Coherence Function Estimates Research Articles

Semiparametric estimation of cross-covariance functions for multivariate random fields.

The prevalence of spatially referenced multivariate data has impelled researchers to develop procedures for joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any arbitrary pair of locations using a multivariate spatial covariance function. However, building a flexible multivariate spatial covariance function that is nonnegative definite is challenging. Here, we propose a semiparametric approach for multivariate spatial covariance function estimation with approximate Matérn marginals and highly flexible cross-covariance functions via their spectral representations. The flexibility in our cross-covariance function arises due to B-spline-based specification of the underlying coherence functions, which in turn allows us to capture nontrivial cross-spectral features. We then develop a likelihood-based estimation procedure and perform multiple simulation studies to demonstrate the performance of our method, especially on the coherence function estimation. Finally, we analyze particulate matter concentrations (PM2.5 ) and wind speed data over the West-North-Central climatic region of the United States, where we illustrate that our proposed method outperforms the commonly used full bivariate Matérn model and the linear model of coregionalization for spatialprediction.

Control-oriented friction modeling of hydraulic actuators based on hysteretic nonlinearity of lubricant film

Friction modeling of hydraulic actuators is essential to system control and the prediction accuracies of friction models (e.g., Stribeck, LuGre) need to be improved when they were applied to hydraulically actuated systems. The test platform for hydraulic cylinders was constructed and the friction forces of plunger type hydraulic actuators were measured under different operating conditions. The dynamic behaviors of film thickness of seal/rod interface were investigated based on elastic hydrodynamic lubrication (EHL) method. The effects of acceleration of rod on friction force were examined using coherence function estimation technique. The Stribeck and LuGre model were modified by incorporating physically based hysteretic film dynamics and Bouc–Wen model into Stribeck function, respectively. It is shown that the Stribeck and LuGre model cannot preferably describe the friction force-velocity behavior, especially for the fluid lubrication regime. The modified Stribeck and LuGre model can simulate the hysteretic behavior of friction force-velocity loop more effectively than unmodified ones and the friction estimation accuracies of Stribeck and LuGre model were improved remarkably.

Generalised framework for nonparametric coherence function estimation

By introducing a nonlinear function of covariance matrix, a generalised framework for the nonparametric coherence function (NPCF) estimation is presented. This framework helps us understand the properties of several existing NPCF estimators more easily, including the single- and multi-window based approaches. Moreover, by properly choosing the parameters of the generalised class of NPCF estimators, a good tradeoff between spectral resolution and variance can be achieved.

Estimation of Time-Varying Coherence Function Using Time-Varying Transfer Functions

We introduce a new method to estimate reliable time-varying coherence functions (TVCF) for causal systems. The technique is based on our previously developed method to estimate time-varying transfer functions (TVTF), known as the time-varying optimal parameter search algorithm [Zou, R., H. Wang, and K. H. Chon. A robust time-varying identification algorithm using basis functions. Ann. Biomed. Eng. 31: 840-853, 2003]. The TVCF is estimated by the multiplication of two TVTFs. The two TVTFs are obtained using signal x as the input and signal y as the output to produce the first TVTF, and signal y as the input and signal x as the output to produce the second TVTF. Demonstration of the feasibility and efficacy of the proposed approach is provided with both simulation examples and application to renal blood flow and pressure data. The proposed approach provides higher time-frequency resolution TVCF than afforded by the short time Fourier transform based TVCF.

Alterations to coherent flow structures and heat transfer due to pulsations in an impinging air-jet

An investigation was carried out to study the effect of flow pulsation on the characteristics of a planar air jet impinging normally on a heated surface. Such information was further utilized to determine the influence of flow characteristics in the plane of impingement on Nusselt number distribution. Time-resolved system properties were investigated with modern instrumentation that allowed instantaneous heat transfer and flow velocity measurements to be performed simultaneously. Based on good coherence function estimates between the signals, heat transfer measurements were used in return to infer flow dynamics near the impingement surface. Experiments were performed for steady and pulsating jets at jet Reynolds numbers of 1 000, 5 500, and 11 000, pulse frequencies up to 82 Hz (corresponding to Strouhal numbers below 0.13), and pulse amplitude at the nozzle exit up to 50 % of the mean flow velocity. Special techniques commonly used for periodically disturbed flow fields elucidated the dynamics of the pulse and associated coherent flow structures. Results indicated the parametric conditions for which alterations are expected in time-averaged heat transfer from the surface. Engineering applications include cooling of electronic packages and heat transfer to gas turbine blades.

