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters subject to positivity constraints on the solution we show that all such linear inverse problems with positivity restrictions lininpos problems for short can be interpreted as statistical estimation problems from incomplete data based on infinitely large samples and that maximum likelihood ml estimation and the em algorithm provide a straightforward method of solution for such problems

dimension rigidity in conformal structures abstract let be the limit set of a conformal dynamical system i e a kleinian group acting on either finite or infinite dimensional real hilbert space a conformal iterated function system or a rational function we give an easily expressible sufficient condition requiring that the limit set is not too much bigger than the radial limit set for the following dichotomy is either a real analytic manifold or a fractal in the sense of mandelbrot i e its hausdorff dimension is strictly greater than its topological dimension our primary focus is on the infinite dimensional case an important component of the strategy of our proof comes from the rectifiability techniques of mayer and urbanski 03 who obtained a dimension rigidity result for conformal iterated function systems including those with infinite alphabets in order to handle the infinite dimensional case both for kleinian groups and for iterated function systems we introduce the notion of pseudorectifiability a variant of rectifiability and develop a theory around this notion similar to the theory of rectifiable sets our approach also extends existing results in the finite dimensional case where it unifies the realms of kleinian groups conformal iterated function systems and rational functions for kleinian groups we improve on the rigidity result of kapovich 09 by substantially weakening its hypothesis of geometrical finiteness moreover our proof based on rectifiability is entirely different than that of kapovich which depends on homological algebra another advantage of our approach is that it allows us to use the demension of stan ko 69 as a substitute for topological dimension for example we prove that any dynamically defined version of antoine s necklace must have hausdorff dimension strictly greater than 1 i e the demension of antoine s necklace

hybrid detectors for subpixel targets subpixel detection is a challenging problem in hyperspectral imagery analysis since the target size is smaller than the size of a pixel detection algorithms must rely solely on spectral information a number of different algorithms have been developed over the years to accomplish this task but most detectors have taken either a purely statistical or a physics based approach to the problem we present two new hybrid detectors that take advantage of these approaches by modeling the background using both physics and statistics results demonstrate improved performance over the well known amsd and ace subpixel algorithms in experiments that include multiple targets images and area types especially when dealing with weak targets in complex backgrounds

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

performance of a class of adaptive detection algorithms in nonhomogeneous environments a two dimensional 2 d adaptive sidelobe blanker asb detection algorithm was developed through experimentation as an extenuate for false alarms caused by undernulled interference encountered when applying the adaptive matched filter amf in nonhomogeneous environments the algorithm s utility has been demonstrated empirically considering theoretic performance analyses of the asb detection algorithm as well as the amf generalized likelihood ratio test glrt and the adaptive cosine estimator ace under nonideal conditions can become fairly intractable rather quickly especially in an adaptive processing context involving covariance estimation in this paper however we have developed and exploited a theoretic framework through which the performance of these algorithms under nonhomogeneous conditions can be examined theoretically it is demonstrated through theoretic analysis that in the presence of undernulled interference the asb is a pliable false alarm regulatory far detector that maintains good target sensitivity a viable method of asb threshold selection is also presented and demonstrated

generalized cfar property and ump invariance for adaptive signal detection in this paper we consider adaptive detection of a signal embedded in additive disturbance whose multivariate distribution belongs to a very general class including many statistical models commonly adopted for radar disturbance we introduce the concept of generalized constant false alarm rate cfar and show that a class of receivers sharing some invariances complies with the quoted property then we devise the generalized likelihood ratio test glrt and prove that under some mild technical conditions it coincides with that obtained under the gaussian assumption for the observations we also deal with the existence of the uniformly most powerful invariant umpi detector either using the wijsman theorem or directly computing the maximal invariant likelihood ratio lr at the analysis stage we focus on a compound matrix variate model for the disturbance component which is a natural generalization of the spherically invariant random vector sirv in this context we assess the performance of some well known invariant decision rules also in comparison with the most powerful invariant mpi detector the results highlight that some among the analyzed receivers exhibit a performance level very close to the mpi test

