Why is the sequential method considered to be more accurate than the direct method?

2 Historical background of sequential procedures

The first strictly sequential method, the sequential probability ratio test, was developed during the Second World War (Wald, 1947). As its main application was the quality control of manufactured materials, its publication was only authorized after the end of the war, in 1947. Another class of sequential test is based on triangular continuation regions (Anderson, 1960). The basic idea on which these methods rely is to constantly use the available information to determine whether the data are compatible with null hypothesis, with alternative hypothesis, or insufficient to choose between these two hypotheses. In the first two cases, the trial is stopped and the conclusion is obtained whereas in the third case the trial continues. The trial is further processed until the data allows a legitimate (or per-protocol) decision between the two hypotheses. An example of a completely sequential trial can be found in Jones et al. (1982).

Armitage (1954) and Bross (1952) pioneered the concept of group sequential methods in medical field (Bross, 1952; Armitage, 1954). At first, these plans were fully sequential and did not gain widespread acceptance perhaps due to the inconvenience in their application. The problems discussed included the fact that response needs to be available soon after the treatment is started and that there would be organizational problems, such as coordination in multicenter trials and a much greater amount of work for the statistician. The shift to group sequential methods for clinical trials did not occur until the 1970s. Elfring and Schultz (1973) specifically used the term ‘group sequential design’ to describe their procedure for comparing two treatments with binary response (Elfring et al., 1973). McPherson(1974) suggested that the repeated significance tests of Armitage et al. (1969) might be used to analyze clinical trial data at a small number of interim analyses (Armitage et al., 1969; McPherson, 1974). Canner (1977) used Monte Carlo simulation to find critical values of a test statistic for a study with periodic analyses of survival endpoint (Canner, 1977). However, Pocock (1977) was the first to provide clear guidelines for the implementation of the GSD attaining particular operating characteristics of type I error and power (Pocock, 1977). He made the case that most investigators do not want to evaluate results every time a couple of new patients are accrued but do want to understand the comparative merit every few months to assess if the trial is worth the time and effort and that continual monitoring does not have a remarkable benefit. More specifically, only a minor improvement is expected with more than five interim looks. A more comprehensive account of this history can be read from the excellent book by Jennison and Turnbull (2000).

3 RESULTS AND DISCUSSION

In order to apply the proposed sequential method for kinetic parameter estimation in the FCC process three feeds were used for MAT experiments: GO-1, a typical FCC feedstock, GO-2, Heavy Vacuum Gas Oil and, GO-3, the typical FCC feedstock plus 5 vol% atmospheric residuum. Characterization of theses feeds were presented in a previous work [5].

The following mass balance, solved with a Runge Kutta Method, was used to evaluate the product yields from a set of kinetic constants for a given kinetic model and for each feedstock and temperature:

(1)dyidz=1WHSVρLρ cri

The minimization of the objective function, based on the sum of square errors between experimental and calculated product yields, was applied to find the best set of kinetic parameters. This objective function was solved using the least squares criterion with a nonlinear regression procedure based on Marquardt’s algorithm [6].

The sequential methodology previously described was followed for kinetic constants estimation in order to decrease the number of parameters estimated simultaneously and to avoid convergence problems.

The mass balance given in equation (1), was solved numerically with the following boundary condition: at z = 0, yA = 1, and yB = yC = yD=yE = 0, in order to calculate gas oil, gasoline, LPG, dry gas and coke yields for each feedstock..

The kinetic parameters obtained by following the proposed methodology for each feedstock are presented in Table 1. Values of kinetic parameter for each model are presented in Ref. [6]. Average values were calculated for those kinetic constants evaluated with more than one model. The kinetic constants for gasoline cracking (kB-C, kB-D and kB-E) show that gasoline gives mainly LPG and dry gas, while the coke is produced only by cracking of gas oil, since the gasoline to coke kinetic constant (kB-E) was many orders of magnitude smaller than the others. Apparent activation energies for each reaction lump involved, which were quite similar for the three feedstocks, are within the range of those reported in the literature (5-20 Kcal/mol).

