noise sensitivity of portfolio selection in constant conditional correlation garch models abstract this paper investigates the efficiency of minimum variance portfolio optimization for stock price movements following the constant conditional correlation garch process proposed by bollerslev simulations show that the quality of portfolio selection can be improved substantially by computing optimal portfolio weights from conditional covariances instead of unconditional ones measurement noise can be further reduced by applying some filtering method on the conditional correlation matrix such as random matrix theory based filtering as an empirical support for the simulation results the analysis is also carried out for a time series of s p500 stock prices

non stationary correlation matrices and noise abstract the exact meaning of the noise spectrum of eigenvalues of the correlation matrix is discussed in order to better understand the possible phenomena behind the observed noise the spectrum of eigenvalues of the correlation matrix is studied under a model where most of the true eigenvalues are zero and the parameters are non stationary the results are compared with real observation of brazilian assets suggesting that although the non stationarity seems to be an important aspect of the problem partially explaining some of the eigenvalues as well as part of the kurtosis of the assets it cannot by itself provide all the corrections needed to make the proposed model fit the data perfectly

universal correlations and power law tails in financial covariance matrices abstract we investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power law tails in order to generate individual eigenvalue distributions a chopping procedure is devised which produces a statistical ensemble of asset price covariances from a single instance of financial data sets local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets they are in good agreement with the universal tracy widom distribution and wigner surmise respectively this suggests a strong degree of robustness especially in the low lying sector of the spectra most relevant for portfolio selections conversely the global spectral density of a single covariance matrix as well as the average over all unfolded nearest neighbour spacing distributions deviate from standard gaussian random matrix predictions the data are in fair agreement with a recently introduced generalised random matrix model with correlations showing a power law decay

large deviations of the maximum eigenvalue in wishart random matrices we analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the wishart laguerre ensemble of positive definite random matrices we show that the probability that all the eigenvalues of a n n wishart matrix w xtx where x is a rectangular m n matrix with independent gaussian entries are smaller than the mean value n c decreases for large n as where 1 2 corresponds respectively to real and complex wishart matrices c n m 1 and x c is a rate sometimes also called large deviation function that we compute explicitly the result for the anti wishart case m n simply follows by exchanging m and n we also analytically determine the average spectral density of an ensemble of wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier numerical simulations are in excellent agreement with the analytical predictions

subspace based algorithm for parameter estimation of polynomial phase signals in this correspondence parameter estimation of a polynomial phase signal pps in additive white gaussian noise is addressed assuming that the order of the pps is at least 3 the basic idea is first to separate its phase parameters into two sets by a novel signal transformation procedure and then the multiple signal classification music method is utilized for joint estimating the phase parameters with second order and above in doing so the parameter search dimension is reduced by a half as compared to the maximum likelihood and nonlinear least squares approaches in particular the problem of cubic phase signal estimation is studied in detail and its simplification for a chirp signal is given the effectiveness of the proposed approach is also demonstrated by comparing with several conventional techniques via computer simulations

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

correlations between eigenvalues of large random matrices with independent entries we derive the connected correlation functions for eigenvalues of large hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group rg inspired approach with the diagrammatic method we obtain a general form for the one two and three point connected green functions for this class of ensembles when matrix elements are identically distributed and then discuss the derivation of higher order functions by the same approach using the rg approach we rederive the one and two point green functions and show they are unchanged by choosing certain ensembles with nonidentically distributed elements throughout we compare the green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes textcopyright 1996 the american physical society

product high order ambiguity function for multicomponent polynomial phase signal modeling parameter estimation and performance analysis issues are studied for multicomponent polynomial phase signals ppss embedded in white gaussian noise identifiability issues arising with existing approaches are described first when dealing with multicomponent pps having the same highest order phase coefficients this situation is encountered in applications such as synthetic aperture radar imaging or propagation of polynomial phase signals through channels affected by multipath and is thus worthy of a careful analysis a new approach is proposed based on a transformation called product high order ambiguity function phaf the use of the phaf offers a number of advantages with respect to the high order ambiguity function haf more specifically it removes the identifiability problem and improves noise rejection capabilities performance analysis is carried out using the perturbation method and verified by simulation results

