a new approach to frequency analysis with amplified harmonics summary for stationary time series having a mixed spectrum we introduce a new method for estimating the discrete frequencies based on amplification of the sine waves in the time series and on minimization of least squares regressions on amplified waves under the assumption that the continuous part of the spectrum arises from a stationary autoregressive moving average process we show that the estimators for the discrete frequencies obtained with this method are strongly consistent more precisely as the serial length n tends to infinity their bias almost surely converges to zero with the rate n 3 2 log n 5 6 2 we also establish a central limit theorem which shows that these estimators have exactly the same asymptotic variance as that given by whittle s method which is known to yield the most accurate frequency estimates however the new technique has two advantages over whittle s method first it yields an algorithm that is both robust against improper starting estimates and computationally efficient secondly it gives rise to a simple tool based on its amplified harmonics for identifying the number of sine waves in a time series a simulation study is also reported that gives statistical results on the properties of the new method on simulated series as well as comparative results with whittle s method

instantaneous frequency estimation using the wigner distribution with varying and data driven window length the estimation of the instantaneous frequency if of a harmonic complex valued signal with an additive noise using the wigner distribution is considered if the if is a nonlinear function of time the bias of the estimate depends on the window length the optimal choice of the window length based on the asymptotic formulae for the variance and bias can be used in order to resolve the bias variance tradeoff however the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the if the goal of this paper is to develop an adaptive if estimator with a time varying and data driven window length which is able to provide quality close to what could be achieved if the smoothness of the if were known in advance the algorithm uses the asymptotic formula for the variance of the estimator only its value may be easily obtained in the case of white noise and relatively high sampling rate simulation shows good accuracy for the proposed adaptive algorithm

revised notation for partial derivatives the principle of small quantities of leibniz s notation for total derivatives is applied logically to partial derivative notation the result is the notation fdx fdy instead of the old customary fdx fdy this improvement removes serious difficulties of the old notation

estimating the frequency of a periodic function summary the problem of estimating the frequency and other parameters of a cyclical oscillation is considered where the data consists of a periodic function observed subject to stationary additive noise an estimation procedure is proposed and the asymptotic properties of the estimators established tests for unknown frequencies to be harmonics of a fundamental frequency are also developed and their asymptotic properties investigated the procedures are applied to observations on the variable star s carinae

nonparametric inference for periodic sequences this article proposes a nonparametric method for estimating the period and values of a periodic sequence when the data are evenly spaced in time the period is estimated by a leave out one cycle version of cross validation cv and complements the periodogram a widely used tool for period estimation the cv method is computationally simple and implicitly penalizes multiples of the smallest period leading to a virtually consistent estimator of integer periods this estimator is investigated both theoretically and by simulation we also propose a nonparametric test of the null hypothesis that the data have constant mean against the alternative that the sequence of means is periodic finally our methodology is demonstrated on three well known time series the sunspots and lynx trapping data and the el nino series of sea surface temperatures

discrete time local polynomial approximation of the instantaneous frequency the local polynomial approximation lpa of the time varying phase is used to develop a new form of the fourier transform and the local polynomial periodogram lpp as an estimator of the instantaneous frequency if spl omega t of a harmonic complex valued signal the lpp is interpreted as a time frequency energy distribution over the t spl omega t spl omega sup 1 t spl omega sup m 1 t space where m is a degree of the lpa the variance and bias of the estimate are studied for the short and long time asymptotic behavior of the if estimates in particular it is shown that the optimal asymptotic mean squared errors of the estimates of spl omega sup k 1 t have orders o n sup 2k 1 and o n sup 2 m k 1 2m 3 k 1 2 m respectively for a polynomial spl omega t of the degree m 1 and arbitrary smooth spl omega t with a bounded mth derivative

application of radial basis function with l stable pade time marching scheme for pricing exotic option in this work a high order l stable method for pricing exotic option has been discussed the spatial discretization is done by radial basis function based local grid free method to achieve first order ordinary differential equation followed by fourth order l stable method numerical study with one and two asset problems for digital option butterfly spread and barrier option is carried out with highly accurate results that are in good agreement with those obtained by other numerical methods in the literature

