robust clustering by detecting density peaks and assigning points based on fuzzy weighted k nearest neighbors clustering by fast search and find of density peaks referred to as dpc was introduced by alex rodriguez and alessandro laio the dpc algorithm is based on the idea that cluster centers are characterized by having a higher density than their neighbors and by being at a relatively large distance from points with higher densities the power of dpc was demonstrated on several test cases it can intuitively find the number of clusters and can detect and exclude the outliers automatically while recognizing the clusters regardless of their shape and the dimensions of the space containing them however dpc does have some drawbacks to be addressed before it may be widely applied first the local density i of point i is affected by the cutoff distance dc and is computed in different ways depending on the size of datasets which can influence the clustering especially for small real world cases second the assignment strategy for the remaining points after the density peaks that is the cluster centers have been found can create a domino effect whereby once one point is assigned erroneously then there may be many more points subsequently mis assigned this is especially the case in real word datasets where there could exist several clusters of arbitrary shape overlapping each other to overcome these deficiencies a robust clustering algorithm is proposed in this paper to find the density peaks this algorithm computes the local density i of point i relative to its k nearest neighbors for any size dataset independent of the cutoff distance dc and assigns the remaining points to the most probable clusters using two new point assignment strategies the first strategy assigns non outliers by undertaking a breadth first search of the k nearest neighbors of a point starting from cluster centers the second strategy assigns outliers and the points unassigned by the first assignment procedure using the technique of fuzzy weighted k nearest neighbors the proposed clustering algorithm is benchmarked on publicly available synthetic and real world datasets which are commonly used for testing the performance of clustering algorithms the clustering results of the proposed algorithm are compared not only with that of dpc but also with that of several well known clustering algorithms including affinity propagation ap density based spatial clustering of applications with noise dbscan and k means the benchmarks used are clustering accuracy acc adjusted mutual information ami and adjusted rand index ari the experimental results demonstrate that our proposed clustering algorithm can find cluster centers recognize clusters regardless of their shape and dimension of the space in which they are embedded be unaffected by outliers and can often outperform dpc ap dbscan and k means

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

sparse kernel regression modeling using combined locally regularized orthogonal least squares and d optimality experimental design the note proposes an efficient nonlinear identification algorithm by combining a locally regularized orthogonal least squares lrols model selection with a d optimality experimental design the proposed algorithm aims to achieve maximized model robustness and sparsity via two effective and complementary approaches the lrols method alone is capable of producing a very parsimonious model with excellent generalization performance the d optimality design criterion further enhances the model efficiency and robustness an added advantage is that the user only needs to specify a weighting for the d optimality cost in the combined model selecting criterion and the entire model construction procedure becomes automatic the value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values

the design of energy transfer filters for energy focus filtering this paper is concerned with extending the energy transfer filters etfs designs to a more general case where the design requirement is to focus signal energy from different input frequency bands into a desired output frequency range etfs are a class of non linear filters recently proposed by the authors to employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new etf designs considered in the present study involve the development of an algorithm for determining output frequencies of non linear systems subject to an input the frequency components of which are located in two separate frequency intervals and a new procedure for the implementation of the filter design the results achieved provide an effective approach to the solution to a more general etf design problem

online identification of the bilinear model expansion on laguerre orthonormal bases this paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input to the output and to the crossed product on three independent laguerre orthonormal bases compared to classical bilinear model the resulting model entitled bilinear laguerre model ensures a significant parameter number reduction as well as simple recursive representation however this reduction is still subject to an optimal choice of the laguerre poles defining the three laguerre bases therefore we propose an analytical solution to optimise the laguerre poles which depend on fourier coefficients defining the bilinear laguerre model and that are identified using the regularised square error the identification procedures of the laguerre poles and fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the bilinear laguerre model the bilinear laguerre model as well as the proposed algorithm are illustrated and tested

domain of stability of synchronous generators by a cell mapping approach in this paper the theory of simple cell mapping is applied to the determination of domains of stability of synchronous generators the generators are represented by two dimensional differential dynamical systems including the effects of variable damping and saliency first the differential equation governing the behaviour of a generator is reduced to a cell to cell mapping a scheme of compactification is introduced in order to obtain a complete cell state space which consists of a finite number of cells the methodology of the simple cell mapping is then used to prepare a computer program using the program all the singularities of the cell mapping and the domain of stability can be determined readily the method is particularly attractive because it can easily take care of any non linearity different systems can be treated by the same program by simply changing the form of the differential equation in the program and or changing the system parameters three kinds of synchronous machine are studied i

extended model set global data and threshold model identification of severely non linear systems abstract new parameter estimation algorithms based on an extended model set a global data model and a threshold model formulation are derived for identifying severely non linear systems it is shown that in each case an integrated structure determination and parameter estimation algorithm based on an orthogonal decomposition of the regression matrix can be derived to provide procedures for identifying parsimonious models of unknown systems with complex structure simulation studies are included to illustrate the techniques discussed

