Sequential probabilities

The probability of observing a distribution of events requires that event xi (with probability pi) occur, andxa (with probability P2) occtrr, and so on. The probability of observing events, (xi,X2,X3,..., x ), is the simultaneous or sequential probability of observing all events in the distribution occtming once, that is, the product of the individual probabilities ... 

Table I contains the results, some of which have been derived previously (11, 12). These equations are appropriate for the case of equal reactivity of both ends of a difunctional molecule and allow for unequal rate constants for the A-B and A-C reactions. These results are presented here in terms of reaction probabilities, p, (the probability that reactant I has reacted with reactant J) where I,J = A, B or C, and p j = Pj - These should be distinguished from the sequential probabilities of...

A sequential procedure was further developed by Tong and Crowe (1996) by applying sequential analysis of the principal component test using the sequential probability ratio test (SPRT). Dunia et al. (1996) also used PCA for sensor fault identification via reconstruction. In that paper it was assumed that one sensor had failed and the remaining sensors are used for reconstruction. Furthermore, the transient behavior of a number of sensor faults in various types of residuals is analyzed, and a sensor validity index is suggested, determining the status of each sensor. 

Cummings JR, Haynes JD, Lipchuck LM, Ronsberg MA (1960) A sequential probability ratio method for detecting compounds with diuretic activity in rats. J Pharmacol Exp Ther 128 414—418... 

Das et al. [37] studied the coefficient of friction signal in order to determine the end-point detection more effectively. As shown in Fig. 4.35, the raw COF data were analyzed and the noise levels were filtered out to more precisely determine the end point of the process. The variance sequential probability ratio test (SPRT) method was adopted to analyze and filter the raw data. The coefficient of friction signal can thus be processed and analyzed to detect the end point more effectively. 

Kulldorff M, Davis RL, Kolczak M. Lewis E, Lieu T, Platt R. 2011. A maximized sequential probability ratio test for drug and vaccine safety surveillance. Sequential Analysis Design, Methods and Applications. 30(l) 58-78. 

Feller [50] has presented an approach which is very similar in concept to the Markov chain approach described above. He works in terms of a matrix M, of sequential probabilities which are different from, but related to, the Markov chain transition probabilities. He shows that for the system A,-B, copolymerizing with A2-B2 (representative of two different hydroxy or amino acids) the CLD may be calculated from ... 

We assume that the sequential errors are not correlated in time, we can write the probability of sampling a sequence of errors as the product of the individual probabilities. We further use the finite time approximation for the delta function and have ... 

J Cornfield. In DL Meyer, RO Collier, eds. The Frequency Theory of Probability, Bayes Theorem, and Sequential Clinical Trials. Bloomington, In Phi Delta Kappa, 1970, pp 1-28. 

The probability of the redundant system failing is the product of the single system time-dependent unavailability which includes approximately linearity between test and repair. There are two ways of performing the surveillance tests on these two systems sequential and staggered testing. 

The process is strictly sequential, so the probability of the stem achieving a length l is alc 1 (the first step must have been a deposition), and the probability of reaching... 

No sequence homologies can be detected. This is, perhaps, not surprising. The X-ray structure analysis of lysozyme by Phillips has shown that the polypeptide chain is folded in a way which puts none of the amino acids in sequential vicinity of the catalytic Asp-52 and Glu-37 that are near to the bound substrate. Comparable folding patterns can probably be realized with widely differing arrangements of amino acids, and thus the apparent lack of homologies. 

A classification decision tree allows one to predict in a sequential way the y value (or corresponding conditional probabilities) that is associated with a particular x vector of values. At the top node of the tree (A in Fig. 3), a first test is performed, based on the value assumed by a particular decision variable (jCj). Depending on the outcome of this test, vector X is sent to one of the branches emanating from node A. A second test follows, being carried out at another node (B), and over the values of the same or a different decision variable (e.g., JCg). This procedure is... 

The Pectolyase Y-23 catalytic activity was studied on several solutions of pectin at pH 3.0 and 6.0 and of polygalacturonic acid at pH 4.1. During pectin depolymerisation, the PL activity is about 20 times lower than the one of the PG and reaches its saturation value after about 30 min whereas the PG activity increases regularly with time. This dual behaviour is probably connected to the competitive hydrolytic action of PE that quickly transforms the pectin into the polygalacturonic acid. Accordingly, the reducing group formation is mainly due to the PG action and the pectin depolymerisation derives from the sequential action of PE and PG. 

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

Bindschaedler and Gurny [12] published an adaptation of the simplex technique to a TI-59 calculator and applied it successfully to a direct compression tablet of acetaminophen (paracetamol). Janeczek [13] applied the approach to a liquid system (a pharmaceutical solution) and was able to optimize physical stability. In a later article, again related to analytical techniques, Deming points out that when complete knowledge of the response is not initially available, the simplex method is probably the most appropriate type [14]. Although not presented here, there are sets of rules for the selection of the sequential vertices in the procedure, and the reader planning to carry out this type of procedure should consult appropriate references. 

The principle of sequential analysis consists of the fact that, when comparing two different populations A and B with pre-set probabilities of risks of error, a and /3, just as many items (individual samples) are examined as necessary for decision making. Thus the sample size n itself becomes a random variable. 

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