There is some asymmetry between the two players in terms of the relative strengths of the algorithms. For player 1, SO-ISMCTS and MO-ISMCTS are on a par while SO-ISMCTS + POM underperforms. For player 2, SO-ISMCTS is outper- formed by SO-ISMCTS + POM which is in turn outperformed by MO-ISMCTS. The three algorithms differ mainly in the assumptions they make about future play. SO-ISMCTS as- sumes that all actions are fully observable, which is both optimistic (I can respond optimally to my opponent’s actions) and pessimistic (my opponent can respond optimally to my actions). SO-ISMCTS hence suffers from strategy fusion, since it is assumed the agent can act differently depending on infor- mation it cannot observe. In a phantom game, SO-ISMCTS + POM optimistically assumes that the opponent plays ran- domly. MO-ISMCTS’s opponent model is more realistic: each opponent action has its own statistics in the opponent tree and so the decision process is properly modeled, but whichever action is selected leads to the same node in the player’s own tree thus preventing the player from tailoring its response to the selected action. This addresses the strategy fusion problem which affects SO-ISMCTS.
Selection - In the selection process, the MCTS algorithm traverses the current tree using a tree policy. A tree pol- icy uses an evaluation function that prioritize nodes with the greatest estimated value. Once a node is reached in the traversal that has children (or moves) left to be added, then MCTS transitions into the expansion step. In figure 2, starting from the root node, the tree policy must make a decision between the 0/1 node and the 2/3 node. Since 2/3 is greater than 0/1, the tree policy will choose the 2/3 node in its traversal. Once at the 2/3 node, the tree policy will then choose the 1/1 node because it is greater than 0/1. This is the first node with children yet to be added, so now MCTS will transition into the expansion step.
However, this problem is intrinsically difficult be- cause it is hard to encode what to say into a sentence while ensuring its syntactic correctness. We propose to use MonteCarlotree search (MCTS) (Kocsis and Szepesvari, 2006; Browne et al., 2012), a stochastic search algorithm for decision processes, to find an optimal solution in the decision space. We build a search tree of possible syntactic trees to generate a sentence, by selecting proper rules through numer- ous random simulations of possible yields.
Abstract— We are addressing the course timetabling problem in this work. In a university, students can select their favorite courses each semester. Thus, the general requirement is to allow them to attend lectures without clashing with other lectures. A feasible solution is a solution where this and other conditions are satisfied. Constructing reasonable solutions for course timetabling problem is a hard task. Most of the existing methods failed to generate reasonable solutions for all cases. This is since the problem is heavily constrained and an e ﬀ ective method is required to explore and exploit the search space. We utilize MonteCarloTree Search (MCTS) in finding feasible solutions for the first time. In MCTS, we build a tree incrementally in an asymmetric manner by sampling the decision space. It is traversed in the best-first manner. We propose several enhancements to MCTS like simulation and tree pruning based on a heuristic. The performance of MCTS is compared with the methods based on graph coloring heuristics and Tabu search. We test the solution methodologies on the three most studied publicly available datasets. Overall, MCTS performs better than the method based on graph coloring heuristic; however, it is inferior compared to the Tabu based method. Experimental results are discussed.
We assume the origin of the asymmetry is statistical. The central limit theory states that the distribution of a sum (or equally, an average) becomes Gaussian- like with increasing number of independent variables (summants) per sum. This is demonstrated in Figure 5.3 as a simplified mathematical exercise. Ten million numbers are drawn from the single hit energy distribution (a) in a toy MonteCarlo experiment. This distribution does not represent actual calorimeter data and is for illustrative purposes only. The numbers are then divided into groups of fixed size and the average of each group is calculated. A group is analogous to an event with the group size given by the number of hits from data. The distribution of the average for a group size of 8 numbers is shown in (b) where the tail towards large energies is clearly visible. On average, 1 GeV electrons have 17 hits per event (c) whereas 6 GeV electrons have 38 hits per event (d). With increasing number of hits per event, the energy sum distribution becomes more Gaussian as stated by the central limit theory.
