Independent error

In the graphs of log k, versus log k2, independent errors of the same magnitude in both directions can be anticipated. Hence, they can be represented by the usual circles. That is, if this radius equals the standard error 6 (or 26), it means that the actual value is situated inside with a probability of. 393 (or. 865) according to the distribution with two degrees of freedom (204). The circles in Figures 4 and 7 correspond to an error of 5% in k. 

For our sample application we assume that the points are measured with independent errors and equal variance. We may thus fit the data points minimizing e e, after which we may estimate as = e e/(n -1). 

Note that for the calculations of precision and standard deviation (equations 38-1 through 38-4), the numerator expression is given as 2(n — 1). This expression is used due to the 2 times error contribution from independent errors found in each independent set (i.e., X and Y) of results. 

Because of the way it is calculated, particularly the way the constant factor is ignored, the expected value of DW is two, when the data does in fact meet all the specified criteria random, independent errors, etc. Nonlinearity will cause the computed value of DW to be statistically significantly less than two. (Homework assignment for the reader what characteristic will make DW be statistically significantly greater then two )... 

Example. Calculate the independent error in the volume of a rectangle resulting from 0.1 cm random error in the measured x,y,z, when... 

Equation G-l presumes that errors in x, y, and z are independent of one another. A common case in which this is not true is when we use the least-squares slope and intercept to compute a new quantity, such as the value of Jt from an observed value of y. In general, uncertainties in the slope and intercept are correlated, so they are not independent errors. 

Jackknifing involves removing one sample from the calibration set, deriving the inference equations based on the remaining set of lakes (i.e., n — 1), and then using the new inference equation to derive an inferred value for the one sample that was removed (i.e., providing an independent error estimate). These steps are repeated until all samples have been left out once from the calibration process and used to calculate a new inferred value. The set of new inferred values is then used in conjunction with the... 

Obtaining realistic errors is one of the most difficult, yet most crucial problems in all flux estimates. Such errors can be approximated through an independent error analysis for several factors that are involved in estimating fresh and altered rock composition. There are uncertainties arising from petrographic observations, in the choices of representative samples, recovery rate biases, and analytical errors. In most cases analytical errors are a relatively minor source of uncertainty, and they are typically rather well documented. Probably the most crucial analytical uncertainty is in acurately determining the titanium concentration that is used as a normalizing factor to account for open-system behavior. This uncertainty directly relates to an error in the fluxes, and thus fluxes are difficult to constrain to better than 1 % of the whole rock abundance of a particular element. 

As noted previously, fluctuations in concentration estimates about the true value arise from multiple, independent, random errors. Each of the independent errors (erf) is statistically additive, such that the total assay error cr = Ecrf. Similarly, the CV% of the assay will be the square root of the sum of the squared CV%... 

To consider the effect of signal averaging on the noise level we must refer to the propagation of errors. The variance associated with the sum of independent errors is equal to the sum of their variances, i.e. 

This condition gives an upper-bound on the number of independent error-inducing Hamiltonians that our method can correct simultaneously and is called the "Hamming bound". 

The whole set of experiments has already been used once to estimate the model parameters. If we assume that there are p parameters in the model, there will be only (n — p) degrees of freedom left to describe the independent errors. If the model is good, then / (n - p) would be an estimate of the experimental... 

An event of this kind, produced by the composition of several events, is said to be a compound event. To throw three aces with three dice at one trial is a compound event dependent on the concurrence of three simple events. Errors of observation are compound events produced by the concurrence of several independent errors. 

From page 600, therefore, the probability of the occurrence of the independent errors, xv x2,. . ., xn is the product of their separate probabilities, or... 

Each Ci = y, — y, error term is a random variable that is assumed independent of all the other values. However, when the error terms are self- or autocorrelated, the error term is not but e, i + That is, e,- (error of the ith value) is composed of the previous error term e, i and a new value called a disturbance, do The di value is the independent error term with a mean of 0 and a variance of 1. 

This philosophy presented here, which naturally must be adapted from case to case, for example, if low-temperature reactions or high-pressure hydrogenations are to be performed, allows for the development of a truly credible worst case. It cannot be compared with a possible worst case scenario constructed with imlimited imagination. But it still has a quality which justifies its title of a worst case. A ctor of simultaneously occurring independent errors of 3 and the additional formal kinetic constraint add up to a process deviation from normal operation, which may be called dramatic with quite some confidence. 

Independent of the precision of the diffraction experiment, if a symmetrical molecule is placed in the crystal lattice in a nonspecific position (i.e., it is not in a position of symmetry required by the space group to which the crystal belongs), then the determined molecular geometry need not follow the symmetry properties obseiwed for the isolated molecule. This is due to an obvioirs experimental condition which holds that each bond length has its own, in principle independent, error of estimation. Therefore, for instance three outer rings in triphenylene (cf Chart 1) may have different (geometry-based) indices of aromaticity, despite C3 symmetry of the isolated molecule. 

Assuming that all errors are independent errors, the total error ET is given by... 

Heat convection seems to enhance turbulent mixing, although the available field measurements lack important details. Model developers have therefore been forced to calibrate their box model entrainment functions by comparing predicted and observed concentration fields, i.e., taking a global attitude to model calibration. This is not quite satisfactory since errors in one part of the dispersion model could be compensated by totally independent errors. 

Clark, G.C. and Cain, J.B. 1981. Error-Correction Coding for Digital Communication. Plenum, New York. Fire, P. 1959. A class of Multiple-Error-Correcting Binary Codes for Non-independent Errors. Sylvania Report No. RSL-E-2, Sylvania Electronic Defense Laboratory, Reconnaissance Systems Division, Mountain View, Calif, March. 

The likelihood function is derived from the probability distribution of the measurement errors relating to the diagnostic functionals. An appropriate distribution is a multivariate Gaussian with independent errors. Thus the joint probability density of the diagnostics and the input is... 

Assuming independent errors in each experiment, the likelihood function for the multiresponse data is... 

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