random matrix theory and fund of funds portfolio optimisation abstract the proprietary nature of hedge fund investing means that it is common practise for managers to release minimal information about their returns the construction of a fund of hedge funds portfolio requires a correlation matrix which often has to be estimated using a relatively small sample of monthly returns data which induces noise in this paper random matrix theory rmt is applied to a cross correlation matrix c constructed using hedge fund returns data the analysis reveals a number of eigenvalues that deviate from the spectrum suggested by rmt the components of the deviating eigenvectors are found to correspond to distinct groups of strategies that are applied by hedge fund managers the inverse participation ratio is used to quantify the number of components that participate in each eigenvector finally the correlation matrix is cleaned by separating the noisy part from the non noisy part of c this technique is found to greatly reduce the difference between the predicted and realised risk of a portfolio leading to an improved risk profile for a fund of hedge funds

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

the coiflet galerkin method for linear volterra integral equations this paper deals with the application of the wavelet galerkin method based on coiflets as a basis for solving linear volterra integral equations vies the main contribution of this work is that some new connection coefficients are introduced and a suitable algorithm is developed for their solution once they have been computed they can be stored and applied to any linear vie the convergence properties of the coiflet galerkin method are analyzed some test examples are presented to illustrate the performance of the method with respect to the error norms and cpu time

cross correlation dynamics in financial time series abstract the dynamics of the equal time cross correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows empirical results for the s p 500 and the dow jones euro stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross correlation matrix over these time windows oppose those of the largest eigenvalue this behaviour is shown to be independent of the size of the time window and the number of stocks examined a basic one factor model is then proposed which captures the main dynamical features of the eigenvalue spectrum of the empirical data through the addition of perturbations to the one factor model leading to a market plus sectors model additional sectoral features are added resulting in an inverse participation ratio comparable to that found for empirical data by partitioning the eigenvalue time series we then show that negative index returns drawdowns are associated with periods where the largest eigenvalue is greatest while positive index returns drawups are associated with periods where the largest eigenvalue is smallest the study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies

random but not so much a parameterization for the returns and correlation matrix of financial time series abstract a parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied this parameterization allows easy introduction of non stationarity and it shows several of the characteristics of the true observed realizations such as fat tails volatility clustering and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of random matrix theory results the predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of brazilian assets and it is shown that those predictions fit the data better than random matrix theory

application of taylor series in obtaining the orthogonal operational matrix in this research first we explicitly obtain the relation between the coefficients of the taylor series and jacobi polynomial expansions then we present a new method for computing classical operational matrices derivative integral and product for general jacobi orthogonal functions polynomial and rational this method can be used for many classes of orthogonal functions

polynomial wigner ville distributions and their relationship to time varying higher order spectra the wigner ville distribution wvd has optimal energy concentration for linear frequency modulated fm signals this paper presents a generalization of the wvd in order to effectively process nonlinear polynomial fm signals a class of polynomial wvd s pwvd s that give optimal concentration in the time frequency plane for fm signals with a modulation law of arbitrary polynomial form are defined a class of polynomial time frequency distributions ptfd s are also defined based on the class of pwvd s the optimal energy concentration of the pwvd enables it to be used for estimation of the instantaneous frequency if of polynomial fm signals finally a link between pwvd s and time varying higher order spectra tvhos is established just as the expected value of the wvd of a nonstationary random signal is the time varying power spectrum the expected values of the pwvd s have interpretations as reduced tvhos

