Random classes

The Canadian Task Force categorized the quality of evidence based on the type of research study. The quality of evidence was organized into three classes Class I evidence comes from procedures having at least one randomized controlled study to support them. Class II is divided into three subclasses, where II-l involves a well-designed controlled study without randomization. Class II-2 evidence comes from well-designed cohort or case-control studies, preferably carried out at more than one research setting. Class II-3 involves uncontrolled research with dramatic results (e.g., penicillin trials in the 1940s). Class III evidence includes the opinions of experts and authorities in the field based on clinical... 

If the nuiiiber of patterns n is less, equal, or only slightly greater than the nuniber of dimensions d, then a linear separability may be found even for a random class assignement to the pattern points. Figure 10 shows this problem for a 2-dimensional pattern space. 

Numerical experiments with random patterns or with chemical patterns but random classes are sometimes used to mark the lowest limit for a classifier performance C94, 251, 288, 303, 3103-... 

There are two classes of solids that are not crystalline, that is, p(r) is not periodic. The more familiar one is a glass, for which there are again two models, which may be called the random network and tlie random packing of hard spheres. An example of the first is silica glass or fiised quartz. It consists of tetrahedral SiO groups that are linked at their vertices by Si-O-Si bonds, but, unlike the various crystalline phases of Si02, there is no systematic relation between... 

The second class, indeterminate or random errors, is brought about by the effects of uncontrolled variables. Truly random errors are as likely to cause high as low results, and a small random error is much more probable than a large one. By making the observation coarse enough, random errors would cease to exist. Every observation would give the same result, but the result would be less precise than the average of a number of finer observations with random scatter. 

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

Suppose that you need to add a reagent to a flask by several successive transfers using a class A 10-mL pipet. By calibrating the pipet (see Table 4.8), you know that it delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since the pipet is calibrated, we can use the standard deviation as a measure of uncertainty. This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL of solution, the volumes actually delivered are randomly scattered around the mean of 9.992 mL. 

Chain-Growth Associative Thickeners. Preparation of hydrophobically modified, water-soluble polymer in aqueous media by a chain-growth mechanism presents a unique challenge in that the hydrophobically modified monomers are surface active and form micelles (50). Although the initiation and propagation occurs primarily in the aqueous phase, when the propagating radical enters the micelle the hydrophobically modified monomers then polymerize in blocks. In addition, the hydrophobically modified monomer possesses a different reactivity ratio (42) than the unmodified monomer, and the composition of the polymer chain therefore varies considerably with conversion (57). The most extensively studied monomer of this class has been acrylamide, but there have been others such as the modification of PVAlc. Pyridine (58) was one of the first chain-growth polymers to be hydrophobically modified. This modification is a post-polymerization alkylation reaction and produces a random distribution of hydrophobic units. 

Pieces of coal are mixtures of materials somewhat randomly distributed in differing amounts. The mineral matter can be readily distinguished from the organic, which is itself a mixture. Coal properties reflect the individual constituents and the relative proportions. By analogy to geologic formations, the macerals are the constituents that correspond to minerals that make up individual rocks. For coals, macerals, which tend to be consistent in their properties, represent particular classes of plant parts that have been transformed into coal (40). Most detailed chemical and physical studies of coal have been made on macerals or samples rich in a particular maceral, because maceral separation is time consuming. 

Random copolymers of vinyl chloride and other monomers are important commercially. Most of these materials are produced by suspension or emulsion polymerization using free-radical initiators. Important producers for vinyl chloride—vinyUdene chloride copolymers include Borden, Inc. and Dow. These copolymers are used in specialized coatings appHcations because of their enhanced solubiUty and as extender resins in plastisols where rapid fusion is required (72). Another important class of materials are the vinyl chloride—vinyl acetate copolymers. Principal producers include Borden Chemicals Plastics, B. F. Goodrich Chemical, and Union Carbide. The copolymerization of vinyl chloride with vinyl acetate yields a material with improved processabihty compared with vinyl chloride homopolymer. However, the physical and chemical properties of the copolymers are different from those of the homopolymer PVC. Generally, as the vinyl acetate content increases, the resin solubiUty in ketone and ester solvents and its susceptibiUty to chemical attack increase, the resin viscosity and heat distortion temperature decrease, and the tensile strength and flexibiUty increase slightly. 

The other class of phenomenological approaches subsumes the random surface theories (Sec. B). These reduce the system to a set of internal surfaces, supposedly filled with amphiphiles, which can be described by an effective interface Hamiltonian. The internal surfaces represent either bilayers or monolayers—bilayers in binary amphiphile—water mixtures, and monolayers in ternary mixtures, where the monolayers are assumed to separate oil domains from water domains. Random surface theories have been formulated on lattices and in the continuum. In the latter case, they are an interesting application of the membrane theories which are studied in many areas of physics, from general statistical field theory to elementary particle physics [26]. Random surface theories for amphiphilic systems have been used to calculate shapes and distributions of vesicles, and phase transitions [27-31]. 

As already mentioned in the Introduction, phenomenological models for amphiphilic systems can be divided into two big classes Ginzburg-Landau models and random interface models. 

Non-Homogeneous CA a characteristic feature of all CA rules defined so far has been that of homogeneity - each cell of the system evolves according to the same rule 0. Hartman and Vichniac [hartSfi] were the first to systematically study a class of inhomogeneous CA (INCA), in which the state-transition rules are allowed to vary from cell to cell. The simplest such example is one where there are only two different 0 s, which are randomly distributed throughout the lattice. Kauffman has studied the other extreme in which the lattice is randomly populated with all 2 possible boolean functions of k inputs. The results of such studies, as well as the relationship with the dynamics of random, mappings, are covered in detail in chapter 8.3. 

Fig. 3.14 Totalistic d=l, fc = 3, r = 2 rules starting with random initial conditions. 3.1.2 Behavioral Classes...

CLASS FINITE SEED RANDOM SEED DIFFERENCE PATTERN PREDICTABILITY... 

Consider two random initial configurations that differ at only one site, so that H t = 0) = 1. The difference plots shown in figure 3.16 suggest that for class cl and c2 rules, H t) rapidly approaches some small fixed value. Class c3 rules, on the other hand, are unstable with respect to such small perturbations H t) generally grows with time. The rate of growth of H t) depends on whether the rules are additive or nonadditive. 

That neither complex nor random-like, behavior, need have a complex dynamical origin is well exemplified by the elementary one-dimensional class c3 rule R30, studied extensively by Wolfram in [wolf84c]. It is defined explicitly by... 

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

A remarkable, but (at first sight, at least) naively unimpressive, feature of this rule is its class c4-like ability to give rise to complex ordered patterns out of an initially disordered state, or primordial soup. In figure 3.65, for example, which provides a few snapshot views of the evolution of four different random initial states taken during the first 50 iterations, we see evidence of the same typically class c4-like behavior that we have already seen so much of in one-dimensional systems. What distinguishes this system from all of the previous ones that we have studied, however, and makes this rule truly remarkable, is that Life has been proven to be capable of universal computation. 

We might mention here in passing that while class-1, class-3 and class-4 (but not class-2) can all be obtained from one another with these two simple rules, class-2 behavior can only be obtained if the system is first quenched [vich86b]. That is, if the lattice is initially randomly populated with AND and XOR rules according to the prescribed value of p and is then frozen for all later times. Such quenched random rules harbor some interesting properties of their own in dimensions d > 1, and are the basis of much of Kauffman s findings on random Boolean networks (see section 8.6). 

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