Tor Arne Øigård

EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution

The heavy-tailed multivariate normal inverse Gaussian (MNIG) distribution is a recent variance-mean mixture of a multivariate Gaussian with a univariate inverse Gaussian distribution. Due to the complexity of the likelihood function, parameter estimation by direct maximization is exceedingly difficult. To overcome this problem, we propose a fast and accurate multivariate expectation-maximization (EM) algorithm for maximum likelihood estimation of the scalar, vector, and matrix parameters of the MNIG distribution. Important fundamental and attractive properties of the MNIG as a modeling tool for multivariate heavy-tailed processes are discussed. The modeling strength of the MNIG, and the feasibility of the proposed EM parameter estimation algorithm, are demonstrated by fitting the MNIG to real world hydrophone data, to wideband synthetic aperture sonar data, and to multichannel radar sea clutter data.

A visual display device for significant features in complicated signals

For many applications the significant features found in a data set depend on the level of resolution for which the data are considered. An excellent example of this is in climatology where the features found on scales of tens and hundreds of years, respectively, may be very different. A natural way to study such data sets is through the scale-space approach. In this paper a new scale-space method, which finds significant features in signals, is proposed. The new method, posterior smoothing, is formulated in a Bayesian framework and utilizes sampling from the posterior density. We compare the new methodology to a successful, existing scale-space technique entitled SiZer (Significant Zero crossings of derivatives). For smooth signals SiZer and the new method have similar performance. In signals containing complicated structures, posterior smoothing is preferable. This is demonstrated by applying the methods to simulated and real data sets. In particular, we show that posterior smoothing has better performance in applications taken from climatology, medical imaging, and fish industry.

Bayesian multiscale analysis for time series data

A recently proposed Bayesian multiscale tool for exploratory analysis of time series data is reconsidered and umerous important improvements are suggested. The improvements are in the model itself, the algorithms to analyse it, and how to display the results. The consequence is that exact results can be obtained in real time using only a tiny fraction of the CPU time previously needed to get approximate results. Analysis of both real and synthetic data are given to illustrate our new approach. Multiscale analysis for time series data is a useful tool in applied time series analysis, and with the new model and algorithms, it is also possible to do such analysis in real time.

The normal inverse Gaussian distribution: a versatile model for heavy-tailed stochastic processes

The normal inverse Gaussian (NIG) distribution is a recent flexible closed form distribution that may be applied as a model of heavy-tailed processes. The NIG distribution is completely specified by four real valued parameters that have natural interpretations in terms of the shape of the resulting probability density function. By choosing the parameters appropriately, one can describe a wide range of shapes of the distribution. We discuss several of the desirable properties of the NIG distribution. In particular, we discuss the cumulant generating function and the cumulants of the NIG-variables. A particularly important property is that the NIG distribution is closed under convolution. Finally, we derive a set of very simple yet accurate estimators of the NIG parameters. Our estimators differ fundamentally from estimators suggested by other authors in that our estimators take advantage of the surprisingly simple structure of the cumulant generating function.

The multivariate normal inverse Gaussian heavy-tailed distribution; simulation and estimation

The normal inverse Gaussian (NIG) distribution is a recent variance-mean mixture of a Gaussian with an inverse Gaussian distribution. The NIG can serve as a model for data that are heavy-tailed (leptokurtic), and the model was first introduced in empirical finance by Bamdorrf-Nielsen in 1995. In this paper, we present the important extension to multivariate NIG (MNIG) distributions, and we discuss some of the basic properties of the MNIG. We furthermore discuss several new and important properties of the MNIG. An important part of the paper deals with the derivation of a fast and accurate method for generating i.i.d. MNIG-distributed variates. We also present a multivariate Expectation-Maximization (EM) algorithm for the estimation of the scalar, vector, and matrix parameters of the MNIG. Finally, we present a fit of the bivariate NIG to an actual multichannel radar data set, where we have applied our EM parameter estimation algorithm. From the insight we have gained, we conclude that the MNIG has numerous potential applications in multivariate data analysis and modeling, and that the simulation and estimation methods described in this paper may serve as important and useful tools in that respect.

