Dispersion variance

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

Table 1 presents a summary of the solutions obtained in this section. Of primary interest at this point is comparison of the forms of the Lagrangian and Eulerian expressions, in particular the relationships between the eddy diffusivities and the plume dispersion variances. For the slender-plume cases, for example, the Lagrangian and Eulerian expressions are identical if... 

There are four major sources of extracolumn dispersion (i) dispersion due to the injection volume, (ii) dispersion due to the volume of the detector cell, (iii) dispersion due to the detector response time, and (iv) dispersion resulting from the volume in the connecting tubing between the injector and the column and also between the column and the detector. Thus, extracolumn dispersion takes place between the injector and the detector, only, and the system volume contributed by the solvent delivery system does not contribute to dispersion. The total permitted extracolumn dispersion (variance) is shared, albeit unequally, between those dispersion sources. A commonly accepted criterion for the instrumental contribution to zone broadening, suggested by Klinkenberg,17 is that it should not exceed 10% of the column variance. 

Another kind of important moment, represented by dispersion variance, are moments referred to the centre. They are defined by... 

For dispersion variance, defined as the mean value of the square of the difference between the variable n and its mean value n,r = 2. 

In statistics, the square root of the dispersion variance is called the standard, or root mean square, deviation. In holds that [50]... 

One. relation or an assembly of relations that contains the condition necessary to impose the absence of important differences between the computed outputs and experimental outputs. This relation or assembly of relations frequently contains the requirement of a minimal dispersion (variance) between computed and experimental process outputs. So we need to minimize the function ... 

During the experiment, the numerical characterization of the population is given by the concentration of the reactant associated to the flow of the material fed into the reactor. Therefore, this reactant s concentration and transformation degree are random variables. As has been explained above (for instance see Chapters 3 and 4), the characterization of random variables can be realized taking into account the mean value, the dispersion (variance) and the centred or non-centred momentum of various degrees. Indeed, the variables can be characterized by the following functions, which describe the density of the probability attached... 

Here p and are, respectively, the mean value and the dispersion (variance) with respect to a population. These characteristics establish all the integral properties of the normal random variable that is represented in our example by the value expected for the species concentration in identical samples. It is not feasible to calculate the exact values of p and because it is impossible to analyse the population of an infinite volume according to a single property. It is important to say that p and show physical dimensions, which are determined by the physical dimension of the random variable associated to the population. The dimension of a normal distribution is frequently transposed to a dimensionless state by using a new random variable. In this case, the current value is given by relation (5.21). Relations (5.22) and (5.23) represent the distribution and repartition of this dimensionless random variable. Relation (5.22) shows that this new variable takes the numerical value of x when the mean value and the dispersion are, respectively, p = 0 and = 1. 

This result is very important because it shows how we compute the values of the elements of the matrix of mean errors M[(B — Br)(B — Br) ]. These elements allow the calculation of the dispersions (variances) that characterize each Pj model coefficient of the process as shown in relation (5.94), which results from combining relations (5.93), (5.90) and (5.89) ... 

A multi-variate probability may also be computed if we include the dispersion (variance) of the reference group by computing the Mahalanobis distances between the unknown and well defined mixtures of vanillins from different origins. 

In a binary mixture of two polymers, A and B, their local concentration (in weight fraction) can be expressed as a(x) + b(x) = 1 (where x denotes vectorial location). Similarly, the sum of the average compositions aav + bav = 1. For such a system, the mixedness can be described by the binary frequency function. For fine dispersions, variance of the composition and the intensity of segregation can be defined as (Hold 1991 Tucker 1991)... 

T CF dispersion (variance) in the sensor volume resulting from Newtonian flow... 

Model Order Reduction (MOR), Figure 6 Dependence of dispersion variance of species band on dimensionless times (n and r )... 

The first important notion in volume-variance relations is the spatial or dispersion variance. The dispersion variance D (a, b) is the variance of values of volume a in a larger volume i. In a geostatistical context, all variances are dispersion variances. A critical relationships in geostatistics is the link between the dispersion variance and the average variogram value ... 

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