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

practical identification of narmax models using radial basis functions a wide class of discrete time non linear systems can be represented by the nonlinear autoregressive moving average narmax model with exogenous inputs this paper develops a practical algorithm for identifying narmax models based on radial basis functions from noise corrupted data the algorithm consists of an iterative orthogonal forward regression routine coupled with model validity tests the orthogonal forward regression routine selects parsimonious radial basistunc tion models while the model validity tests measure the quality of fit the modelling of a liquid level system and an automotive diesel engine are included to demonstrate the effectiveness of the identification procedure

model selection and validation methods for non linear systems the theory of hypothesis testing is used to select a model with the correct structure and the relation of such a method to the aic and fpe criteria is investigated parametric validation and correlation validation methods are developed for non non linear difference equation models several shortcomings of traditional methods especially when applied to non linear systems are described

automorphisms of free groups have finitely generated fixed point sets an automorphism of a finitely generated free group f extends to a homeomorphism of the end completion e of f the set of fixed points of this homeomorphism is finitely generated in a certain sense in particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense the emphasis on e instead of f in this paper is analogous to thurston s study of surface groups where measured laminations are studied instead of simple closed curves the techniques in this paper arose from a study of the behaviour of an automorphism under iteration that is the dynamics of the automorphism see 11 i thank bill thurston for explaining some of his ideas to me for analysing automorphisms of free groups in particular the result on bounded cancellation below is due to thurston and matt grayson peter scott asked whether the fixed subgroup is always finitely generated in 3 and gersten first answered this question in 2 using quite different methods

output frequency response function of nonlinear volterra systems an expression for the output frequency response function ofrf which defines the explicit analytical relationship between the output spectrum and the system parameters is derived for nonlinear systems which can be described by a polynomial form differential equation model an effective algorithm is developed to determine the ofrf directly from system simulation or experimental data simulation studies demonstrate the significance of the ofrf concept and verify the effectiveness of the algorithm which evaluates the ofrf numerically these new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain

dynamic strain mapping and real time damage state estimation under random fatigue loading this paper presents a passive sensing technique for real time structural health monitoring in the proposed technique two reference or healthy state dynamic models are estimated using a nonlinear statistical pattern recognition technique known as the gaussian process the nonlinear dynamic models are estimated by mapping real time dynamic strain measurements at two different locations with corresponding load histories the strain measurements are performed using strain gauge rosettes placed on opposite sides of a probable damage path using the estimated reference model with the new loading information at any given instant of time the corresponding dynamic strains are predicted the predicted strains are compared with the actual strain measurements and used to estimate the corresponding damage states via correlation analysis the state estimation approach is demonstrated on an aluminum 2024 cruciform specimen fatigued under random biaxial loading the numerical study shows that the proposed approach can estimate real time damage states caused by both stage ii and stage iii fatigue cracks

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

application of a general multi model approach for identification of highly nonlinear processes a case study an identification method for highly nonlinear processes is proposed based on a multi model approach and kolmogorov gabor polynomials owing to the large number of possible terms in this general model structure the significant terms are selected by several statistical test procedures leading automatically to a minimal order model realization the performance of this method is evaluated in an in depth case study using a simulated ph neutralization process the effects of important variables such as range of operating conditions signal noise ratio and data length are discussed

nonlinear system identification and prediction using orthogonal functions we describe a systematic scheme for the nonlinear adaptive filtering of signals that are generated by nonlinear dynamical systems the complete filter consists of three sections a signal independent standard orthonormal expansion a scaling derived from an estimate of the vector probability density function pdf and an adaptive linear combiner the orthonormal property of the expansions has two significant implications for adaptive filtering first model order reduction is trivial since the contribution of each term to the mean squared error is directly related to the coefficient in the final linear combiner and second consistent and rapid convergence of stochastic gradient algorithms is assured a technique based on the inverse fourier transform for obtaining a pdf estimate from the characteristic function is also presented the prediction and identification performance of this nonlinear structure is examined for a number of signals and it is contrasted with common radial basis function and linear networks

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

a suboptimal bootstrap method for structure detection of non linear output error models with application to human ankle dynamics identification of non linear systems involves estimating unknown parameters and structure detection selection of a subset of candidate terms that best describe the observed output this is a necessary procedure to compute an efficient system description which may afford greater insight into the functionality of the system or a simpler controller design for non linear systems simple output additive noise can generate multiplicative terms between the input output and noise the terms associated with noise need to be modelled to obtain unbiased parameter estimates significantly increasing the number of candidate terms to be estimated and considered in special cases it may be possible to use an output error oe model structure and the instrumental variable iv estimator to obtain unbiased parameters without modelling the noise this significantly reduces the dimensionality of the structure computation problem therefore in this paper we propose a suboptimal bootstrap structure detection sobsd algo

identification of non linear parametrically varying models using separable least squares in this paper a novel identification algorithm for a class of non linear possibly parameter varying models is proposed the algorithm is based on separable least squares ideas these models are given in the form of a linear fractional transformation lft where the forward part is represented by a conventional linear regression and the feedback part is given by a non linear map which can take into account scheduling variables available for measurement the non linear part of the model can be parameterized according to various paradigms like e g neural network nn or general nonlinear autoregressive exogenous narx models the estimation algorithm exploits the separability of the criterion used to estimate the parameters when using a nn it is possible the explicit computation of the frechet derivative needed to implement a separable least square algorithm

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

fast orthogonal identification of nonlinear stochastic models and radial basis function neural networks a new fast orthogonal estimation algorithm is derived for a wide class of nonlinear stochastic models including training radial basis function neural networks the selection of significant regressors and the estimation of unknown parameters in the presence of nonlinear noise sources are considered and simulated examples are included to demonstrate the efficiency of the new procedure

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples