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Monte Carlo Simulation
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Kevin Cronin, James P. Gleeson
The integration domain may be transformed to (0,1) by a change of variables, e.g., σ = 1/tan(πz). Figure 16.8 shows the fractional error of a Monte Carlo calculation of the mean of y, (circles) compared with a calculation using the quasi-random Halton sequence of base 2 (squares). Note the parameters used are D = 1, t = 1. The Monte Carlo error is averaged over 100 realizations , each using n random evaluation points; the fractional error decays as n−1/2 in accordance with Equation 16.46. The Halton sequence is a deterministic sequence in which successive points “fill in” gaps left by previous points; it clearly converges more quickly to the exact value, at a rate on the order of n−1.
Performance and Feasibility Model Generation Using Learning-Based Approach
Published in Soumya Pandit, Chittaranjan Mandal, Amit Patra, Nano-Scale CMOS Analog Circuits, 2018
Soumya Pandit, Chittaranjan Mandal, Amit Patra
Once the data range is fixed, samples need to be extracted within this range. There are two approaches for the extraction of sample data: static sampling and dynamic sampling [95]. The static sampling process remains ignorant of how a sample will be used and applies some fixed criterion for the sampling process. Once the sampling is completed, the generated dataset corresponding to extracted samples is used to construct the ANN model. On the other hand, in the dynamic sampling process, the sampling process is controlled by a so-called learning machine, or learner. A static sampler does not take into account the shape of the function under scrutiny, whereas a dynamic sampler strives to reduce modeling error through strategic placement of sample points in the design space. The commonly used static sampling methods are uniform grid distribution, pseudo-random and quasi-random sampling technique, nonuniform grid distribution, design of experiments (DOE) technique [19]. In the uniform distribution, samples for each input design variable αi are collected at equal intervals. Suppose the number of grids along the input dimension αi is ni. The total number of a α¯ – ρ̅ samples is given by Πi=1nni. Pseudo-random sampling is typically produced by pseudo-random number generators which are available in all programming languages. The pseudorandom number generator technique requires a seed, a number on which to base all future generated numbers. Once the seed is chosen, the procedure for the generation of numbers becomes deterministic. However, with change in the seed value, the sequence becomes altogether different. The distribution of numbers produced is usually uniform over the interval [0,1], although more specialized generators are available for producing Gaussian, or normal distributions. The sample points obtained through the pseudo-random technique are often found to be clustered in localized areas within the sample space and a small number of samples is distributed near the boundaries. The quasi-random sample generation technique, on the other hand, produces samples that are more uniformly distributed in the sample space; however, maintaining the random nature. However, the quasi-random generators lack the concept of a seed and will always produce the same sequence of numbers. An example of the quasi-random number generation technique is the Halton sequence generation technique [105].
How salutogenic workplace characteristics influence psychological and cognitive responses in a virtual environment
Published in Ergonomics, 2023
Lisanne Bergefurt, Rianne Appel-Meulenbroek, Theo Arentze
A mixed-multinomial logit model (MMNL) was run to determine the probability that a particular alternative is chosen by an individual. The utility of the null-alternative ‘no preference’ was represented in a constant. A random component was estimated for each attribute, to capture the unobserved heterogeneity in the model. To estimate the parameters of the model, 150 Halton draws were used. The use of Halton sequences increases the accuracy of the model and reduces the computation time. The influence of personal- and physical workplace characteristics on respondents’ preferences were also analysed, by stepwise including control variables that were also questioned in the survey (e.g. age, gender). The control variables were entered into the model as non-random parameters. The package ‘GMNL’ in R was used to run the MMNL.
An improved particle swarm optimization based on the reinforcement of the population initialization phase by scrambled Halton sequence
Published in Cogent Engineering, 2020
Pouriya Amini Digehsara, Saeed Nezamivand Chegini, Ahmad Bagheri, Masoumeh Pourabd Roknsaraei
There are various ordinary Monte Carlo methods that employ random numbers. Quasi-Monte Carlo is one of them produced highly uniform quasi-random numbers in the structure of Monte Carlo pseudorandom numbers. These methods are widely applied in many subjects especially in multidimensional domains (Niederreiter, 1992; Spanier & Maize, 1994). Halton sequence is one of the renowned low-discrepancy sequences and used in many Quasi-Monte Carlo applications. So, the Halton sequence could employ random numbers that have a mathematical structure and this strengthens the exploration phase and covers the search space by a logical method. Ganesha et al. (2016) applied optimal Halton sequences to only for 4 benchmark functions with PSO to understand and obtained some results and data to make a comparison between Halton sequences family. They did not apply this method to any engineering problems.
Efficient g-function approximation with artificial neural networks for a varying number of boreholes on a regular or irregular layout
Published in Science and Technology for the Built Environment, 2019
Bernard Dusseault, Philippe Pasquier
The general principle behind the construction of Halton sequences is to start with as many coprime numbers as there are dimensions in a dataset (see example in Table 2). Each coprime number will serve as a base in its own dimension and will be used as a multiple of the denominators to assemble a series of fractions ranging from 0 to 1. Once all dimensions have their series of fractions based on their own coprime numbers, the coordinates of the sampling points are created by combining fractions from all of these series to pinpoint to specific locations. These newly constructed coordinates, all ranging from 0 to 1, can then be modified by simple multiplications to fill the available real multidimensional space.