Exponential Weighting Research Articles

Discrete adaptive observer with exponential weighting properties

An adaptive observer based on the exponentially weighted least-squares method is studied, and a method of selecting the state variable filters is proposed. By this method, all the past measurement data are weighted exponentially both with weighting coefficient and the state variable filters. The proposed scheme guarantees fast convergence and is very stable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On- and off-line identification of linear state-space models

Abstract A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-varying systems using windowing or exponential weighting. These results are illustrated by examples, including the identification of an industrial plant.

On a new functional form for the dispersive flux in porous media

A recently developed second‐order model for local dispersive transport in porous media has been simplified to yield a new, closed‐form relationship for the dispersive flux. In situations characterized by negligible velocity gradients, the flux can generally be represented as a convolution or “memory” integral over time of previous concentration gradients. The strength of this memory is controlled by an exponential weighting factor related to the magnitudes of the velocity and local molecular diffusive flux. The form of this result is consistent with other models of diffusive and dispersive transport phenomena over various spatial scales. In circumstances where the memory strength is small, the integral can be simplified and cast in the form of a standard Fickian relationship with apparent time‐dependent dispersivity functions that grow to finite, asymptotic values. This specific formulation can be manipulated to yield a one‐equation transport balance law in the form of a telegraph equation. Nonphysical effects, such as spurious upstream dispersion and instantaneous propagation of mass to extremely distant points predicted with a Fickian law, are reduced or eliminated. Although the importance of the new result in transport simulations will depend on the spatial and temporal scales of interest, it should provide some insight in the interpretation and design of new experiments.

Characterization of the Weights of Least Squares Adaptive Polynomials

Passi and Morel have provided a concept of adaptive least squares polynomials for time series data using exponential weighting $w_k = a^{ - k} w_0 ,0 < a < 1$. Given the least squares polynomial $p(t)$ obtained from data $Y_k ,k\leqq 0$, they derived a computationally efficient recursion algorithm to update the polynomial coefficients when a new data value is received. They showed that the exponential weighting possesses the “update property,” i.e., the polynomial-update is invariant to new data value $Y_1 $ if $Y_1 = p(1)$. It was this update property that made the recursive algorithm possible. In this paper we explore the weights ${\bf w}$ which possess this update property for a more general class of functions that include the polynomials as a special case. In this general case we give a necessary and sufficient condition for a weight function to satisfy the update property. In the case of polynomials it is shown that the exponential weights are the only weight functions of practical significance which...

On the application of hinged finite elements to eddy current problems

A Finite Element method that uses a hinged weighting function is applied to 1-D and 2-D eddy current problems. A method of choosing the hinge parameters such that the resulting piecewise linear function approximates an exponential weighting function is developed. A special purpose quadrilateral element that incorporates the hinged weighting function is derived. The hinged elements are used to solve 1-D and 2-D eddy current test problems. Computed results that illustrate the advantages of the hinged F.E.M. are presented.

Parameter estimation with variable sampling time via generalized block pulse functions

Abstract Parameter estimation with variable sampling time is developed in this paper, using a continuous model. The input‐output variables are approximated by using generalized block pulse series expansion. A method of determining the sampling interval is proposed. The algorithm depends on the variations of the input and output variables. Parameter estimation is carried out by using a least‐squares estimation with exponential data weighting. Two examples are presented to demonstrate that the method exhibits satisfactory results.

Memory and the efficient use of information

We consider the problem of how an animal's memory should be designed in order to cope with a stochastic and changing environment. In particular we consider the problem of forming the best estimate of an unknown and possibly changing environmental parameter. Under the simple model we consider, the effect of an observation is to update this estimate using a linear operator. Two models of a changing environment are analysed. For each model we show how estimates change as a function of time elapsed and observations taken. The effect of a regular sequence of observations is also considered, and it is shown that an exponential weighting of past observations is a sufficient statistic on which to base decisions. The weighting factors are different in the two model environments considered, but each is shown to be a function of the rate at which the environment is changing.