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

the statistical performance of some instantaneous frequency estimators the authors examine the class of smoothed central finite difference scfd instantaneous frequency if estimators which are based on finite differencing of the phase of the analytic signal these estimators are closely related to if estimation via the periodic first moment with respect to frequency of discrete time frequency representations tfrs in l cohen s 1966 class the authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero crossing estimator and one based on linear regression on the signal phase it is found that the regression if estimator is biased and exhibits a large threshold for much of the frequency range by replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic scfd pscfd estimator which is unbiased and has a frequency independent variance yet retains the optimal performance and simplicity of the original estimator

repeated spatial extrapolation an extraordinarily efficient approach for option pricing various finite difference methods for option pricing have been proposed in this paper we demonstrate how a very simple approach namely the repeated spatial extrapolation can perform extremely better than the finite difference schemes that have been developed so far in particular we consider the problem of pricing vanilla and digital options under the black scholes model and show that if the payoff functions are dealt with properly then errors close to the machine precision are obtained in only some hundredths of a second

conditional mean and maximum likelihood approaches to multiharmonic frequency estimation the performance of an extended kalman filter ekf applied to the problem of estimating the assumed constant parameters fundamental frequency harmonic phases and amplitudes of a complex multiharmonic signal measured in noise is shown to be asymptotically i e as the number of measurements tends to infinity efficient the cramer rao cr bounds associated with the estimation problem are derived for the case where the measurements commence at an arbitrary time distinct from zero

the relationship between instantaneous frequency and time frequency representations the relationship between instantaneous frequency estimation based on the derivative of the phase of the analytic signal and the first moment of the general time frequency representation from cohen s 1989 class is given in the continuous and discrete time domains application of the standard linear definition of first moment to discrete time frequency representations leads to biased instantaneous frequency estimators with high variance to correct this problem it is shown that periodic circular definitions of moments must be used to account for the periodization of the frequency variable due to sampling

limitations on snr estimator accuracy we consider the samples of a pure tone in additive white gaussian noise awgn for which we wish to determine the signal to noise ratio snr defined here to be spl alpha a sup 2 2 spl sigma sup 2 where a is the tone amplitude and spl sigma sup 2 is the noise variance a and spl sigma sup 2 are assumed to be deterministic but unknown a priori if the variance of an unbiased estimator of spl alpha is spl sigma sub spl alpha spl circ sup 2 we show that at high snr the normalized standard a deviation satisfies the cramer rao lower bound crlb according to spl sigma sub spl alpha spl circ spl alpha spl ges spl radic 2 n where n is the number of independent observables used to obtain the snr estimate spl sigma spl circ

semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals we propose a semiparametric approach to fundamental frequency estimation of an unknown periodic signal in additive white noise based on model selection our estimator maximizes a penalized version of the cumulated periodogram and is proved to be consistent and asymptotically efficient under very general conditions when the number of observations is fixed an implementation of this estimation method is proposed and illustrated on specific synthetic signals which arise in laser vibrometry we extend this method for estimating the fundamental frequencies of two periodic functions having different fundamental frequencies when the data consist of their sum and additive white noise we also compare the performances of our procedure with the so called microdoppler technique which is commonly used for laser vibrometry signals analysis we show on simulated data that the penalized cumulated periodogram yields an accurate estimation of the frequencies at very low signal to noise ratios

nonparametric estimation of a periodic function summary motivated by applications to brightness data on periodic variable stars we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times it is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known also first order properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known numerical simulations and applications to real data show the method to work well in practice

nonparametric methods for deconvolving multiperiodic functions summary multiperiodic functions or functions that can be represented as finite additive mixtures of periodic functions arise in problems related to stellar radiation there they represent the overall variation in radiation intensity with time the individual periodic components generally correspond to different sources of radiation and have intrinsic physical meaning provided that they can be deconvolved from the mixture we suggest a combination of kernel and orthogonal series methods for performing the deconvolution and we show how to estimate both the sequence of periods and the periodic functions themselves we pay particular attention to the issue of identifiability in a nonparametric sense of the components this aspect of the problem is shown to exhibit particularly unusual features and to have connections to number theory the matter of rates of convergence of estimators also has links there although we show that the rate of convergence problem can be treated from a relatively conventional viewpoint by considering an appropriate prior distribution for the periods

