An Application of the Wiener-Kolmogorov Smoothing Theory to Matrix Inversion

Foster, M. R.

doi:10.1137/0109031

Cited by 120 publications

(38 citation statements)

References 5 publications

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“…Equation (12) gives the most general expression for a generalized pseudoinverse (Foster 1961;Strand and Westwater 1968). The model using the pseudoinverse of Eq.…”

Section: Derivation Of a Class Of Integrable Pseudoinversesmentioning

confidence: 99%

A comparison of models explaining muscle activation patterns for isometric contractions

Bolhuis

Gielen

1999

Biological Cybernetics

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One of the main problems in motor-control research is the muscle load sharing problem, which originates from the fact that the number of muscles spanning a joint exceeds the number of degrees of freedom of the joint. As a consequence, many different possibilities exist for the activation of muscles in order to produce a desired joint torque. Several models describing muscle activation have been hypothesized over the last few decades to solve this problem. This study presents theoretical analyses of the various models and compares the predictions of these models with new data on muscle activation patterns for isometric contractions in various directions. None of the existing models fitted the experimental data in all aspects. The best fit was obtained by models based on minimization of the squared sum of muscle forces ( summation operator(m)φ(2)(m), which is almost equivalent to the Moore-Penrose pseudo-inverse solution), muscle stress sigma ( summation operator(m)sigma(m)(2)) or muscle activation alpha ( summation operator(m)alpha(m)(2)). Since muscle activation patterns are different for isometric contractions and for movements, it could well be that other models or optimization criteria are better suited to describe muscle activation patterns for movements. The results of our simulations demonstrate that the predicted muscle activation patterns do not depend critically on the parameters in the model. This may explain why muscle activation patterns are highly stereotyped for all subjects irrespective of differences between subjects in many neuro-anatomical aspects, such as, for example, in the physiological cross-sectional area of muscle.

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“…Equation (12) gives the most general expression for a generalized pseudoinverse (Foster 1961;Strand and Westwater 1968). The model using the pseudoinverse of Eq.…”

Section: Derivation Of a Class Of Integrable Pseudoinversesmentioning

confidence: 99%

A comparison of models explaining muscle activation patterns for isometric contractions

Bolhuis

Gielen

1999

Biological Cybernetics

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“…We can, however attempt to minimize the mean square deviation of the estimated profile from the true profile in a statistical sense. A formulation of this type has been given for general linear systems by Foster (1961) and for the temperature inversion problem by Rodgers (1966) and by Strand and Westwater (1968). The procedure below follows the general derivation given by these authors.…”

Section: 246mentioning

confidence: 99%

On the estimation of relative humidity profiles from medium-resolution infrared spectra obtained from a satellite

Conrath

1969

J. Geophys. Res.

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Measurements of radiation emitted by the atmosphere of the earth in an infrared absorption band of water vapor, obtained with satellite‐borne instrumentation, contain information on atmospheric relative humidity. Two methods are developed for estimating tropospheric relative humidity profiles from infrared spectral measurements for which the spectral resolution elements are narrow in comparison with the total width of the absorption band but wide in comparison with a single absorption line. The methods, which are essentially complementary, consist of a direct estimation technique, which requires a minimum of a priori knowledge of the behavior of the relative humidity profile, and a statistical estimation technique, which can make full use of a knowledge of the statistics of tropospheric humidities in situations where such knowledge exists. An analysis of the propagation of errors in the measured spectral intensities indicates that meaningful estimates should be obtained from the 6.3‐μ water vapor band in the presence of realistic instrumental noise for most types of atmospheres. One exception is the polar winter atmosphere, where strong error propagation occurs because of the behavior of the temperature profile. An examination of the effects on the inferred relative humidity profile of errors in the temperature profile employed in the estimation reveals that the relative humidities inferred in the lowest layers of the troposphere are extremely sensitive to errors in the surface temperature, which may prove to be the limiting factor in obtaining complete relative humidity profiles. Examples of applications of the techniques to synthetic data from model atmospheres and to real data obtained with a balloon‐borne infrared interferometer spectrometer are given.

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“…However, Hoerl and Kennard (1970a) show that when (ATA) is nearly singular, as it can be in many geophysical problems, the average difference between liP estimated and AP true becomes very large. The ridge regression method (Levenberg 1944;Foster 1961;Marquardt 1963Marquardt , 1966 seeks to reduce this difference during the iteration process by damping the diagonal terms of (ATA). The ridge regression estimate of I' IPRR is where I is the density matrix and K > 0.…”

Section: ) Ridge Regressionmentioning

confidence: 99%

Comparison of Five Least‐squares Inversion Techniques in Resistivity Sounding*

Hoversten

Dey

Morrison

1982

Geophysical Prospecting

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A brief history of the development of the inverse problem in resistivity sounding is presented with the development of the equations governing the Least Squares Inverse. Five algorithms for findinq the minimum of least square problem are described and their speed of convergence is compared on data from two planar earth models. Of the five algorithms studied, the ridge-regression algorithm required the fewest numbers of forward problem evaluations to reach a desired minimum.

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An Application of the Wiener-Kolmogorov Smoothing Theory to Matrix Inversion

Cited by 120 publications

References 5 publications

A comparison of models explaining muscle activation patterns for isometric contractions

A comparison of models explaining muscle activation patterns for isometric contractions

On the estimation of relative humidity profiles from medium-resolution infrared spectra obtained from a satellite

Comparison of Five Least‐squares Inversion Techniques in Resistivity Sounding*

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