Variables redefined

PCA finds a smaller number of factors that describe the majority of the variability or spread in the dataset. Using these factors often called principal components, the row space is transformed or mapped into a new coordinate system in which the principal components, which can combine the information from several variables, redefine the axes based upon the factors, and these new axes describe the degree of variation or spread in the dataset. 

Parameter estimates are tested on whether they differ significantly from zero at a certain probability level. If not, the parameter should be skipped from the model and the model should be redefined even if the test for model adequacy was positive. When the errors have constant variance, the random variable... 

If we only include the first r latent variables we have to redefine our relationships between data, latent vectors and latent values ... 

A reserved variable, or a dummy argument of a function, or a variable set by an earlier CONSTANT statement has been redefined as an array. 

Often you want to redefine an already existing variable within a SAS DATA step. As simple as this may sound, it can lead to unexpected results if not done carefully. The following example displays some unexpected behavior that may occur when you redefine a variable within a DATA step. In this example you want to flag the subject who had the Fatal MI adverse event as having died (death =1). 

Program 4.12 Redefining a Variable within a DATA Step... 

Even after reading the SAS documentation Combining SAS Data Sets and understanding how the program data vector works, it can still be a bit confusing as to when it may be safe to redefine a pre-existing data set variable in place. The good news is that there is a safe and simple way to avoid the unplanned retention of variables Do... 

As described by Brogan ( ) the addition of state variable feedback to the system of Figure 1 results in the control scheme shown in Figure 5. The matrix K has been added. This redefines the input vector as... 

The strategy is to redefine the relevant dimensionless variables by combining the original groups in such a way that the unknown variable appears in one group. For example, / and /VRe can be combined to cancel the unknown (Q) as follows ... 

We redefined the sense of the optimization to be maximization. The optimal objective value of this problem is a lower bound on the MINLP optimal value. The MILP subproblem involves both the x and y variables. At iteration k, it is formed by linearizing all nonlinear functions about the optimal solutions of each of the subproblems NLP (y ),/ = 1,. .., , and keeping all of these linearizations. If x solves NLP(yl), the MILP subproblem at iteration k is... 

The eigenvalue/eigenvector decomposition of the covariance matrix thus allows us to redefine the problem in terms of Nc independent, standard normal random variables 0in. 

As soon as observations are considered as samples of random variables, we must redefine the concepts of distance and projection. Let us consider in three-dimensional space a vector y of one observation of three random variables Yj, Y2, and Y3 with its density of probability function fy. The statistical distance c of the vector. p to another point y can be defined by the non-negative scalar c2, which has already been met a few times, e.g., in equations (5.2.1) and (5.3.7), and such that... 

As discussed in the introduction to this chapter, examining the row space of a matrix is an effective way of investigating the relationship between samples. However, this is only feasible when the number of measurement variables (columns) is less than three. Principal components analysis is a mathematical manipulation of a data matrix where the goal is to represent the variation present in many variables using a small number of "factors. A new row space is constructed in which to plot the samples by redefining the axes using factors rather than the original measurement variables. Tliesc new axes, referred to as factors or principal components (PCs), allow the analyst to probe matrices... 

For three-dimensional diffusion, if there is spherical symmetry (i.e., concentration depends only on radius), the diffusion equation can be transformed to a one-dimensional type by redefining the concentration variable w = rC. This transformation would work for a solid finite sphere, a spherical shell, an infinite sphere with a spherical hole in the center, or an infinite sphere. 

This is easily shown by redefining the variable of integration. Convolution also obeys associativity,... 

Redefining the independent variable in terms of the Gaussian half-width, q = xJ n2/Ax0, we may write this convolution as... 

For a detailed analysis of monomer reactivity and of the sequence-distribution of mers in the copolymer, it is necessary to make some mechanistic assumptions. The usual assumptions are those of binary, copolymerization theory their limitations were discussed in Section III,2. There are a number of mathematical transformations of the equation used to calculate the reactivity ratios and r2 from the experimental results. One of the earliest and most widely used transformations, due to Fineman and Ross,114 converts equation (I) into a linear relationship between rx and r2. Kelen and Tudos115 have since developed a method in which the Fineman-Ross equation is used with redefined variables. By means of this new equation, data from a number of cationic, vinyl polymerizations have been evaluated, and the questionable nature of the data has been demonstrated in a number of them.116 (A critique of the significance of this analysis has appeared.117) Both of these methods depend on the use of the derivative form of,the copolymer-composition equation and are, therefore, appropriate only for low-conversion copolymerizations. The integrated... 

It is convenient to redefine the pertinent variables in the terms used by Loeb, Wiersema, and Overbeek so that the available tabular quantities can be used directly as input to the computer. These terms are q0 = a, x = l/ r, and a0 = (ckT/47re)I(q0, u0). These substitutions give rise to Equation 9b. 

The requirement that a function be single valued, for a given input value for the independent variable, will hold for the majority of associations between one number and another. However, the association between any real number and its square root always yields both a positive and a negative result. For example, the two square roots of 9 are +3, and so we say that 9 is associated with both -3 and 3. Thus, if we write this association as y = x1/2, then we cannot define the function y =J x) = xl/2. However, if we explicitly limit the values of y to the positive (or negative) roots only, then we can redefine the association as a single-valued function. Alternatively, if we square both sides to yield y2 = x, we can take the association between x now as a dependent variable and y as an independent variable and define the function x=g(x)=y2, for which there is only one value of x for any value of y. 

This unitary transformation redefines the variables of the field (ak, E, A, etc.) and displaces the momentum operator of each charge. Retaining only the electronic dipolar terms, the total hamiltonian becomes... 

If F(t) is redefined such that F(0) = 0, then F(0) disappears. An example would be a dimensionless concentration variable... 

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