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Statistics, probability and probability distributions
Published in Stephen A. Thompson, Hydrology for Water Management, 2017
A theoretical probability distribution must be fit to a sample of data. There are three ways to fit a theoretical distribution to a sample. The most fundamental method is to estimate the parameters of the distribution using the sample statistics. Two methods for doing this are the method of moments and the method of maximum likelihood. The method of moments is generally easier but the resulting parameter estimates are less accurate, especially for small samples. The moments for the normal distribution are the mean μ and variance σ2 (see Table 3.4). The maximum likelihood parameter estimation is a more accurate, but a more complex method. In practice simplicity often triumphs over accuracy and the method of moments is more often used. With the parameters in hand probability is estimated using the either PDF or CDF. The PDF was used in Example 3.2. In Chapter 4 the technique is demonstrated using the CDF for both the normal and gamma distributions.
Queuing Theory
Published in Paul J. Fortier, George R. Desrochers, Modeling and Analysis of Local Area Networks, 1990
Paul J. Fortier, George R. Desrochers
The method of moments is useful when we think we know the distribution of the sample but do not know what the distribution parameters are. Suppose the distribution whose parameters we wish to estimate has n parameters. In this method, we first find the first n distribution moments as described earlier in Chapter 4. Next, we calculate the first n sample moments and equate the results to the moments found earlier. From this we get n equations in n unknowns, which can then be solved simultaneously for the desired parameters. We derive the kth sample moment for a sample size of m samples as () Mk=∑i=1mXikm
Flood Forecasting
Published in Saeid Eslamian, Faezeh Eslamian, Flood Handbook, 2022
Priyanka Sharma, Pravin Patil, Saeid Eslamian
The method of maximum likelihood is the most theoretically correct method of fitting probability distributions to data in the sense that it produces the most efficient parameter estimates – those which estimate the population parameters with the least average error. But, for some probability distributions, there is no analytical solution for all the parameters in terms of sample statistics and the log-likelihood function must then be numerically maximized, which may be quite difficult. In general, the method of moments is easier to apply than the method of maximum likelihood and is more suitable for practical hydrologic analysis.
Improved assessment of maximum streamflow for risk management of hydraulic infrastructures. A case study
Published in International Journal of River Basin Management, 2023
Ana Margarida Bento, Andreia Gomes, João Pedro Pêgo, Teresa Viseu, Lúcia Couto
Eight different return periods were considered: T = 2, 5, 10, 20, 50, 100, 500 and 1000 years. For estimating the parameters of each probability distribution, the method of moments (MoM), the L-moments method and the method of maximum likelihood (MLE) are the most advisable by hydrologists (Kite, 2019). While the method of moments (MoM) is the easiest method of application, it is also less robust when compared with the estimators of the maximum likelihood (MLE) method, particularly for distributions of three or more parameters. However, for small samples, frequent in hydrology, MoM estimators can have comparable or even higher attributes than other estimators (Naghettini & Pinto, 2007). Thus, the method of moments (MoM) was selected as preferable to estimate the parameters of the aforementioned distributions. The adoption of the method of moments (MoM) in the design floods estimation was also supported by the limited 10 years of real streamflow data. It should be noted that the remaining values for completing the minimum 30 hydrological years (required for a frequency analysis) were estimated by the herein developed fill-in missing data technique.
Moment-based travel time reliability assessment with Lasserre’s relaxation
Published in Transportmetrica B: Transport Dynamics, 2019
Xiangfeng Ji, Xuegang (Jeff) Ban, Jian Zhang, Bin Ran
In practice, the support information (i.e. the upper bound and the lower bound), and the moments of the total link travel time of each link can be estimated from the gathered data in ITS. If sufficient data (e.g. loop detector data) can be obtained, the proper value of the parameters can be better estimated. One potential way to do the estimation is that we can select some archived data in a relatively long time (e.g. a year or longer) and record the maximum travel times (i.e. the upper bounds) of each link in each day. The 95% percentile of all these upper bounds can be used as the upper bound of the link travel time. Using the same data, we can get the mean travel time on the link via averaging all the link travel times. Therefore, the value of is the quotient of estimated upper bound and mean of the link travel time. The value of can be obtained in a similar way. The moments can be estimated with certain statistical methods (e.g. the method of moments; Casella and Berger 2002). However, we cannot determine the proper values for these parameters without the information. Therefore, in the following, we set different values for the parameters to demonstrate the performance of our method. The similar RUBI analysis shown in Table 4 is omitted hereinafter and the readers can make similar comparisons if necessary.
A new method for solving population balance equations using a radial basis function network
Published in Aerosol Science and Technology, 2020
Kaiyuan Wang, Suyuan Yu, Wei Peng
The discretization methods are a collection of several sub methods including the sectional method (Gelbard, Tambour and Seinfeld 1980; Kumar et al. 2006; Park and Rogak 2004), the finite volume method (Filbet and Laurencot 2004; Kumar, Saha, and Tsotsas 2015; Qamar and Warnecke 2007), and the finite element method (Ganesan 2012; Rigopoulos and Jones 2003). These methods discretize the continuous number density function into a finite number of sections, pivots, finite elements, or finite volumes. Then the PBE is converted to a system of ordinary differential equations (ODEs) that describe the changes of particle quantities (e.g., number, volume, or surface area) in each section. The discretization methods can be sufficiently accurate in predicting the time evolution of the number density function with relatively small sections (Landgrebe and Pratsinis 1990). The method of moments is a highly efficient method as it resolves a finite set of moments rather than the detailed number density function. This method has been widely used in practical problems due to its simplicity and low computational cost (Yu and Lin 2010). The Monte Carlo method differs from previous deterministic methods because it solves the PBE using a probabilistic interpretation. This method can generate accurate results even considering multiple physicochemical processes, which is often used to validate other models (Salenbauch et al. 2019). However, the application of the Monte Carlo method is limited due to the high computational cost. The spectral methods receive less attention as compared with the former three classes of methods. The idea is to write the solution as a sum of basis functions with time-dependent coefficients and then to convert the PBE into a system of ODEs for the coefficients based on collocation. The representative orthogonal collocation method has been successfully used to solve univariate and multivariate particle coagulation and growth problems (Arias-Zugasti 2006, 2012). This method can provide stable and accurate results for the local properties (e.g., values or derivatives) and global properties (e.g., moments) of the number density function.