Combining Variables

Many analytical measures cannot be represented as a time-series in the form of a spectrum, but are composed of discrete measurements, e.g. compositional or trace analysis. Data reduction can still play an important role in such cases. The interpretation of many multivariate problems can be simplified by considering not only the original variables but also linear combinations of them. That is, a new set of variables can be constructed each of which contains a sum of the original variables each suitably weighted. These linear combinations can be derived on an ad hoc basis or more formally using established mathematical techniques. Whatever the method used, the aim is to reduce the number of variables considered in subsequent analysis and obtain an improved representation of the original data. The number of variables measured is not reduced. 

An important and commonly used procedure which generally satisfies these 

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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. 

In some cases, one needs to combine variables of mixed types (binary, ordinal or continuous). The usual way to do this is to eliminate the effect of varying ranges by scaling. All variables are transformed, so that they take values from 0 to 1 using range scaling for the continuous variables or the procedure for scaling described... 

When we consider the multivariate situation, it is again evident that the discriminating power of the combined variables will be good when the centroids of the two sets of objects are sufficiently distant from each other and when the clusters are tight or dense. In mathematical terms this means that the between-class variance is large compared with the within-class variances. 

The distribution cost index combines variable transportation and warehousing costs as well as capital costs for local and transit inventories. 

The production cost index combines variable and fixed production costs. 

This suggests that the group y/V(vr) can be used as a combined variable. For convenience, a factor of 2 may be introduced and the combined variable tj defined as... 

These boundary conditions can be expressed in terms of the combined variable 77 as... 

Most probabilistic assessments have tended to combine variability and parameter uncertainty, and not consider model or decision rule uncertainty. Recent guidance from the US National Academy of Sciences (NRC 1994), USEPA (1997), US DOE (Bechtel Jacobs Company 1998), and others (Hattis and Burmaster 1994 Hoffman and Hammonds 1994) has emphasized the importance of tracking variability and parameter uncertainty separately. Indeed, the USEPA (2000) states that the risk assessor should strive to distinguish between variability and uncertainty. Two major advantages of tracking variability and parameter uncertainty separately in an uncertainty analysis are... 

In vernier structures the modulation in the composition occurs on the basic stmcture thus combining variable composition with ordering (Anderson 1973). The essential feature of this structure is that the crystal unit can be resolved into two independent sublattices, one with a constant periodicity (based on that of the... 

A full three factorial matrix on the 11 variables in the cure cycle shown in Figure 15.1 would mean 177,147 individual trials. A full two factorial design would still mean 2048 trials. Such a design, however, assumes that all interactions, even between all 11 variables, will be important. DOE provides an ordered means of combining variables to reduce the total number of trials. The assumption made is that high-order interactions (i.e., interactions of three or more variables) are rare and/or insignificant. There are several methods for combining variables by DOE. A detailed discussion of these methods is the subject of another book [9]. 

Often it is useful to combine variables that affect physical phenomenon into dimensionless parameters. For example, the transition from laminar to turbulent flow in a pipe depends on the Reynolds number, Re = pLv/p, where p is the fluid density, I is a characteristic dimension of the pipe, v is the velocity of flow, and // is the viscosity of the fluid. Experiments show that the transition from laminar to turbulent flow occurs at the same value of Re for different fluids, flow velocities, and pipe sizes. Analyzing dimensions is made easier if we designate mass as M, length as L, time as t, and force as F. With this notation, the dimensions of the variables in Re are ML 3 for p, (L) for L, (L/t) for v, and (FL 2t) for //. Combining these it is apparent that Re = pLu/p, is dimensionless. 

Let the reader excuse us for such a word-combination "variable constant , but we have to use... 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

FIG. 29-75 Combined variable-speed and motor drive. Reeves Pulley Co. from Kent, Mechanical Engineers Handbook, 12th ed., Wiley, New York, 1961.)... 

By considering the combined variable z = x — xj2, we remove the mixed partial differential term from Eq. (4.293). The transformation obtained is the hyperbolic partial differential equation (4.294). This equation represents a new form of the stochastic model of the deep bed filtration and has the characteristic univocity conditions given by relations (4.295) and (4.296). The univocity conditions show that the suspension is only fed at times higher than zero. Indeed, here, we have a constant probability for the input of the microparticles ... 

When we introduce the combined variable and the dimensionless temperature 0( ) into Eq. (6.123) we have ... 

Table 2 shows the results. In each case, multiple regression was used to combine variables from the CAT tasks to predict IQ. The multiple correlations obtained ranged from 0.63 to 0.93 with a mean of 0.79 and a standard deviation of 0.11. There are many ways to compute multiple regression with the wealth of data provided by CAT. Table 2 shows that several different methods were used including cross validation. Even when we use variables that have correlations of 0.25 or less with intelligence, we obtain a multiple correlation with IQ of 0.63. Cross validation yields a multiple correlation almost as high as that obtained in the original sample. 

Factor analysis is the name given to eigen analysis of a data matrix with the intended aim of reducing the data set of n variables to a specified number, p, of fewer linear combination variables, or factors, with which to describe the data. Thus, p is selected to be less than n and, hopefully, the new data matrix will be more amenable to interpretation. The final interpretation of the meaning and significance of these new factors lies with the user and the context of the problem. 

Parabolic partial differential equations are solved using the similarity solution technique in this section. This method involves combining the two independent variables (x and t) as one (rj). For this purpose, the original initial and boundary conditions should become two boundary conditions in the new combined variable (rj). The methodology involves converting the governing equation (PDF) to an ordinary differential equation (ODE) in the combined variable (rj). This variable transformation is very difficult to do by hand. In this chapter, we will show how... 

Next, the velocity variables u and v (i.e., derivatives of the stream function) are expressed in terms of the combined variable and f ... 

Both the Laplace transform and the similarity solution techniques are powerful techniques for partial differential equations in semi-infinite domains. The Laplace transform technique can be used for all linear partial differential equations with all possible boundary conditions. The similarity solution can be used only if the independent variables can be combined and if the boundary conditions in x and t can be converted to boundary conditions in the combined variable. In addition, unlike the Laplace transform technique, the similarity solution technique cannot handle partial differential equations in which the dependent variable appears explicitly. The Laplace transform cannot handle elliptic or nonlinear partial differential equations. The similarity solution can be used for elliptic and for a few nonlinear partial differential equations as shown in section 4.6. There are thirteen examples in this chapter. 

We use the method of combination of variables, with the combined variable f = z/yf. ... 

In our analysis of the multicomponent penetration model we used the combined variable i = z/yfit. 

Because three-dimensional diagrams of mineral stability planes are difficult to show and interpret visually, the number of axes is usually reduced from three to two either by fixing one of the variables or, more commonly, by representing two variables on one axis. For aluminosilicates, the combined variable, pH — l pAl, is often plotted against pSi OH)S. In that case, each mineral s stability field is represented by a straight line. A typical diagram of this sort is shown in Figure 6.15, where the sol-... 

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