Spike train data find a growing list of applications in computational neuroscience, streaming data and finance. Statistical analysis of spike trains is based on various probabilistic and neural network models. The statistical approach relies on parametric or nonparametric specifications of the underlying model. In this paper we consider the nonparametric classification problem for a class of spike train data characterized by nonparametricaly specified intensity functions. We derive the optimal Bayes rule and next formulate the plug-in non-parametric kernel classifiers. Asymptotical properties of the rules are established including the limit with respect to the increasing observation time interval and the size of a training set. The obtained results are supported by a finite sample simulation studies.

This study adapts the highly-versatile siamese neural network model to the event data domain. We introduce a supervised training framework for optimizing Earth Mover's Distance (EMD) between spike trains with spiking neural networks (SNN). We train this model on images of the MNIST dataset converted into spiking domain with novel conversion schemes. The quality of the siamese embeddings of input images was evaluated by measuring the classifier performance for different dataset coding types. The models achieved performance similar to existing SNN-based approaches (F1-score of up to 0.9386) while using only about 15% of hidden layer neurons to classify each example. Furthermore, models which did not employ a sparse neural code were about 45% slower than their sparse counterparts. These properties make the model suitable for low energy consumption and low prediction latency applications.

Given noisy samples of a signal, the problem of testing whether the signal belongs to a restricted class of parametrically specified class of signals is considered. Using a nonparametric kernel-based approach for reconstructing the signal we compare the parametric signal class to a broad alternative of signal classes that cannot be parametrized. For such a setup, we introduce testing procedures relying on nonparametric kernel-type sampling reconstruction algorithms properly adjusted for noisy data. First, we propose the testing procedures utilizing the L2-distance between the kernel estimate and signals from the target parametric class. Then, we make use of the maximum likelihood ratio test extended to the posed nonparametric signal specification problem. Central limit theorems of the test statistics are derived yielding consistent testing methods. Hence, we obtain testing procedures that keep asymptotically the desired level of the probability of false alarm and showing a power tending to one as the number of samples increases. The finite sample size properties of the introduced tests are also given.

The nonlinear characteristic in a Hammerstein system, i.e., a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade, is recovered with the nonparametric nearest neighbor regression estimate. The a priori information is nonparametric, both the nonlinear characteristic and the impulse response are completely unknown and can be of any form. Local and global properties of the estimate are examined. Whatever the probability density of the input signal, the estimate converges at every continuity point of the characteristic as well as in the global sense. We derive the asymptotic bias and variance of the proposed estimate. As a result, the optimal rate of convergence is established that additionally is independent of the shape of the input density. Results of numerical simulations are also presented.

A nonparametric version of the sequential signal detection problem is studied. Our signal model includes a class of time-limited signals for which we collect data in the sequential fashion at discrete points in the presence of correlated noise. For such a setup we introduce a novel signal detection algorithm relying on the postfiltering smooth correction of the classical Whittaker–Shannon interpolation series. Given a finite frame of noisy samples of the signal, we design a detection algorithm being able to detect a departure from a reference signal as quickly as possible. Our detector is represented as a normalized partial-sum continuous time stochastic process, for which we obtain a functional central limit theorem under weak assumptions on the correlation structure of the noise. Particularly, our results allow for noise processes such as ARMA and general linear processes as well as$ \alpha $-mixing processes. The established limit theorems allow us to design monitoring algorithms with the desirable level of the probability of false alarm and able to detect a change with probability approaching one.

In this paper we examine a class of multiple-input, single output (MISO) nonlinear systems of the block-oriented structure. In particular, we focus on MISO Hammerstein systems being the cascade connection of a multivariate nonlinearity with a linear dynamical subsystem. In order to alleviate an apparent curse of dimensionality occurring in the problem of estimating the nonlinearity, we propose to a semi-parametric strategy for identification of the nonlinear system. This is carried out by projecting the d-dimensional input signal onto one dimensional subset which, in turn, is mapped by a uni variate nonparametric function to an internal unobserved signal of the system. Such a parsimonious representation allows us to overcome the curse of dimensionality as the accuracy of our identification algorithms is independent of d. We identify the system via the semi-parametric version of the least squares. The statistical accuracy of the resulting estimates is obtained via the theory of M- estimation. These theoretical findings are verified in numerous simulation studies.

This paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions. A nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box. As such, it will be coined as the Vertical Box Control Chart (V-Box Chart). Its implementation requires merely to count the number of data points which fall into the box attached to the last available observation. Noa prioriknowledge of data distributions is required and proper tuning of the box size provides a quick detection technique. This is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the V-Box Chart. These theoretical results are verified by simulation studies which indicate that the V-Box Chart may provide quick detection with zero delay for jumps of moderate sizes. Its averaged run length to detection is more favorable than the one for the classical EWMA method. By comparison with the classical Shewhart chart, which was optimized for normal errors, our method provides comparable or better performance.

Statistical tests are introduced for checking whether an image functionf(x,y) defined on the unit discD={(x,y):x2+y2les 1} is invariant under certain symmetry transformations ofD, given that discrete and noisy data are observed. Invariance under reflections or under rotations by rational angles is considered, as well as rotational invariance. These symmetry relations can be naturally expressed as restrictions for the Zernike moments off(x,y). Therefore, the test statistics are based on theL2distance between Zernike series estimates of the image function itself and its version obtained after applying the symmetry transformation. The asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives is derived. Furthermore, the quality of the asymptotic approximations via simulation studies is investigated. The usefulness of our theory is verified by examining an important problem in confocal microscopy, i.e., possible imprecise alignments in the optical path of the microscope are investigated. For optical systems with rotational symmetry, the theoretical point-spread function (PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. The tests are used to investigate whether the required symmetries can indeed be detected in the empirical PSF.

An algorithm for high-precision numerical computation of Zernike moments is presented. The algorithm, based on the introduced polar pixel tiling scheme, does not exhibit the geometric error and numerical integration error which are inherent in conventional methods based on Cartesian coordinates. This yields a dramatic improvement of the Zernike moments accuracy in terms of their reconstruction and invariance properties. The introduced image tiling requires an interpolation algorithm which turns out to be of the second order importance compared to the discretization error. Various comparisons are made between the accuracy of the proposed method and that of commonly used techniques. The results reveal the great advantage of our approach.

The paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the Whittaker-Shannon (WS) sampling interpolation formula. A class of band-limited signals as well as signals which are non-band-limited are taken into consideration. The proposed reconstruction method is a smooth post-filtering correction of the classical WS interpolation series. We assess both the point-wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes. This includes short and long-memory stationary errors being independent of the sampling rate. We also examine a class of noise processes for which the correlation function depends on the sampling rate. Whereas the short-memory errors have relatively small influence on the reconstruction accuracy, the long-memory errors can greatly slow down the convergence rate. In the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed. We give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero. We argue that the obtained rates are close to be optimal. In fact, in a number of special cases they agree with known optimal min-max rates. The problem of the limit distribution of the L2distance of the proposed reconstruction algorithm is also addressed. This result allows us to tackle an important problem of designing non- parametric lack-of-fit tests. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case.

Zernike moment (ZM) is a useful image feature because of its distinguishing properties such as the magnitude invariance to image rotation. However, we find that for digital images, the invariant property of ZMs is far from ideal when they are computed with the commonly used Cartesian method, which inevitably brings about geometric error and integral error. In this paper, we propose a polar coordinate based algorithm for the computation of ZMs, which avoids both kinds of errors. We provide solutions to the key issues in ZM computation under polar coordinate system, including the derivation of computation formulas, the polar pixel arrangement scheme, and the interpolation-based image conversion etc. Simulation results show that the proposed polar approach improves the rotational invariance of ZMs significantly.

The post-filtering modification of the Whittaker-Shannon interpolation series in the presence of long-memory errors is examined. The pointwise and global rates of convergence of the reconstruction algorithm are evaluated.

We consider the extension of the Whittaker-Shannon (WS) reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited. Observing that in this situation the classical sampling expansion yields inconsistent reconstruction, we introduce a class of signal recovery methods with a smooth correction of the interpolation series. Two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing. We assess the accuracy of the methods by the global L/sub 2/ error. Both band-limited and non-band-limited signals are considered. A general class of correlated noise processes is taken into account. The weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed. The influence of noise memory and its moment structure on the accuracy is thoroughly examined.

The paper deals with the problem of reconstruction of nonlinearities in a certain class of nonlinear dynamical systems of the multichannel form. A prior information about the system is very limited excluding the standard parametric approach to the problem. The system structure consists of nonlinearities being embedded in a block oriented structure containing dynamic linear subsystems and other “nuisance” nonlinearities. The multiresolution idea, being the fundamental concept of the modern wavelet theory, is adopted and multiscale expansions associated with a large class of scaling functions are applied to construct nonparametric identification techniques of the nonlinearities. The pointwise convergence properties of the proposed identification algorithms are established. Conditions for the convergence are given; and for nonlinearities satisfying a local Lipschitz condition, the rate of convergence is evaluated.

