Basic evaluation measures from the confusion matrix

We introduce basic performance measures derived from the confusion matrix through this page. The confusion matrix is a two by two table that contains four outcomes produced by a binary classifier. Various measures, such as error-rate, accuracy, specificity, sensitivity, and precision, are derived from the confusion matrix. Moreover, several advanced measures, such as ROC and precision-recall, are based on them.

After studying the basic performance measures, don’t forget to read our introduction to precision-recall plots (link) and the section on tools (link). Also take note of the issues with ROC curves and why in such cases precision-recall plots are a better choice (link).

Test datasets for binary classifier

A binary classifier produces output with two class values or labels, such as Yes/No and 1/0, for given input data. The class of interest is usually denoted as “positive” and the other as “negative”.

A binary classifier with input and output data.
A binary classifier produces output with two classes for given input data.

Test dataset for evaluation

A dataset used for performance evaluation is called a test dataset. It should contain the correct labels (observed labels) for all data instances. These observed labels are used to compare with the predicted labels for performance evaluation after classification.

A test dataset with two labels.
In binary classification, a test dataset has two labels; positive and negative.

Predictions on test datasets

The predicted labels will be exactly the same if the performance of a binary classifier is perfect, but it is uncommon to be able to develop a perfect binary classifier that is practical for various conditions.

A perfect binary classifier.
The performance of a binary classifier is perfect when it can predict the exactly same labels in a test dataset.

Hence, the predicted labels usually match with part of the observed labels.

Predicted labels of a classifier.
The predicted labels of a classifier match with part of the observed labels.

Confusion matrix from the four outcomes

A confusion matrix is formed from the four outcomes produced as a result of binary classification.

Four outcomes of classification

A binary classifier predicts all data instances of a test dataset as either positive or negative. This classification (or prediction) produces four outcomes – true positive, true negative, false positive and false negative.

  • True positive (TP): correct positive prediction
  • False positive (FP): incorrect positive prediction
  • True negative (TN): correct negative prediction
  • False negative (FN): incorrect negative prediction
Four outcomes of classification.
Classification of a test dataset produces four outcomes – true positive, false positive, true negative, and false negative.

Confusion matrix

A confusion matrix of binary classification is a two by two table formed by counting of the number of the four outcomes of a binary classifier. We usually denote them as TP, FP, TN, and FN instead of “the number of true positives”, and so on.

Predicted
Positive Negative
Observed Positive TP (# of TPs) FN (# of FNs)
Negative FP (# of FPs) TN (# of TNs)

Basic measures derived from the confusion matrix

Various measures can be derived from a confusion matrix.

First two basic measures from the confusion matrix

Error rate (ERR) and accuracy (ACC) are the most common and intuitive measures derived from the confusion matrix.

Error rate

Error rate (ERR) is calculated as the number of all incorrect predictions divided by the total number of the dataset. The best error rate is 0.0, whereas the worst is 1.0.

Error rate calculation.
Error rate is calculated as the total number of two incorrect predictions (FN + FP) divided by the total number of a dataset (P + N).
  • \mathrm{ERR = \displaystyle \frac{FP + FN}{TP + TN + FN + FP} = \frac{FP + FN}{P + N}}

Accuracy

Accuracy (ACC) is calculated as the number of all correct predictions divided by the total number of the dataset. The best accuracy is 1.0, whereas the worst is 0.0. It can also be calculated by 1 – ERR.

Accuracy calculation.
Accuracy is calculated as the total number of two correct predictions (TP + TN) divided by the total number of a dataset (P + N).
  • \mathrm{ACC = \displaystyle \frac{TP +TN}{TP + TN + FN + FP} = \frac{TP + TN}{P + N}}

Other basic measures from the confusion matrix

Error costs of positives and negatives are usually different. For instance, one wants to avoid false negatives more than false positives or vice versa. Other basic measures, such as sensitivity and specificity, are more informative than accuracy and error rate in such cases.

Sensitivity (Recall or True positive rate)

Sensitivity (SN) is calculated as the number of correct positive predictions divided by the total number of positives. It is also called recall (REC) or true positive rate (TPR). The best sensitivity is 1.0, whereas the worst is 0.0.

Sensitivity calculation.
Sensitivity is calculated as the number of correct positive predictions (TP) divided by the total number of positives (P).
  • \mathrm{SN = \displaystyle \frac{TP}{TP + FN} = \frac{TP}{P}}

Specificity (True negative rate)

Specificity (SP) is calculated as the number of correct negative predictions divided by the total number of negatives. It is also called true negative rate (TNR). The best specificity is 1.0, whereas the worst is 0.0.