persymmetric adaptive detectors in homogeneous and partially homogeneous environments this paper addresses the problem of detecting a signal in partially homogeneous and homogeneous environments in partially homogeneous environments i e both the test data and training data share the same noise covariance matrix structure up to an unknown scaling factor a persymmetric adaptive coherence estimator per ace detector is proposed by exploiting the persymmetric structure of the covariance matrix the per ace can reduce training data requirements furthermore the expressions for the probabilities of false alarm and detection are derived along with the distribution of the loss factor in homogeneous environments a persymmetric adaptive matched filter per amf detector has been presented however its probability of detection has not been obtained yet thus we derive the expression for the probability of detection for both the per ace and per amf numerical results of these proposed expressions are confirmed with those of monte carlo trials in addition simulation results show that the proposed per ace outperforms the conventional ace in training limited scenarios

maximal invariants and performance of some invariant hypothesis tests for an adaptive detection problem maximal invariants for adaptive detection of a signal in unknown interference from multiple observations is derived given coherent samples from p sets of observations it is shown that a maximal invariant statistic for the detection problem is a 2p 1 dimensional vector comprising the eigenvalues of two hermitian positive definite matrices obtained from the data set two invariant detectors well known for p 1 are generalized for the case of multiple observations and closed form expressions for the probability of detection and probability of false alarm are derived along with the distributions of the signal to interference plus noise loss factors several novel invariant detectors are constructed from the maximal invariants and the receiver operating characteristics of the detectors compared

theoretical performance analysis of the w abort detector in a recent paper we introduced a modification of the adaptive beaniformer orthogonal rejection test abort for adaptive detection of signals in unknown noise by supposing under the null hypothesis the presence of signals orthogonal to the nominal steering vector in the whitened observation space we will refer to this new receiver as the whitened adaptive beamformer orthogonal rejection test w abort through monte carlo simulations this new detector was shown to provide better rejection capabilities of mismatched e g sidelobe signals than existing ones like abort or the adaptive coherence estimator ace but at the price of a certain loss in terms of detection of matched i e mainlobe signals the aim of this paper is to provide a theoretical validation of this fact we consider both the case of distributed targets and point like targets we provide a statistical characterization of the w abort test statistic under the null hypothesis and for matched and mismatched signals under the alternative hypothesis for distributed targets the probability of false alarm and the probability of detection can only be expressed in terms of multi dimensional integrals and are thus very complicated to obtain in contrast for point like targets such probabilities can be easily calculated by numerical integration techniques the theoretical expressions derived herein corroborate the simulation results obtained previously

rao test for adaptive detection in gaussian interference with unknown covariance matrix this paper deals with the problem of detecting a signal known up to a scaling factor in the presence of gaussian disturbance with unknown covariance matrix a new detector based on the rao test criterion is introduced and its invariance properties and constant false alarm rate cfar behavior are studied at the analysis stage the performance of the new receiver is assessed also in comparison with some classical adaptive radar detectors both in the matched as well as in the mismatched signal case remarkably the rao test may achieve a matched detection performance which is commensurate with that of the generalized likelihood ratio test glrt based detectors if a sufficient number of training data is available moreover it also exhibits better rejection capabilities of mismatched signals than the counterparts in the last part of the work a two stage detector whose second stage coincides with the rao test is devised it represents a suitable means to restore the detection performance of the plain rao test in the presence of a small number of training data finally the performance of the aforementioned two stage processor is analyzed in closed form and the cfar behavior is proved

kernel spectral matched filter for hyperspectral imagery in this paper a kernel based nonlinear spectral matched filter is introduced for target detection in hyperspectral imagery which is implemented by using the ideas in kernel based learning theory a spectral matched filter is defined in a feature space of high dimensionality which is implicitly generated by a nonlinear mapping associated with a kernel function a kernel version of the matched filter is derived by expressing the spectral matched filter in terms of the vector dot products form and replacing each dot product with a kernel function using the so called kernel trick property of the mercer kernels the proposed kernel spectral matched filter is equivalent to a nonlinear matched filter in the original input space which is capable of generating nonlinear decision boundaries the kernel version of the linear spectral matched filter is implemented and simulation results on hyperspectral imagery show that the kernel spectral matched filter outperforms the conventional linear matched filter

adaptive detection of a signal known only to lie on a line in a known subspace when primary and secondary data are partially homogeneous this paper deals with the problem of detecting a signal known only to lie on a line in a subspace in the presence of unknown noise using multiple snapshots in the primary data to account for uncertainties about a signal s signature we assume that the steering vector belongs to a known linear subspace furthermore we consider the partially homogeneous case for which the covariance matrix of the primary and the secondary data have the same structure but possibly different levels this provides an extension to the framework considered by bose and steinhardt the natural invariances of the detection problem are studied which leads to the derivation of the maximal invariant then a detector is proposed that proceeds in two steps first assuming that the noise covariance matrix is known the generalized likelihood ratio test glrt is formulated then the noise covariance matrix is replaced by its sample estimate based on the secondary data to yield the final detector the latter is compared with a similar detector that assumes the steering vector to be known

