Least mean squares

Figure 3. Least mean-square slope of a wavefront.

Figure 3-3 Examples of (a) cup-and-bob viscosity (torque vs. speed) and (b) viscosity versus shear rate. (Lines are least-mean-square fit.)...

The data lie on a straight line only for Plot (1), the graph of [HI] vs. t. Therefore, the reaction is zero order with respect to HI. The slope of the line = -0.00546 mM-s-1, using a least mean square regression fitting program. However, the slope can be estimated from any two points on the line. If we use the first and last points ... 

FIG. 27. A comparison between the in-plane dimensions, determined by SXS and STM, of the electrocompressive Pb upd adlayer formed on Ag(lll). The open circles are the STM data and the solid line is a least-mean-square fit to the data, while the dotted line is derived from SXS measurements (k = A/a when A is the period of the Moire pattern while a is the lattice spacing of the overlayer). (From Ref. 343.)... 

The seven group correlation contributions El, Eb, Ell. Elb. .. were then determined by a least-mean-squares fit of Ec to the ab initio valence correlation energies of the afore-mentioned sets of molecules. They were found to have the following values 38) ... 

Nonmodel-based controllers, such as the least mean square (LMS) and artificial neural network back-propagation adaptive controllers, employ iterative approaches to update control parameters in real time [14-17]. However, those methods may encounter difficulties of numerical divergence and local optimiza-... 

The outputs of the sensors were used in two closed-loop control strategies developed for combustor performance optimization [7]. The objective of the first strategy, based on an adaptive least-mean squares (LMS) algorithm, was to maximize the magnitude and coherence of temperature oscillations at the forcing frequency /o in the measured region. The LMS algorithm was used to determine... 

Other modifications are possible to the same basic approach of seeking a filter that is optimum in the sense of least mean-square error. Backus and Gilbert (1970), for example, derive a linear filter by minimizing a sum of terms in which noise and resolution criteria are separately weighted. Frieden (1975) discusses variations of this technique. 

To obtain the distribution extrapolated to zero concentration, the distribution at each concentration is divided into a number of zones within the weight fraction zone 0 to 1. Then for each zone a plot of s or 1/s versus the sample concentration is made and extrapolated to obtained the sedimentation coefficient at zero concentration, sq A plot of weight fraction versus so is the corrected integral distribution at zero concentration. The differential distribution, dc/ds, can be obtained by fitting groups of points with a sliding least mean squares cubic fit. 

Since the integral distribution patterns for the 80 and 100 min. patterns were nearly the same, only the results from the former are shown in Figure 3. The differential patterns (not shown) obtained from the sliding 15 point least mean squares cubic fit were also nearly the same, with considerable noise for the two solutions of lowest concentration. 

The results of the sliding 7 point least mean squares fit are shown in Figure 5. The accuracy of the data is indicated by the closeness of the actual data points (squares) to the smoothed points (continuous line) for the differential distribution curve given in Figure 6. One cannot tell whether the small convolutions which are revealed in the corresponding differential distribution pattern are real, or represent artifacts or errors in the data. 

Spectra like the ones shown in Fig. 3.10 may be readily decomposed into their line profiles. As an example, we show that the low-temperature measurement may be accurately represented by three identical profiles. Using the so-called BC model profile with three adjustable parameters and centering one at zero frequency (the Qo(l) line), another one at 354 cm-1 (the H2 So(0) line) and the third one at 587 cm-1 (the So(l) line), one may fit the measurement using least mean squares techniques, Fig. 3.11. The superposition (heavy line type) of the three profiles (thin... 

If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

From these data, it was found that 11 leading A coefficients, Eq. 4.18, could be determined by least mean squares techniques with sufficient numerical significance, namely the Ax r R) with subscripts... 

It is a straightforward matter to fit various model profiles to realistic, exact computed profiles, selecting a greater or lesser portion near the line center of the exact profile for a least mean squares fit. In this way, the parameters and the root mean square errors of the fit may be obtained as functions of the peak-to-wing intensity ratio, x = G(0)/G(comax)- As an example, Fig. 5.8 presents the root mean square deviations thus obtained, in units of relative difference in percent, for two standard models, the desymmetrized Lorentzian and the BC shape, Eqs. 3.15 and 5.105, respectively. 

Numerical problems arising from the use of the Lorentzian for fitting spectra have also been reported [69]. These are related to the non-dif-ferentiable profile at the points where the exponential wings are attached to the Lorentzian core. Partial derivatives with respect to the line shape parameters are usually needed in least mean squares fitting procedures. 

In order to determine the ion pair dissociation constant Kd, of a salt it is necessary therefore to measure X as a function of C and obtain a roughly extrapolated value for X0. Calculation of the variables F(z)/X and f 2 F(z)CX is usually accomplished with a small computer program, and hence a more accurate value for X0 and a first value for Kd obtained from a straight line plot of these functions. It is, however, more convenient to carry out the whole process by computer with iteration accompanied by a least mean square calculation to obtained the most accurate value for X0 and Kd. For solvents of low dielectric constant, and if sufficiently dilute solutions are not examined, Fuoss plots deviate downward at higher concentrations, because of triple ion formation. This can lead to an excessively low estimate for X0 and too high a value for Kd. 

All attempts to find a deactivation rate law relating k to total hexane fed failed. The most satisfactory rectifying plot was found to be log k vs. the cumulative amount of hexane actually cracked (designated Y) in any given run. The plots of log k vs. Y are shown in Figure 1. (The scale for Y is shown at the top of Figure 1 for samples 1, 4, and 5, and at the bottom for samples 2 and 3.) The deactivation behavior is well fitted for samples 1, 3, and 4, somewhat less well for sample 5, and poorly for sample 2. The lines were obtained by least-mean-squares fitting of the data. 

A conventional linear least-mean-square fit of (57) yields the value of k. ... 

A graph of log- N against the reciprocal of the absolute temperature (Fig. 1 was obtained using a Least Mean Squares Program on the Hewlett-Packard 9810A Calculator this has a gradient of -2.09 x 10 K with a correlation coefficient of -0.977. Hence the energy of activation for the nucleation of latex particles, EM = 40.0 kJ mol . ... 

The full set of equations was used to model experiments from the literature using numerical methods. In one of these experiments [3], a clay sample in a flexible wall permeameter was subjected to a salt concentration gradient and salinity and pressure profiles were measured. In [4], a scripted finite element solver was used to provide numerical simulations. Using a least mean squares fit, the storage parameter and the reflection coefficient were inferred from the experimental data. Relevant parameters for this experiment are shown in Table 2. 

One of the simplest approaches is to scale by the same factor all the calculated frequencies in order to obtain the least mean-squared deviation between the experimental and the theoretical frequencies [14], This procedure is equivalent to homogeneous scaling of all the elements in Fl m. The advantage of this method is that it uses only one adjustable parameter. However, this is payed out by the necessity of very extensive quantum-mechanical calculations large atomic basis sets, and appropriate account for the electron correlation effects. If simpler theoretical schemes are used the homogeneous scaling may result in improper assignment of the experimental frequencies. 

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