Linear dependences, between variables

Cases of linear dependence between variables present problems that are often difficult to handle. A frequent treatment of this situation is a compression along principal components, after which one or more of the smaller eigenvalues is dropped. It is not always recognized that this procedure has implications for the model parameters that may or may not be physically realistic and that it may be worthwhile in such a case to redesign the experiment. 

Equations of the First Degree. Equations diat define functions showing a linear dependence between variables are known as equations of the first degree or first-degree equations. These functions describe a dependence commonly called the... 

The correlation coefficient measures the linear dependence between the two variables X and Y. Let us assume that they are perfectly correlated, i.e., Y = aX + b with a and b constant. The linearity of the expectation operator amounts to... 

According to all things said for the coefficient of determination, the correlation coefficient itself is a measure of the strength of relationship and it takes values between -1 and +1. When the correlation coefficient nears one the linear relationship between variables is strong, and when it is close to zero it means that there is no linear relationship between variables. This, however, does not mean that there is no relationship between variables, which might even be strong, of a certain curved shape. We point out that the correlation coefficient is an indefinite number, i.e. it does not depend on the units the variables have been expressed in. 

Only if assuming a linear dependence between the variables should a linear regression model be applied. 

As seen in Equation 8.10, there is a linear dependence between the input variables or controlled factors that create a nonunique solution for the regression coefficients if calculated by the usual polynomials. To avoid this problem, Scheffe [3] introduced the canonical form of the polynomials. By simple transformation of the terms of the standard polynomial, one obtains the respective canonical forms. The most commonly used mixture polynomials are as follows ... 

This method is frequently used for filtering, smoothing and identifying parameters in the case of a dynamic time process. It has been developed taking into account the following conditions (i) acceptance of the gaussian distribution of the disturbances and exits of the variables of the process (ii) there is a local linear dependence between the exit vector and the state vector in the mathematical model of the process. 

If we once more consider the example studied throughout this chapter, we can use the statistical data presented in Table 5.3 in order to compute the value of the correlation coefficient. However, before carrying out this calculation, we can observe an important dependence between variables x and y due to the physical meaning of the results in this table. The value obtained for the correlation coefficient confirms our a priori assumption because the cov has a value near unity. It shows that a linear relationship can be established between process variables. The results of these calculations are shown in Table 5.7. 

We can eliminate all the false dependent variables from the statistical model thanks to the correlation analysis. When we obtain = 1 for a process with two dependent variables (yj, y2), we have a linear dependence between these variables. Then, in this case, both variables exceed the independence required by the output process variables. Therefore, yj or can be eliminated from the list of the dependent process variables. 

Different symbols have been used for the degree of advancement and the concentrations depending on the conditions. Furthermore vectors have been symbolised by bold letters with arrows, matrices by bold letters, variables as italics. Stoichiometric coefficients and the matrix p are written in Greek symbols. In addition, the concentrations of reactants to be determined and the number of degrees of advancement can be reduced because of the law of conservation of mass and linear dependencies between different steps of the reaction, respectively. Therefore some indices have been introduced to characterise the different variables. The different symbols are summarised in Table 2.2. 

The Free Wilson model was in its original formulation [16] not as simple. No reference compound was selected and so-called symmetry equations were generated to avoid the problem of linear dependences between the variables. 

A linear correlation was found between the binding energies and the sum of the Hammett substituent constants for X-TAZ---Y complexes. Linear dependences between electron densities at BCP and the appropriate interatomic distances are satisfied, but the slopes of these lines are different, indicating variable sensitivity of the noncovalent interactions to the substituent effects (see Figure 6). 

This part of the research work focuses on the impact of the fibre thickness on the durability of the fibres. Durability is represented quantitatively by the number of cycles of the flex abrasion test. The impact is presented in form of the Spearman s rank correlation coefficient. The correlation coefficient gives a value between -1 and 1 inclusively, where -1 is total negative, 1 is total positive and 0 is no correlation. Spearman s rank correlation is a nonparametric tool which describes the linear dependency between two variables, and can be defined as ... 

The presence of conserved elements and conserved moieties cause linear dependence between the rows of the stoichiometric matrix p and decrease the rank of the stoichiometric matrix. In most cases, the number of species Ns is much less than the number of reaction steps N, that is, Ns < Wr. If the stoichiometric matrix p has N rows and Ns columns, and conserved properties are not present, then the rank of the stoichiometric matrix is usually Ns - If Nq conserved properties are present, then the rank of the stoichiometric matrix isN = Ns— Nq- In this case, the original system of ODEs can be replaced by a system of ODEs having N variables, since the other concentrations can be calculated from the computed concentrations using algebraic relations related to the conserved properties. 

As we saw above, one of the most convenient ways to represent the functional dependence of the variables of the system is by the use of coordinate systems. This is because each set of numbers is easily represented by a coordinate axis, and the graphs that result give an immediate visual representation of the behavior. In this section we shall explore several types of graphical representation of functions. We begin with functions that describe a linear dependence between the variables. 

Some variables often have dependencies, such as reservoir porosity and permeability (a positive correlation) or the capital cost of a specific equipment item and its lifetime maintenance cost (a negative correlation). We can test the linear dependency of two variables (say x and y) by calculating the covariance between the two variables (o ) and the correlation coefficient (r) ... 

Multiple linear regression (MLR) models a linear relationship between a dependent variable and one or more independent variables. 

Whenever a linear relationship between dependent and independent variables (ordinate-resp. abscissa-values) is obtained, the straightforward linear regression technique is used the equations make for a simple implementation, even on programmable calculators. 

As an extension of perceptron-like networks MLF networks can be used for non-linear classification tasks. They can however also be used to model complex non-linear relationships between two related series of data, descriptor or independent variables (X matrix) and their associated predictor or dependent variables (Y matrix). Used as such they are an alternative for other numerical non-linear methods. Each row of the X-data table corresponds to an input or descriptor pattern. The corresponding row in the Y matrix is the associated desired output or solution pattern. A detailed description can be found in Refs. [9,10,12-18]. 

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... 

Figure 1 shows the non linear dependence of 03 production efficiency on NOx and non methane hydrocarbon (NMHC) levels. The 03 production efficiency is defined as the number of 03 molecules produced per molecule of NOx removed from the atmosphere. This non linear behavior of chemistry in the atmosphere reflects the occurrence of catalytic cycles. It implies the consideration of the spatial variability of short-lived 03 precursors which present important concentration gradients between continental and oceanic areas. 

---