预测使用说criminant Analysis Models

预测uses three quantities to classify observations:posterior probability,prior probability, and成本。

预测classifies so as to minimize the expected classification cost:

$\hat{y} = \underset{y = 1, 。.., K}{\arg \min} \sum_{k = 1}^{K} \hat{P} (k | x) C (y | k),$

where

$\hat{y}$ 是个预测ed classification.
K是个number of classes.
$\hat{P} (k | x)$ 是个posterior probability of classkfor observationx。
$C (y | k)$ 是个成本的classifying an observation asywhen its true class isk。

The space ofXvalues divides into regions where a classificationYis a particular value. The regions are separated by straight lines for linear discriminant analysis, and by conic sections (ellipses, hyperbolas, or parabolas) for quadratic discriminant analysis. For a visualization of these regions, seeCreate and Visualize Discriminant Analysis Classifier。

Posterior Probability

The posterior probability that a pointxbelongs to classk是个product of theprior probability和the multivariate normal density. The density function of the multivariate normal with 1-by-d意思是μ._k和d-by-dcovariance Σ_kat a 1-by-d观点xis

$P (x | k) = \frac{1}{{({(2 π.)}^{d} | Σ_{k} |)}^{1 / 2}} \exp (- \frac{1}{2} (x - {μ.}_{k}) Σ_{k}^{- 1} {(x - {μ.}_{k})}^{T}),$

where $| Σ_{k} |$ 是个determinant of Σ_k, and $Σ_{k}^{- 1}$ 是个inverse matrix.

让P(k) represent the prior probability of classk。Then the posterior probability that an observationx是课堂kis

$\hat{P} (k | x) = \frac{P (x | k) P (k)}{P (x)},$

whereP(x) is a normalization constant, namely, the sum overk的P(x|k)P(k).

Prior Probability

现有概率是三种选择之一：

'uniform'— The prior probability of classkis 1 over the total number of classes.
'empirical'— The prior probability of classk是个number of training samples of classk除以培训样本总数。
A numeric vector — The prior probability of classk是个jth element of thePriorvector. Seefitcdiscr.。

After creating a classifierobj, you can set the prior using dot notation:

obj.Prior = v;

wherev是表示每个元素发生的频率的正元素的矢量。在设置新的先前时，您无需重新定制分类器。

成本

There are two costs associated with discriminant analysis classification: the true misclassification cost per class, and the expected misclassification cost per observation.

True Misclassification Cost per Class

成本(i,j)是个成本的classifying an observation into classjif its true class isi。By default,成本（i，j）= 1if我〜= j, and成本（i，j）= 0if我= J.。换句话说,是成本0正确分类，和1for incorrect classification.

您可以在创建分类器时设置您喜欢的任何成本矩阵。通过成本矩阵成本name-value pair infitcdiscr.。

After you create a classifierobj, you can set a custom cost using dot notation:

obj.Cost = B;

Bis a square matrix of sizeK-by-Kwhen there areKclasses. You do not need to retrain the classifier when you set a new cost.

Expected Misclassification Cost per Observation

Suppose you have谈判observations that you want to classify with a trained discriminant analysis classifierobj。Suppose you haveKclasses. You place the observations into a matrixXnewwith one observation per row. The command

[label,score,cost] = predict(obj,Xnew)

returns, among other outputs, a cost matrix of size谈判-by-K。Each row of the cost matrix contains the expected (average) cost of classifying the observation into each of theKclasses.成本(n,k)is

$\sum_{i = 1}^{K} \hat{P} (i | X (n)) C (k | i),$

where

K是个number of classes.
$\hat{P} (i | X (n))$ 是个posterior probability的classifor observationXnew(n).
$C (k | i)$ 是个成本的classifying an observation askwhen its true class isi。