Variable linear relationship between

Use words to report a linear relationship between variables rather than graphing a single straight line, for example, A linear relationship was observed between A and B (r=0.97). ... 

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. 

In this example, k is the Charles s law constant. Graphical representation of a direct proportion results in a straight-line (linear) relationship between variables ... 

Over the last decade there has been a growing interest in using ANNs for source confirmation modeling of environmental contaminants. ANNs are able to map the non-linear relationships between variables that are characteristic of a common pollution source. ANNs require no a priori assumptions about the model in terms of mathematical relationships or distribution of data. Therefore ANNs have the potential to discover useful models where domain knowledge of original sources is limited. 

The utilization degree where the production costs reach the revenues is the break-even point (for the given example shown in Fig. 5.5.5 65%). For projects with a certain risk of investment, the break-even point should be less than 70%. For detailed calculation of the influence of the utilization degree on the production costs, the assumption of a linear relationship between variable costs and utilization should no longer be used, and we obtain cost curves such as shown in Figure 5.5.6. 

Before proceeding with the techniques, used to determine a fuzzy model, it is necessary to realize what one tries to achieve in fuzzy modeling. Generally, one tries to approximate a non-linear relationship that exists between an output variable and one or more input variables. This is done by dividing the data space into regions in which a linear relationship exists between the output variable and input variables. In the regions where two relationships overlap, an appropriate weighting is made of these two relationships to produce an accurate approximation of the measured data. A fuzzy model consists therefore of multiple linear relationships between variables. 

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

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

The coefficient of determination is the fraction of the variation that is explained by a linear relationship between two variables and is given by... 

Correlation coefficient. In order to establish whether there is a linear relationship between two variables xx and the Pearson s correlation coefficient r is used. 

It must be noted, however, that a value of r close to either + 1 or — 1 does not necessarily confirm that there is a linear relationship between the variables. It is sound practice first to plot the calibration curve on graph paper and ascertain by visual inspection if the data points could be described by a straight line or whether they may fit a smooth curve. 

Christiansen (123) supposed f(T) in a special form e l /T. When now A, B, and a are constant throughout the series and only C is variable, a linear relationship between A Hx and A Sx arises. However, the assumptions seem rather unrealistic. 

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. 

Having done so, we mix the seven (7) separate products. We could then plot the particle distribution as shown in 5.4.2., also given on the next page, where we have plotted the log of the size of particles in microns vs the log of the number of particles created. Obviously, there is a linear relationship between the two variables. The other factor to note is that this distribution consists of specific (discrete) sizes of pcuticles. 

There are several good reasons to focus on linear models. Theory may indicate that a linear relation is to be expected, e.g. Lambert-Beer s law of the linear relationship between concentration and absorbance. Even when a linear relation does not hold strictly it can be a sufficiently good local approximation. Finally, one may try and find a transformation of the individual variables (e.g. a logarithmic transformation), in order to obtain an acceptable linear model for the transformed variables. Thus, we simplify eq. (36.1) to... 

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]. 

It should be noted that the analysis of covariance assumes a linear relationship between the standard value and the variable value, for a given treatment, and assumes independence between an effect of a given treatment and the value of the standard. The relationship between these may be more complex in many drift studies. Therefore, the standard treatment should be near the median of the variables being investigated. The only alternative involves a great deal of patience and time to obtain very similar... 

Assuming a linear relationship between the output vector and the state variables (y = Cx), the above equation becomes... 

On occasion you need to obtain correlation coefficients between two variables. Correlation coefficients are a way of measuring linear relationships between two variables. A correlation coefficient of 1 or -1 indicates a perfect linear relationship, and a coefficient of 0 indicates no strong linear relationship. Pearson correlation coefficients are useful for continuous variables, while Spearman correlation coefficients are useful for ordinal variables. For example, look at the following SAS code ... 

The main concept addressed in this new multi-part series is the idea of correlation. Correlation may be referred to as the apparent degree of relationship between variables. The term apparent is used because there is no true inference of cause-and-effect when two variables are highly correlated. One may assume that cause-and-effect exists, but this assumption cannot be validated using correlation alone as the test criteria. Correlation has often been referred to as a statistical parameter seeking to define how well a linear or other fitting function describes the relationship between variables however, two variables may be highly correlated under a specific set of test conditions, and not correlated under a different set of experimental conditions. In this case the correlation is conditional and so also is the cause-and-effect phenomenon. If two variables are always perfectly correlated under a variety of conditions, one may have a basis for cause-and-effect, and such a basic relationship permits a well-defined mathematical description. 

Figure 68-1 (a) Artificial data representing a linear relationship between the two variables. This data represents a linear, one-variable calibration, (b) The same artificial data extended in a linear manner. The extrapolated calibration line (broken line) can predict the data beyond the range of the original calibration set with equivalent accuracy, as long as the data itself is linear. 

While physical chemistry can appear to be horribly mathematical, in fact the mathematics we employ are simply one way (of many) to describe the relationships between variables. Often, we do not know the exact nature of the function until a later stage of our investigation, so the complete form of the relationship has to be discerned in several stages. For example, perhaps we first determine the existence of a linear equation, like Equation (1.1), and only then do we seek to measure an accurate value of the constant c. 

Figure 1.4 In using a thermometer, we assume the existence of a linear response between the length l of the mercury and the controlled variable temperature T. Trace (a) shows such a relationship, and trace (b) shows a more likely situation, in which there is a close approximation to a linear relationship between length l and temperature T...

Using the same inlet/initial conditions as were employed for the one-step reaction, this reaction system can be written in terms of two reaction-progress variables (Fi, Y2) and the mixture fraction f. A linear relationship between c and (co, Y, f) can be derived starting from (5.162) with y = Y2 = A0B0/(A0 + B0) ... 

It has been observed that while normal, rabbit serum failed to bind labelled phenobarbital, the serum from immunized rabbits bound 75 to 80% of the added pentobarbital and there exists a linear relationship between 14C-phenobarbital and the concentration of added antibody. Besides, when variable quantities of 14C-pentobarbital are added to a constant quantity of antibody, there exists a linear relationship between added and bound 14C-phenobarbital as depicted in Figure 32.4. 

Using cyclohexylamine, as M ef, the data for various enantiomeric mixtures ofl-(l-naphthyl)ethylamine (M) display a linear relationship between RPIj /RPP and ee. Enantiomeric impurieties as small as about 2% can currently be detected with this method." " Variable-temperature FT-ICR-MS measurements of the ligand... 

The basic reason for using different control-valve trims is to keep the stability of the control loop fairly constant over a wide range of flows. Linear-trim valves are used, for example, when the pressure drop over the control valve is fairly constant and a linear relationship exists between the controlled variable and the flow rate of the manipulated variable. Consider the flow of steam from a constant-pressure supply header. The steam flows into the shell side of a heat exchanger. A process liquid stream flows through the tube side and is heated by the steam. There is a linear relationship between the process outlet temperature and steam flow (with constant process flow rate and inlet temperature) since every pound of steam provides a certain amount of heat. 

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