Browsing by Subject "wavelets"
Now showing items 1-6 of 6
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(2004)The non-stationary character of stock market returns manifests itself through the volatility clustering effect and large jumps. An efficient way of representing a time series with such complex dynamics is given by wavelet methodology. With the help of a wavelet basis, the discrete wavelet transform (DWT) is able to break a time series with respect to a time-scale while preserving the time dimension and energy unlike the traditional Fourier transform which 'trades' time for frequency. Time-scale specific information is important if one accepts the view that stock market consists of heterogenous investors operating at different time-scales. In that case considerable more insight into the volatility dynamics is gained by looking at the data at several time-scales. At small time-scales, in particular, the locality of wavelet analysis allows one to fully exploit high-frequency data. Wavelet transforms are also fast to calculate, so they are ideally suited for analyzing large data sets. The 'large-scale aim' of this licentiate thesis is to first introduce wavelet methodology to econometricians and then to analyze stock market volatility with it. In more detail, the data consists of 5-minute observations of the liquid Nokia Oyj stock at the Helsinki Stock Exchange (HEX). Several microstructure problems have to be dealt with, some characteristic of the HEX. Pre-filtered volatility series is then being analyzed by the 'maximal overlap' DWT to study both the global and local scaling laws in a turbulent 'IT-bubble' (1999 - 2000) and its calmer aftermath period (2001 - 2002). Significant time-scale specific differences between these two periods are found. The global scaling laws may not be time-invariant as usually claimed. The bubble period also experienced stronger long-memory in volatility than its aftermath. Thus long-memory may be time-varying as well. Such a finding can be applied in a locally stationary stochastic volatility model. Finally, the effects of the intraday volatility periodicity are studied and they are also found to be significant.
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(2017)A new method is proposed for solving the glottal inverse filtering (GIF) problem. The goal of GIF is to separate an acoustical speech signal into two parts: the glottal airflow excitation and the vocal tract filter. To recover such information one has to deal with a blind deconvolution problem. This ill-posed inverse problem is solved under a deterministic setting, considering unknowns on both sides of the underlying operator equation. A stable reconstruction is obtained using a double regularization strategy, alternating between fixing either the glottal source signal or the vocal tract filter. This enables not only splitting the nonlinear and nonconvex problem into two linear and convex problems, but also allows the use of the best parameters and constraints to recover each variable at a time. This new technique, called alternating minimization glottal inverse filtering (AM-GIF), is compared with two other approaches: Markov chain Monte Carlo glottal inverse filtering (MCMC-GIF), and iterative adaptive inverse filtering (IAIF), using synthetic speech signals. The recent MCMC-GIF has good reconstruction quality but high computational cost. The state-of-the-art IAIF method is computationally fast but its accuracy deteriorates, particularly for speech signals of high fundamental frequency (F0). The results show the competitive performance of the new method: With high F0, the reconstruction quality is better than that of IAIF and close to MCMC-GIF while reducing the computational complexity by two orders of magnitude.
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(2015)The total least squares (TLS) method is a successful approach for linear problems if both the right-hand side and the operator are contaminated by some noise. For ill-posed problems, a regularisation strategy has to be considered to stabilise the computed solution. Recently a double regularised TLS method was proposed within an infinite dimensional setup and it reconstructs both function and operator, reflected on the bilinear forms Our main focuses are on the design and the implementation of an algorithm with particular emphasis on alternating minimisation strategy, for solving not only the double regularised TLS problem, but a vast class of optimisation problems: on the minimisation of a bilinear functional of two variables.
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(2021)We propose a novel convolutional neural network (CNN), called Psi DONet, designed for learning pseudodifferential operators (Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are Psi DOs or Fourier integral operators.
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(2017)Prominences and boundaries are the essential constituents of prosodic struc- ture in speech. They provide for means to chunk the speech stream into linguis- tically relevant units by providing them with relative saliences and demarcating them within utterance structures. Prominences and boundaries have both been widely used in both basic research on prosody as well as in text-to-speech syn- thesis. However, there are no representation schemes that would provide for both estimating and modelling them in a unified fashion. Here we present an unsupervised unified account for estimating and representing prosodic promi- nences and boundaries using a scale-space analysis based on continuous wavelet transform. The methods are evaluated and compared to earlier work using the Boston University Radio News corpus. The results show that the proposed method is comparable with the best published supervised annotation methods.
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(2018)Region of interest (ROI) tomography has gained increasing attention in recent years due to its potential to reducing radiation exposure and shortening the scanning time. However, tomographic reconstruction from ROI-focused illumination involves truncated projection data and typically results in higher numerical instability even when the reconstruction problem has unique solution. To address this problem, both ad hoc analytic formulas and iterative numerical schemes have been proposed in the literature. In this paper, we introduce a novel approach for ROI tomographic reconstruction, formulated as a convex optimization problem with a regularized term based on shearlets. Our numerical implementation consists of an iterative scheme based on the scaled gradient projection method and it is tested in the context of fan-beam CT. Our results show that our approach is essentially insensitive to the location of the ROI and remains very stable also when the ROI size is rather small.