Tracking properties and steady-state performance of RLS adaptive filter algorithms

Adaptive signal processing algorithms derived from LS (least squares) cost functions are known to converge extremely fast and have excellent capabilities to "track" an unknown parameter vector. This paper treats analytically and experimentally the steady-state operation of RLS (recursive least squares) adaptive filters with exponential windows for stationary and nonstationary inputs. A new formula for the "estimation-noise" has been derived involving second- and fourth-order statistics of the filter input as well as the exponential windowing factor and filter length. Furthermore, it is shown that the adaptation process associated with "lag effects" depends solely on the exponential weighting parameter λ. In addition, the calculation of the excess mean square error due to the lag for an assumed Markov channel provides the necessary information about tradeoffs between speed of adaptation and steady-state error. It is also the basis for comparison to the simple LMS algorithm, in a simple case of channel identification, it is shown that the LMS and RLS adaptive filters have the same tracking behavior. Finally, in the last part, we present new RLS restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise, namely, finite precision arithmetic.

Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

AbstractFor a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocityu(“vertically” averaged), ice thicknessh, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) valuesuo,h0, andα0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of∆h(x) and ∆α(x),xbeing the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of localh(x) and α(x) on the flowu(x):where the integration is over the lengthLof the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variablex′, represents the influence of localh(x′),α(x′), and channel-shape factorf(x′), at longitudinal coordinatex′, on the flowuat coordinatex, the influence being weighted by the “influence transfer function” exp (−|x′ −x|/ℓ) in the integral.The quantityℓthat appears as the scale length in the exponential weighting function is called thelongitudinal coupling length. It is determined by rheological parameters via the relationship, wherenis the flow-law exponent,ηthe effective longitudinal viscosity, andηthe effective shear viscosity of the ice profile,ηis an average of the local effective viscosityηover the ice cross-section, and (η)–1is an average of η−1that gives strongly increased weight to values near the base. Theoretically, the coupling lengthℓis generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer,ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations inhor α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called thelongitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length4lthat gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding toℓ/h≈ 2 in both cases.Ifℓvaries withx, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation,ℓin the averaging integral isℓ(x) but is not a function ofx′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.

Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

AbstractFor a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocity u (“vertically” averaged), ice thickness h, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) values uo, h0, and α0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of ∆h(x) and ∆α(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and α(x) on the flow u(x): where the integration is over the length L of the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x′, represents the influence of local h(x′), α(x′), and channel-shape factor f(x′), at longitudinal coordinate x′, on the flow u at coordinate x, the influence being weighted by the “influence transfer function” exp (−|x′ − x|/ℓ) in the integral.The quantity ℓ that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. It is determined by rheological parameters via the relationship , where n is the flow-law exponent, η the effective longitudinal viscosity, and η the effective shear viscosity of the ice profile, η is an average of the local effective viscosity η over the ice cross-section, and (η)–1 is an average of η−1 that gives strongly increased weight to values near the base. Theoretically, the coupling length ℓ is generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer, ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called the longitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length 4l that gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding to ℓ/h ≈ 2 in both cases.If ℓ varies with x, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation, ℓ in the averaging integral is ℓ(x) but is not a function of x′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.

NMR data processing using the maximum entropy method

NMR spectroscopists have become accustomed to the Fourier transform for converting free induction decays into spectra. Indeed the FT is now so familiar that there is a tendency to ignore its deficiencies and forget that alternative data processing techniques do exist and may sometimes be superior. Perhaps the most important failing of the discrete F’T is its inability to handle satisfactorily a free induction decay the duration of which is less than about 3TT. Fourier transformation of such a truncated decay produces “sine wiggle” artifacts in the spectrum while apodization prior to transformation, for example by exponential weighting, gives an unwelcome loss in spectral resolution. Sine wiggles are a direct consequence of the unwarranted assumption, implicit in the Fourier analysis of truncated signals, that the unrecorded data points are either nonexistent or equal to zero. While these problems can usually be avoided simply by extending the acquisition time, truncated responses are often unavoidable or even desirable in two-dimensional NMR and NMR imaging. An undesirable feature of some two-dimensional NMR spectra is the “phase twist” lineshape produced by Fourier transformation of a 2D data matrix in which the FiDs in t2 are phase modulated as a function of the other time variable, t,. Unless the two quadrature amplitude modulated signals are available separately, it is impossible by Fourier transformation alone to separate the two-dimensional absorption and dispersion lineshapes. So there are situations in which the FT is less than ideal and an alternative procedure would seem desirable. The properties required of such a technique are (a) absence of prejudice concerning missing data points, (b) no requirement to perform a IFourier transform on the experimental data with its inherent imperfections, and (c) that no feature should appear in the final spectrum for which there is not su:t&ient evidence in the data. As will be shown below a data handling technique known as the maximum entropy method (I-3) (MEM) meets these conditions and so circumvents some of the pitfalls of Fourier transformation. Maximum entropy has enjoyed widespread use in recent years in such fields as radioastronomy, crystallography, and medical tomography, but has barely been exjplored in the context of magnetic resonance. Van Ormondt et al. (4, 5) were the first, using the method for processing electron spin-echo spectra. More recently, Sibisi et al. (6, 7) have applied maximum entropy to NMR, claiming simultaneous improvements in both resolution and signal-to-noise, as well as seemingly spectacular

