Functions unimodal

Optimization problems in crystallographic structure refinement are seldom convex, that is very rarely characterized by a unimodal function/(x). Regularization of a two-atom model is an example of such a unimodal function. Fig. 11.2a. in contrast. Fig. 11.2b shows a profile of a function for modelling an amino acid side chain - the peaks correspond to the possible rotamers. In this case, the shape of the function/(x) is called multimodal. Such functions arise naturally in structural macromolecular optimization problems and possess a highly complex multiminima energy landscape that does not lend itself favourably to standard robust optimization techniques. 

A procedure to estimate and remove any bias due to assuming a unimodal functional form has been incorporated into the LUT. The bias is determined by testing the LUT performance with extinction spectra calculated from measured bi- and tri-modal size distributions obtained from wire-impactor and dustsonde measurements which coincide in space and time with the SAGE D/CLAES composites. The bi- and tri-modal distributions were reported by Pueschel et al. [9], Goodman et al. [10] and Deshler et al. [11,12], The calculated bias is then subtracted from the retrieved Reff values to obtain bias-corrected values of RtJf. Similar procedures are used to estimate and remove bias in retrieved S and V. Figure 5 compares a measured post-Pinatubo bimodal... 

To illustrate how this works, let us suppose that there is only one operating variable, of which the right-hand side of (8) is always a unimodal function. Special as this may seem, it is the important case in Chapter 7. Let us denote df/dT by/r, dfidui by/,-, and use the summation convention on the repeated index i = I,. .. n. Then Eq. (8) may take one of three forms ... 

All of the methods discussed in this section are best suited for unimodal functions, i.e., functions with no more than one maximum or minimum within the bounded range. 

The BzzMinimizationMono class is designed to solve one-dimensional minimization problems for unimodal functions. This class can be employed when the function is continuous and defined on the overall interval of uncertainty. 

As the minimum has to be found in a set of minima, one initial essential modification is required. In the case of unimodal functions, only the values of abscissas f and of functions/ = F(fi) to define the next iteration are collected, whereas when the function is multimodal, they have to be collected into two dedicated vectors, t and f. This is the only way to check for the existence of intervals that may contain a local minimum. 

It can be observed that the tuned version (indicated by TPSO) is able to improve the results of the standard versions on the majority of the functions, including those which were not within the ones that were tuned. This supremacy is also confirmed, and in some cases improved, when the dimensionality of the problem increases to 30 and 50, showing that our approach is able to find parameter sets that generalize well. In particular, the standard versions perform comparably on the unimodal functions, while the tuned version is constantly better on the multimodal and composition ones. 

Thesetwofunctions (for examples see Fig. l)representunimodal, bellshaped distributions. Other types, such as two-peaked curves, can be constructed by adding two or more suitably chosen unimodal functions 12,13). 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

The nature of the relationships and constraints in most design problems is such that the use of analytical methods is not feasible. In these circumstances search methods, that require only that the objective function can be computed from arbitrary values of the independent variables, are used. For single variable problems, where the objective function is unimodal, the simplest approach is to calculate the value of the objective function at uniformly spaced values of the variable until a maximum (or minimum) value is obtained. Though this method is not the most efficient, it will not require excessive computing time for simple problems. Several more efficient search techniques have been developed, such as the method of the golden section see Boas (1963b) and Edgar and Himmelblau (2001). 

The unimodal association areas in turn project to multimodal sensory association areas that integrate information about more than one sensory modality. The highest level of cognitive brain function takes place in these areas. These areas process, integrate, and interpret sensory information and then link these data to the planning of movement and goal-directed action. 

The inset shows a unimodal distribution of relaxation times r = I 1 obtained by a CONTIN analysis. Besides CONTIN there is a number of alternative techniques [51] for the determination of the distribution from the correlation function. Detailed discussions of this topic have been given by Stock and Ray [52] and by Stepanek [50]. 

Is the following function unimodal (only one extremum) or multimodal (more than one extremum) ... 

For any of the three procedures outlined in this section, in minimization you assume the function is unimodal, bracket the minimum, pick a starting point, apply the iteration formula to get xk+l (or jc ) from xk (or xP and xP), and make sure that fixk+l) [Pg.161]

The theory necessary for understanding two-station tracer measuring techniques is outlined in Appendix 1. An arbitrary, but unimodal, impulse of tracer is created in a system inlet and the outlet response recorded, see Fig. 21 (Appendix 1). Then, the mean, Mj, of that which resides between the points at which inlet and outlet pulses are observed and recorded is equal to the difference in means of these two signals. Similarly, the variance, T2, and the skewness, T3 are equal to the differences in these respective moments between inlet and outlet. This enables the system transfer function to be defined in terms of a few low-order moments via eqns. (A.5) or (A.9) of Appendix 1, this in turn defining the system RTD. Recall that system moments and moments of the system RTD are one and the same. 

Fuzzy arithmetic Fuzzy arithmetic is the arithmetic embodied in operations snch as addition, subtraction, multiplication, and division of fnzzy nnmbers. Fnzzy nnmbers are unimodal distribution functions of the real line that grade all real numbers according to the possibility that each might be a valne the fnzzy number could take on. The minimum of the function is 0, which represents impossible values, and the maximum is 1, which represents those... 

Similarly to the most robust methods of solving nonlinear equations, we start with bracketing. Assume that the interval [xy, X ] contains a single minimum point r, i.e., the function f is decreasing up to r and increasing afterwards. Then the function is said to be unimodal on the interval [xy, Xy], This property is exploited in cut-off methods, purported to reduce the length of the interval which will, however, include the minimum point in all iterations. 

Distribution functions for very short n-alkane chains are calculated. The results obtained implicate the discrete nature of the RIS approximation as the origin of the multimodal nature of the short-chain distribution, and suggest that the best representation of such results would be a simple unimodal curve averaging out all of the minima and retaining only the most prominent maximum. 

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