calculation of a constant q spectral transform the frequencies that have been chosen to make up the scale of western music are geometrically spaced thus the discrete fourier transform dft although extremely efficient in the fast fourier transform implementation yields components which do not map efficiently to musical frequencies this is because the frequency components calculated with the dft are separated by a constant frequency difference and with a constant resolution a calculation similar to a discrete fourier transform but with a constant ratio of center frequency to resolution has been made this is a constant q transform and is equivalent to a 1 24 oct filter bank thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range this transform against log frequency to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted this is compared to the conventio

time dependent roc curves for censored survival data and a diagnostic marker roc curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker x for a binary disease variable d however many disease outcomes are time dependent d t and roc curves that vary as a function of time may be more appropriate a common example of a time dependent variable is vital status where d t 1 if a patient has died prior to time t and zero otherwise we propose summarizing the discrimination potential of a marker x measured at baseline t 0 by calculating roc curves for cumulative disease or death incidence by time t which we denote as roc t a typical complexity with survival data is that observations may be censored two roc curve estimators are proposed that can accommodate censored data a simple estimator is based on using the kaplan meier estimator for each possible subset x c however this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in x an alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of x t where t represents survival time akritas m j 1994 annals of statistics 22 1299 1327 we present an example where roc t is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the roc t curve displays the impact of modifying eligibility criteria for sample size and power in hiv prevention trials

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

estimation of bivariate and marginal distributions with censored data consider a pair of random variables both subject to random right censoring new estimators for the bivariate and marginal distributions of these variables are proposed the estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring such a method of estimation is proposed the weak convergence of the estimators proposed is obtained a small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the kaplan meier estimator for the marginal distribution and the estimators of pruitt and van der laan for the bivariate distribution the use of the estimators in practice is illustrated by the analysis of a data set

nonparametric estimators of the bivariate survival function under simplified censoring conditions summary new bivariate survival function estimators are proposed in the case where the dependence relationship between the censoring variables are modelled specific examples include the cases when censoring variables are univariate mutually independent or specified by a marginal model large sample properties of the proposed estimators are discussed the finite sample performance of the proposed estimators compared with other fully nonparametric estimators is studied via simulations a real data example is given

an analytic wavelet transform with a flexible time frequency covering we develop a rational dilation wavelet transform for which the dilation factor the q factor and the redundancy can be easily specified the introduced transform contains hilbert transform pairs of atoms therefore it is also suitable for oscillatory signal processing the transform may be modified to obtain a tight chirplet frame for discrete time signals a fast implementation that makes use of an equivalent filter bank makes the transform suitable for long signals examples on natural signals are provided to demonstrate the utility of the transform

a four parameter atomic decomposition of chirplets a new four parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components the four parameter chirplet atom is obtained from the unit gaussian function by successive applications of scaling fractional fourier transform frft and time shift and frequency shift operators the application of the frft operator results in a rotation of the wigner distribution of the gaussian in the time frequency plane by a specified angle the decomposition is realized by using the matching pursuit algorithm for this purpose the four parameter space is discretized to obtain a small but complete subset in the hilbert space a time frequency distribution tfd is developed for clear and readable visualization of the signal components it is observed that the chirplet decomposition and the related tfd provide more compact and precise representation of signal inner structures compared with the commonly used time frequency representations

wigner based formulation of the chirplet transform using the wigner distribution we derive and analyze a matrix formulation for the chirplet transform a signal analysis tool that generalizes the wavelet and short time fourier transforms the formulation expresses the translations scalings and shears of the chirplet transform in terms of affine matrix transformations on the time frequency plane our approach leads naturally to several new signal transforms which we derive analyze and extend

shear madness new orthonormal bases and frames using chirp functions the proportional bandwidth and constant bandwidth time frequency signal decompositions of the wavelet gabor and wilson orthonormal bases have attracted substantial interest for representing nonstationary signals however these representations are limited in that they are based on rectangular tessellations of the time frequency plane while much effort has gone into methods for designing nice wavelet and window functions for these frameworks little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time frequency tilings the authors take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear or chirp in the time frequency plane in addition to translate and scale the new scale shear fan bases and shift shear chevron bases are obtained by operating on an existing wavelet gabor 1946 or wilson basis set with two special unitary warping transformations in addition to the theoretical benefit of broadening the class of valid time frequency plane tilings these new bases could possibly also be useful for representing certain types of signals such as chirping and dispersed signals

