View source for Discriminant function

A statistic for the construction of a rule for distinguishing in problems of [[Discriminant analysis|discriminant analysis]] involving two distributions. The discrimination problem for two distributions may be stated as follows. Let an observed object with vector of measurements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332301.png" /> belong to one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332303.png" />, and let it be unknown to which one. The problem is to find a rule which would make it possible to assign the object to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332304.png" /> if the observed value of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332305.png" /> is known (the discrimination rule). The construction of such a rule is based on subdivision of the sampling space for the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332306.png" /> into domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332308.png" />, such that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d0332309.png" /> is found in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323010.png" />, it is reasonable (from the point of view of the chosen principle for an optimal solution) to assign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323012.png" />. If the discrimination rule is based on the subdivision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323016.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323017.png" />, the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323018.png" /> is called the discriminant function, while the domain in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033230/d03323019.png" /> is called the uncertainty zone. Linear discriminant functions, which are simple to handle, are particularly important. If the distributions are normal and have equal covariance matrices, the discriminant function will be linear if the optimality requirements of the discrimination rule are reasonable. The concept of a [[Discriminant informant|discriminant informant]] is introduced in the problem of discriminant analysis with multiple distributions when the [[Bayesian approach|Bayesian approach]] is adopted.