the phase of a vector perturbed by gaussian noise and differentially coherent receivers some results are presented regarding the asymptotic distribution of the phase of a vector perturbed by gaussian noise it is shown that for large signal to noise ratio the asymptotic distribution of the phase is of the tikhonov type this framework is then used for the synthesis of differentially coherent receiver structures one for m ary phase shift keying mpsk and the other for minimum shift keying msk the first structure bridges the performance gap between coherent and differentially coherent demodulation of mpsk the msk receiver uses matched filtering with differential demodulation

branched channel free streamlines abstract two configurations of free streamlines in a branched channel in plane incompressible potential flow are analysed for three different branch angles these cases were studied before by modi et al and sinha et al but their results are found to be incorrect the hodograph method is used producing a schwarz christoffel like integral over a semi disk region the integration is carried out using convolution number algebra which produces the potential flow in analytical form the unknown parameters in the integral are solved by iteration using the newton raphson method with the jacobian matrix approximated by a finite difference quotient matrix it is shown how a further conformal mapping of the semi disk to the full circular disk leads to the transformation of elcrat and trefethen

analysis of the variance threshold of kay s weighted linear predictor frequency estimator a theoretical approximation for the variance of kay s weighted linear predictor frequency estimator is derived from this expression an inequality describing the variance threshold of the estimator is found the window weights are then optimized to improve the variance numerical simulations demonstrate that the variance approximations are valid for medium to high signal to noise ratios or for large numbers of samples

cyclic barankin type bounds for non bayesian periodic parameter estimation in many practical periodic parameter estimation problems the appropriate performance criteria are periodic in the parameter space the existing mean square error mse lower bounds such as cramer rao bound crb and barankin type bounds do not provide valid lower bounds in such problems in this paper cyclic versions of the crb and the barankin type bounds hammersley chapman robbins and mcaulay seidman are derived for non bayesian parameter estimation the proposed bounds are lower bounds on the mean cyclic error mce of any cyclic unbiased estimator where the cyclic unbiasedness is defined by using lehmann unbiasedness these mce lower bounds can be readily obtained from existing mse lower bounds and thus can be easily calculated the cyclic barankin type bounds and the performance of the maximum likelihood ml estimator are compared in terms of mce in von mises distributed measurements problem and for frequency and amplitude estimation with gaussian noise in these problems the ml estimator is found to be cyclic unbiased

a unified approach to the stft tfds and instantaneous frequency cohen s class of time frequency representations tfrs is reformulated into a discrete time discrete frequency computer implementable form it is shown how in this form many of the properties of the continuous time continuous frequency formulation are either lost or altered intuitions applicable in the continuous time case do not necessarily carry over to the discrete time case examined the properties of the discrete variable formulation examined are the presence and form of cross terms instantaneous frequency estimation and relationships between cohen s class of tfrs a parameterized class of distributions which is a blending between the short time fourier transform stft and the wigner ville distribution the two main conclusions are that all tfrs of cohen s class implementable in the given form which includes all commonly used tfrs possess cross terms and that instantaneous frequency estimation using periodic moments of these tfrs is purposeless since simpler methods obtain the same result

nonparametric estimation of a periodic sequence in the presence of a smooth trend in this paper we study a nonparametric regression model including a periodic component a smooth trend function and a stochastic error term we propose a procedure to estimate the unknown period and the function values of the periodic component as well as the nonparametric trend function the theoretical part of the paper establishes the asymptotic properties of our estimators in particular we show that our estimator of the period is consistent in addition we derive the convergence rates as well as the limiting distributions of our estimators of the periodic component and the trend function the asymptotic results are complemented with a simulation study that investigates the small sample behaviour of our procedure finally we illustrate our method by applying it to a series of global temperature anomalies