overlaps between eigenvectors of correlated random matrices we obtain general exact formulas for the overlaps between the eigenvectors of large correlated random matrices with additive or multiplicative noise these results have potential applications in many different contexts from quantum thermalization to high dimensional statistics we find that the overlaps only depend on measurable quantities and do not require the knowledge of the underlying true noiseless matrices we apply our results to the case of empirical correlation matrices that allow us to estimate reliably the width of the spectrum of the true correlation matrix even when the latter is very close to the identity we illustrate our results on the example of stock returns correlations which clearly reveal a nontrivial structure for the bulk eigenvalues we also apply our results to the problem of matrix denoising in high dimensions

a fast algorithm for estimating the parameters of a quadratic fm signal this paper describes a fast algorithm that can be used for estimating the parameters of a quadratic frequency modulated fm signal the proposed algorithm is fast in that it requires only one dimensional 1 d maximizations the optimal maximum likelihood method by contrast requires a three dimensional 3 d maximization which can only be realized with an exhaustive 3 d grid search asymptotic statistical results are derived for all the estimated parameters the amplitude estimate is seen to be optimal whereas the phase parameters are in general suboptimal of the four phase parameter estimates two approach optimality as the signal to noise ratio snr tends to infinity the other two have mean square errors that are within 50 of the theoretical lower bounds for high snr simulations are provided to support the theoretical results extensions to multiple components and higher order fm signals are also discussed

efficient algorithms to solve singular integral equations of abel type in the present paper we obtain the approximate solution of abel s integral equation by using the following powerful efficient but simple methods i he s homotopy perturbation method hpm ii modified homotopy perturbation method mhpm iii adomian decomposition method adm and iv modified adomian decomposition method madm the validity and applicability of these techniques are illustrated through various particular cases which demonstrate their efficiency and simplicity in solving these types of integral equations compared with the other existing methods

emergence of correlations between securities at short time scales abstract the correlation matrix is the key element in optimal portfolio allocation and risk management in particular the eigenvectors of the correlation matrix corresponding to large eigenvalues can be used to identify the market mode sectors and style factors we investigate how these eigenvalues depend on the time scale of securities returns in the u s market for this purpose one minute returns of the largest 533 u s stocks are aggregated at different time scales and used to estimate the correlation matrix and its spectral properties we reveal the emergence of several dominant eigenvalues as the time scale increases a simple lead lag factor model is proposed to capture and reproduce the observed time scale dependence of eigenvalues using this model the relaxation time of the eigenvalues emergence is estimated to be around one minute for all the dominant eigenmodes including the market mode as a consequence the use of five minute returns time series for inferring correlations between stocks turns out to be a good compromise between statistical abundance of data points and well established correlations our findings evidence that the underlying economic and financial mechanisms determining the correlation structure of securities depend as well on time scales

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

signal and noise in correlation matrix using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator this relation can be used in practice to compute eigenvalue invariants of the covariance correlation matrix results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom like for instance those in statistical physics lattice measurements of field theory genetics quantitative finance and other applications of multivariate statistics

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

polynomial phase signal analysis based on the polynomial derivatives decompositions an exact decomposition of the derivatives of any order of a polynomial spl phi t is proposed in terms of spl phi t t sub 0 spl phi t t sub n this result allows us to introduce generalized time frequency distributions for studying signals having a polynomial phase and a constant amplitude in order to determine the degree and the coefficients of the corresponding phase the relationships between these distributions and the already known polynomial distributions i e the polynomial phase transform and the polynomial wigner ville distribution are discussed illustrations by example are proposed

correlation functions in disordered systems recently we found that the correlation between the eigenvalues of random hermitian matrices exhibits universal behavior here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary we compute the current current correlation function discuss various generalizations and compare our work with the work of other authors we study the distribution of eigenvalues of hamiltonians consisting of a sum of a deterministic term and a random term the correlation between the eigenvalues when the deterministic term is varied is calculated

on the analysis of cross correlations in south african market data we report briefly on an application of random matrix theory to the analysis of sa financial market data an analysis of cross correlations in south african financial market data e print cond mat 0402389 correlation matrices c are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange from january 1993 to december 2002 spectral properties of c are tested against random matrix predictions we highlight some quantitative differences which arise when treating prices as existing only when measured as opposed to interpolating missing or illiquid trading days with a zero order hold

