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Probability Distributions of Univariate Data
Published in Nong Ye, Data Mining, 2013
The mode of a probability distribution for a variable x is located at the value of x that has the maximum probability density. When a probability density function has multiple local maxima, the probability distribution has multiple modes. A large probability density indicates a cluster of similar data points. Hence, the mode is related to the clustering of data points. A normal distribution and a skewed distribution are examples of unimodal distributions with only one mode, in contrast to multimodal distributions with multiple modes. A uniform distribution has no significant mode since data are evenly distributed and are not formed into clusters. The dip test (Hartigan and Hartigan, 1985) determines whether or not a probability distribution is unimodal. The mode test in the R statistical software (www.r-project.org) determines the significance of each potential mode in a probability distribution and gives the number of significant modes.
Statistical Inference I
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
If sample data are measured on the interval or ratio scale, then all three measures of centrality (mean, median, and mode) are defined, provided that the level of measurement precision does not preclude the determination of a mode. When data are symmetric and unimodal, the mode, median, and mean are approximately equal (the relative positions of the three measures in cases of asymmetric distributions is discussed in Section 1.4). Finally, if the data are qualitative (measured on the nominal or ordinal scales), using the mean or median is senseless, and the mode must be used. For nominal data, the mode is the category that contains the largest number of observations.
Descriptions and Quantifications of Univariate Samples: Numerical Summaries
Published in P. A. W. Lewis, E. J. Orav, Simulation Methodology for Statisticians, Operations Analysts, and Engineers, 2017
One of the most basic attributes of the distribution of a random variable X is its location or center. Often we will be considering symmetric, unimodal distributions, examples of which are the Normal, logistic, and Laplace distributions. (See Table 6.1.2 for details on these distributions.) A distribution FX(x) (or by convention a random variable X) is said to be symmetric about a point c if FX(c − x) = P(X ≤ c − x) = P(X > c + x) = 1 − FX(c + x).
Combined Emission Economic Load Dispatch with Renewable Energy Sources Employing Hybrid Statistical Multiswarm Particle Swarm Optimizer-Sine Cosine Algorithm
Published in Electric Power Components and Systems, 2023
Rinki Keswani, Harish Kumar Verma, Shailendra Kumar Sharma
Owing to the uncertainty in generated power of solar plant due to the intermittency in solar irradiance, factors for underestimation and overestimation of solar power are included in the model [30]. The output of solar power plant primarily depends on irradiations. The probability distribution of irradiations for the typical location follows a Bimodal probability distribution. The unimodal probability distribution function is modeled by Log-normal, Beta and Weibull Probability Distributed Functions (PDF). The Bimodal Weibull distribution function is used in Eqs.(12–23) as: and are the scale and shape factors, its range for is in between 0 and 1. The cumulative distribution function (CDF) for Weibull PDF is given by: The linear transformation of the solar irradiation random variable () to solar power random variable is given as: where g is a transformation function. It is given as, where
Gas emissions and particulate matter of non-road diesel engine fueled with F-T diesel with EGR
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2019
Wang Zhong, Jiahui Yang, Li Ruina, Liu Shuai
In the exhaust process of diesel engines, collisions, coagulation, and other processes occur between the particles, resulting in changes of particle size. Diesel engine exhaust particles can be divided into PM10 (DP< 10 μm), fine particles (DP< 2.5 μm), ultrafine particles (DP< 100 nm), and nanoparticles (DP< 50 nm) according to the particle size. It can be divided into nuclear particles (5 nm < DP < 50 nm), aggregation particles (50 nm < DP < 1,000 nm), and coarse-grinding particles (DP> 1 μm) according to the formation mechanism (Tian and Cai et al. 2018). Nuclear particle size is small, the main components of which are soluble organics and sulfates. The aggregated particles have a larger particle size, the main components of which are aggregated soot and its adsorbed material (Tan, Lou, et al. 2010). Non-road diesel engines use conventional piston pumps. Its injection pressure is small, atomization effect of which is poor, promoting the generation of large-size particles. Particle size distribution is usually a unimodal normal distribution (Liu, Ge, et al. 2009).
A data-driven approach for the optimisation of product specifications
Published in International Journal of Production Research, 2019
Lei Zhang, Xuening Chu, Hansi Chen, Bo Yan
The module-level customer satisfaction can be obtained from the distribution of the performance indicators constructed in Section 4.1(i.e. multi-core score, APCPD, ASCPD and TRO), and the product-level satisfaction can be calculated from the module-level satisfaction and their corresponding weights. The data of multi-core score, APCPD, ASCPD are all unimodal distributed and can be transformed to normal distribution. The customer satisfaction for the CPU, battery and storage then can be calculated by Equations (4)–(14). While the TRO is multi-peak distributed and the customer satisfaction can be calculated by Equations (15)–(18). Table 5 shows the customer satisfaction calculated by Equations (4)–(18) for the customer C1.