We consider the problem of estimating a function f (x, y) on the unit disk f {(x, y): x/sup 2/+y/sup 2//spl les/1}, given a discrete and noisy data recorded on a regular square grid. An estimate of f (x, y) based on a class of orthogonal and complete functions over the unit disk is proposed. This class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate. For-radial functions, the orthogonal set has a particularly simple form being related to the classical Legendre polynomials. We give the statistical accuracy analysis of the proposed estimate of f (x, y) in the sense of the L/sub 2/ metric. It is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle. In fact, the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain.

We give a detailed analysis of the accuracy of Zernike moments in terms of their discretization errors and the reconstruction power. It is found that there is an inherent limitation in the precision of computing the Zernike moments due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to a celebrated problem in analytic number theory of evaluating the lattice points within a circle.

The problem of recovering a signal in the class of band-limited functions is studied. We consider a situation when discrete data points are first grouped to the points of an uniform grid and then the reconstruction is carried out from such a reduced data set. The data grouping is common for computer rounding errors and may also be viewed as a data compression process. The accuracy of the proposed grouping techniques is examined. These results are used to provide an understanding of the number of grid points required to achieve a given level of accuracy.

Research has been performed investigating the use of moments for pattern recognition in recent years. The basic problem of the influence of discretization and noise on moment accuracy as object descriptors, has been barely investigated. In this paper, the detailed error analysis involved in the moment method is discussed. Several new techniques to increase the accuracy and efficiency of moment descriptor are proposed. We utilize these results for the problem of image reconstruction from the orthogonal Legendre moments computed from discrete and noisy data. The automatic selection of an optimal number of moments is also discussed.

We consider the problem of recovering a band-limited signal f(t) from noisy data yk=f(k/spl tau/)+/spl Gt/epsilon//sub k/, where /spl tau/ is the sampling rate. Starting from the truncated Whittaker-Shannon cardinal expansion with or without sampling windows (both cases yield inconsistent estimates of f(t)) we propose estimators that are convergent to f(t) in the pointwise and uniform sense. The basic idea is to cut down high frequencies in the data and to use suitable oversampling /spl tau//spl les//spl pi///spl Omega/, /spl Omega/ being the bandwidth (maximum frequency) of f(t). The simplest estimator we propose is given by f/spl circ//sub n/(t)=/spl tau/ /spl Sigma//|t-k/spl tau/|/spl les/n/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/n/spl tau/. Generalizations of f/spl circ//sub n/ including sampling windows are also examined. The main aim is to examine the mean squared error (MSE) properties of such estimators in order to determine the optimal choice of the sampling rate /spl tau/ yielding the fastest possible rate of convergence. The best rate for the MSE we obtain is O(In(n)/n).

We describe alternative learning schemes for radial basis function (RBF) networks both in the context of classification and function recovering problems. The connection between RBF networks and well established kernel-type regression estimators is exhibited. We show that the asymptotic accuracy (convergence and rate of convergence) of the proposed RBF nets can be equivalent to the kernel technique.

Systems consisting of linear dynamic and memory-less nonlinear subsystems are identified. The paper deals with systems in which the nonlinear element is followed by a linear element, as well as systems in which the subsystems are connected in parallel. The goal of the identification is to recover the nonlinearity from noisy input-output observations of the whole system; signals interconnecting the elements are not measured. Observed values of the input signal are rearranged in increasing order, and coefficients for the expansion of the nonlinearity in trigonometric series are estimated from the new sequence of observations obtained in this way. Two algorithms are presented, and their mean integrated square error is examined. Conditions for pointwise convergence are also established. For the nonlinearity satisfying the Lipschitz condition, the error converges to zero. The rate of convergence derived for differentiable nonlinear characteristics is insensitive to the roughness of the probability density of the input signal. Results of numerical simulation are also presented.>

The posterior-probability estimate of the classification error rate of some nonparametric classification rules is studied. The variance of the estimator is shown to have same remarkable distribution-free properties for the k-nearest neighbor, kernel, and histogram rules. We also investigate the bias of the estimate and establish its consistency and upper bounds. The version of the estimate calculated from an independent set of unclassified patterns is also considered.>