identifiability of bivariate survival curves from censored data abstract we show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made the common assumption has been mutual independence of t 1 t 2 and z 1 z 2 where t 1 t 2 is the true survival vector z 1 z 2 is a nuisance censoring vector and bivariate right censored data is observed we show that the distribution of t 1 t 2 is identifiable under weaker conditional independence assumptions for distributions with full support bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple possibly related times are of interest the mutual independence model has become popular as a nonparametric way of analyzing such data analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

covariance and survivor function estimation using censored multivariate failure time data summary the covariance between counting process martingales is used to characterize the dependence between two failure time variates a representation of the bivariate survivor function is obtained in terms of the marginal survivor functions and this covariance function a closely related representation expresses the bivariate survivor function in terms of marginal survivor functions and a conditional covariance function leading to a new nonparametric survivor function estimator generalizations to higher dimensional failure time variates are also given simulation evaluations of the survivor function estimator are presented and generalizations to regression problems are outlined

wavelet transform with tunable q factor this paper describes a discrete time wavelet transform for which the q factor is easily specified hence the transform can be tuned according to the oscillatory behavior of the signal to which it is applied the transform is based on a real valued scaling factor dilation factor and is implemented using a perfect reconstruction over sampled filter bank with real valued sampling factors two forms of the transform are presented the first form is defined for discrete time signals defined on all of z the second form is defined for discrete time signals of finite length and can be implemented efficiently with ffts the transform is parameterized by its q factor and its oversampling rate redundancy with modest oversampling rates e g three to four times overcomplete being sufficient for the analysis synthesis functions to be well localized

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

overcomplete discrete wavelet transforms with rational dilation factors this paper develops an overcomplete discrete wavelet transform dwt based on rational dilation factors for discrete time signals the proposed overcomplete rational dwt is implemented using self inverting fir filter banks is approximately shift invariant and can provide a dense sampling of the time frequency plane a straightforward algorithm is described for the construction of minimal length perfect reconstruction filters with a specified number of vanishing moments whereas in the nonredundant rational case no such algorithm is available the algorithm is based on matrix spectral factorization the analysis synthesis functions discrete time wavelets can be very smooth and can be designed to closely approximate the derivatives of the gaussian function

perfect reconstruction filter banks with rational sampling factors an open problem namely how to construct perfect reconstruction filter banks with rational sampling factors is solved such filter banks have n branches each one having a sampling factor of p i qi and their sum equals one in this way the well known theory of filter banks with uniform band splitting is extended to allow for nonuniform divisions of the spectrum this can be very useful in the analysis of speech and music the theory relies on two transforms the first transform leads to uniform filter banks having polyphase components as individual filters the other results in a uniform filter bank containing shifted versions of same filters this in turn introduces dependencies in design and is left for future work as an illustration several design examples for the 2 3 1 3 case are given filter banks are then classified according to the possible ways in which they can be built it is shown that some cases cannot be solved even with ideal filters with real coefficients

an efficient algorithm for the calculation of a constant q transform an efficient method of transforming a discrete fourier transform dft into a constant q transform where q is the ratio of center frequency to bandwidth has been devised this method involves the calculation of kernels that are then applied to each subsequent dft only a few multiples are involved in the calculation of each component of the constant q transform so this transformation adds a small amount to the computation in effect this method makes it possible to take full advantage of the computational efficiency of the fast fourier transform fft graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin

the chirplet transform physical considerations we consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis we propose the use of quadratic chirp functions which we will call q chirps for short giving rise to a parameter space that includes both the time frequency plane and the time scale plane as 2 d subspaces the parameter space contains a time frequency scale volume and thus encompasses both the short time fourier transform as a slice along the time and frequency axes and the wavelet transform as a slice along the time and scale axes in addition to time frequency and scale there are two other coordinate axes within this transform space shear in time obtained through convolution with a q chirp and shear in frequency obtained through multiplication by a q chirp signals in this multidimensional space can be obtained by a new transform which we call the q chirplet transform or simply the chirplet transform the proposed chirplets are generalizations of wavelets related to each other by 2 d affine coordinate transformations translations dilations rotations and shears in the time frequency plane as opposed to wavelets which are related to each other by 1 d affine coordinate transformations translations and dilations in the time domain only

