Application of higher order statistics/spectra in biomedical signals : a review

For many decades correlation and power spectrum have been primary tools for digital signal processing applications in the biomedical area. The information contained in the power spectrum is essentially that of the autocorrelation sequence; which is sufficient for complete statistical descriptions of Gaussian signals of known means. However, there are practical situations where one needs to look beyond autocorrelation of a signal to extract information regarding deviation from Gaussianity and the presence of phase relations. Higher order spectra, also known as polyspectra, are spectral representations of higher order statistics, i.e. moments and cumulants of third order and beyond. HOS (higher order statistics or higher order spectra) can detect deviations from linearity, stationarity or Gaussianity in the signal. Most of the biomedical signals are non-linear, non-stationary and non-Gaussian in nature and therefore it can be more advantageous to analyze them with HOS compared to the use of second order correlations and power spectra. In this paper we have discussed the application of HOS for different bio-signals. HOS methods of analysis are explained using a typical heart rate variability (HRV) signal and applications to other signals are reviewed.

Cardiac health diagnosis using higher order spectra and support vector machine

The Electrocardiogram (ECG) is an important bio-signal representing the sum total of millions of cardiac cell depolarization potentials. It contains important insight into the state of health and nature of the disease afflicting the heart. Heart rate variability (HRV) refers to the regulation of the sinoatrial node, the natural pacemaker of the heart by the sympathetic and parasympathetic branches of the autonomic nervous system. The HRV signal can be used as a base signal to observe the heart's functioning. These signals are non-linear and non-stationary in nature. So, higher order spectral (HOS) analysis, which is more suitable for non-linear systems and is robust to noise, was used. An automated intelligent system for the identification of cardiac health is very useful in healthcare technology. In this work, we have extracted seven features from the heart rate signals using HOS and fed them to a support vector machine (SVM) for classification. Our performance evaluation protocol uses 330 subjects consisting of five different kinds of cardiac disease conditions. We demonstrate a sensitivity of 90% for the classifier with a specificity of 87.93%. Our system is ready to run on larger data sets.

Removal of toxic arsenic and antimony from groundwater Spiro Tunnel Bulkhead in Park City Utah using colloidal iron hydroxide : comparison with reverse ssmosis

Verification testing of two model technologies in pilot scale to remove arsenic and antimony based on reverse osmosis and chemical coagulation/filtration systems was conducted in Spiro Tunnel Water Filtration Plant located in Park City, Utah, US. The source water was groundwater in abandoned silver mine, naturally contaminated by 60-80 ppb of arsenic and antimony below 10 ppb. This water represents one of the sources of drinking water for Park City and constitutes about 44% of the water supply. The failure to remove antimony efficiently by coagulation/filtration (only 4.4% removal rate) under design conditions is discussed in terms of the chemistry differences between Sb (III, V) and As (III, V). Removal of Sb(V) at pH > 7, using coagulation/filtration technology, requires much higher (50 to 80 times) concentration of iron (III) than As. The stronger adsorption of arsenate over a wider pH range can be explained by the fact that arsenic acid is tri-protic, whereas antimonic acid is monoprotic. This difference in properties of As(V) and Sb(V) makes antimony (V) more difficult to be efficiently removed in low concentrations of iron hydroxide and alkaline pH waters, especially in concentration of Sb < 10 ppb.

Application of higher order spectra to identify epileptic EEG

Epilepsy is characterized by the spontaneous and seemingly unforeseeable occurrence of seizures, during which the perception or behavior of patients is disturbed. An automatic system that detects seizure onsets would allow patients or the people near them to take appropriate precautions, and could provide more insight into this phenomenon. Various methods have been proposed to predict the onset of seizures based on EEG recordings. The use of nonlinear features motivated by the higher order spectra (HOS) has been reported to be a promising approach to differentiate between normal, background (pre-ictal) and epileptic EEG signals. In this work, we made a comparative study of the performance of Gaussian mixture model (GMM) and Support Vector Machine (SVM) classifiers using the features derived from HOS and from the power spectrum. Results show that the selected HOS based features achieve 93.11% classification accuracy compared to 88.78% with features derived from the power spectrum for a GMM classifier. The SVM classifier achieves an improvement from 86.89% with features based on the power spectrum to 92.56% with features based on the bispectrum.

Pattern recognition using invariants defined from higher order spectra : 2-D image inputs

A new algorithm for extracting features from images for object recognition is described. The algorithm uses higher order spectra to provide desirable invariance properties, to provide noise immunity, and to incorporate nonlinearity into the feature extraction procedure thereby allowing the use of simple classifiers. An image can be reduced to a set of 1D functions via the Radon transform, or alternatively, the Fourier transform of each 1D projection can be obtained from a radial slice of the 2D Fourier transform of the image according to the Fourier slice theorem. A triple product of Fourier coefficients, referred to as the deterministic bispectrum, is computed for each 1D function and is integrated along radial lines in bifrequency space. Phases of the integrated bispectra are shown to be translation- and scale-invariant. Rotation invariance is achieved by a regrouping of these invariants at a constant radius followed by a second stage of invariant extraction. Rotation invariance is thus converted to translation invariance in the second step. Results using synthetic and actual images show that isolated, compact clusters are formed in feature space. These clusters are linearly separable, indicating that the nonlinearity required in the mapping from the input space to the classification space is incorporated well into the feature extraction stage. The use of higher order spectra results in good noise immunity, as verified with synthetic and real images. Classification of images using the higher order spectra-based algorithm compares favorably to classification using the method of moment invariants