glrt based adaptive detection algorithms for range spread targets we address adaptive detection of a range spread target or targets embedded in gaussian noise with unknown covariance matrix to this end we assume that cells referred to in the following as secondary data that are free of signal components are available those secondary data are supposed to possess either the same covariance matrix or the same structure of the covariance matrix of the cells under test in this context we design detectors relying on the generalized likelihood ratio test glrt and on a two step glrt based design procedure remarkably both criteria lead to receivers ensuring the constant false alarm rate cfar property with respect to the unknown quantities a thorough performance assessment of the proposed detection strategies together with the evaluation of their processing cost highlights that the two step design procedure is to be preferred with respect to the plain glrt in fact the former leads to detectors that achieve satisfactory performance under several situations of practical interest and are simpler to implement than those designed resorting to the latter

cfar matched direction detector in a previously published paper by besson et al we considered the problem of detecting a signal whose associated spatial signature is known to lie in a given linear subspace in the presence of subspace interference and broadband noise of known level we extend these results to the case of unknown noise level more precisely we derive the generalized likelihood ratio test glrt for this problem which provides a constant false alarm rate cfar detector it is shown that the glrt involves the largest eigenvalue and the trace of complex wishart matrices the distribution of the glrt is derived under the null hypothesis numerical simulations illustrate its performance and provide a comparison with the glrt when the noise level is known

cfar detection of multidimensional signals an invariant approach the paper deals with constant false alarm rate cfar detection of multidimensional signals embedded in gaussian noise with unknown covariance we attack the problem by resorting to the principle of invariance which proves a valuable statistical tool for ensuring a priori namely at the design stage the cfar property in this context we determine a maximal invariant statistic with respect to a proper group of transformations that leave unaltered the hypothesis testing problem under study devise the optimum invariant detector and show that no uniformly most powerful invariant umpi test exists thus we establish the conditions an invariant detector must fulfill in order to ensure the cfar property finally we discuss several suboptimal implementable invariant receivers and remarkably show that the generalized likelihood ratio test glrt detector is a member of this class the performance analysis which has been carried out in the presence of a gaussian signal array shows that the proposed detectors exhibit a quite acceptable loss with respect to the optimum neyman pearson detector

eremenko points and the structure of the escaping set much recent work on the iterates of a transcendental entire function f has been motivated by eremenko s conjecture that all the components of the escaping set i f are unbounded we prove several general results about the topological structure of i f including the fact that if i f is disconnected then it contains uncountably many pairwise disjoint unbounded continua all of which are subsets of ar f the core of the fast escaping set we also show that for some r 0 the set ar f is connected and has the structure of an infinite spider s web or it contains uncountably many unbounded connected f sets there are analogous results for the intersections of these sets with the julia set when multiply connected wandering domains are not present but very different results when such wandering domains are present in proving these we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior indeed uncountably many

beyond the binary boltzmann machine the definition of the usual boltzmann machine is extended to allow for neurons with polytomous multicategory rather than simply binary responses updating rules are defined along with the associated stationary distributions and an alternating minimization method is described for the purposes of training emphasis is placed on the relevance of statistical ideas including polytomous logistic regression the iterative proportional fitting procedure and the em algorithm

iterative generalized likelihood ratio test for mimo radar we consider a multiple input multiple output mimo radar system where both the transmitter and receiver have multiple well separated subarrays with each subarray containing closely spaced antennas because of this general antenna configuration both the coherent processing gain and the spatial diversity gain can be simultaneously achieved we compare several spatial spectral estimators including capon and apes for target detection and parameter estimation we introduce a generalized likelihood ratio test glrt and a conditional generalized likelihood ratio test cglrt for the general antenna configuration based on glrt and cglrt we then propose an iterative glrt iglrt procedure for target detection and parameter estimation via several numerical examples we show that iglrt can provide excellent detection and estimation performance at a low computational cost

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

constructing entire functions by quasiconformal folding we give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f among other applications we construct a function f with bounded singular set whose fatou set contains a wandering domain