matrix tensor notation part ii skew and curved coordinates in part i a notation called matrix tensor notation was introduced for rectilinear orthogonal coordinates part ii discusses how the same notation is equally efficient for vectors and 2nd order tensors in skew bases and skew curved coordinates without considering coordinates first vectors and 2nd order tensors are described in skew bases covariant and contravariant descriptions being distinguished similar to tensor notation a consistent interpretation of deformation and strain in terms of skew bases is given the use of the transpose symbol complicates the matrix algebra but integrates it with tensor algebra which makes it possible to interpret customary tensor equations as relations in space with all the advantages that an isomorphism with euclidean space has it is demonstrated how different metrics can be assigned arbitrarily to define higher dimensional orthogonal vectors in abstract higher dimensional space a consistent notation is given to distinguish between subspace and subbase to write higher order tensors as matrices or 2nd order tensors as vectors partial transpose is introduced which is a formal stretching operation to write any higher order tensor product operation as matrix vector product the description of the variable bases in generally curved coordinates is given in this notation together with the corresponding notation for the partial differentiation rules the vector operations grad div and curl are discussed and an almost physical notation for the christoffel symbols is given however the replacement of tensor notation for higher than second order tensors by matrix tensor notation is not feasible a version of matrix tensor notation that merges with the customary printed form is presented in an appendix further applications are deferred to part iii applying the new notation to curved coordinates in function space and the equations of mechanics projected on finite dimensional subspaces

non bayesian periodic cramer rao bound the cramer rao bound crb is one of the most important tools for performance analysis in parameter estimation in many practical periodic parameter estimation problems the appropriate criterion is periodic in the parameter space however the crb does not provide a valid lower bound in such problems in this paper the periodic crb for non bayesian periodic parameter estimation is derived the proposed periodic crb is a lower bound on the mean square periodic error mspe of any periodic unbiased estimator where the periodic unbiasedness is defined by using lehmann unbiasedness it is shown that if there exists a periodic unbiased estimator which achieves the bound then the maximum likelihood produces it the periodic crb and the performance of some periodic unbiased estimators are compared in terms of mspe in a linear gaussian problem with modulo measurements and for phase estimation problems

a fast efficient technique for the estimation of frequency summary a technique is presented for the estimation of the frequency of a sinusoid in the presence of noise the technique is based on fitting arma 2 2 models iteratively in a special way the estimator is shown to be strongly consistent and as efficient as the least squares estimator of the frequency or the periodogram maximizer a simple accelerated version of the technique is shown to converge in a small number of iterations the number depending on the accuracy of the initiator of the procedure the results of a number of simulations are reported

kernel estimation of the instantaneous frequency considers kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise the expected estimation error consists of two terms the systematic bias error grows as the kernel halfwidth increases while the random error decreases for a nonmodulated signal g t the kernel halfwidth that minimizes the expected error is proportional to h spl sim spl sigma sup 2 n spl part sub t sup 2 g 2 sup 1 5 where spl sigma sup 2 is the noise variance and n is the number of measurements per unit time the author shows that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal a t exp i spl phi t for instantaneous frequency estimation the halfwidth which minimizes the expected error is larger h sub 1 3 spl sim spl sigma sup 2 a sup 2 n spl part sub t sup 3 e sup i spl phi t sup 2 sup 1 u7 since the optimal halfwidths depend on derivatives of the unknown function the authors initially estimate these derivatives prior to estimating the actual signal

accurate and efficient pricing of vanilla stock options via the crandall douglas scheme in this paper we apply the crandall douglas scheme quarterly of applied mathematics 13 1955 318 320 journal of mathematics and physics 35 1956 145 151 for the heat equation to the computation of vanilla stock option prices this method has the advantage of being second order accurate in time and fourth order accurate in space i e asset price and can even be made sixth order accurate if we restrict the size of the time step although there is an exact solution for european options we study their numerical approximation so as to determine suitable discretization parameters and permit reasonable placement of the far field boundary we then turn to american options where there are no analytical pricing formulae discretization of the associated free boundary problem when expressed as a variational inequality leads to a linear complementarity problem at each time step these subproblems are solved in linear time by the elliott ockendon algorithm weak and variational methods for moving boundary problems pitman 1982 pp 115 117 the net result is an option pricing methodology which is both very accurate and highly efficient we present the application of our numerical scheme to calls paying continuous dividends as well as to puts in both cases permitting early exercise of the option

