Sigmoid functionA sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as an alias for the logistic function.
Weighted arithmetic meanThe weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean.
Bonferroni correctionIn statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem. The method is named for its use of the Bonferroni inequalities. An extension of the method to confidence intervals was proposed by Olive Jean Dunn. Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple hypotheses are tested, the probability of observing a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.
Wilks' theoremIn statistics Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals for maximum-likelihood estimates or as a test statistic for performing the likelihood-ratio test. Statistical tests (such as hypothesis testing) generally require knowledge of the probability distribution of the test statistic. This is often a problem for likelihood ratios, where the probability distribution can be very difficult to determine.
Normal probability plotThe normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw data, residuals from model fits, and estimated parameters. In a normal probability plot (also called a "normal plot"), the sorted data are plotted vs. values selected to make the resulting image look close to a straight line if the data are approximately normally distributed.
Checking whether a coin is fairIn statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory.
Linear separabilityIn Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are linearly separable if there exists at least one line in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if the line is replaced by a hyperplane.
Additive smoothingIn statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth categorical data. Given a set of observation counts from a -dimensional multinomial distribution with trials, a "smoothed" version of the counts gives the estimator: where the smoothed count and the "pseudocount" α > 0 is a smoothing parameter. α = 0 corresponds to no smoothing. (This parameter is explained in below.
