smooth blockwise iterative thresholding a smooth fixed point estimator based on the likelihood s block gradient the proposed smooth blockwise iterative thresholding estimator sbite is a model selection technique defined as a fixed point reached by iterating a likelihood gradient based thresholding function the smooth james stein thresholding function has two regularization parameters and and a smoothness parameter s it enjoys smoothness like ridge regression and selects variables like lasso focusing on gaussian regression we show that sbite is uniquely defined and that its stein unbiased risk estimate is a smooth function of and for better selection of the two regularization parameters we perform a monte carlo simulation to investigate the predictive and oracle properties of this smooth version of adaptive lasso the motivation is a gravitational wave burst detection problem from several concomitant time series a nonparametric wavelet based estimator is developed to combine information from all captors by block thresholding multiresolution coefficients we study how the smoothness parameter s tem

sequential change point detection methods for nonstationary time series we present two new spectral based methods for detection of changes in autocorrelation structure in a continuous valued time series in an online process monitoring setting our methods are based on the idea that changes in the autocorrelation structure are reflected by changes in the fourier or wavelet based spectrum and can be detected by comparing estimated spectra of adjacent blocks of the series to be effective for slowly changing spectral structure the methods are extended to allow information from more than one past block to be used in determining whether a change has occurred in such a way as to minimize computational burden through simulation we evaluate the performance of our methods and find that they can provide reliable and timely detection of changes in covariance structure in an online monitoring framework we illustrate the methods using electroencephalogram traces brain waves and run time computer performance metrics

eta invariants torsion forms and flat vector bundles we present a new proof as well as a c q extension and also certain c z extension of the riemann roch grothendieck theorem of bismut lott for flat vector bundles the main techniques used are the computations of the adiabatic limits of invariants associated to the so called sub signature operators we further show that the bismut lott analytic torsion form can be derived naturally from transgressions of forms appearing in the adiabatic limit computations

wavelet variance based estimation for composite stochastic processes this article presents a new estimation method for the parameters of a time series model we consider here composite gaussian processes that are the sum of independent gaussian processes which in turn explain an important aspect of the time series as is the case in engineering and natural sciences the proposed estimation method offers an alternative to classical estimation based on the likelihood that is straightforward to implement and often the only feasible estimation method with complex models the estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances wv estimates and the model based wv indeed the wv provides a decomposition of the variance process through different scales so that they contain the information about different features of the stochastic model we derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the mle and the lse with

wavelet generalized least squares a new blu estimator of linear regression models with 1 f errors long memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors classical autoregressive moving average arma methods can adequately address the problem of linear time invariant short memory errors but may be inefficient and or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function here we introduce a novel method called wavelet generalized least squares wls which is to a good approximation the best linear unbiased blu estimator of regression model parameters in the context of long memory errors the method also provides maximum likelihood ml estimates of the hurst exponent which can be readily translated to the fractal dimension or spectral exponent characterizing the correlational structure of the errors and the error variance the algorithm exploits the whitening or kar

wavelet variance analysis for random fields on a regular lattice there has been considerable recent interest in using wavelets to analyze time series and images that can be regarded as realizations of certain 1 d and 2 d stochastic processes on a regular lattice wavelets give rise to the concept of the wavelet variance or wavelet power spectrum which decomposes the variance of a stochastic process on a scale by scale basis the wavelet variance has been applied to a variety of time series and a statistical theory for estimators of this variance has been developed while there have been applications of the wavelet variance in the 2 d context in particular in works by unser in 1995 on wavelet based texture analysis for images and by lark and webster in 2004 on analysis of soil properties a formal statistical theory for such analysis has been lacking in this paper we develop the statistical theory by generalizing and extending some of the approaches developed for time series thus leading to a large sample theory for estimators of 2 d wavelet variances we apply our theory to simulated data from gaussian random fields with exponential covariances and from fractional brownian surfaces we demonstrate that the wavelet variance is potentially useful for texture discrimination we also use our methodology to analyze images of four types of clouds observed over the southeast pacific ocean

statistical properties and uses of the wavelet variance estimator for the scale analysis of time series abstract many physical processes are an amalgam of components operating on different scales and scientific questions about observed data are often inherently linked to understanding the behavior at different scales we explore time scale properties of time series through the variance at different scales derived using wavelet methods the great advantage of wavelet methods over ad hoc modifications of existing techniques is that wavelets provide exact scale based decomposition results we consider processes that are stationary nonstationary but with stationary dth order differences and nonstationary but with local stationarity we study an estimator of the wavelet variance based on the maximal overlap undecimated discrete wavelet transform the asymptotic distribution of this wavelet variance estimator is derived for a wide class of stochastic processes not necessarily gaussian or linear the variance of this distribution is estimated using spectral methods simulations confirm the theoretical result