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

design of higher order polynomial wigner ville distributions the polynomial wigner ville distribution pwvd is a time frequency signal analysis tool for representing polynomial frequency modulated fm signals we present an optimization procedure to design pwvds and discuss the need for higher order pwvds

a wavelet based computational method for solving stochastic ito volterra integral equations this paper presents a computational method based on the chebyshev wavelets for solving stochastic ito volterra integral equations first a stochastic operational matrix for the chebyshev wavelets is presented and a general procedure for forming this matrix is given then the chebyshev wavelets basis along with this stochastic operational matrix are applied for solving stochastic ito volterra integral equations convergence and error analysis of the chebyshev wavelets basis are investigated to reveal the accuracy and efficiency of the proposed method some numerical examples are included

signal and noise in financial correlation matrices abstract using random matrix theory one can derive exact relations between the eigenvalue spectrum of the covariance matrix and the eigenvalue spectrum of its estimator experimentally measured correlation matrix these relations will be used to analyze a particular case of the correlations in financial series and to show that contrary to earlier claims correlations can be measured also in the random part of the spectrum implications for the portfolio optimization are briefly discussed

joint model order selection and parameter estimation of chirps with harmonic components we consider the problem of jointly determining the number of harmonic components of a fundamental linear chirp and estimating its parameters i e its initial frequency and frequency rate given time samples of the observed signal common model order criteria select the number of harmonics based on the maximum likelihood estimator we develop exact and approximated maximum likelihood estimators of these parameters to avoid an exhaustive search in the initial frequency frequency rate space involved by those estimators we propose an alternative low complexity two step estimation method the first step separates the signal to its harmonic components then in the second step the parameters of interest are estimated using least squares method given the phases of the harmonic components the method is compared to the exact and approximated maximum likelihood estimators and to the well known high order ambiguity function based method numerical simulations and real data examples demonstrate that the proposed low complexity method can successfully replace the maximum likelihood estimator in the model order criteria at moderate to high signal to noise ratio since the estimates obtained by the proposed method achieve the cramer rao lower bound at these signal to noise ratios

spectrum estimation from samples we consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution equivalently the problem of approximating the population spectrum given access to samples drawn from the distribution we consider this recovery problem in the regime where the sample size is comparable to or even sublinear in the dimensionality of the distribution first we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix we then leverage this accurate moment recovery via a wasserstein distance argument to accurately reconstruct the vector of eigenvalues together this yields an eigenvalue reconstruction algorithm that is asymptotically consistent as the dimensionality of the distribution and sample size tend toward infinity even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero in addition to our theoretical results we show that our approach performs well in practice for a broad range of distributions and sample sizes

a transform based covariance differencing approach for some classes of parameter estimation problems a method of parameter estimation is presented for a class of problems in which the desired signal is embedded in colored noise with unknown covariance the proposed algorithm does not require multiple estimates of the signal covariance matrix instead it uses a priori knowledge of the structure of the noise covariance matrix to transform the array covariance matrix in a way that leaves its noise component unchanged while the signal component is transformed in some appropriate manner the noise component is then eliminated by forming the difference of the original and transformed covariance differencing methods to be applied to a wider class of problems than was previously possible this is illustrated by applying the algorithm to the problems of bearing estimation resolution of overlapping echos and transient response analysis simulation results demonstrate that the technique affords significant improvement in certain situations over conventional methods

