Error summation

Another general effect observed for a cycle is robustness of stationary rate and relaxation time. For multiscale systems with random constants, the standard deviation of constants that determine stationary rate (the smallest constant for a cycle) or relaxation time (the second in order constant) is approximately n times smaller than the standard deviation of the individual constants (where n is the cycle length). Here we deal with the so-called order statistics. This decrease of the deviation as n is much faster than for the standard error summation, where it decreases with increasing n as... 

The feedback block, on the other hand, consists of the voltage divider (if present) and the compensated error amplifier. Note that we may prefer to visualize the error amplifier block as two cascaded stages — one that just computes the error (summation node), and another that accounts for the gain (and its associated compensation network). Note that the basic principle behind the pulse width modulator stage (which determines the shape of the pulses driving the switch), is explained in the next section, and in Figure 7-11. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

D[X t) is used to denote a path integral. Hence, equation (14) corresponds to a summation of all paths leading from X(0) to X t). The same expression is used for the Brownian trajectories and for Newtonian s trajectories with errors. The action is of course different in both cases. 

The term on the left-hand side is a constant and depends only on the constituent values provided by the reference laboratory and does not depend in any way upon the calibration. The two terms on the right-hand side of the equation show how this constant value is apportioned between the two quantities that are themselves summations, and are referred to as the sum of squares due to regression and the sum of squares due to error. The latter will be the smallest possible value that it can possibly be for the given data. 

Heats of Vapon a/ion andFusion. A simple linear summation of most of the Lyderson groups (187) has been proposed for heat of vaporization at the normal boiling point and heat of fusion at atmospheric pressure for a wide variety of organic compounds (188). Average errors of 1.2 and 4.3% for group contribution-based estimations of heats of vaporization for selected n- and iso-alkanes, respectively, have been reported (215). 

The summated current is the sum of all the CT secondary currents of the different circuits. The rating of the instrument connected on the secondary of the summation CT should be commensurate with the summated current. The error of measurement is now high, as the errors of all individual CTs will also add up vectorially. 

Any main CT that is itndeiioaded will also add to the error in the measurement. Similarly, if provision is tnade in the primary of the summation CT to accommodate fntnre circuits but is not being utilized it tnust be left open, otherwise it will also add to the error. The impedance of the shorting terminals will add to the impedance of the circuit and will increase the total error. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Fig. 4—Comparison of relative error for different schemes, (a) A comparison of relative errors for a uniform pressure on a rectangle area 2a X 2b, in which the multi-summation is calculated via DS, FFT, and MLMI, and 1C is determined through bilinear interpolation based scheme, (b) A comparison of relative errors for a uniform pressure on a rectangle area 2ax2fa, in which the multisummation is calculated via DS and 1C is determined through the Green, constant, and bilinear-based schemes, (c) A comparison of relative errors for a Hertzian pressure on a circular region in radius a, in which the multi-summation is calculated via DS, and 1C is determined through the Green, constant, and bilinear-based schemes.

The component-of-varlance analysis Is based upon the premise that the total variance for a particular population of samples Is composed of the variance from each of the Identified sources of error plus an error term which Is the sample-to-sample variance. The total population variance Is usually unknown therefore. It must be estimated from a set of samples collected from the population. The total variance of this set of samples Is estimated from the summation of the sum of squares (SS) for each of the Identified components of variance plus a residual error or error SS. For example ... 

This is the velocity form algorithm which is considered to be more attractive than the position form. The summation of error is not computed explicitly and thus the velocity form is not as susceptible to reset windup. 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

The reader may be surprised to learn that for the selected data the slope using either method computes to a value of 1.93035714285714, while the intercept for both methods of computation have values of 1.51785714285715 (summation notation method) versus 1.51785714285714 for the Miller and Miller cited method (this, however, is the probable result of computational round-off error). 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

Summation of absolute differences (I) results in an ME in which all differences have the same statistical weight. Summation of squared differences (II) is the more common practice and gives an MSE in which large deviations have higher weight than small ones. In order to make the metric independent of the number N of observations, the error sum must be related to N or an equivalent sum of the observations ... 

The use of E magnitudes and the limits we shall impose on the reflections entering the summation mean that the electron density is only approximate (At the very least there are serious series termination errors.) but hopefully is sufficient to reveal stmctural features so that model building can begin. 

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