Specificity calculation.
Specificity is calculated as the number of correct negative predictions (TN) divided by the total number of negatives (N).
  • \mathrm{SP = \displaystyle \frac{TN}{TN + FP} = \frac{TN}{N}}

Precision (Positive predictive value)

Precision (PREC) is calculated as the number of correct positive predictions divided by the total number of positive predictions. It is also called positive predictive value (PPV). The best precision is 1.0, whereas the worst is 0.0.

Precision calculation.
Precision is calculated as the number of correct positive predictions (TP) divided by the total number of positive predictions (TP + FP).
  • \mathrm{PREC = \displaystyle \frac{TP}{TP + FP}}

False positive rate

False positive rate (FPR) is calculated as the number of incorrect positive predictions divided by the total number of negatives. The best false positive rate is 0.0 whereas the worst is 1.0. It can also be calculated as 1 – specificity.

False positive rate calculation.
False positive rate is calculated as the number of incorrect positive predictions (FP) divided by the total number of negatives (N).
  • \mathrm{FPR = \displaystyle \frac{FP}{TN + FP} = 1 - SP}

Correlation coefficient and F-score

Mathews correlation coefficient and F-score can be useful, but they are less frequently used than the other basic measures.

Matthews correlation coefficient

Matthews correlation coefficient (MCC) is a correlation coefficient calculated using all four values in the confusion matrix.

  • \mathrm{MCC = \displaystyle \frac{TP \cdot TN - FP \cdot FN}{\sqrt{(TP + FP)(TP + FN)(TN + FP)(TN + FN)}}}

F-score

F-score is a harmonic mean of precision and recall.

  • \mathrm{F_{\beta} = \displaystyle \frac{(1 + \beta^2) (PREC \cdot REC)}{(\beta^2 \cdot PREC + REC)}}

β is commonly 0.5, 1, or 2.

  • \mathrm{F_{0.5} = \displaystyle \frac{1.25 \cdot PREC \cdot REC}{0.25 \cdot PREC + REC}}
  • \mathrm{F_{1} = \displaystyle \frac{2 \cdot PREC \cdot REC}{PREC + REC}}
  • \mathrm{F_{2} = \displaystyle \frac{5 \cdot PREC \cdot REC}{4 \cdot PREC + REC}}

An example of evaluation measure calculations

Let us assume that the outcome of some classification results in 6 TPs, 4 FNs, 8 TNs, and 2 FPs.

An example of four outcomes of binary classification.
This example shows that a binary classifier has produced 6 TPs, 4 FNs, 2 FPs, and 8 TNs.

First, a confusion matrix is formed from the outcomes.

Predicted
Positive Negative
Observed Positive 6 4
Negative 2 8

Then, the calculations of basic measures are straightforward once the confusion matrix is created.

measure calculated value
Error rate ERR 6 / 20 = 0.3
Accuracy ACC 14 / 20 = 0.7
Sensitivity
True positive rate
Recall
SN
TPR
REC
6 / 10 = 0.6
Specificity
True negative rate
SP
TNR
8 / 10 = 0.8
Precision
Positive predictive value
PREC
PPV
6 / 8 =0.75
False positive rate FPR 2 / 10 = 0.2

Other evaluation measures

Please see the following pages for more advanced evaluation measures.

28 thoughts on “Basic evaluation measures from the confusion matrix”

  1. “False positive rate (FPR) is calculated as the number of incorrect negative predictions… ” Should not be: incorrect positive predictions?

    Like

  2. very smooth explanation. i have a small question. for multiple classes, how am i going to calculate the error rate. according to this, i calculate the class.error for each classes but the general “OOB estimate of error rate” is different from what the algorithm calculates. What could have been the thing I miss? Many thanks

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    1. The calculation of the error rate is still the same even for multi-class classifications as: (# of misclassified instances) / (# of total instances). Some of machine learning methods that use bootstrap resampling require no validation datasets during training because they can use OOB instead. For instance, you can calculate an error rate for each subsample, and then you can aggregate all of them to calculate the mean error rate. You still need a test data set to evaluate your final model, though.

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    1. No, not really. You need to consider other factors, such as your problem domain, test data sets and so on, to estimate whether the performance of your model is good or bad from these metrics. But, you can always use them for comparisons among multiple models and hypotheses.

      Like

  3. A.A

    Sir! i am using Weka tool and run DecisionTable classifier model and get confusion Matrix but i need to Label as a TP,TN,FP and FN

    a b Class

    2781 0 a = No

    26 425 b = Yes

    So, please help me sirr

    Regards

    Like

    1. Assume that the labels Yes (b) and No (a) respectively represent positive and negative, and the matrix is as follows:

      a . b . <– predicted/classified labels
      2781 . 0 . (a = No in your test set)
      26 . 425 . (b = Yes in your test set)

      Then, TPs = 425, TNs = 2781, FPs = 0 and FNs = 26.

      Like

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