A review of homomorphic deconvolution

The purpose of this paper is to review the definition, to outline advantages and disadvantages, and to present both solved and unsolved problems of homomorphic deconvolution. The advantages of homomorphic deconvolution are that it does not require the assumptions of a minimum‐phase wavelet and of a white random reflection coefficient series. Recognized disadvantages of the method have been difficulties in unwrapping the phase, in dealing with band‐limited signals, and in handling mixed‐phase reflection coefficient series. These difficulties may be overcome by using an “adaptive numerical integration algorithm,” frequency transformations, and exponential weighting of the signal, respectively. Two unresolved problems in homomorphic deconvolution are deciding the cutoff quefrencies in liftering convolutional components and eliminating the effect of additive noise. No theoretical means of recovering one convolutional component of an arbitrary real seismogram which is not contaminated by the other component has been discovered. Additive noise plays an important role in homomorphic deconvolution, so that it is unreliable when the spectral amplitudes of the signal are very small over certain frequency bands in signals of relatively small signal‐to‐noise ratios.

Comparison of least-squares and stochastic gradient lattice predictor algorithms using two performance criteria

The least-squares (LS) and stochastic gradient (SG) lattice prediction algorithms are compared using two different performance criteria. These are a) output mean squared error and b) the accuracy of the autoregressive, estimate obtained from the mean values of the lattice coefficients, assuming a stationayinput. It is found that the second performance criterion is more sensitive than the first. This spectral performance criterion is a measure of the accuracy of the estimatcd autoregressive model coefficients. Bias in the LS and SG coefficient estimates can cause significant deviation of the asymptotic estimates from the actual input spectrum: The similarly between the LS and SG lattice algorithms enables comparative simulations with analogous initial conditions and exponential weighting constants. For both types of comparisons, the LS algorithm offers a modest performance improvement over the SG algorithms simulated. This improvement is more noticeable when the input is highly correlated. It is also found that slight changes in the SG lattice algorithm may significantly affect its performance.

Homomorphic deconvolution

The advantages of homomorphic deconvolution are that it does not require the assumptions of minimum‐phase wavelet and of a white random reflection coefficient series. Disadvantages of the method which have been recognized in the public domain are difficulties in unwrapping the phase, in dealing with band‐limited signals, and in handling mixed‐phase reflection coefficient series. These difficulties may be respectively overcome by using an “adaptive numerical integration algorithm” (Tribolet, 1977), frequency transformations (Tribolet, 1979), and exponential weighting of the signal (Tribolet, 1979). There seems to have been some understanding in the literature and among exploration researchers that additive noise would affect the performance of homomorphic deconvolution. However, to the best of our knowledge there have not appeared in the literature any analytical expressions or experiments conclusively showing how additive noise affects homomorphic deconvolution. Analytic and experimental analyses demonstrated that additive noise plays a critical role in homomorphic deconvolution such that homomorphic deconvolution is unreliable whenever the spectral amplitudes of the signal are very small over certain frequency bands and even a small amount of noise is present. This unreliability of the method overshadows its advantages.

The Discrete Laguerre Polynomials and Their Use in Exponential Smoothing

Abstract The formulation and methodology of curve fitting polynomials by discounted least squares can be simplified through the use of the discrete Laguerre polynomials, which are mutually orthogonal with respect to an exponential weighting function. One advantage of this formulation is that the first k+ 1 coefficients of a fitting polynomial of degree m > k are the same as those of a fitting polynomial of degree k. The Laguerre polynomials are introduced and their properties are derived. The recursive method for computing coefficients is shown to have a simple structure that is independent of the degree k of the fitting polynomials. In addition, some properties of the method of smoothing and forecasting are discussed.

