Variables ranked

Variable Ranking Variable Ranking Variable Ranking... 

Sometimes, several feature selection methods are used for a given analysis. For example, an analyst might reduce chromatogram to a peak table, selecting a series of candidate variables of interest and then perform further variable ranking and optimization on the integrated peak table, especially in the case of multidimensional separations where hundreds, if not thousands of compounds can be resolved (Felkel et al., 2010). 

Brereton, R. G. and Elbergali, A. K. (1994) Use of double windowing, variable selection, variable ranking and resolvability indices in window factor analysis. Journal of Chemometrics, 8, 423-37. 

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

One significant feature of the Parametric Method is that it indicates, through the (1 + K 2) value, the relative contribution of each variable to the uncertainty in the result. Subscript i refers to any individual variable. (1 + K ) will be greater than 1.0 the higher the value, the more the variable contributes to the uncertainty in the result. In the following example, we can rank the variables in terms of their impact on the uncertainty in UR. We could also calculate the relative contribution to uncertainty. 

Figure 6.12 Ranking of impact of variables on uncertainty in reserves...

The plot immediately shows whioh of the parameters the 10% NPV is most sensitive to the one with the steepest slope. Consequently the variables can be ranked in order of their relative impact. 

Theorem 1. The number of products in a complete set of B-numbers associated with a physical phenomenon is equal to n — r, where n is the number of variables that are involved in the phenomenon and ris the rank of the associated dimensional matrix. 

To show the equivalence of Theorems 1 and 3, it is only necessary to demonstrate that the maximum number of the variables that will not form a dimensionless product is equal to the rank of the dimensional matrix D. 

According to Theorem 5, the transpose 3 complete B-matrix. Since there are five variables and since the rank of Dis 3, Theorem 1 reveals... 

Let m be the rank of the Ot matrix. Then p = n — m is the number of dimensionless groups that can be formed. One can choose m variables [Pj] to be the basis and express the other p variables in terms of them, givingp dimensionless quantities. 

Here m < 5, n = 8, p > 3. Choose D, V, i, k, and as the primary variables. By examining the 5x5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5 thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, p, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

Significance of risk contribution may be done by ordering the risk contributors from most-to-least (rank order), but because of the arbitrariness of variation of the variables, this may be meaningless A more systematic approach is to calculate the fractional change in risk or reliability for a fractional change in a variable. 

It has been shown [56] that if we measure the areas under the approach and retract curves of the force-distance plot we can get quantitative values of the resilience. Resilience is closely related to the ability of the polymer chain to rotate freely, and thus will be affected by rate and extent of deformation, as well as temperature. Different materials will respond differently to changes in these variables [46] hence, changing the conditions of testing will result in a change in absolute values of resilience and may even result in a change in ranking of the materials. Compared to more traditional methods of resilience measurement such as the rebound resiliometer or a tensUe/compression tester. 

Iman RL, Helton JC, Campbell JE. An approach to sensitivity analysis of computer models Part II—Ranking of input variables, response surface validation, distribution effect and technique synopsis. / Quality Technol 1981 13 232-40. 

The number of singular vectors r is at most equal to the smallest of the number of rows n or the number of columns p of the data table X. For the sake of simplicity we will assume here that p is smaller than n, which is most often the case with measurement tables. Hence, we can state here that r is at most equal to p or equivalently that rindependent measurements in X. Independent measurements are those that cannot be expressed as a linear combination or weighted sum of the other variables. 

Basically, we make a distinction between methods which are carried out in the space defined by the original variables (Section 34.4) or in the space defined by the principal components. A second distinction we can make is between full-rank methods (Section 34.2), which consider the whole matrix X, and evolutionary methods (Section 34.3) which analyse successive sub-matrices of X, taking into account the fact that the rows of X follow a certain order. A third distinction we make is between general methods of factor analysis which are applicable to any data matrix X, and specific methods which make use of specific properties of the pure factors. 

Reduced rank regression (RRR), also known as redundancy analysis (or PCA on Instrumental Variables), is the combination of multivariate least squares regression and dimension reduction [7]. The idea is that more often than not the dependent K-variables will be correlated. A principal component analysis of Y might indicate that A (A m) PCs may explain Y adequately. Thus, a full set of m... 

An alternative and illuminating explanation of reduced rank regression is through a principal component analysis of Y, the set of fitted F-variables resulting from an unrestricted multivariate multiple regression. This interpretation reveals the two least-squares approximations involved projection (regression) of Y onto X, followed by a further projection (PCA) onto a lower dimensional subspace. 

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