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Classification Techniques
Published in Harry G. Perros, An Introduction to IoT Analytics, 2021
The logistic distribution f(x) of a random variable X is given by the expressionfx=e−x−x0/ss1+e−x−x0/s2,where x0 is the mean of the distribution and s is a scale parameter. Its shape is similar to the normal distribution, except that its tails are fatter. The standard logistic distribution is obtained by setting x0 = 0 and s = 1, i.e.,fx=e−x1+e−x2.The cumulative distribution of the standard logistic distribution, referred to as the standard logistic function, is given by the expressionFx=11+e−xand shown in Figure 9.16. The logistic function dates back to the 19th century and it has applications in many different fields, including Machine Learning.
Estimating Common Scale Parameter of Two Logistic Populations: A Bayesian Study
Published in American Journal of Mathematical and Management Sciences, 2020
Nadiminti Nagamani, Manas Ranjan Tripathy, Somesh Kumar
The problem of estimating “common parameter” has not received much attention when the underlying statistical models are other than normal or exponential. This may be due to the complicated nature of the density function or the nonexistence of the closed form of the distributions of the sufficient statistics. However, in practice there are certain situations where the data sets can be satisfactorily modeled by other distributions, for example, by a logistic distribution. This motivates one to consider the problem of estimating the common scale parameter of two logistic populations when the location parameters are unknown and possibly unequal. Though the shape of the logistic distribution is similar to that of a normal distribution, it is more peaked in the center and has heavier tails than the normal distribution.
Investigation of ride comfort limits on urban asphalt concrete pavements
Published in International Journal of Pavement Engineering, 2018
Ufuk Kırbaş, Mustafa Karaşahin
Logistic regression analysis determines the relationship between the categorical-dependent variable and independent variables by estimating probabilities using a logistic function, which is the cumulative logistic distribution (Hosmer Jr et al.2013). There are two types of logistic regressions, which are binary logistic regression and multinomial logistic regression. Binary logistic regression is typically used when the dependent variable is dichotomous and the independent variables are either continuous or categorical variables (Harrell 2015). When the dependent variable is comprised of more than two cases, a multinomial logistic regression is employed (Menard 2002). The maximum likelihood estimation procedure is used to obtain the estimates of the regression coefficients by maximising the value of log-likelihood function through an iterative process with the aim of making the likelihood of observed data greater (Hassan et al.2015). In logistic regression analysis, the effect of descriptive variables on the dependent variables is determined as probabilities, which allows the risk factor probabilities to be determined. The said model is shown below:
Monitoring serially dependent categorical processes with ordinal information
Published in IISE Transactions, 2018
Jian Li, Jiakun Xu, Qiang Zhou
Notice that Equation (2) requires the specific form of the PDF f(z*) and the CDF F(z*) of the latent continuous variable Z*. This raises the question of how to specify them. In reality we cannot observe the numerical values of Z*, and we have no idea about f(z*) or F(z*). Instead, we have only the probabilities of factor Z in each level πk (k = 1, …, h). Here we borrow the idea in Li et al. (2014b) that we might as well suppose that Z* is subject to a logistic distribution, which has PDF and CDF expressed as Here μ and σ are the location parameter and the scale parameter, respectively, and Z* has mean μ and variance π2σ2/3. Selecting a standard logistic distribution for Z* creates the following advantages. First, the logistic distribution has a similar bell shape to the normal distribution but with heavier tails. Even if the true distribution of Z* is not logistic, simulation results later will show its robustness to other types of distributions. Second, a logistic distribution leads to a simple form of f(z*) and F(z*), which considerably facilitates the computation for online monitoring.