optimal allocation of trend following strategies abstract we consider a portfolio allocation problem for trend following tf strategies on multiple correlated assets under simplifying assumptions of a gaussian market and linear tf strategies we derive analytical formulas for the mean and variance of the portfolio return we construct then the optimal portfolio that maximizes risk adjusted return by accounting for inter asset correlations the dynamic allocation problem for n assets is shown to be equivalent to the classical static allocation problem for n 2 virtual assets that include lead lag corrections in positions of tf strategies the respective roles of asset auto correlations and inter asset correlations are investigated in depth for the two asset case and a sector model in contrast to the principle of diversification suggesting to treat uncorrelated assets we show that inter asset correlations allow one to estimate apparent trends more reliably and to adjust the tf positions more efficiently if properly accounted for inter asset correlations are not deteriorative but beneficial for portfolio management that can open new profit opportunities for trend followers these concepts are illustrated using daily returns of three highly correlated futures markets the e mini s p 500 euro stoxx 50 index and the us 10 year t note futures

the discrete polynomial phase transform the discrete polynomial phase transform dpt is a new tool for analyzing constant amplitude polynomial phase signals the main properties of the dpt are its ability to identify the degree of the phase polynomial and to estimate its coefficients the transform is robust to deviations from the ideal signal model such as slowly varying amplitude additive noise and nonpolynomial phase the authors define the dpt derive its basic properties and use it to develop computationally efficient estimation and detection algorithms a statistical accuracy analysis of the estimated parameters is also presented

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

localized short range correlations in the spectrum of the equal time correlation matrix we suggest a procedure to identify those parts of the spectrum of the equal time correlation matrix mathbf c where relevant information about correlations of a multivariate time series is induced using an ensemble average over each of the distances between eigenvalues all nearest neighbor distributions can be calculated individually we present numerical examples where a information about cross correlations is found in the so called bulk of eigenvalues which generally is thought to contain only random correlations and where b the information extracted from the lower edge of the spectrum of mathbf c is statistically more significant than that extracted from the upper edge we apply the analysis to electroencephalographic recordings with epileptic events

estimating signal parameters using the nonlinear instantaneous least squares approach a novel method for signal parameter estimation is presented termed the nonlinear instantaneous least squares nils estimator the basic idea is to use the observations in a sliding window to compute an instantaneous short term estimate of the amplitude used in the separated nonlinear least squares nlls criterion the effect is a significant improvement of the numerical properties in the criterion function which becomes well suited for a signal parameter search for small sized sliding windows the global minimum in the nlis criterion function is wide and becomes easy to find for maximum size windows the nils is equivalent to the nlls estimator which implies statistical efficiency for gaussian noise a blind signal parameter search algorithm that does not use any a priori information is proposed the nils estimator can be interpreted as a signal subspace projection based algorithm moreover the nils estimator can be interpreted as an estimator based on the prediction error of a structured linear predictor hereby a link is established between nlls signal subspace fitting and linear prediction based estimation approaches the nils approach is primarily applicable to deterministic signal models specifically polynomial phase signals are studied and the nils approach is evaluated and compared with other approaches simulations show that the signal to noise ratio snr threshold is significantly lower than that of the other methods and it is confirmed that the estimates are statistically efficient just as the nlls approach the nils estimator can be applied to nonuniformly sampled data

an effective scheme for solving system of fractional volterra fredholm integro differential equations based on the muntz legendre wavelets abstract in this study a class of wavelet techniques is used for finding approximate solutions of systems of fractional integro differential volterra fredholm fidvf equations based on the muntz legendre wavelets mlw for the suggested method operational matrices of the riemann liouville fractional rlf integral and caputo fractional cf derivative operators are obtained and used for converting the system of the integral equations into a system of linear or nonlinear algebraic equations using the lipschitz s condition for multivariate functions and the fixed point theorem the existence and uniqueness of the solution are shown and also convergence stability and error bound of the solution in interval 0 1 are investigated in this work at the end three examples are indicated and the results of the proposed method are compared with the first and second kind wavelet chebyshev methods