phase based time domain estimation of the frequency and phase of a single sinusoid in awgn the role and applications of the additive observation phase noise model this paper presents the theoretical foundation for time domain phase based estimation of the frequency and phase of a single sinusoid in additive white gaussian noise awgn analogous to the theoretical foundation provided by rife and boorstyn for frequency domain fourier transform based estimation it is shown from the maximum a posteriori probability map and the maximum likelihood ml estimation principles that with the additive observation phase noise aopn due to the awgn being described by its a posteriori distribution conditioned on the received signal magnitude the received signal phase is a sufficient statistic for estimating the single sinusoid angle parameters using a geometric approach the exact statistical model for the aopn is derived where the a posteriori probability density function pdf and the corresponding a priori pdf are given by explicit closed form expressions that are valid for arbitrary signal to noise ratios snrs the a posteriori pdf is tikhonov and is of particular interest as it establishes the aopn model for phase based frequency phase map ml estimation in the time domain it is further illustrated that the results derived can yield various aopn models as special cases and the underlying physical insights and interconnections that exist among these models are revealed it is shown that the model derived by tretter is an ultimate specialization in the high snr limit of the aopn models developed here for high snr the a posteriori tikhonov pdf can be accurately approximated by a gaussian distribution which leads to the best linearized aopn model the applications of these aopn models to the design of linear estimators including the linear minimum mean square error lmmse estimator the linear minimum variance estimator and the lmmse implementation of the weighted phase averager are presented and their estimation performances are compared through computer simulations with the cramer rao lower bound crlb and the bayesian crlb as the benchmark to facilitate estimator design the a priori statistical models of the frequency and phase are proposed from the information theoretic perspective and an improved phase unwrapping algorithm over that given by fu and kam is presented it is shown that by incorporating all the information available in the aopn the estimation accuracy can be much improved

nonlinear processing and analysis of angular signals physical quantities referring to angles like vector direction color hue etc exhibit an inherently periodic nature due to this periodicity digital filters and edge operators proposed for data on the line cannot be applied on such data we introduce filters for angular signals circular mean circular median circular a trimmed mean circular modified trimmed mean particular emphasis is given to the circular median filter for which some interesting properties are derived we also use estimators of circular dispersion to introduce edge detectors for angular signals three variations for the extension of quasirange to circular data are proposed and expressions for their output pdf are derived these circular quasiranges have good and user controlled properties as edge detectors in noisy angular signals the performance of the proposed edge operators is evaluated on angular edges using certain quantitative criteria finally a series of experiments featuring one dimensional 1 d angular signals and hue images is used to illustrate the operation of the new filters and edge detectors

estimating frequency by interpolation using fourier coefficients the periodogram of a time series that contains a sinusoidal component provides a crude estimate of its frequency parameter the maximizer over the fourier frequencies being within o t sup 1 of the frequency as the sample size t increases in the paper a technique for obtaining an estimator that has root mean square error of order t sup 3 2 is presented which involves only the fourier components of the time series at three frequencies the asymptotic variance of the estimator varies between roughly the asymptotic variance of the maximizer of the periodogram over all frequencies the cramer rao lower bound and three times this variance the advantage of the new estimator is its computational simplicity

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

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

detecting periodic components in a white gaussian time series abstract a family of tests for periodic components in a white gaussian series is proposed the test is based on a statistic which is proportional to the ratio of the maximum periodogram to the trimmed mean of the periodograms the asymptotic distribution of the statistic is obtained it is shown that the test proposed and fisher s test have the same asymptotic powers at the alternative hypotheses that the series contains a single periodic component at a non zero fourier frequency the tests are applied to detect the eigenfrequencies of the earth the proposed test detects some peaks that fisher s test fails to detect

performance of quadratic time frequency distributions as instantaneous frequency estimators general performance analysis of the shift covariant class of quadratic time frequency distributions tfds as instantaneous frequency if estimators for an arbitrary frequency modulated fm signal is presented expressions for the estimation bias and variance are derived this class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear if however when the if is not a linear function of time then the estimate is biased cases of white stationary and white nonstationary additive noises are considered the well known results for the wigner distribution wd and linear fm signal and the spectrogram of signals whose if may be considered as a constant within the lag window are presented as special cases in addition we have derived the variance expression for the spectrogram of a linear fm signal that is quite simple but highly signal dependent this signal is considered in the cases of other commonly used distributions such as the born jordan and the choi williams distributions it has been shown that the reduced interference distributions outperform the wd but only in the case when the if is constant or its variations are small analysis is extended to the if estimation of signal components in the case of multicomponent signals all theoretical results are statistically confirmed