Shape discrimination using invariants defined from higher-order spectra

An approach to pattern recognition using invariant parameters based on higher-order spectra is presented. In particular, bispectral invariants are used to classify one-dimensional shapes. The bispectrum, which is translation invariant, is integrated along straight lines passing through the origin in bifrequency space. The phase of the integrated bispectrum is shown to be scale- and amplification-invariant. A minimal set of these invariants is selected as the feature vector for pattern classification. Pattern recognition using higher-order spectral invariants is fast, suited for parallel implementation, and works for signals corrupted by Gaussian noise. The classification technique is shown to distinguish two similar but different bolts given their one-dimensional profiles

A general procedure for the derivation of principal domains of higher-order spectra

A general procedure to determine the principal domain (i.e., nonredundant region of computation) of any higher-order spectrum is presented, using the bispectrum as an example. The procedure is then applied to derive the principal domain of the trispectrum of a real-valued, stationary time series. These results are easily extended to compute the principal domains of other higher-order spectra

Position, rotation, and scale invariant recognition of images using higher-order spectra

A new approach to recognition of images using invariant features based on higher-order spectra is presented. Higher-order spectra are translation invariant because translation produces linear phase shifts which cancel. Scale and amplification invariance are satisfied by the phase of the integral of a higher-order spectrum along a radial line in higher-order frequency space because the contour of integration maps onto itself and both the real and imaginary parts are affected equally by the transformation. Rotation invariance is introduced by deriving invariants from the Radon transform of the image and using the cyclic-shift invariance property of the discrete Fourier transform magnitude. Results on synthetic and actual images show isolated, compact clusters in feature space and high classification accuracies

The molecular structure of the mineral sarmientite Fe2(AsO4,SO4)2(OH)6•5H2O : implications for arsenic accumulation and removal

Sarmientite is an environmental mineral; its formation in soils enables the entrapment and immobilisation of arsenic. The mineral sarmientite is often amorphous making the application of X-ray diffraction difficult. Vibrational spectroscopy has been applied to the study of sarmientite. Bands are attributed to the vibrational units of arsenate, sulphate, hydroxyl and water. Raman bands at 794, 814 and 831 cm−1 are assigned to the ν3 (AsO4)3− antisymmetric stretching modes and the ν1 symmetric stretching mode is observed at 891 cm−1. Raman bands at 1003 and 1106 cm−1 are attributed to vibrations. The Raman band at 484 cm−1 is assigned to the triply degenerate (AsO4)3− bending vibration. The high intensity Raman band observed at 355 cm−1 (both lower and upper) is considered to be due to the (AsO4)3−ν2 bending vibration. Bands attributed to water and OH stretching vibrations are observed.

Pattern recognition using invariants defined from higher order spectra - one-dimensional inputs

A new approach to pattern recognition using invariant parameters based on higher order spectra is presented. In particular, invariant parameters derived from the bispectrum are used to classify one-dimensional shapes. The bispectrum, which is translation invariant, is integrated along straight lines passing through the origin in bifrequency space. The phase of the integrated bispectrum is shown to be scale and amplification invariant, as well. A minimal set of these invariants is selected as the feature vector for pattern classification, and a minimum distance classifier using a statistical distance measure is used to classify test patterns. The classification technique is shown to distinguish two similar, but different bolts given their one-dimensional profiles. Pattern recognition using higher order spectral invariants is fast, suited for parallel implementation, and has high immunity to additive Gaussian noise. Simulation results show very high classification accuracy, even for low signal-to-noise ratios.

Higher-Order spectra of nonlinear polynomial models for Chua's circuit

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

Raman spectroscopy of selected tsumcorite Pb(Zn,Fe3+)2(AsO4)2(OH,H2O) minerals –implications for arsenate accumulation

The presence of arsenic in the environment is a hazard. The accumulation of arsenate by a range of cations in the formation of minerals provides a mechanism for the accumulation of arsenate. The formation of the tsumcorite minerals is an example of a series of minerals which accumulate arsenate. There are about twelve examples in this mineral group. Raman spectroscopy offers a method for the analysis of these minerals. The structure of selected tsumcorite minerals with arsenate and sulphate anions were analysed by Raman spectroscopy. Isomorphic substitution of sulphate for arsenate is observed for gartrellite and thometzekite. A comparison is made with the sulphate bearing mineral natrochalcite. The position of the hydroxyl and water stretching vibrations are related to the strength of the hydrogen bond formed between the OH unit and the AsO43- anion. Characteristic Raman spectra of the minerals enable the assignment of the bands to specific vibrational modes.