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

numerical solution of differential equations by using chebyshev wavelet operational matrix of integration in this paper we present operational matrix of integration omi of chebyshev wavelets basis and the product operation matrix pom of it some comparative examples are included to demonstrate the superiority of operational matrix of chebyshev wavelets to those of legendre wavelets

a wavelet based approach for vibration analysis of framed structures in this paper an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions initially the differential equation of vibration governing sdof single degree of freedom systems has been solved by wavelet operators and later the proposed approach has been generalized for mdof multi degrees of freedom systems for this purpose two different types of wavelet functions have been exemplified including complex chebyshev wavelet functions and simple haar wavelet functions in the proposed approach a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition emphasizing on frequency domain approximation a simple step by step algorithm has been implemented and improved to calculate the response of mdof systems moreover stability and accuracy of results have been evaluated the effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of newmark b wilson q and central difference method in all the procedures computation time involved has also been considered finally it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach particularly in large scaled systems

a fast and accurate single frequency estimator a novel frequency estimator for a single complex sinusoid in complex white gaussian noise is proposed the estimator is more computationally efficient that the optimal maximum likelihood estimator yet attains equally good performance at moderately high signal to noise ratios the estimator is shown to be related to the linear prediction estimator this relationship is used to reveal why the linear prediction estimator does not attain the cramer rao bound even at high signal to noise ratios

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

estimating the doa and the polarization of a polynomial phase signal using a single polarized vector sensor this paper introduces a novel algorithm to estimate the direction of arrival doa and the polarization of a completely polarized polynomial phase signal of an arbitrary degree the algorithm utilizes a polarized vector sensor comprising a spatially collocated six component electromagnetic vector sensor a dipole triad or a loop triad this esprit based algorithm is based on a time invariant matrix pencil pair derived from the time delayed data sets collected by a single polarized vector sensor the high order difference function of the signal s phase constructs the invariant factor used in the esprit algorithm the steering vector is estimated from the signal subspace eigenvector of the data correlation matrix following which the closed form doa and polarization can be obtained given the degree of the polynomial phase signal this approach resolves the two dimensional azimuth elevation angle and the polarization of the source and requires neither a priori knowledge of the polynomial phase signal s coefficients nor a priori knowledge of the polynomial phase signal s frequency spectrum the efficacy of the proposed algorithm is verified by monte carlo simulations estimation accuracies of the doa and the polarization parameters are evaluated by the closed form cramer rao bounds which are independent of the polynomial coefficients the degree of the polynomial phase signal and the azimuth angle of the source

estimating the frequency of a noisy sinusoid by linear regression corresp the estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated at high signal to noise ratios the additive noise is viewed as an equivalent phase noise suggesting frequency and phase estimation by linear regression on the signal phase the variances of the regression estimates are shown to achieve the cramer rao bounds a formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum a simple formula for the variance with 1 f 2 phase noise is presented

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

discretization algorithm for fractional order integral by haar wavelet approximation abstract a discretization algorithm is proposed by haar wavelet approximation theory for the fractional order integral in this paper the integration time is divided into two parts one presents the effect of the past sampled data calculated by the iterative method and the other presents the effect of the recent sampled data at a fixed time interval calculated by the haar wavelet this method can reduce the amount of the stored data effectively and be applied to the design of discrete time fractional order pid controllers finally several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

rotational invariant estimator for general noisy matrices we investigate the problem of estimating a given real symmetric signal matrix c from a noisy observation matrix m in the limit of large dimension we consider the case where the noisy measurement m comes either from an arbitrary additive or multiplicative rotational invariant perturbation we establish using the replica method the asymptotic global law estimate for three general classes of noisy matrices significantly extending previously obtained results we give exact results concerning the asymptotic deviations called overlaps of the perturbed eigenvectors away from the true ones and we explain how to use these overlaps to clean the noisy eigenvalues of m we provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of bayesian statistics