An evaluation of approximate methods for calculating anisotropic flux escape and transmission probabilities for cylindrical annular regions

In the interface current method for solving the integral transport equation, most of the computer time is spent in the precalculation of various region transmission, escape and collision probabilities. Therefore, any reduction in computer time used to calculate these probabilities will directly affect the efficiency of this method. The normal methods such as the Trapezoidal rule or Gauss-quadrature formula for calculating the various probabilities require the evaluation of many Bickley functions per integral, which is time consuming. This paper discusses efficient methods for calculating the various probabilities corresponding to anisotropic terms of the angular flux expansion at region interfaces for cylindrical annular geometry. The paper first discusses the application of Bonalumi's Pseudo-Linear (P-L) approximation and the second-order correction to it which requires the evaluation of one and two Bickley functions, respectively, per integral. The accuracy of the second-order approximation is of the order of 0.1% for all the probabilities except the transmission probability from outer to outer surface ( P ρv ∞) of the region (where ρ and v denote the order of anisotropy of the flux entering and leaving the outer boundary of the annular region). The paper then discusses a Gauss-quadrature formula with exponential weighting and its modified form which is suitable for calculating P ρv ∞ efficiently.

Implementation of self-tuning regulators with variable forgetting factors

A modified version of the self-tuning regulator having limited adaptability has been successfully implemented on a large-scale chemical pilot plant. The new algorithm uses a least-squares estimator with variable weighting of past data; at each step a weighting factor is chosen to maintain constant a scalar measure of the information content of the estimator. It is shown that, for nearly deterministic systems, such an approach enables the parameter estimates to follow both slow and sudden changes in the plant dynamics. Furthermore, the use of a variable forgetting factor with correct choice of information bound can avoid one of the major difficulties associated with constant exponential weighting of past data—namely, ‘blowing-up’ of the covariance matrix of the estimates and subsequent unstable control. Accordingly, the control algorithm described here may be well suited to the regulation of plants which would otherwise require periodic re-tuning of control constants.

&amp;lt;tex&amp;gt;L_infty&amp;lt;/tex&amp;gt;-instability of large-scale interconnected systems using orthogonal decomposition and exponential weighting

In this paper, we give an explicit characterization of the orthogonal complement of the "set of stabilizing inputs" for an unstable convolution operator. This characterization involves the right-coprimne factorization of the transfer function matrix of the convolution operator. Using this characterization, we then explicitly demonstrate specific inputs that would cause <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_2</tex> -instability in a given large-scale interconnected system, using the technique of orthogonal decomposition of the input space. Finally, using the technique of exponential weighting, we give a general result that can be used to establish the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{\infty}</tex> -instability of a given system by proving that an associated system is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_2</tex> -unstable. In this way, one can obtain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{\infty}</tex> -instability criteria corresponding to virtually every known criterion for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_2</tex> -instability that uses the technique of orthogonal decomposition.

New&amp;lt;tex&amp;gt;L_2&amp;lt;/tex&amp;gt;-instability criteria for large-scale interconnected systems

In this paper, we derive several new criteria for the L_2 - Instability of large-scale interconnected systems. These fall into two categories: the first set of results is based on the idea of dissipativity, while the second set is based on an orthogonal decomposition of the input space. The latter results have the advantage that they can be readily converted into L_{\infty} -instabiity criteria using the technique of exponential weighting. The former results are more general, but provide only L_2 -instability criteria and no L_{\infty} -instability criteria.

Homomorphic analysis and modeling of ECG signals.

Homomorphic analysis and pole-zero modeling of electrocardiogram (ECG) signals are presented in this paper. Four typical ECG signals are considered and deconvolved into their minimum and maximum phase components through cepstral filtering, with a view to study the possibility of more efficient feature selection from the component signals for diagnostic purposes. The complex cepstra of the signals are linearly filtered to extract the basic wavelet and the excitation function. The ECG signals are, in general, mixed phase and hence, exponential weighting is done to aid deconvolution of the signals. The basic wavelet for normal ECG approximates the action potential of the muscle fiber of the heart and the excitation function corresponds to the excitation pattern of the heart muscles during a cardiac cycle. The ECG signals and their components are pole-zero modeled and the pole-zero pattern of the models can give a clue to classify the normal and abnormal signals. Besides, storing only the parameters of the model can result in a data reduction of more than 3:1 for normal signals sampled at a moderate 128 samples/s.