frequency estimation of a single complex sinusoid using a generalized kay estimator this letter introduces a generalized version of kay s estimator for the frequency of a single complex sinusoid in complex additive white gaussian noise the kay estimator is a maximum likelihood ml estimator at high signal to noise ratio snr based on differential phase measurements with a delay of one symbol interval in this letter the corresponding ml estimator with an arbitrary delay in the differential phase measurements is derived the proposed estimator reduces the variance at low snr compared with kay s original estimator for certain delay values explicit expressions for the window function and the corresponding high snr variance of the proposed generalized kay gk estimator are presented furthermore for some delay values the window function is nearly uniform and the implementation complexity is reduced compared with the original kay estimator for a delay value of two we show that the variance at asymptotically high snr approaches the cramer rao bound as the sequence length tends to infinity we also explore the effect of exchanging the order of summation and phase extraction for reduced complexity reasons the resulting generalized weighted linear predictor estimator and the gk estimator are compared with both autocorrelation based and periodogram based estimators in terms of computational complexity estimation range and performance at both low and high snrs

frequency estimation by phase unwrapping single frequency estimation is a long studied problem with application domains including radar sonar telecommunications astronomy and medicine one method of estimation called phase unwrapping attempts to estimate the frequency by performing linear regression on the phase of the received signal this procedure is complicated by the fact that the received phase is wrapped modulo and therefore must be unwrapped before the regression can be performed in this paper we propose an estimator that performs phase unwrapping in the least squares sense the estimator is shown to be strongly consistent and its asymptotic distribution is derived we then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem we derive a polynomial time algorithm that computes the least squares estimator the results of various simulations are described for different values of sample size and snr

an exact direct method of sinusoidal parameter estimation derived from finite fourier integral of differential equation in this paper we propose a novel method for estimating the parameters frequency amplitude and phase of real sinusoids to derive the estimator we start from the characteristic differential equation of a sinusoid to remove differentials and obtain an algebraic relation for frequency we introduce finite period weighted integrals of the differential equation which become equivalent to the differential equation when a sufficient number of weight functions are applied as weight functions we show that fourier kernels have excellent properties terms related to integral boundaries are readily eliminated observations are provided by fourier coefficients and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced we solve the obtained equations in two ways one is for approaching to the cramer rao lower bound crlb and the other is for enhancing the interference rejection capability also methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy proposed algorithms are examined under noise and sinusoidal interference error variances are compared with the crlb and other fast fourier transform fft based methods

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

semiparametric bayesian inference for time series with mixed spectra summary a bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms the periodic terms may have known or unknown frequencies the advantage of our approach is that different features of the data such as the regression parameters the spectral density unknown frequencies and missing observations are combined in a hierarchical bayesian framework and estimated simultaneously a bayesian test to detect deterministic components in the data is also constructed by using an asymptotic approximation to the likelihood the computation is carried out efficiently using the markov chain monte carlo method in o mn operations where n is the sample size and m is the number of iterations we show empirically that our approach works well on real and simulated samples

using the periodogram to estimate period in nonparametric regression summary properties of the periodogram are seldom studied in the setting of nonparametric regression although that is the context in which the periodogram is widely applied in astronomy there it is a competitor with more recent least squares methods the periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem however as we show in the present paper it also has drawbacks the estimator that it produces has somewhat higher variance than its least squares counterpart and a periodogram based approach is more prone to suffer difficulties caused by periodicity of the observation schedule while the periodogram remains a very attractive tool the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting this aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset

semiparametric curve alignment and shift density estimation for biological data we observe a large number of signals all of them with identical although unknown shape but with a different random shift the objective is to estimate the individual time shifts and their distribution such an objective appears in several biological applications like neuroscience or ecg signal processing in which the estimation of the distribution of the elapsed time between repetitive pulses with a possibly low signal noise ratio and without a knowledge of the pulse shape is of interest we suggest an m estimator leading to a three stage algorithm we first split our data set in blocks then the shift estimation in each block is done by minimizing a cost function based on the periodogram the estimated shifts are eventually plugged into a standard density estimator we show that under mild regularity assumptions the density estimate converges weakly to the true shift distribution the theory is applied both to simulations and to alignment of real ecg signals the proposed approach outperforms the standard methods for curve alignment and shift density estimation even in the case of low signal to noise ratio and is robust to numerous perturbations common in ecg signals