Group delay based magnitude square coherence estimation by an ARMA model

In this paper, a new method of estimation of the magnitude square coherence function (MSC) by an ARMA model is proposed. The estimation is achieved by modeling the periodogram estimate of the MSC and the ARMA model has been realized by the pole-zero decomposition property of the group delay function (GDF). Its performance has been found to be superior to that of the periodogram MSC estimate in terms of normalized sum of the sample mean square error (NSSMSE) and maximum sample root mean square error (RMSE). Further, its performance is comparable with those of the existing ARMA based estimation methods and requires less computations. The proposed ARMA-MSC estimation is simple and does not involve any explicit solution of a system of equations.

Linear systems analysis of cutaneous Type I mechanoreceptors

Linear system transfer functions of Type I mechanoreceptor domes in hairy skin were extracted for study. The system input, defined in units of indentation depth, was a punctate, 85 Hz bandlimited colored noise indenting stimulus. The system output was a dome's afferent AP event stream. After digitally low-pass filtering the AP event stream to prevent aliasing, transfer and coherence function estimates were generated from spectral and cross spectral estimates of the sampled input and output signals. Transfer function magnitudes (receptor sensitivity) beyond 10 Hz could be well fit by a fractional exponent (0.35 +/- 0.10) of frequency, indicating fractional order differentiation. Transfer function phases were near 120 degrees for frequencies less than 10 Hz and decreased progressively beyond 10 Hz. Coherence function estimates were low as a result of inherent nonlinearities, primarily rectification. Single dome results closely matched those reported by French and Kuster [6] for the tactile spine of the cockroach, suggesting that in terms of spectral sensitivity the receptors have similar transduction mechanisms. However, nonlinear systems modeling techniques are required for a more complete system characterization.

Analysis of auditory evoked potentials by magnitude-squared coherence.

Evoked potentials are usually analyzed in the time domain (voltage versus time). The most familiar frequency-domain measure, the power spectral density function, displays power as a function of frequency but doesn't distinguish signal power from noise power. The coherence function estimates, for each frequency, the ratio of signal power to total (signal plus noise) power and, thus, indicates the degree to which system output (scalp potential) is determined by input (acoustic stimulus). Coherence ranges from 0 to 1; values above specified critical values can be accepted as demonstrating statistically significant system response. In this paper, we present coherence analysis of human scalp responses to clicks and amplitude-modulated tones. In both cases, this analytic method provides insight into the spectral character of the response (for example, assisting in specifying desirable filter characteristics). Threshold sensitivity is also improved: statistically significant responses can be detected at lower intensity by coherence analysis than by inspection of time-domain waveforms.

Relations between time- and frequency-domain measures of signal transmission from cutaneous afferents to dorsal horn neurons

In pentobarbitone-anesthetized cats, the spike sequences of dorsal horn neurons were recorded in response to random stimulation of branches of the suralis nerve. Combined frequency- and time-domain analysis was performed on the stimulus and spike trains. Coherence function estimates computed by spectral analysis with peristimulus time histograms (PSTHS). The cell responses were divided into 4 main types: PSTHs with a single high and narrow peak were associated with broad-range high coherence; PSTHs with two (or sometimes 3) distinct peaks concurred with a coherence which was high at low frequencies, low at intermediate ones and higher again at high frequencies; broad unstructured PSTH peaks of varying height concurred with coherence declining from high values at low frequencies to lower values at higher ones; and small and broad PSTH peaks were associated with generally low coherence. Thus, the variation of coherence with frequency depends on the precise pattern of cell discharge.