collapsing and the differential form laplacian the case of a smooth limit space we analyze the limit of the p form laplacian under a collapse with bounded sectional curvature and bounded diameter to a smooth limit space as an application we characterize when the p form laplacian has small positive eigenvalues in a collapsing sequence

wavelet based estimation for seasonal long memory processes we introduce the multiscale analysis of seasonal persistent processes that is time series models with a singularity in their spectral density function at one or more frequencies in 0 1 2 the discrete wavelet packet transform dwpt and a nondecimated version of it known as the maximal overlap dwpt modwpt are introduced as alternative methods to fourier based techniques for analyzing time series that exhibit seasonal long memory the approximate log linear relationship between the wavelet packet variance and frequency is used to produce a least squares estimator of the fractional difference parameter approximate maximum likelihood estimation is performed by replacing the variance covariance matrix with a diagonalized matrix based on the dwpt simulations are performed to compare the wavelet based techniques with the spectral estimate based techniques for both least squares and maximum likelihood procedures an application of this methodology to atmospheric and economic time series is used for demon

on estimation of the wavelet variance summary the wavelet variance decomposes the variance of a time series into components associated with different scales we consider two estimators of the wavelet variance the first based upon the discrete wavelet transform and the second called the maximal overlap estimator based upon a filtering interpretation of wavelets we determine the large sample distribution for both estimators and show that the maximal overlap estimator is more efficient for a class of processes of interest in the physical sciences we discuss methods for determining an approximate confidence interval for the wavelet variance we demonstrate through monte carlo experiments that the large sample distribution for the maximal overlap estimator is a reasonable approximation even for the moderate sample size of 128 observations we apply our proposed methodology to a series of observations related to vertical shear in the ocean

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

differentiating intraday seasonalities through wavelet multi scaling it is well documented that strong intraday seasonalities may induce distortions in the estimation of volatility models these seasonalities are also the dominant source for the underlying misspecifications of the various volatility models therefore an obvious route is to filter out the underlying intraday seasonalities from the data in this paper we propose a simple method for intraday seasonality extraction that is free of model selection parameters which may affect other intraday seasonality filtering methods our methodology is based on a wavelet multi scaling approach which decomposes the data into its low and high frequency components through the application of a non decimated discrete wavelet transform it is simple to calculate does not depend on a particular model selection criterion or model specific parameter choices the proposed filtering method is translation invariant has the ability to decompose an arbitrary length series without boundary adjustments is associated with a zero phase filter and is circular being circular helps to preserve the entire sample unlike other two sided filters where data loss occurs from the beginning and the end of the studied sample

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

optimal boundary control for the evolutionary navier stokes system the three dimensional case optimal boundary control problems for the three dimensional evolutionary navier stokes equations in the exterior of a bounded domain are studied control is effected through the dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with almost minimal possible regularity the control objective is to minimize the drag functional the existence of an optimal solution is proved a strong form of an optimality system of equations is derived on the basis of regularity results established in this work for the adjoint oseen equations with regular initial data which do not satisfy the compatibility conditions

determination of changes in streamflow variance by means of a wavelet based test 1 changes in the variance of streamflow over time may possibly be caused by land cover changes or climate changes knowledge of these possible changes in the time series structure is desirable for validation of modeling efforts detection is not always straightforward however we present the use of a wavelet based statistical test for the detection of changes in the variance of a time series applied to records of streamflow from 12 rivers in the united states from the period 1954 1999 the test is able to show during which time periods and at what wavelet scales changes in the variance are taking place it is found that the major similarities in variance change for all the streamflow time series exist when the largest time domain wavelet is used and also when a wavelet of the second largest frequency scale is used sampling considerations prevent exact localization of the time and frequency of the change s results of the test indicate that for the data analyzed in this study a change in the variance of the streamflow time series most likely did take place over the time period examined

wavelet spectrum and its characterization property for random processes the wavelet spectrum of a random process comprises the variances of the wavelet coefficients of the process computed within each scale this paper investigates the possibility of using the wavelet spectrum obtained from a continuous wavelet transform cwt to uniquely represent the second order statistical properties of random processes particularly stationary processes and long memory nonstationary processes as is well known the fourier spectrum of a stationary process is mathematically equivalent to the autocovariance function acf and thus uniquely determines the second order statistics of the process this characterization property is shown to be possessed also by the wavelet spectrum under very mild regularity conditions that are easily satisfied by many widely used wavelets it is also shown that under suitable regularity conditions the characterization property remains valid for processes with stationary increments including 1 f noise

