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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
Any probability distribution can be characterized by its parameters, which in turn can be estimated from the corresponding sample statistics. Generally, a probability distribution function has three parameters that can be geometrically interpreted as defining the location, scale, and shape of the distribution. A location parameter represents the position of the distribution on the x-axis by specifying an abscissa such as the minimum value or the average of the distribution. Changing the location parameter shifts the distribution left or right along the x-axis. The scale parameter represents the width or dispersion of the distribution (such as the standard deviation in the normal distribution). Changing this parameter compresses or expands the distribution without changing its basic shape. The shape parameter represents the shape of distribution usually characterized by the skewness of the distribution. Note that not all distributions have a shape parameter and some have more than one (such as the beta distribution).
Statistical methods in hydrology
Published in Amithirigala Widhanelage Jayawardena, Fluid Mechanics, Hydraulics, Hydrology and Water Resources for Civil Engineers, 2021
Amithirigala Widhanelage Jayawardena
The location parameter gives the location relative to some specific point. For some distributions, the location parameter is given by the mean value. Two probability distribution functions can have the same location parameter but different scale parameters. The shape parameter determines the geometrical configuration. Some distributions have more than one shape parameter, while some others have none.
Influence and robustness
Published in Raymond J. Carroll, David Ruppert, Transfor mation and Weighting in Regression, 2017
Raymond J. Carroll, David Ruppert
Extreme outliers are often gross errors. Hampel et al. (1986) discuss the need for robust estimators that completely reject extreme outliers instead of merely bounding their influence. For location parameters extreme outliers can be automatically rejected by using a redescending ‘psi function’.
An Enhanced Hybrid Rhino Herd–PSO Optimizer for Optimal Technical and Economic Operation of Power Systems Considering Environmental Concerns
Published in Electric Power Components and Systems, 2023
Nawal Taleb, Bachir Bentouati, Saliha Chettih, Harrouz Abdelkader, Korhan Kayisli
In Thirunavukkarasu et al. and Kang et al. [32, 35], a hybrid power system is a clean and effective energy solution whose output power is dependent on the flow rate (Q/h) and pressure head (H/h). The rate of water flow is determined by the Gumbel probability distribution. It is believed that the Gumbel probability density function (PDF) for water flow rate with scale parameter γ and location parameter λ can be calculated analytically as follows: The frequency distribution of water flow rate and the fitting of the Gumbel and Weibull distributions for wind power units are explained in Figures 1 and 2, respectively. These results were obtained by simulating 8000 Monte Carlo scenarios, with all parameters assigned realistic values, many of which are considered to be practically excellent. In Qin et al. [47], the hydropower output based on water flow rate and pressure head is described as follows:
Deep learning-based underground object detection for urban road pavement
Published in International Journal of Pavement Engineering, 2020
Namgyu Kim, Kideok Kim, Yun-Kyu An, Hyun-Jong Lee, Jong-Jae Lee
Figures 3(a) and 4(a) show examples of typical A-scan waveforms of GPR data. The maximum absolute amplitudes of each A-scan waveform are plotted in Figures 3(b) and 4(b). Figures 3(c–d) and 4(c–d) are the histograms of the maximum absolute amplitudes of GPR data. Here, totally 588,880 and 366,200 A-scan waveforms are used to plot the histograms, respectively. The histograms are then fitted with Gamma and Gumbel distributions. Gumbel distribution, also known as type I extreme value distribution, is generally used to model extreme cases such as the distribution of the maximum or the minimum of a value of samples. The probability density function of Gumbel distribution can be defined aswhere μ is the location parameter, and σ (σ > 0) is the scale parameter. From the fitted distributions, it has been found that Gumbel distribution can better reflect the distribution of maximum absolute amplitudes of GPR data than Gamma distribution. Table 1 shows the obtained location and scale parameters of Gumbel distribution for all field surveying regions. It has been found that the location parameters are almost 3 times of scale parameters regardless of the scanning distance.
Higher-order normal approximation approach for highly reliable system assessment
Published in IISE Transactions, 2020
Zhaohui Li, Dan Yu, Jian Liu, Qingpei Hu
The proposed expansion on is applied to the log-location-scale family, which is the most widely used lifetime model (Hong and Meeker, 2014). A location-scale family is a family of distributions that are parameterized by a location parameter and a non-negative scale parameter. For example, the log-normal distribution and Weibull distribution both belong to the log-location-scale family. Suppose are independently and identically distributed (i.i.d.) with a CDF, where are its location parameter and scale parameter, respectively. Without loss of generality, suppose the first- and second-order moments of are zero and one. It is easy to derive that The moment estimation is immediately followed by which immediately gives the MoM estimation of reliability: where are the i.i.d sample with CDF are usually called pivots because they are a function of and are distributed free of parameters.