covariance sequence approximation for parametric spectrum modeling parametric methods of spectrum analysis are founded on finite dimensional models for covariance sequences rational spectrum approximants for continuous spectra are based on autoregressive ar moving average ma or autoregressive moving average arma models for covariance sequences line spectrum approximants to discrete spectra are based on cosinusoidal models for covariance sequences in this paper we make the point that a wide variety of spectrum types admit to modal analysis wherein the modes are characterized by amplitudes frequencies and damping factors the associated modal decomposition is appropriate for both continuous and discrete components of the spectrum the domain of attraction for the decomposition includes arma sequences harmonically or nonharmonically related sinusoids damped sinusoids white noise and linear combinations of these the parametric spectrum analysis problem now becomes one of identifying mode parameters this we achieve by solving two modified least squares problems numerical results are presented to illustrate the identification of mode parameters and corresponding spectra from finite records of perfect and estimated covariance sequences the results for sinusoids and sinusoids in white noise are interpreted in terms of in phase and quadrature effects attributable to the finite record length

iterated filter banks with rational rate changes connection with discrete wavelet transforms some properties of two band filter banks with rational rate changes rational filter banks are first reviewed focusing then on iterated rational filter banks compactly supported limit functions are obtained in the same manner as previously done for dyadic schemes allowing a characterization of such filter banks these functions are carefully studied and the properties they share with the dyadic case are highlighted they are experimentally observed to verify a shift property strictly verified in the dyadic ease up to an error which can be made arbitrarily small when their regularity increases in this case the high pass outputs of an iterated filter bank can be very close to samples of a discrete wavelet transform with the same rational dilation factor straightforward extension of the formalism of multiresolution analysis is also made finally it is shown that if one is ready to put up with the loss of the shift property rational iterated filter banks can be used in the same manner as if they were dyadic filter banks with the advantage that rational dilation factors can be chosen closer to 1

transfer of tail information in censored regression models consider a heteroscedastic regression model y m x sigma x varepsilon where the functions m and sigma are smooth and varepsilon is independent of x the response variable y is subject to random censoring but it is assumed that there exists a region of the covariate x where the censoring of y is light under this condition it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light censoring to regions of heavy censoring crucial for this transfer is the estimator of the distribution of varepsilon based on nonparametric regression residuals whose weak convergence is obtained the idea of transferrring tail information is applied to the estimation of the conditional distribution of y given x x with information on the upper tail borrowed from the region of light censoring and to the estimation of the bivariate distribution p x leq x y leq y with no regions of undefined mass the weak convergence of the two estimators is obtained by products of this investigation include the uniform consistency of the conditional kaplan meier estimator and its derivative the location and scale estimators and the estimators of their derivatives

inference on multivariate m estimators based on bivariate censored data consider two random variables subject to random right censoring for example the survival times of twins or two recurrence times of a certain disease recorded on the same individual of interest is the estimation of a multivariate m functional of these two random variables asymptotic results for the proposed estimator are established under general conditions on the criterion function of the m estimator the obtained results include the asymptotic normality of the estimator its local power and construction of a confidence region our primary example of interest is the bivariate l1 median we compare the small sample behavior of this estimator with that of the vector of marginal medians by means of a simulation study finally we consider a dataset on kidney dialysis patients and estimate the median time to two different infections for these individuals

the product limit estimator and the bootstrap some asymptotic representations the product limit estimator and its quantile process are represented as i i d mean processes with a remainder of ordern 3 4 logn 3 4 a s corresponding bootstrap versions of these representations are given which can help one visualize how the bootstrap procedure operates in this set up

theory implementation and applications of nonstationary gabor frames signal analysis with classical gabor frames leads to a fixed time frequency resolution over the whole time frequency plane to overcome the limitations imposed by this rigidity we propose an extension of gabor theory that leads to the construction of frames with time frequency resolution changing over time or frequency we describe the construction of the resulting nonstationary gabor frames and give the explicit formula for the canonical dual frame for a particular case the painless case we show that wavelet transforms constant q transforms and more general filter banks may be modeled in the framework of nonstationary gabor frames further we present the results in the finite dimensional case which provides a method for implementing the above mentioned transforms with perfect reconstruction finally we elaborate on two applications of nonstationary gabor frames in audio signal processing namely a method for automatic adaptation to transients and an algorithm for an invertible constant q transform

improving resolution for autoregressive spectral estimation by decimation in this paper we present a method for efficiently improving the resolution of autoregressive spectral estimation algorithms we derive the exact autoregressive spectrum for k complex sinusoids in additive white noise from this equation resolution boundaries are constructed which give the resolution in terms of the model order and the signal to noise ratio simulation results are used to compare the resolution boundaries for decimated and undecimated spectra our results demonstrate that decimation by d with a model order m yields the same resolution as a model order md used with the undecimated signal and that decimation reduces the computation