comparison of eigeninference based on one and two point green s functions we compare two methods of eigeninference from large sets of data our analysis points at the superiority of our eigeninference method based on one point green s functions and pade approximants over a method based on fluctuations and two point green s functions the first method is orders of magnitude faster than the second one moreover we found a source of potential instability of the second method and identified it as arising from the spurious zero and negative modes of the estimator for the variance operator of a certain multidimensional gaussian distribution inherent for that method we also present eigeninference based on spectral moments of negative orders for strictly positive spectra finally we compare the cases of eigeninference of real valued and complex valued correlated wishart distributions reinforcing our conclusions on the advantage of the one point green s function method

spectrum estimation for large dimensional covariance matrices using random matrix theory estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics the eigenvalues of covariance matrices play a key role in many widely used techniques in particular in principal component analysis pca in many modern data analysis problems statisticians are faced with large datasets where the sample size n is of the same order of magnitude as the number of variables p random matrix theory predicts that in this context the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance we propose to use a fundamental result in random matrix theory the marcenko pastur equation to better estimate the eigenvalues of large dimensional covariance matrices the marcenko pastur equation holds in very wide generality and under weak assumptions the estimator we obtain can be thought of as shrinking in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues inspired by ideas of random matrix theory we also suggest a change of point of view when thinking about estimation of high dimensional vectors we do not try to estimate directly the vectors but rather a probability measure that describes them we think this is a theoretically more fruitful way to think about these problems our estimator gives fast and good or very good results in extended simulations our algorithmic approach is based on convex optimization we also show that the proposed estimator is consistent

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

chebyshev wavelets approach for nonlinear systems of volterra integral equations in this paper a new approach for solving nonlinear systems of volterra integral equations has been proposed the method is based on chebyshev wavelets approximations the method is described and after that the error is analyzed at the end some examples are presented to illustrate the ability and simplify of the method and the results reveal the effectiveness of the technique

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

an analysis of cross correlations in an emerging market abstract we apply random matrix theory to compare correlation matrix estimators c obtained from emerging market data the correlation matrices are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange jse from january 1993 to december 2002 we test the spectral properties of c against random matrix predictions and find some agreement between the distributions of eigenvalues nearest neighbour spacings distributions of eigenvector components and the inverse participation ratios for eigenvectors we show that interpolating both missing data and illiquid trading days with a zero order hold increases agreement with rmt predictions for the more realistic estimation of correlations in an emerging market we suggest a pairwise measured data correlation matrix for the data set used this approach suggests greater temporal stability for the leading eigenvectors an interpretation of eigenvectors in terms of trading strategies is given as opposed to classification by economic sectors

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

a new class of multilinear functions for polynomial phase signal analysis this paper introduces a new class of multilinear functions which can be used for analyzing a signal with time varying frequency the new class subsumes a number of existing functions including the higher order ambiguity functions hafs the polynomial wigner ville distributions pwvds and the higher order phase hp functions as well as establishing a link between these existing functions the new class provides a formalism which allows for the creation of useful new multilinear functions a number of new functions are derived from the class

theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane emden type equations abstract in this article a new method is generated to solve nonlinear lane emden type equations using legendre hermite and laguerre wavelets we are interested to note that these wavelets will give same solutions with good accuracy theorems on convergence analysis are stated and proved on the spaces which are created by legendre hermite and laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis the main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients a very high level of accuracy reflects the reliability of this scheme for such problems

on polynomial phase signals with time varying amplitudes we address the parameter estimation problem for a class of nonstationary signals modeled as polynomial phase signals with time varying amplitudes exponentially damped polynomial phase signals are treated as a special case and are analyzed in detail high order instantaneous moments provide the basic analytical tool but links are shown to exist with either the usually employed fft based technique or the high resolution kumaresan tufts 1982 music and matrix pencil methods asymptotic properties of the relevant estimators are established cramer rao lower bounds on the amplitude and phase parameter estimates are derived and computer simulations are carried out to evaluate the performance of various schemes we focus on parametric modeling of am fm signals mainly because parametric techniques offer parsimony and allow for theoretically unlimited resolution