Probability density curves

The area (probability, integral) under the probability density curve, p, from -oo to x can be calculated by... 

An informative description of the prediction errors is a visual representation, for instance by a histogram or a probability density curve these plots, however, require a reasonable large number of predictions. For practical reasons, the error distribution can be characterized by a single number, for instance the standard deviation. 

However, square of wave function, y2, is the measure of probability of finding the electron in a unit volume around a particular point and is called probability density. A graph of y2 as a function of distance from the nucleus is called probability density curve. The probability density curves for Is and 2s orbitals are shown in following figure ... 

Fig (a) Probability density curve of Is orbital, (b) Probability density curve of 2s orbital... 

FIGURE 26-1 The gaussian-distribu-tion or probability-density curve. 

In the graphs in Figure 5-20, the electron probability density at a given distance from the nucleus is plotted against distance from the nucleus, for s orbitals. It is found that the electron probability density curve is the same regardless of the direction in the atom. We... 

Figure 12-12. Probability density curves for Gaussian distribution of peak heights and ordinate heights. A Probability density of an ordinate being a peak at height y, where y = fi/d and the sampling interval...

I = 2.3 B - B Probability density curve for the Gaussian distribution of ordinates where y = (i/tf. Data by Whitehouse and Archard [10]. 

A surface can be characterized by many, many parameters. In fact it is easier to define a new parameter than to come with a thorough analysis of the usefulness of the already existing parameters. Parameters are defined in many ISO standards (see references). They can be separated in 2-D and 3-D parameters, and further separation is possible in amplitude parameters, spacing parameters, hybrid parameters, parameters derived from integrated probability density curves, and topological parameters. 

Parameters Derived from Probability Density Curves... 

The integrated probability density curve of the ordinates is also known as the... 

Figure 1. Structure resistance and load response - probability density curves.

It is recommended to use histograms instead of probability density curves for the calculation (Fig. 3). The probability density,/(x), is described as follows ... 

Instead of two probability density curves for each feature only one function q(x ) for each feature has to be stored for classifications of unknowns. 

This algorithm utilizes simplified probability density curves and has... 

Figure 38 shows class-dependent probability density curves for a... 

FIGURE 38. Probability density curves for feature i in a two-class problem. Overlap of the curves is possible in four different arrangements. ... 

The algorithm is simple and fast. Difficulties arise in the determination of the boundaries if large data sets are applied. The feet of the probability density curves spread out and minimum and maximum values of the overlapping region become useless. 

Complete information on the importance of a single feature i for the separation of classes is shown by class-conditional probability density curves (Figure 46). A small overlap of both curves indicates a high discrimination power of that feature. The overlap may be measured by the common area, by the Bhattacharyya coefficient, by the transinformation, or some other criteria (Chapter 11.6 and 11.7). 

FIGURE 46. Probability density curves of feature i for a binary classification problem. p(x. m)Ax is the probability that a pattern belonging to class m has the value of feature i in the interval Ax. 

This means that the total area under the probability density curve is always equal to unity. 

The integral, which can be obtained from tables of error functions, has the value of 0 to 1/2 for x = 0 to oo and similarly for x 0 to — oo. This merely shows that whatever one has lost from one side of the boundary will have been gained on the other side. The plot of concentration gradient against x (obtained from refractive index changes) gives an integral probability density curve... 

The values of the error function erf(kA/f2) are given in Table 4.2. The reverse relation of A multiples relative to same most used values of the area under the probability density curve are given in Table 4.3. 

A single variable is often insufficient for describing the relevant data and information of objects. Consider for instance how the origin of fruit should be classified by chemical-analytical methods. An assumption may be that the concentrations of trace elements in soil are reflected in the corre.sponding concentrations in the fruits. Let assume an investigation of a set of samples originating from two different sites (class A and B) shows some difference of the class means for the concentrations of an element 1 . However, no complete separation of the two classes is possible because of an overlap of the probability density curves. The concentrations of another element 2 unfortunately may show the same behavior. Obviously a... 

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