Table 1. Kinetic parameters of the 5-lump model for each feedstock at 500 °C

Figure 3 shows a comparison between experimental and predicted gasoline yields as a function of gas oil conversion in the range of 480-520 °C for GO-2 feedstock. Conversion is defined as the sum of gasoline, LPG, dry gas and coke yields. The effect of the type of feedstock on gasoline yield at 500 °C is shown in Figure 4.

It can be observed from these figures that the application of the proposed sequential method to estimate the rate constants of the 5-lump kinetic model predicts sufficiently well the experimental data with average deviations less than 2%.

5 CONCLUSIONS

It has been shown that the catalytic properties of P-containing CoMo/Al2O3 catalysts were greatly modified by the impregnation method (sequential or co-impregnation) and by the amount of P added. Introduction of P prior to Mo and Co (D.I.) led to badly dispersed active phases, because P occupied a part of the anchoring sites of the molybdate anions on the support. Hence, a decrease in activity was observed, which was still aggravated by an increase of the P content. When P, Mo and Co were impregnated in one step (C.I.), homogeneously dispersed catalysts were obtained, leading to increased activity. The increased dispersion was rationalised through the formation of (Co containing) phosphomolybdate anions, capable of penetrating more deeply into the pore system of Al2O3 and the lesser competition between P and Mo for adsorption on the support.

1 Introduction

Many problems in computational geometry come from application areas (such as pattern recognition, computer graphics, operations research, computer-aided design, robotics, etc.) that require real-time speeds. This need for speed makes parallelism a natural candidate for helping achieve the desired performance. For many of these problems, we are already at the limits of what can be achieved through sequential computation. The traditional sequential methods can be inadequate for those applications in which speed is important and that involve a large number of geometric objects. Thus, it is important to study what kinds of speed-ups can be achieved through parallel computing. As an indication of the importance of this research direction, we note that four of the eleven problems used as benchmark problems to evaluate parallel architectures for the DARPA Architecture Workshop Benchmark Study of 1986 were computational geometry problems.

In parallel computation, it is the rule rather than the exception that the known sequential techniques do not translate well into a parallel setting; this is also the case in parallel computational geometry. The difficulty is usually that these techniques use methods which either seem to be inherently sequential, or would result in inefficient parallel implementations. Thus new paradigms are needed for parallel computational geometry. The goal of this chapter is to give a detailed look at the currently most successful techniques in parallel computational geometry, while simultaneously highlighting some open problems, and discussing possible extensions to these techniques. It differs from [33] in that it gives a much more detailed coverage of the shared-memory model at the expense of the coverage of networks of processors. Since it is impossible to describe all the parallel geometric algorithms known, our focus is on general algorithmic techniques rather than on specific problems; no attempt is made to list exhaustively all of the known deterministic parallel complexity bounds for geometric problems. For more discussion of parallel geometric algorithms, the reader is referred to [15,139].

The rest of the chapter is organized as follows. Section 2 briefly reviews the PRAM parallel model and the notion of efficiency in that model, Section 3 reviews basic subproblems that tend to arise in the solutions of geometric problems on the PRAM, Section 4 is about inherently sequential (i.e. non-parallelizable) geometric problems, Section 5 discusses the parallel divide and conquer techniques, Section 6 discusses the cascading technique, Section 7 discusses parallel fractional cascading, Section 8 discusses cascading with labeling functions, Section 9 discusses cascading in the EREW model, Section 10 discusses parallel matrix searching techniques, Section 11 discusses a number of other useful PRAM techniques, and Section 12 concludes.

9.4 Social sequence analysis

Social sequence analysis refers to a set of methods, suitable for the analysis of sequences of events or activities or other phenomena pertaining to problems from social sciences; the interested reader may refer to Abbott (1995), Elzinga and Studer (2015), or the book by Cornwell (2015), and the edited volume by Ritschard and Studer (2018). Although literature review indicates that it is only during the last four decades that the sequential methods have become popular in the field of social sciences, it appears that there is an increasing interest in such methods recently; this can be readily seen from the books of Cornwell (2015), and Ritschard and Studer (2018). This could perhaps be attributed to the current great availability of longitudinal data sources or real-time social data, and by the readily available software packages for performing the respective analysis. Nowadays, sequential analysis plays a key role, among others, in the study of life course, occupational careers and health successive conditions (trajectories). One of the extensively studied problems under social sequence analysis is the computation of pairwise dissimilarities (distances) between sequences and subsequent use of these distances for creating clusters. For example, the most frequently used distance between sequences is the optimal matching metric; it expresses the cost of making the two sequences exactly the same based on three operations: replacement, deletion, and insertion.

Apart from this kind of problems, there are also cases wherein the role of runs, usually referred to as “spells” or “episodes” in social sciences, is very crucial; for example, it is of importance to study the time (length of run) one spends in a specific state (e.g., married-no child or employment) before transitioning into a different state (e.g., married-with child or unemployment). Moreover, the computation of the probability of occurrence of runs of given lengths under specific distributional assumptions offers a deeper insight into the nature of the problems under study.

Thus we believe that the criteria used in start-up demonstration theory may further contribute to social sciences research; for example, although a run of the state “unemployment” may be important due to economic or other problems that it may cause (see Van Belle et al., 2018), one may however have to take into account whether this run came after a relatively long run of the state “employment.” In addition, monitoring the current state of occupation and whether a run of “unemployment” came much faster than any expected (under market conditions) run of “employment” may provide useful tools for establishing policies and intervention strategies for reducing (or the impact of) unemployment. Therefore the theoretical tools offered, for example, by models such as CSCF, might be proved quite useful in this regard.

4 IMPLEMENTATION

The implementation of the code follows the same approach as [5]. Improvement in the serial performance of the code is possible by using the periodic nature of the solution. This is done simply by using the initial guess at time level n + 1 to be the solution at time level n + 1 − c, where c is the number of timesteps per cycle. The use of extrapolation as an initial guess for the next time level has been found when strong moving shocks are present to casue robustness problems.

This approach means that full solutions must be saved to disk at every iteration. This amounts to tens of Megabytes per iteration even for modest problem sizes. The sequential method of storing all the data in one file and having each processor read the whole file can have a large impact on run times as the number of processors increases. Table 1 shows the effect of reading in the data located on a single file server. In the single file case each processor reads in the whole file while in the multiple file case each processor just reads the information that its needs. Files can be written in either ASCII or binary format. In can be seen for the ASCII single file the read times remain constant until 8 nodes where it takes ≈ 10% longer. This is due either to the saturation of the network bandwidth or the accessing of data from the hard disk. For binary reads from a single file this saturation starts much earlier. It is always faster to read a binary file but the advantage is reduced from a ratio 8 : 1 for a single node to less than 3 : 1 for 8 nodes. If multiple files are used the ASCII mode shows some favourable parallel speedup where the binary file times remain constant. It should be noted however that it is still always faster to read the binary files even though they scale like 0(1).

Table 1. IO Performance with no caching

It is possible to remove the effects of the disk access time and data transfer rate in the file server by caching all the data into memory. This means all the times in table 2 are just limited by network saturation and/or congestion. It can be seen that all the conclusions still hold true with the only marked difference being that the multiple binary reads for more than one processor are approximately halved.

Table 2. IO Performance with file server caching

In table 3 the best solution is illustrated, which is to use any local disk which is available. Here the results are as expected with the multiple file reads scaling perfectly while the single reads remaining constant.

Table 3. IO Performance with local disk usage

In the above nothing was mentioned about writing the data to disk but it should be noted that writing multiple files is also much cheaper as you do not have to communicate the data back to a master node for writing. However, having a process only write data to the local disk means that if a process is migrated to another node either a large amount of local disk information must also be migrated, the local disks must all be mountable on any node or the last solution is used.