Having established several variants of the stochastic simulation algorithm for chemi- cally reacting systems, we now turn our attention to how kinetic MonteCarlo methods may be applied to epitaxial systems. In order to describe the evolution of a crystalline surface, we define the position of atoms according to a lattice structure and utilize a height array to indicate the surface configuration at any given state. Furthermore, a solid-on-solid approach is assumed in which the crystalline film grows such that one atom can only be accommodated by another atom, thereby prohibiting structural defects like overhangs and vacancies. The surface evolves from state-to-state through the movement of a single atom with the transition dependent on the local crystal configuration.
update. The corresponding model is called measurement model, denoted as | . It describes the formation process by which sensor measurements are generated in the physical world . Many different sensors are used in robots, such as tactile sensors, range sensors, and cameras. The specifics of the model depend on the sensors used in robots. The details of all the sensors equipped on the robot used in our experiments will be given in chapter 4. Here we just discuss the sensors used in measurement model: bumper sensors, wall sensor, and infrared sensor. The first two types of sensors are uses to detect the landmarks such as wall or any other obstacles in the environment. If any of these sensors return a positive value then we will assign high probability (high weight) to particles who are around walls and obstacles, and low probability to the rest of particles. The weighted particles are then used in the resampling phase. The infrared sensor is used to detect other robots. We put a virtual wall (a standard Infrared remote transmitter) on the top of each robot. If one robot’s infrared sensor (virtual wall sensor) returns a positive value, it will consider it has detected another robot. Then according to our proposed approach, it will come to a decision of whether to allow an information exchange happen.
In 1960s, United Kingdom began to use CBA with the application of the technique to the London – Birmingham highway. In 1967 a UK Government White Paper gave formal recognition to the existence of cost-benefit analysis and assigned it a limited role for nationalized industries (UK Government, 1967). In the late 1960s CBA was extended to less developed counties with the publication or a Manual of Industrial Project Analysis (Little and Mirrlees, 1969). The Manual was prepared for the Organization for Economic Co-operation and Development (OECD). In 1975, the World Bank’s guidelines came which were heavily relied on the earlier work of Little and Mirrlees (Squire and Tak, 1975). From then on the CBA became a useful tool for executive decision making used in many areas and CBA also gained additional impetus with the environmental revolution.
The most commonly considered model output is the ICER, typically the cost per quality-adjusted life year (QALY) gained. If the decision maker requires only an estimate of the mean ICER, the number of simulations should be just sufficient for this statistic to converge. Additional simulations may be required beyond the number required for the ICER statistic to converge if outputs other than the mean ICER are of interest to the decision maker [11,12]. For example, the decision maker may be interested in a confidence interval for the ICER estimate, the - effectiveness falls below a certain threshold, or the expected value of perfect information (EVPI) of an intervention. Alternatively it may be important that the estimated distributions of the model outputs are accurate between PSAs, and not merely the mean ICER alone. This may require additional simulations to ensure convergence towards the edges of each distribution which are inherently less stable than the mean.
parametric configuration given by the LFC for k = 8 giving 2 populations on each of the four boundaries. Note that this configuration, denoted as (2,2,2,2), is the Least favorable Configuration under which there are an equal number of populations on all the four boundaries as shown in (2.2). Note that the average sample size n ¯ is fairly close to the unknown optimal sample size n ∗ for all the cases which we considered. Also, note that from Theorem 3, that the second-order expansion provides that asymptotically the difference between the value of n ¯ and n ∗ should be β. From Table (2.3) one obtains that asymptotically this difference should be 0.3762. That is, the purely sequential procedure (2.3.12) over-samples by a third of a sample asymptotically. The simulated values in Table (2.4) confirm this asymptotic difference between the n ¯ and n ∗ . Also, note that the average value of the probability of correct decision P ¯ matches the target value of 0.95 in all the cases considered. The findings in Table (2.4), confirm the theoretical results derived in Theorem 2 and 3 for the purely sequential procedure.
The NPV criterion is usually calculated using point estimations for the input parameters, thus providing a single value outcome. However, this is just a possibility among many others, due to the range of values that the input parameters may take. A more complete approach, that is suggested hereafter, is to define the uncertainty factors. That is, the variables (specific costs or revenues) which take part in the NPV calculation and may receive more than one value. Having done that, the decision maker may feed the probabilistic NPV model with data according to the different available scenarios (alternatives) and come up with a probability curve for the NPV of each scenario. The tangible result is that the decision maker may now compare n curves (where n is the number of available scenarios) indicating the range of possible outcomes rather than comparing n deterministic values.
We think that jackknife standard errors should be routinely used in reporting MonteCarlo results. They are much easier to obtain than those from the standard delta method approach, especially when using the MonteCarlo descriptive ratio statistics discussed in Section 3. The F -statistic studied in Section 3, based on individual estimators and their jackknife variance estimate in each sample, is a potentially useful method for testing equality of parameters from k populations.
