The Most Probable Symmetric Shape

Given n points in 2D whose positions are given as normal probability distributions Q At(P ,A() i = 0... n -1, we find the C -symmetric configuration of points P( q i which is most probable. Denote by CO the center of mass of P.  

Having that / Jq are Cn-symmetric, the following must be satisfied, 

given the measurements Q0. Qn x ye need to find the most probable P0 and CO. We find P0 and (0 that maximize ProbGP,- 11 to, 0 under the symmetry constraints of Eq. 8. 

Thus we need to find those parameters which maximize, 

Note that when all A are equal (i.e., all points have the same uncertainty, which is equivalent to the cases in the previous sections where point location is known with no uncertainty), Eqs. 9-10 reduce to Eqs. 5-6 in Section IV. 

---

What is the most probable symmetric shape represented by the data ... 

Figure 18a shows a configuration of points whose locations are given by a normal distribution function (marked as rectangles having width and length proportional to the standard deviation). In this section we show a method of evaluating the most probable symmetric shape closest to the data. For simplicity we derive the method with respect to rotational symmetry of order n (Cn-symmetry). The solution for mirror symmetry is similar (see Appendix D). 

Several examples are shown in Figure 18 where for a given set of measurements (Figure 18a), the most probable symmetric shapes are shown (Figure 18b-e for C2, C3, C6 and mirror symmetry respectively). 

Figure 19 shows an example of varying the probability distribution of a measurement on the resulting symmetric shape. Figure 19a shows the most probable C2-symmetric shape for the set of measurements of Figure 18a. Figures 19b-d show the most probable C2-symmetric shape after varying the distribution of the bottom measurement.

Appendix D The Most Probable Mirror Symmetric Shape... 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

The distribution of random errors should follow the Gaussian or normal curve if the number of measurements is large enough. The shape of Gaussian distribution was given in Chapter 3 (Fig. 3.4). It can be characterized by two variables—the central tendency and the symmetrical variation about tjie central tendency. Two measures of the central tendency are the mean, X, and the median. One of these values is usually taken as the correct value for an analysis, although statistically there is no correct value but rather the most probable value. The ability of an analyst to determine this most probable value is referred to as his accuracy. 

Both Equation 5.37 and the plots in Figure 5.19 show that the most probable speed, like the rms speed, (uj np, is proportional to the square root of the temperature and inversely proportional to molecnlar (or molar) mass. Because the speed distribution is not symmetric aronnd the most probable value, the rms speed ((M)rms) and the most probable speed ((uj np) will be similar, but not identical. The shape of the distribution is such that (M)rms is larger than (u) p. 

The columnar phases of oligobenzoate Hekates are most probably formed from -shaped conformers which pack disordered in hexagonal columnar phases and for symmetric mesogens they can tilt and aggregate in highly ordered helical arrangements to form the 3D columnar structures (see Fig. 10a, b). Indications... 

By far the most widely assumed probability distribution applicable to biological data is the Normal Probability Distribution, or Gaussian distribution. When plotted, this distribution forms the familiar bell-shaped curve that is symmetrical about the mean. The math atical expression describing this distribution is... 

Since it is silica-based, glass fiber for polymer reinforcement could be thought of as a cousin to mineral fillers. But glass fiber is more carefully produced in controlled, uniform, and symmetrical shapes with extremely high aspea ratios, with particle dimensions that are (usually) visible to the human eye. Glass-fiber reinforcement is probably the most cost-effective and most proven way of reinforcing polymers to inaease tensile and flexural modulus and strength. 

Normal distribution is probably the most commonly used distribution in practice. However, due to it symmetric shape, its applicadon in some areas, such as the financial industry, is limited. In a lot of cases, other asymmetric distributions are used such as lognormal distribution. As a much more flexible distribution, generalized hyperbolic distribution can be used for N-type occurrence function. 

Most of the antiparallel /3 domains have their sheets wrapped around into a cylinder, or barrel, shape. None of the antiparallel barrels has as symmetrical or as continuously hydrogen-bonded a cylindrical sheet as the singly wound parallel fi barrels of triosephos-phate isomerase and pyruvate kinase dl however, antiparallel barrels are very much more common. Because of gaps in the hydrogenbonding, some of these structures have been described as two /3 sheets facing each other (e.g., Schiffer et al., 1973 Blake et al., 1978 Harrison et al., 1978). Our reasons for treating them all as barrels are that the gap positions are sometimes different in domains that are probably related, and that the barrel description yields very much simpler and more unified topologies. 

---