PROPERTIES OF COHERENCE FUNCTIONS AND MODIFICATION OF COMPUTATIONAL METHOD

The difference of seismic waves observed on piers at both ends of a long span bridge or at locations along an underground pipe often becomes a critical factor and is taken into consideration for aseismic design of such structures. The characteristics of the effect of the difference may be described by a coherence function which represents the dependency of two time series, and it is calculated from the power and cross-spectral density functions. However, in its estimation, it is often smoothed in the frequency domain. In this paper, the effects of the smoothing on the coherence function estimate are analyzed quantitatively, in the case that a time delay is observed between the two records or that they are statistically independent. Then, a modified method of calculating coherence function estimates is proposed.

Experimental modal analysis of non-linear systems: A feasibility study

Experimental modal analysis techniques are based on the theory of linear systems. In this paper the feasibility of their application to non-linear systems is examined. A three degree of freedom non-linear mechanical system was modeled on an analog computer and system responses due to band limited random excitation were examined by using a two channel FFT analyzer. Frequency response functions, natural frequencies and model responses were studied experimentally for a number of example cases. This study shows that the measured transfer function plots can alert one to the fact that the structure is non-linear; measured modal response data base can still give a rough idea about the system behavior. In addition, measured coherence function estimates can be used as the “warning singls” for inherent non-linearities.

The Accuracy of Frequency Response Function Measurements Using FFT-Based Analyzers With Transient Excitation

The accuracy of the frequency response measurement obtained using impact excitation and a Fast Fourier Transform based spectrum analyzer has been investigated. It has been shown that with impact excitation, provided the impacts are reproducible and the extraneous noise level is low, the coherence estimates obtained from the analyzer are unity, irrespective of the frequency resolution employed. Hence the H1 (Sxy/Sxx) and H2 (Syy/Sxy*) frequency response function estimates are identical. However, these frequency response function estimates are affected by a bias error caused by inadequate frequency resolution so unity coherence does not necessarily imply accurate results. The results with impact excitation are compared with those obtained using random excitation where both the coherence and frequency response function estimates are affected by bias error. The bias error in the frequency response function estimates with impact excitation is intermediate between that in the H1 and H2 estimates when random excitation is used. The theoretical predictions have been verified by tests on an analogue computer and on a built-up structure.

Transformed coherence functions for multivariate studies

In this paper, transformed coherence function estimates are defined which display several desirable properties when compared with the conventional forms; 1) their probability distribution functions are more nearly normal, 2) their mean values are normalized to a value of unity for totally uncorrelated data, and 3) their variances are independent of the true values.

Effects of time window on the estimation of coherence function for the system with delay.

The effects of the shape of time window on the estimation of the coherence function for the system having group delay are investigated. It is well known that the time-windowing determines the spectral resolution and the sidelobes, and that the suitable time window should be selected from that point of view. Time-windowing has another effect on the estimation when the system has group delay. Since the cross power spectrum obtained from DFT's corresponds to the circular convolution of the input and output sequences in time domain, the windowed input and output include the uncorrelated components in the initial and final parts of the windows due to group delay. Consequently, the estimation of the coherence function is biased toward zero. The estimation is improved by using the Hanning window as far as the delay is sufficiently shorter than the window duration, but the rectangular window becomes superior to the Hanning window when the delay is longer than about 0.18 times of the window duration. Theoretical and experimental results indicate the effects of time window on the estimation of the coherence function.

Statistical errors in measurement of coherence functions and input/output quantities

Practical statistical error analysis formulas are developed for evaluating ordinary, partial and multiple coherence function estimates, and associated quantities which occur in analysis of single input/output problems and in multiple input/output problems. These include estimates of coherent output spectra and estimates of gain and phase factors of related frequency response functions. Models are either single input/output models or conditioned input/output models, where the latter models can be created from arbitrary measured multiple input/output data. Guidelines are presented to reduce statistical errors in such computed quantities. These results are essential for applications to identify different multiple noise sources, to determine multiple system properties, to measure response effects due to the sources acting singly or in various combinations, to simulate multiple random environments, and to predict system failures from response measurements only.