Statistical techniques to select detection thresholds for peak signals in ice-core data

Five statistical techniques to determine peaks in ice-core time series are presented and compared. The ice-core time series, representing different signal characteristics, comprise electrical conductivity measurements (ECM), dielectric properties (DEP) and sulphate. Three techniques (I-III) utilize all the data in the time series to estimate significant thresholds for identifying peaks. Technique IV applies a moving window and conducts a statistical inference within the defined window. In technique V, a family of smoothed estimates of the ice-core time series is produced, and statistical tests are performed on the significant changes in the derivative of the estimates. The correction of the significance level, � , due to multiple tests is introduced and implemented in techniques II-V. The threshold obtained by techniques I-III is determined by the influence of the error term on the global variance estimate, whereas the threshold of IV is determined by the data within the window. The success of identifying peaks with technique V is dependent on the redundancy in the data, i.e. the sampling rate. It is concluded that techniques II and III are superior to the other techniques due to their simplicity and robustness.

Time-frequency and dual-frequency representation of fractional brownian motion

Fractional Brownian motion (fBm) is a useful non-stationary model for certain fractal and long-range dependent processes of interest in telecommunications, physics, biology, and finance. Conventionally, the power spectrum of fBm is claimed to be a fractional power-law. However, fBm is not a wide-sense stationary process, so the precise meaning of this spectrum is unclear. In this paper, we model and analyze fBm in the context of harmonizable random processes. We derive and interpret exact expressions for novel useful complex valued second-order moment functions for fBm. These moment functions are time-frequency and dual-frequency correlation functions, connecting the random process to its infinitesimal random Fourier generator. In particular, we derive and discuss the time-frequency Rihaczek spectrum, and the dual-frequency Loeve spectrum. Our main finding is that the dual-frequency spectrum of fBm has its spectral support confined to three discrete lines. This leads to the surprising conclusion that for fBm, the DC component of the infinitesimal Fourier generator is correlated with ail other frequencies of the Fourier generator. We propose and apply multitaper based estimators for the moment functions, and numerical estimates based on synthetic fBm data and real world earthquake data confirm our theoretical results

Multivariate-multidimensional Rihaczek spectra and associated canonical correlations

Harmonizable processes constitute an important class of non-stationary stochastic processes. In this paper we study the important extension to multivariate harmonizable random fields. We derive the multivariate-multidimensional Rihaczek spectrum and show that it determines a complex time-frequency varying Wiener filter for approximating a multivariate random field from its infinitesimal Fourier generator. We derive the time-frequency coherence function, and generalize it to canonical correlations between a time domain subspace and a frequency domain subspace. We show how to construct estimators, and we finally demonstrate the theoretical concepts by the analysis of synthetic data.

Spectral, bispectral, and dual-frequency analysis of tube amplified electric guitar sound

We have analyzed the sound of an electric guitar that has been amplified by a high-quality all-tube amplifier, and emitted by means of a speaker cabinet. We re-amplified a recording of a clean guitar through a state-of-the-art all-tube amplifier at three different preamplifier gain settings: one clean, one half-distorted, and one massively distorted. Spectral analysis of recordings of the three signals exhibited a remarkably rich overtone spectrum, and we observed that only the high frequency part of the spectrum was boosted by an increase in the distortion levels. A bispectral analysis of the amplified guitar sound showed that quadratic nonlinearities are responsible for coherent phase coupling among the partials, and that the fraction of the total power which is due to quadratic nonlinearities is larger for the clean sound than for the distorted sound. Finally, a dual-frequency analysis showed that the sound, even for the sustained part of a single string pluck, is in fact a nonstationary random process. Our analysis showed that the guitar tone should be classified as an (almost) cyclostationary random process.