Variations of the Method

If one class is more frequent in the training set than the other, the resulting classifier may recognize the more frequent class well but is unsatisfactory for the other class C157, 2293- To overcome this problem it is useful to construct a fictitious sample with equal numbers of patterns in both classes C248, 2513. For this purpose, the expected values E in equations (57) and (58) are split into two class-dependent parts (expected value for class 1) and E (for class 2). If p(1) is defined as the probability of class 1 in the training set, then 

To obtain the expected value E for a fictitious sample with equal probabilities of both classes, p(1) is given the value 0.5. Instead of equations (57) and (58) the following equations are used to determine the coefficients of the set of Linear equations (56). 

Instead of the scalar product s in equation (49) the tanh(s) is used andy according to the least-squares principle the function to be minimized becomes 

The weight vector can be found by an iterative procedure C118, 128, 229, 260, 3033. 

Other methods of linear regression for chemical applications of pattern recognition have been used by Bos and Jasink C213, Kowalski and Reilly C1613, and Volkmann C3183. 

---

The energy conserving method (6) is a close variation of the method... 

The deterrnination of hydrogen content of an organic compound consists of complete combustion of a known quantity of the material to produce water and carbon dioxide, and deterrnination of the amount of water. The amount of hydrogen present in the initial material is calculated from the amount of water produced. This technique can be performed on macro (0.1—0.2 g), micro (2—10 mg), or submicro (0.02—0.2 mg) scale. Micro deterrninations are the most common. There are many variations of the method of combustion and deterrnination of water (221,222). The oldest and probably most reUable technique for water deterrnination is a gravimetric one where the water is absorbed onto a desiccant, such as magnesium perchlorate. In the macro technique, which is the most accurate, hydrogen content of a compound can be routinely deterrnined to within 0.02%. Instmmental methods, such as gas chromatography (qv) (223) and mass spectrometry (qv) (224), can also be used to determine water of combustion. 

Triglyeerides The loss of NADH is followed at 340 nm. This is one of over 20 variations of the method. 

An increased hydrogen ion concentration, that is a considerably greater amount of acid than the theoretical two equivalents of Scheme 2-1, is necessary in the diazotization of weakly basic amines. The classic example of this is the preparation of 4-nitrobenzenediazonium ions 4-nitroaniline is dissolved in hot 5-10 m HC1 to convert it into the anilinium ion and the solution is either cooled quickly or poured onto ice. In this way the anilinium chloride is precipitated before hydrolysis to the base can occur. On immediate addition of nitrite, smooth diazotization can be obtained. The diazonium salt solution formed should be practically clear and should not become cloudy on standing in the dark. Some practice is necessary, and details can be found in the books emphasizing preparative aspects (Fierz-David and Blangey, 1952 Saunders and Allen, 1985 in Houben-Weyl, Vol. E 16a, Part II, the chapter written by Engel, 1990). These books give a series of detailed prescriptions for specific examples and a useful review of the principal variations of the methods of diazotization. Such reviews have also been written by Putter (1965) and Schank (1975). 

Major emphasis in studies of N-nitroso compounds in foods has been placed upon volatile nitrosamines, in part because these compounds are relatively easy to isolate from complex matrices by virtue of their volatility. Procedures utilizing atmospheric pressure or vacuum distillation have been used by most investigators, with variations of the method of Fine e al. (2) being among the most popular. This procedure employs vacuum distillation of a mineral oil suspension of the sample with optional addition of water to improve nitrosamine recovery from low moisture content samples (6) The usual approach to prevention of nitrosamine formation during analysis involves adding sulfamic acid or ascorbate to destroy residual nitrite at an early stage of sample preparation. 

The algorithm of entropy maximization is nonlinear, and must therefore be applied iteratively. It is possible to solve for both X(n + 1) and pf n + 1), starting from X(n) and p/n) at each iteration it. The starting values are A(0) % 0, and p/0) equal to the prior density. Achieving convergence involves a two-step process, in which first the y2 — IV constraint is satisfied, and subsequently the entropy S is maximized. In a variation of the method, X is kept fixed at a small value adequate to ensure convergence (Sakata and Sato 1990). 