It is known that the relationship between the spherical wave coeﬃcients of the incident and scattered waves can be linearly represented by the T -matrix . In [14, 15], the T -matrix of each tree element, such as leaf, branch, and trunk, was determined by the hybrid approach. First, the scattered ﬁelds of a single scatterer induced by the multiple incident wave directions are determined using the conventional electromagnetic solvers. In [14, 15], the FEKO software , which relies on a MOM solver, was used. Second, the spherical wave coeﬃcients of such a single scatterer are determined using point matching approach  in which the coeﬃcients are computed to match the obtained scattered ﬁelds at several matching points. Using the known spherical wave coeﬃcients of the incident wave, the T -matrix of a single scatterer can be ﬁnally determined. The authors in [14, 15] showed that the total electromagnetic scattered ﬁeld in the far-ﬁeld region of the vegetation structure such as tree can be approximated by the superposition of the scattered ﬁeld of the ﬁrst-order scattering of each individual tree element determined by using its T -matrix.
This is particularly challenging for Bayesian statistics since the posterior dis- tribution can become too computationally expensive to evaluate or simulate from exactly, and has lead researchers to develop a range of new approximate algorithms; see [Angelino et al., 2016] for an overview. Bardenet et al.  also offer a discus- sion of solutions in the MonteCarlo literature, whilst Hoffman et al.  discuss this issue in the context of variational inference. Another direction of research in the tall data setting has been to consider methods to summarise large datasets with a subset of representative weighted samples. This is called a coreset [Bachem et al., 2017; Huggins et al., 2016; Campbell and Broderick, 2017] and can be used instead of the entire dataset to reduce the computational cost associated with evaluating likelihoods. However, these methodologies are still in their infancy and further de- velopments are required.
optimal error estimates for the resulting semi-discrete scheme which then provide corresponding error estimates for expectation values and Monte-Carlo approxima- tions. Application of efficient solution techniques, such as adaptivity , multigrid methods , and Multilevel Monte-Carlo techniques [3, 9, 10] is very promising but beyond the scope of this paper. In our numerical experiments we investigate a corresponding fully discrete scheme based on an implicit Euler method and observe optimal convergence rates.
The SDM elements extracted from the data will not represent the true SDM elements. The less than perfect angular resolution, the finite selection efficiency and the acceptance of the OPAL detector are just some of the factors th a t will have an effect on the SDM elements. The data sample is expected to contain some background events and these will also cause deviation from the true SDM elements. The problem is increased further when you include such effects as ISR and the finite W width. Figure 6.1 indicates the extent of the problem detector effects cause. It shows the SDM elements extracted from a sample of fully detector simulated MonteCarlo events containing all possible signal and background processes. They have been passed through the same selection and reconstruction as is used for the data. Overlaid is the theoretical prediction for the Standard Model calculated from the purely analytical expression of the process e+e" -> W+W~
known as single nucleotide polymorphisms (SNPs). The data (which is freely available from http://pubmlst.org/saureus/) used in our examples consists of seven “multi-locus sequence type” (MLST) genes of 25 Staphylococcus aureus sequences, which have been chosen to provide a sample representing the worldwide diversity of this species (Everitt et al. 2014). We make the assumption that the population has had a constant size over time, that it evolves clonally and that SNPs are the result of mutation. Our task is to infer the clonal ancestry of the individuals in the study, i.e., the tree describing how the individuals in the sample evolved from their common ancestors, and [additional to Dinh et al. (2018)] the rate of mutation in the population. We describe a TSMC algorithm for addressing this problem in Sect. 4.2, before presenting results in Sect. 4.3. In the remainder of this section, we intro- duce a little notation.
Having estimated the Weibull probability distribution for measured degradation rates, MonteCarlo simulation was used to predict a number of service lifetimes in different sized AC pipes subjected to the combined internal pressure and external loading. Using the variables detailed shown in Table 3 and 4, The MonteCarlo simulation sampled failure times by repeatedly generating random numbers for degradation rate, e.g. 10,000, and using these to predict the time to failure for a set of trials. Pipe lifetime was determined iteratively by calculating the limit state, Eq. (6), for each year of age until failure occurs. Pipe failure frequency (expressed as failures per year) was determined by dividing the number of pipes that fail in any years by the total number of simulated pipes. In this study, the total number of random simulated data is set to be 10,000.