The problem of reconstruction of band-limited signals from discrete and noisy data is studied. The reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined. In particular, the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated. The main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently. This distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise

Nonparametric kernel classification rules derived from incomplete (missing) data are studied. A number of techniques of handling missing observation in the training set are taken into account. In particular, the straightforward approach of designing a classifier only from available data (deleting missing values) is considered. The class of imputation techniques is also taken into consideration. In the latter case, one estimates missing values and then calculates classification rules from such a completed training set. Consistency and speed of convergence of proposed classification rules are established. Results of simulation studies are presented.>

Studies the detection and size measurement of discontinuities from sampled noisy data of an univariate function. The proposed tests and estimators are of the form of linear convolution filters with characteristics employing series expansions. In particular, the standard and conjugate Fourier series are taken into consideration. The convergence properties (consistency and rate of convergence) of the proposed estimates are established. The asymptotic results model the performance of the estimates on grids which become increasingly fine, i.e., when the signal resolution increases. Unlike other techniques in the literature, no specific form of the underlying function is assumed, and behavior of the function in the neighborhood of the discontinuity need not be in the form of a step function.>

Examines the reconstruction properties of orthogonal moment descriptors. An asymptotic expansion for the global reconstruction error is established. This reveals mutual relationships between a number of moments, the image smoothness, sampling rate and noise model characteristics. The problem of a data-driven selection of an optimal number of moments is addressed.>

The problem of reconstruction of an image from discrete and noisy data by the method of moments is examined. The set of orthogonal moments based on Legendre polynomials is employed. A general class of signal-dependent noise models is taken into account. An asymptotic expansion for the global reconstruction error is established. This reveals mutual relationships between a number of moments, the image smoothness, sampling rate, and noise model characteristics. The problem of an automatic (data-driven) section of an optimal number of moments is studied. This is accomplished with the help of cross-validation techniques.>

The nonparametric kernel classification rule derived from incomplete data is studied. Methods of designing kernel decision rules possessing optimal asymptotic properties are proposed. Consistency and rates of convergence are examined. It is argued that the replacement methods using the regression approach can lead to the inconsistency of resulting decision rules. On the other hand, a method employing the concept of predictive density yields asymptotically optimal classification rules.>

A discrete-time nonlinear Hammerstein system is identified, and the correlation and frequency-domain methods for identification of its linear subsystem are presented. The main results concern the estimation of the nonlinear memoryless subsystem. No conditions concerning the functional form of the transform characteristic of the subsystem are made, and an algorithm for estimation of the characteristic is given. The algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data. It is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable. Pointwise as well as global consistencies are established. For Lipschitz characteristics the rate of the convergence in probability is O(n/sup -1/3/).>

It is shown that, for a nonparametric recursive kernel classification rule,\sum^{n}_{i=1}h^{d}(i)I_{ \{h(i) > \epsilon \} } / \sum^{n}_{j=1} h^{d} (j) \rightarrow 0 {\rm as} n \rightarrow \infty,all\epsilon > 0and\sum^{\infty}_{i=1}h^{d}(i)= \inftyconstitute a set of conditions which are not only sufficient but also necessary for weak and strong Bayes risk consistency of the rule. In this way, weak and strong consistencies are shown to be equivalent.

The nonparametric estimate derived from the Hermite orthogonal system of the functionalI=\int f^{2}(x) dxwherefis an unknown probability density, is studied. Sufficient conditions for the weak and strong consistency of the estimate are presented, and the rate of convergence is given. In particular, under mild assumptions onf, the rate of mean-square error convergence isO(n^{-1}), whereas for almost complete convergence it isO((n^{-1} \log n)^{1/2}). Moreover, several possible applications in the area of nonparametric inference of the estimate are indicated.

It is shown that the kernel estimate of the regressionE(Y|X = x)is weakly or strongly consistent for almost allx(\mu), where\muis the probability measure ofX. The result is valid for any distribution ofX. The asymptotical optimality of classification rules derived from the estimate is examined. The optimality is independent of class distributions, i.e., it is distribution-free.

It is shown that the recursive kernel estimate of the regression functionE(Y|X = x)is consistent at almost everyx(\mu)regardless of the distribution\muofX. Thus the result is distribution-free. From this we show that the risk for a suitable classification rule derived from the estimate converges to Bayes' risk, no matter what the class distributions are.