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

kernel methods and flexible inference for complex stochastic dynamics abstract approximation theory suggests that series expansions and projections represent standard tools for random process applications from both numerical and statistical standpoints such instruments emphasize the role of both sparsity and smoothness for compression purposes the decorrelation power achieved in the expansion coefficients space compared to the signal space and the reproducing kernel property when some special conditions are met we consider these three aspects central to the discussion in this paper and attempt to analyze the characteristics of some known approximation instruments employed in a complex application domain such as financial market time series volatility models are often built ad hoc parametrically and through very sophisticated methodologies but they can hardly deal with stochastic processes with regard to non gaussianity covariance non stationarity or complex dependence without paying a big price in terms of either model mis specification or computational efficiency it is thus a good idea to look at other more flexible inference tools hence the strategy of combining greedy approximation and space dimensionality reduction techniques which are less dependent on distributional assumptions and more targeted to achieve computationally efficient performances advantages and limitations of their use will be evaluated by looking at algorithmic and model building strategies and by reporting statistical diagnostics

long run wavelet based correlation for financial time series abstract the asset allocation decision often relies upon correlation estimates arising from short run data short run correlation estimates may however be distorted by frictions in this paper we introduce a long run wavelet based correlation estimator distinguishing between long run common behavior and short run singular events using generated data we demonstrate a reduction in bias and error of up to 84 2 and 38 9 respectively relative to a traditional subsampled approach exploiting the wavelet decomposition into short and long run components we develop a model to help understand the sources of any heterogeneity in correlation the implication is that short run correlation may be downward biased by frictions the latter manifesting as serial and cross serial correlation in the raw time series in an empirical application to g7 international equity markets we present evidence of increasing correlations at longer run horizons the significance for the asset allocation decision are examined using a minimum variance framework highlighting distinct optimal allocation weights at short and long run horizons

a wavelet based estimator of the degrees of freedom in denoised fmri time series for probabilistic testing of functional connectivity and brain graphs connectome mapping using techniques such as functional magnetic resonance imaging fmri has become a focus of systems neuroscience there remain many statistical challenges in analysis of functional connectivity and network architecture from bold fmri multivariate time series one key statistic for any time series is its effective degrees of freedom df which will generally be less than the number of time points or nominal degrees of freedom n if we know the df then probabilistic inference on other fmri statistics such as the correlation between two voxel or regional time series is feasible however we currently lack good estimators of df in fmri time series especially after the degrees of freedom of the raw data have been modified substantially by denoising algorithms for head movement here we used a wavelet based method both to denoise fmri data and to estimate the effective df of the denoised process we show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over type i error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom we also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes assign a p value to each edge and then iteratively add edges to the graph in order of increasing p these probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations finally we show that time windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts wavelet despiking is both an algorithm for fmri time series denoising and an estimator of the effective df of denoised fmri time series accurate estimation of df offers many potential advantages for probabilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non stationary noise code for wavelet despiking seed correlational testing and probabilistic graph construction is freely available to download as part of the brainwavelet toolbox at www brainwavelet org

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

scaling properties of foreign exchange volatility in this paper we investigate the scaling properties of foreign exchange volatility our methodology is based on a wavelet multi scaling approach which decomposes the variance of a time series and the covariance between two time series on a scale by scale basis through the application of a discrete wavelet transformation it is shown that foreign exchange rate volatilities follow different scaling laws at different horizons particularly there is a smaller degree of persistence in intra day volatility as compared to volatility at one day and higher scales therefore a common practice in the risk management industry to convert risk measures calculated at shorter horizons into longer horizons through a global scaling parameter may not be appropriate this paper also demonstrates that correlation between the foreign exchange volatilities is the lowest at the intra day scales but exhibits a gradual increase up to a daily scale the correlation coefficient stabilizes at scales one day and higher therefore the benefit of currency diversification is the greatest at the intra day scales and diminishes gradually at higher scales lower frequencies the wavelet cross correlation analysis also indicates that the association between two volatilities is stronger at lower frequencies

multi scale tests for serial correlation this paper introduces a new family of portmanteau tests for serial correlation using the wavelet transform we decompose the variance of the underlying process into the variance of its low frequency and of its high frequency components and we design a variance ratio test of no serial correlation in the presence of dependence such decomposition can be carried out iteratively each wavelet filter leading to a rich family of tests whose joint limiting null distribution is a multivariate normal we illustrate the size and power properties of the proposed tests through monte carlo simulations