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Accuracy measure

The accuracy of a classifier can be measured in any set: training, evaluation, and test. Let $ Z$ be any one of these sets and $ N$ be the number of samples in $ Z$. The accuracy $ Acc$ is measured by taking into account that the classes may have different sizes in $ Z$. If there are two classes, for example, with very different sizes and a classifier always assigns the label of the largest class, its accuracy will fall drastically due to the high error rate on the smallest class.

Let $ N(i)$, $ i=1,2,\ldots,c$, be the number of samples in $ Z$ from each class $ i$. We define

$\displaystyle e_{i,1}=\frac{FP(i)}{\vert Z\vert-\left\vert N(i)\right\vert} $    and $\displaystyle e_{i,2}=\frac{FN(i)}{\left\vert N(i)\right\vert}, i=1,\ldots,c$ (1)

where $ FP(i)$ and $ FN(i)$ are the false positives and false negatives, respectively. That is, $ FP(i)$ is the number of samples from other classes that were classified as being from the class $ i$ in $ Z$, and $ FN(i)$ is the number of samples from the class $ i$ that were incorrectly classified as being from other classes in $ Z$. The errors $ e_{i,1}$ and $ e_{i,2}$ are used to define

$\displaystyle E(i)=e_{i,1}+e_{i,2},$ (2)

where $ E(i)$ is the partial sum error of class $ i$. Finally, the accuracy $ Acc$ of the classification is written as

$\displaystyle Acc=\frac{2c-\sum_{i=1}^{c}E(i)}{2c}=1-\frac{\sum_{i=1}^{c}E(i)}{2c}.$ (3)

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Joao Paulo Papa 2009-09-30