Difference between revisions of "Discriminant informant"
(TeX) |
m (dots) |
||
| Line 1: | Line 1: | ||
{{TEX|done}} | {{TEX|done}} | ||
| − | A term in [[Discriminant analysis|discriminant analysis]] denoting a variable used to establish a rule for assigning an object with measurements $x=(x_1,\ | + | A term in [[Discriminant analysis|discriminant analysis]] denoting a variable used to establish a rule for assigning an object with measurements $x=(x_1,\dotsc,x_p)$, drawn from a mixture of $k$ sets with distribution densities $p_1(x),\dotsc,p_k(x)$ and a priori probabilities $q_1,\dotsc,q_k$, to one of these sets. The $i$-th discriminant informant of the object with measurement $x$ is defined as |
| − | $$S_i=-[q_1p_1(x)r_1i+\ | + | $$S_i=-[q_1p_1(x)r_1i+\dotsb+q_kp_k(x)r_{ki}],\quad i=1,\dotsc,k,$$ |
where $r_{ij}$ is the loss due to assigning an element from the distribution $i$ to the distribution $j$. The rule for assigning an object to the distribution with the largest discriminant informant has minimum mathematical expectation of the loss. In particular, if all $k$ distributions are normal and have identical covariance matrices, all discriminant informants are linear. Then, if $k=2$, the difference $S_1-S_2$ is Fisher's linear [[Discriminant function|discriminant function]]. | where $r_{ij}$ is the loss due to assigning an element from the distribution $i$ to the distribution $j$. The rule for assigning an object to the distribution with the largest discriminant informant has minimum mathematical expectation of the loss. In particular, if all $k$ distributions are normal and have identical covariance matrices, all discriminant informants are linear. Then, if $k=2$, the difference $S_1-S_2$ is Fisher's linear [[Discriminant function|discriminant function]]. | ||
Latest revision as of 12:24, 14 February 2020
A term in discriminant analysis denoting a variable used to establish a rule for assigning an object with measurements $x=(x_1,\dotsc,x_p)$, drawn from a mixture of $k$ sets with distribution densities $p_1(x),\dotsc,p_k(x)$ and a priori probabilities $q_1,\dotsc,q_k$, to one of these sets. The $i$-th discriminant informant of the object with measurement $x$ is defined as
$$S_i=-[q_1p_1(x)r_1i+\dotsb+q_kp_k(x)r_{ki}],\quad i=1,\dotsc,k,$$
where $r_{ij}$ is the loss due to assigning an element from the distribution $i$ to the distribution $j$. The rule for assigning an object to the distribution with the largest discriminant informant has minimum mathematical expectation of the loss. In particular, if all $k$ distributions are normal and have identical covariance matrices, all discriminant informants are linear. Then, if $k=2$, the difference $S_1-S_2$ is Fisher's linear discriminant function.
References
| [1] | C.R. Rao, "Linear statistical inference and its applications" , Wiley (1965) |
Discriminant informant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discriminant_informant&oldid=44583