Some variations of the method have been used to prepare cyclopropyl and cyclobutyl halides. Simultaneous addition of bromine and 3-bromocyclobutanecarboxylic acid to the suspension of mercuric oxide gives 1,3-dibromocyclobutane in good yield.7 Similarly, cyclopropanecarboxylic acid gives bromo-cyclopropane,9 and 3-(bromomethyl)cyclobutanecarboxylic acid gives 3-(bromomethyl)cyclobutyl bromide.10 In the latter reaction, it was found desirable to remove the water from the reaction as it is formed in order to obtain high yields. Another variation is the addition of a mixture of the acid and mercuric oxide to excess bromine in bromotrichloromethane.6... 

A novel variation of the method of adding an allylie alcohol to a reactive a-fluoro carbonyl compound, as shown in the previous example, has been developed by Shi and Cai in a general strategy for the synthesis of /h/i-difluoro-a-amtno acids via the a-oxo esters 53.20 Ethyl tri-fluoropyruvate reacts with primary allylie alcohols 49 in benzene to form the stable hemiacetals... 

A variation of the method utilizes a laser as the heat source.52,53 This nonequilibrium technique involves fast growth and rapid heating/cooling rates (100 000 K s-1) in the reaction zone. Ochoa et al. (chapter 27), provide a synopsis of the laser pyrolysis method and describe an Fe3C product used for catalysis. 

Reduction to diol 16 was achieved with a variation of the method of Luchc 5 involving sodium borohydnde and cerium trichloride. The initial product is cyclic hemiacetai 51, which is opened and reduced via intermediate 52 ultimately to diol 16. 

If we can t have both, it is more important for a method to be size-consistent than variational. Of the methods we have seen in this book ... 

This method can be used to compensate for inhibition of a biochemical pathway which results in a deficiency of an essential metabolic product. Detailed variations of the method are provided by Dayan et al.7 and Amagasa et al.1 The inhibitor concentration should be no higher than that required for strong herbicidal effect. Metabolite concentrations should be below that which is phytotoxic. For example, certain amino acids such at methionine, are growth inhibitors at relatively low concentrations. So, in preliminary work, dose-response studies should be done with amino acids to find the maximum concentrations that do not inhibit growth. Then, seeds of test plants should be imbibed in solutions of the phytotoxin with and without metabolite solutions. Amino acids, tricarboxylic acid cycle intermediates, vitamins, nucleotides, and reducing agents have all been used in complementation studies to elucidate modes of action of a variety of phytotoxins. Examples of each of these is provided by Dayan et al.7... 

There exists a substantial history of interest in flow and deformation properties of monolayers. Perhaps, the first is the theoretical formulation of hydrodynamic coupling between the monolayer and subphase by Harkins and Kirkwook in 1938 [129], in determination of steady shear viscosity of mono-layers, which has since been augmented by Hansen [130] and Goodrich [131]. A variation of the method based on the Maxwell model was proposed by Mannheimher and Schechter [132] in an oscillatory mode in a canal. Experimentally, the method was implemented by joint efforts in our laboratories for determinations of steady shear viscosity of monolayers through the canal... 

A variation of the method was described in 1884 by Pape, who used a mixture of silicon tetrachloride and propyl iodide with metallic zinc instead of preparing the zinc dipropyl separately. Another variation, exploited by Friedel and Ladenburg, eliminated the sealed tube by introducing metallic sodium along with the zinc alkyl and silicate ester ... 

The authors state that the detection limit for a-tocopherol was of the order of 1.0 ng at a signal to noise ratio of 5, and the recovery of a-tocopherol added prior to homogenization was >90% the coefficient of variation of the method was 4.7%. 

A variation of the method for determining ° Pb- ° Pb dates of whole single zircons by TIMS analysis that eliminates the need for zircon dissolution and chemical separation of lead was introduced by Kober (1986), with a slight, yet important, modification described a year later (Kober, 1987). In the modified method, usually referred to as the zircon evaporation method, a single zircon crystal is placed into a folded side rhenium filament (the evaporation filament), which is positioned opposite of a blank rhenium ionization filament. The evaporation filament is heated for a short time to evaporate lead onto the target ionization filament. The current to the evaporation filament is then mrned off and the ionization filament is heated until Pb ionization begins and lead isotopic ratios are measured in the normal fashion. Current to the ionization filament... 

A variation of the method uses geminal a-dibromoketones as a route to monoalkylcyc-lopropenones (equation 28) . ... 

---