spectrum manipulation for improved resolution a novel approach to improving the frequency resolution of parametric power spectrum estimation for signals with rational spectra is discussed specifically the effect of errors in the calculated autocorrelation values is considered analysis and numerical simulation reveal that the error sensitivity becomes acute when the poles are closely spaced demonstrating one of the difficulties encountered when high resolution spectrum estimates are required to remedy this problem two modified spectrum estimation procedures are proposed the first applies to situations where discrete time techniques are used to estimate the spectra of continuous time processes it involves selecting the autocorrelation function sampling period to separate the poles the second procedure involves desampling an autocorrelation sequence to separate closely spaced poles artificially numerical examples verify improved resolution for both procedures

a new design algorithm for two band orthonormal rational filter banks and orthonormal rational wavelets we present a new algorithm for the design of orthonormal two band rational filter banks owing to the connection between iterated rational filter banks and rational wavelets this is also a design algorithm for orthonormal rational wavelets it is basically a simple iterative procedure which explains its exponential convergence and adaptability under various linear constraints e g regularity although the filters obtained from this algorithm are suboptimally designed they show excellent frequency selectivity after an in depth account of the algorithm we discuss the properties of the rational wavelets generated by some designed filters in particular we stress the possibility to design almost shift error free wavelets which allows the implementation of a rational wavelet transform

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

a dual tree rational dilation complex wavelet transform in this correspondence we introduce a dual tree rational dilation complex wavelet transform for oscillatory signal processing like the short time fourier transform and the dyadic dual tree complex wavelet transform the introduced transform employs quadrature pairs of time frequency atoms which allow to work with the analytic signal the introduced wavelet transform is a constant q transform a property lacked by the short time fourier transform which in turn makes the introduced transform more suitable for models that depend on scale also the frequency resolution can be as high as desired a property lacked by the dyadic dual tree complex wavelet transform which makes the introduced transform more suitable for processing oscillatory signals like speech audio and various biomedical signals

a modified tls prony method using data decimation the paper introduces a modified tls prony method that incorporates data decimation the use of data decimation results in the reduction in the computational complexity because one high order estimation is replaced by several low order estimations the authors present an analysis of pole variance statistics for this modified tls prony method this analysis provides a quantitative comparison of the parameter estimation accuracy as a function of decimation factors the authors show that by using decimation one can obtain comparable statistical performance results at a fraction of the computational cost when compared with the conventional tls prony algorithm

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

separation of multiple time delays using new spectral estimation schemes the problem of estimating multiple time delays in presence of colored noise is considered this problem is first converted to a high resolution frequency estimation problem then the sample lagged covariance matrices of the resulting signal are computed and studied in terms of their eigenstructure these matrices are shown to be as effective in extracting bases for the signal and noise subspaces as the standard autocorrelation matrix which is normally used in music and the pencil based methods frequency estimators are then derived using these subspaces the effectiveness of the method is demonstrated on two examples a standard frequency estimation problem in presence of colored noise and a real world problem that involves separation of multiple specular components from the acoustic backscattered from an underwater target

frequency domain design of overcomplete rational dilation wavelet transforms the dyadic wavelet transform is an effective tool for processing piecewise smooth signals however its poor frequency resolution its low q factor limits its effectiveness for processing oscillatory signals like speech eeg and vibration measurements etc this paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied the new wavelet transform can attain higher q factors desirable for processing oscillatory signals or the same low q factor of the dyadic wavelet transform the new wavelet transform is modestly overcomplete and based on rational dilations like the dyadic wavelet transform it is an easily invertible constant q discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for l2 r the wavelet can be made to resemble a gabor function and can hence have good concentration in the time frequency plane the construction of the new wavelet transform depends on the judicious use of both the transform s redundancy and the flexibility allowed by frequency domain filter design

fast matching pursuit with a multiscale dictionary of gaussian chirps we introduce a modified matching pursuit algorithm called fast ridge pursuit to approximate n dimensional signals with m gaussian chirps at a computational cost o mn instead of the expected o mn sup 2 logn at each iteration of the pursuit the best gabor atom is first selected and then its scale and chirp rate are locally optimized so as to get a good chirp atom i e one for which the correlation with the residual is locally maximized a ridge theorem of the gaussian chirp dictionary is proved from which an estimate of the locally optimal scale and chirp is built the procedure is restricted to a sub dictionary of local maxima of the gaussian gabor dictionary to accelerate the pursuit further the efficiency and speed of the method is demonstrated on a sound signal