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Matthews correlation coefficient

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The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975. It takes into account true and false positives and negatives and is generally regarded as a balanced measure which can be used even if the classes are of very different sizes. The MCC is in essence a correlation coefficient between the observed and predicted binary classifications; it returns a value between −1 and +1. A coefficient of +1 represents a perfect prediction, 0 no better than random prediction and −1 indicates total disagreement between prediction and observation. The statistic is also known as the phi coefficient. MCC is related to the chi-square statistic for a 2×2 contingency table

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| MCC | = χ 2 n

where n is the total number of observations.

While there is no perfect way of describing the confusion matrix of true and false positives and negatives by a single number, the Matthews correlation coefficient is generally regarded as being one of the best such measures. Other measures, such as the proportion of correct predictions (also termed accuracy), are not useful when the two classes are of very different sizes. For example, assigning every object to the larger set achieves a high proportion of correct predictions, but is not generally a useful classification.

The MCC can be calculated directly from the confusion matrix using the formula:

MCC = T P × T N − F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N )

In this equation, TP is the number of true positives, TN the number of true negatives, FP the number of false positives and FN the number of false negatives. If any of the four sums in the denominator is zero, the denominator can be arbitrarily set to one; this results in a Matthews correlation coefficient of zero, which can be shown to be the correct limiting value.

The original formula as given by Matthews was:

N = T N + T P + F N + F P S = T P + F N N P = T P + F P N MCC = T P / N − S × P P S ( 1 − S ) ( 1 − P )

This is equal to the formula given above. As a correlation coefficient, the Matthews correlation coefficient is the geometric mean of the regression coefficients of the problem and its dual. The component regression coefficients of the Matthews correlation coefficient are Markedness (Δp) and Youden's J statistic (Informedness or Δp'). Markedness and Informedness correspond to different directions of information flow and generalize Youden's J statistic, the deltap statistics and (as their geometric mean) the Matthews Correlation Coefficient to more than two classes.

Confusion Matrix

Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

Multiclass Case

The Matthews correlation coefficient has been generalized to the multiclass case. This generalization was called the R K statistic (for K different classes) by the author, and defined in terms of a K × K confusion matrix C .

MCC = ∑ k ∑ l ∑ m C k k C l m − C k l C m k ∑ k ( ∑ l C k l ) ( ∑ k ′ | k ′ ≠ k ∑ l ′ C k ′ l ′ ) ∑ k ( ∑ l C l k ) ( ∑ k ′ | k ′ ≠ k ∑ l ′ C l ′ k ′ )

When there are more than two labels the MCC will no longer range between -1 and +1. Instead the minimum value will be between -1 and 0 depending on the true distribution. The maximum value is always +1.

References

Matthews correlation coefficient Wikipedia
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