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The false positive rate (false alarm rate) is = + [1]. where is the number of false positives, is the number of true negatives and = + is the total number of ground truth negatives.
The probability of detection (POD), a function of the NDI method, accessibility, and crack size, can be modeled by the equation below. = {() / ()} In this equation, a 0 is defined as the crack size below which detection is impossible.
The cumulants satisfy a recursion formula + = (). The geometric distributions , (number of failures before one success with probability p of success on each trial). The cumulant generating function is K ( t ) = log( p / (1 + ( p − 1)e t )) .
Given that the probability distributions for both true positive and false positive are known, the ROC curve is obtained as the cumulative distribution function (CDF, area under the probability distribution from to the discrimination threshold) of the detection probability in the y-axis versus the CDF of the false positive probability on the x-axis.
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
If time can take on any positive value, then the cumulative distribution function F(t) is the integral of the probability density function f(t). For the air-conditioning example, the graph of the CDF below illustrates that the probability that the time to failure is less than or equal to 100 hours is 0.81 , as estimated using the exponential ...
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is [ 2 ] [ 3 ] f ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2 ...
Cumulative frequency distribution, adapted cumulative probability distribution, and confidence intervals. Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called frequency of non-exceedance.