The Analysis of Closed-Loop Dynamic-Stochastic Systems

Certain types of non-linear transformations of stochastic signal processes have proven useful in engineering applications. A form of processing called hard clipping (or hard limiting) has frequently been used in electronic systems which work in “real time” in order to reduce the number of information bits which the system has to process. With hard clipping only one binary digit is used per sample value of an input signal X(t). However, information is lost about X(t) and the spectrum which is calculated from the clipped signal is a distorted form of the original signal spectrum. By a sine transformation of the sample covariance of the clipped process, a consistent estimator is derived for the spectral density of X(t). The asymptotic variance-covariance function is calculated for this estimator.     

A DIAGNOSTIC TEST FOR NONLINEAR SERIAL DEPENDENCE IN TIME SERIES FITTING ERRORS

Abstract. Time series analysts have begun to consider the applicability of nonlinear models. In order for nonlinear models to be accepted by practitioners, practicai tests must be avilable to test for the presence of nonlinearity in both raw time series and in the residuals from fitted models. A diagnostic test, based on the bispectrum, for the presence of nonlinear serial dependence in these time series is investigated here using artificial data. Detection of such nonlinear dependence is taken to indicate that nonlinear modelling methods are necessary. The theory behind the test is reviewed and simulations driven by pseudorandom numbers are presented for a variety of models and sample sizes. The simulations indicate that the test has substantial power for many models. In addition, theoretical and empirical results are presented which show that the bispectral diagnostic test is equally powerful for both the source series and for the fitting errors from a line& model. Thus, while the test is suitable for use as a diagnostic test on the fitting errors of linear time series models, prior linear modeling of the time series is not required.     

On the principal domain of the discrete bispectrum of a stationarysignal

The paper presents a simplifying, yet general approach to determining the symmetry structure of a bispectrum. The principal domain (PD) of the bispectrum and its region of positive support are derived in a way that illuminates the controversy surrounding the triangle in the PD, which is called the outer triangle (OT) by Hinich and Wolinsky (1988), where the bispectrum is zero for a stationary random sampled process that is not aliased. The basic statistical issues of testing for nonzero bispectral structure are reviewed     

Time delay estimation using the cross bispectrum

The cross bispectrum phase can be effectively used to estimate the time required for a nonGaussian signal to propagate between a pair of spatially separated sensors in the presence of highly correlated Gaussian noise. The authors present a consistent estimator of the phase of the cross bispectrum, derive the exact distribution of the phase of a complex Gaussian sample bispectrum, and show that in most cases the exact distribution can be approximated by a Gaussian distribution. Using this Gaussian approximation, the authors derive the variance of the time delay estimate computed from the sample cross bispectrum of a signal in additive correlated noise. These results allow the performance of time delay estimators based on the cross bispectrum phase to be quantified as a function of the sample size, the skewness of the signal, the signal-to-noise ratio (SNR), and the noise correlation     

UDP-glucose dehydrogenase from bovine liver: primary structure and relationship to other dehydrogenases

The primary structure of bovine liver UDP-glucose dehydrogenase (UDPGDH), a hexameric, NAD(+)-linked enzyme, has been determined at the protein level. The 52-kDa subunits are composed of 468 amino acid residues, with a free N-terminus and a Ser/Asn microhetergeneity at one position. The sequence shares 29.6% positional identity with GDP-mannose dehydrogenase from Pseudomonas, confirming a similarity earlier noted between active site peptides. This degree of similarity is comparable to the 31.1% identity vs. the UDPGDH from type A Streptococcus. Database searching also revealed similarities to a hypothetical sequence from Salmonella typhimurium and to "UDP-N-acetyl-mannosaminuronic acid dehydrogenase" from Escherichia coli. Pairwise identities between bovine UDPGDH and each of these sequences were all in the range of approximately 26-34%. Multiple alignment of all 5 sequences indicates common ancestry for these 4-electron-transferring enzymes. There are 27 strictly conserved residues, including a cysteine residue at position 275, earlier identified by chemical modification as the expected catalytic residue of the second half-reaction (conversion of UDP-aldehydoglucose to UDP-glucuronic acid), and 2 lysine residues, at positions 219 and 338, one of which may be the expected catalytic residue for the first half-reaction (conversion of UDP-glucose to UDP-aldehydoglucose). A GXGXXG pattern characteristic of the coenzyme-binding fold is found at positions 11-16, close to the N-terminus as with "short-chain" alcohol dehydrogenases.(ABSTRACT TRUNCATED AT 250 WORDS)     

TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES

Abstract. Stable autoregressive (AR) and autoregressive moving average (ARMA) processes belong to the class of stationary linear time series. A linear time series { } is Gaussian if the distribution of the independent innovations {ε( t )} is normal. Assuming that E ε( t ) = 0, some of the third-order cumulants c_xxx= Ex ( t ) x ( t + m ) x ( t + n ) will be non-zero if the ε( t ) are not normal and E ε³( t )≠O. If the relationship between { x ( t )} and {ε( t )} is non-linear, then { x ( t )} is non-Gaussian even if the ε( t ) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of { c _xxx( m, n )}. This sample bispectrum is used to construct a statistic to test whether the bispectrum of { x ( t )} is non-zero. A rejection of the null hypothesis implies a rejection of the hypothesis that { x ( t )} is Gaussian. Another test statistic is presented for testing the hypothesis that { x ( t )} is linear. The asymptotic properties of the sample bispectrum are incorporated in these test statistics. The tests are consistent as the sample size N →-∞     

Higher order cumulants and cumulant spectra

An exposition of joint cumulants and cumulant spectra is presented. A distinction is emphasized in this paper between the cumulant spectrum of a time series and its stationary version, here called apolyspectrum. The variance and covariance of the sample bispectrum is then derived using a relationship between cumulant spectra of the finite Fourier transform for the 2nd and 4th cumulant function, and the bispectrum and trispectrum of the time series.This work was supported by the Office of Naval Research under contract N00014-91-J-1276.     

Fitting a Nonstationary Model of Ionospheric Motions

This paper presents a non standard analysis of systematic nonstationarity in data on ionospheric motions. The methodology is applicable to any analysis of ambient back-ground noise in communication systems which use the atmosphere as a medium.

The data used in this paper were obtained from an array of radar-like devices called phase-path sounders, which measure the velocity of the vertical motion of the neutral gases in the ionosphere. The model fits between 36% and 5%, of the variation of the log spectrum for frequencies less than 3 CPH and greater than 6 CPH using a period of 24 hours (diurnal) and the first harmonic of 12 hours (semidiurnal).     

A NEW DIAGNOSTIC TEST OF MODEL INADEQUACY WHICH USES THE MARTINGALE DIFFERENCE CRITERION

Abstract. Let {x(t)} denote a discrete-time random process. Given a sample of increments e (t) = x(t) - x(t - 1) from the time series, we wish to test formally whether the sample is consistent with the assumption that {e(t)} is a martingale difference. It is shown that the martingale criterion is more general than the white noise criterion in analyzing fitted residuals for signs of model inadequacy. In this paper we present such a test which approximately achieves a given type 1 error probability for samples. We assume that (1) the process is strictly stationary, (2) all its k th-order cumulant functions exist and (3) the k th-order cumulants are absolutely summable and satisfy a mixing condition. The martingale assumption implies that most third-order cumulants of the increment process are zero, and thus the third-order cumulant sequence is sparse. This result is used to derive test statistics based on a modified sample bispectrum. The test can be regarded as a two-dimensional portmanteau test of serial dependence. The large-sample results are demonstrated through the use of artificial data. Finally, the test is applied to a daily financial series.     

Development and application of a statistically based feature extraction algorithm for monitoring tool wear in Circuit Board Assembly

Vibroacoustic signals of rotating machinery are composed of sums of modulated periodicities, broadband random components, and occasionally a set of transient responses. These signals are not ergodic as the modulated periodicities are partially coherent. Progressive wear of the rotating machine causes the nonlinear structure of the received signal to intensify, and nonlinearity results in transfer of energy between harmonics of the signal's periodic components. Statistics developed from bispectrum and second-order cumulant spectrum estimates of the measured signal are combined with power spectrum amplitudes as feature inputs for standard multivariate classifiers. The higher-order statistics measure, respectively, the extent of nonlinearity and intermodulation of the received signal. Classification results of simulated and actual incipient wear data collected from a controlled experiment drilling circuit boards illustrate the potential of this novel statistical signal processing approach.