Abstract

We use a sequential method to allocate software test cases among partitions of a software to minimize the expected loss incurred by the Bayes estimator of the overall software reliability. The Bayesian approach allows us to take advantage of the previous information obtained from testing. We will show that the myopic sampling scheme has advantages over the optimal fixed in terms of the expected loss incurred when the overall reliability is estimated by its Bayes estimator. Theoretical results and numerical are provided for the comparison. This myopic scheme shows a great promise in software reliability estimation.

6.2 Kriging metamodels for optimization

Efficient global optimization (EGO) is a well-known sequential method that uses Kriging to balance local and global search; i.e., it balances exploitation and exploration. When EGO selects a new (standardized) combination x0, it estimates the maximum of the expected improvement (EI) comparing w(x0) and — in minimization — min iw(xi) with i=1,…,n. We saw below (41) that s2{ y^(x0)} increases as x0 lies farther away from xi. So, EI reaches its maximum if either y^ is much smaller than min w(xi) or s2{y^(x0)} is relatively large so y^ is relatively uncertain. We present only basic EGO for deterministic simulation; also see the classic EGO reference, Jones, Schonlau, and Welch (1998).

Note: There are many EGO variants for deterministic and random simulations, constrained optimization, multi-objective optimization including Pareto frontiers, the “admissible set” or “excursion set”, robust optimization, estimation of a quantile, and Bayesian approaches; see, Preuss et al. (2012) and the many references in Kleijnen (2015, p. 267–269) plus the file with “Corrections and additions” for Kleijnen (2015, p. 267) available on https://sites.google.com/site/kleijnenjackpc/home/publications.

We start with a pilot sample, typically selected through LHS. To the resulting simulation I/O data (X, w), we fit a Kriging metamodel y(x) Next we find fmin = min1 ≤ i ≤ nw(xi). This gives

(49)EI(x)=E[max(fmin−y(x),0)].

Jones et al. (1998) derives the following closed-form expression for the estimator of EI:

(50)EI^(x)=(fmin−y^(x))Φ(fmin−y^(x)s{y^(x0)})+s{ y^(x0)}ϕ(fmin−y^(x)s{y^(x0)})

where Φ and ϕ denote the cumulative distribution function (CDF) and the PDF of the standard Normal distribution. Using (50), we find x^opt, which denotes the estimate of x that maximizes EI^(x). (To find x^opt, we should apply a global optimizer; a local optimizer is undesirable because s{y^(xi)}=0 so EI(xi)=0. Alternatively, we use a set of candidate points selected through LHS.) Next we run the simulation with this x^ opt, and obtain w(x^opt). Then we fit a new Kriging model to the augmented I/O data (Kamiński, 2015) presents methods for avoiding re-estimation of the Kriging parameters). We update n and return to (50) — until we satisfy a stopping criterion; e.g., EI^(x^opt) is “close” to 0.

For the constrained nonlinear random optimization problem already formalized in (48) we may derive a variant of EGO; see the Note immediately preceding (49). Kleijnen et al. (2010), however, derives a heuristic called Kriging and integer mathematical programming (KrIMP) for solving (48) augmented with constraints on z. These constraints are necessary if z includes resources such as the number of employees with prespecified expertise and the number of trunk lines in a call-center simulation. Altogether, (48) is augmented with s constraints fg for z and the constraint that z must belong to the set of nonnegative integers N, so KrIMP tries to solve

(51)minzE(w(1)|z)E(w(l′)|z)≥cl′(l′=2,…,r)fg(z)≥cg(g=1,…,s) zj∈N(j=1,…,k).

KrIMP combines the following three methods: (i) sequentialized DOE to specify the next combination, like EGO does; (ii) Kriging to analyze the resulting I/O data, and obtain explicit functions for E(w(l)|z) (l=1,…,r), like EGO does; (iii) integer nonlinear programming (INLP) to estimate the optimal solution from these explicit Kriging models, unlike EGO. Experiments with KrIMP and OptQuest (OptQuest has already been mentioned in the beginning of Section 6) suggest that KrIMP requires fewer simulated combinations and gives better estimated optima.