map ml estimation of the frequency and phase of a single sinusoid in noise the problem of estimating the frequency and carrier phase of a single sinusoid observed in additive white gaussian noise is addressed much of the work in the literature considers maximum likelihood ml estimation however the ml estimator given by the location of the peak of a periodogram in the frequency domain shown in d c rife and r r boorstyn single tone parameter estimation from discrete time observations ieee transactions on information theory vol it 20 pp 591 598 sep 1974 has a very high computational complexity this paper derives the explicit structure of the ml estimator for data processing in the time domain assuming only reasonably high signal to noise ratio snr the result of this approximate ml estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the estimator and the phase data alone as assumed in s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and s kay a fast and accurate single frequency estimator ieee transactions on acoustics speech and signal processing vol 39 pp 1203 1205 may 1991 is not a sufficient statistic the sample by sample iterative processing nature of the estimator enables us to propose a novel recursive phase unwrapping algorithm that allows the estimator to be implemented efficiently to facilitate the performance analysis a new linearized observation model for the instantaneous signal phase that is more accurate than that of s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and of s kay a fast and accurate single frequency estimator vol 39 pp 1203 1205 may 1991 is proposed this new model explains physically why the phase data are weighted by the magnitude information in the ml estimator moreover by incorporating a priori knowledge via the a priori probability density function of the unknown frequency and the carrier phase the explicit structure of the approximate maximum a posteriori probability map estimator is derived and the bayesian cramer rao lower bound bcrlb on the mean square error mse is obtained our analysis shows that the mse performance of the map estimator can approach the bcrlb very closely

autoregressive frequency estimation summary presume that an autoregressive model of given order is fitted to data obtained from a realization of a sinusoidal signal embedded in white noise this paper investigates the statistical properties of the estimate of the frequency of the signal obtained by minimizing the transfer function of the autoregression strong consistency and convergence in distribution of the estimate are established when the order of the autoregression is allowed to increase with sample size at an appropriate rate it is also shown that the operational characteristics of commonly used order determination procedures are such as to produce values of the selected order compatible with the application of the asymptotic theory the outcome of some simulation experiments and an empirical example illustrating the practical impact of the results obtained are also presented

a parameter estimation approach to estimation of frequencies of sinusoids it is shown that with uniform sampling a sample of the sum of m sinusoids at time nt can be uniquely expressed as a linear combination of the 2m samples at times n 1 t n 2m t the 2m coefficients of linear combination are also the coefficients of a 2m order polynomial whose roots are e pmj omega i omega i being the frequencies of the sinusoids given the samples of sinusoids plus noise a consistent estimate of the 2m coefficients are obtained by the instrumental variable method of parameter estimation it is also possible to track time varying frequencies by a recursive algorithm that exponentially weighs out the past data simulation examples of one and two sinusoids are given

high order accurate implicit methods for barrier option pricing abstract this paper deals with a high order accurate implicit finite difference approach to the pricing of barrier options in this way various types of barrier options are priced including barrier options paying rebates and options on dividend paying stocks moreover the barriers may be monitored either continuously or discretely in addition to the high order accuracy of the scheme and the stretching effect of the coordinate transformation the main feature of this approach lies on a probability based optimal determination of boundary conditions this leads to much faster and accurate results when compared with similar pricing approaches the strength of the present scheme is particularly demonstrated in the valuation of discretely monitored barrier options where it yields values closest to those obtained from the only semi analytical valuation methods available the scheme is also applied to the analysis of greeks data such as delta and gamma

efficient and high accuracy pricing of barrier options under the cev diffusion binomial and trinomial lattices are popular techniques for pricing financial options these methods work well for european and american options but for barrier options the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price a finite difference implementation is simpler and we propose a fourth order numerical scheme for continuously and discretely monitored barriers we demonstrate the superior performance of our technique over existing procedures for the black scholes model and we then price barriers under constant elasticity of variance cev diffusion continuously monitored barriers under cev admit an analytical solution but evaluation via this formula is not straightforward furthermore discretely monitored barriers have to be priced numerically our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under cev and we provide several numerical examples to illustrate the merit of the new technique