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Confusion matrix

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Predicted condition Sources: [1][2][3][4][5][6][7][8]
Total population
= P + N 		Predicted positive Predicted negative Informedness, bookmaker informedness (BM)
= TPR + TNR − 1 		Prevalence threshold (PT)
= ⁠√TPR × FPR − FPR/TPR − FPR⁠
Actual condition
 Positive (P) [a] True positive (TP),
hit[b] 		False negative (FN),
miss, underestimation 		True positive rate (TPR), recall, sensitivity (SEN), probability of detection, hit rate, power
= ⁠TP/P⁠ = 1 − FNR 		False negative rate (FNR),
miss rate
type II error [c]
= ⁠FN/P⁠ = 1 − TPR
Negative (N)[d] False positive (FP),
false alarm, overestimation 		True negative (TN),
correct rejection[e] 		False positive rate (FPR),
probability of false alarm, fall-out
type I error [f]
= ⁠FP/N⁠ = 1 − TNR 		True negative rate (TNR),
specificity (SPC), selectivity
= ⁠TN/N⁠ = 1 − FPR
Prevalence
= ⁠P/P + N⁠ 		Positive predictive value (PPV), precision
= ⁠TP/TP + FP⁠ = 1 − FDR 		Negative predictive value (NPV)
= ⁠TN/TN + FN⁠ = 1 − FOR 		Positive likelihood ratio (LR+)
= ⁠TPR/FPR⁠ 		Negative likelihood ratio (LR−)
= ⁠FNR/TNR⁠
Accuracy (ACC)
= ⁠TP + TN/P + N⁠ 		False discovery rate (FDR)
= ⁠FP/TP + FP⁠ = 1 − PPV 		False omission rate (FOR)
= ⁠FN/TN + FN⁠ = 1 − NPV 		Markedness (MK), deltaP (Δp)
= PPV + NPV − 1 		Diagnostic odds ratio (DOR)
= ⁠LR+/LR−⁠
Balanced accuracy (BA)
= ⁠TPR + TNR/2⁠ 		F1 score
= ⁠2 PPV × TPR/PPV + TPR⁠ = ⁠2 TP/2 TP + FP + FN⁠ 		Fowlkes–Mallows index (FM)
= √PPV × TPR 		phi or Matthews correlation coefficient (MCC)
= √TPR × TNR × PPV × NPV - √FNR × FPR × FOR × FDR 		Threat score (TS), critical success index (CSI), Jaccard index
= ⁠TP/TP + FN + FP⁠
  1. ^ the number of real positive cases in the data
  2. ^ A test result that correctly indicates the presence of a condition or characteristic
  3. ^ Type II error: A test result which wrongly indicates that a particular condition or attribute is absent
  4. ^ the number of real negative cases in the data
  5. ^ A test result that correctly indicates the absence of a condition or characteristic
  6. ^ Type I error: A test result which wrongly indicates that a particular condition or attribute is present

In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as an error matrix,[9] is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one (in unsupervised learning it is usually called a matching matrix). Each row of the matrix represents the instances in a predicted class while each column represents the instances in an actual class, or vice versa – both variants are found in the literature.[10] The name stems from the fact that it makes it easy to see whether the system is confusing two classes (i.e. commonly mislabeling one as another).

It is a special kind of contingency table, with two dimensions ("actual" and "predicted"), and identical sets of "classes" in both dimensions (each combination of dimension and class is a variable in the contingency table).

Example

Given a sample of 12 pictures, 8 of cats and 4 of dogs, where cats belong to class 1 and dogs belong to class 0,

actual = [1,1,1,1,1,1,1,1,0,0,0,0],

assume that a classifier that distinguishes between cats and dogs is trained, and we take the 12 pictures and run them through the classifier, and the classifier makes 9 accurate predictions and misses 3: 2 cats wrongly predicted as dogs (first 2 predictions) and 1 dog wrongly predicted as a cat (last prediction).

prediction = [0,0,1,1,1,1,1,1,0,0,0,1]

With these two labelled sets (actual and predictions) we can create a confusion matrix that will summarize the results of testing the classifier:

Predicted
class
Actual class
 Cat Dog
Cat 6 2
Dog 1 3

In this confusion matrix, of the 8 cat pictures, the system judged that 2 were dogs, and of the 4 dog pictures, it predicted that 1 were cats. All correct predictions are located in the diagonal of the table (highlighted in bold), so it is easy to visually inspect the table for prediction errors, as they will be represented by values outside the diagonal.

In terms of sensitivity and specificity, the confusion matrix is as follows:

Predicted
class
Actual class
 P N
P TP FN
N FP TN

Table of confusion

Comparing mean accuracy and percent of false negative (overestimation) of five machine learning (multi-class) classification models. Models #1, #2 and #4 have a very similar accuracy but different false negative or overestimation levels.[11]

In predictive analytics, a table of confusion (sometimes also called a confusion matrix) is a table with two rows and two columns that reports the number of false positives, false negatives, true positives, and true negatives. This allows more detailed analysis than mere proportion of correct classifications (accuracy). Accuracy will yield misleading results if the data set is unbalanced; that is, when the numbers of observations in different classes vary greatly. For example, if there were 95 cats and only 5 dogs in the data, a particular classifier might classify all the observations as cats. The overall accuracy would be 95%, but in more detail the classifier would have a 100% recognition rate (sensitivity) for the cat class but a 0% recognition rate for the dog class. F1 score is even more unreliable in such cases, and here would yield over 97.4%, whereas informedness removes such bias and yields 0 as the probability of an informed decision for any form of guessing (here always guessing cat). Confusion matrix is not limited to binary classification and can be used in multi-class classifiers as well.[11]

According to Davide Chicco and Giuseppe Jurman, the most informative metric to evaluate a confusion matrix is the Matthews correlation coefficient (MCC).[12]

Assuming the confusion matrix above, its corresponding table of confusion, for the cat class, would be:

Predicted
class
Actual class
 Cat Non-cat
Cat 6 true positives 2 false negatives
Non-cat 1 false positive 3 true negatives

The final table of confusion would contain the average values for all classes combined.

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 confusion matrix, as follows:

Predicted condition Sources: [13][14][15][16][17][18][19][20]
Total population
= P + N 		Predicted positive Predicted negative Informedness, bookmaker informedness (BM)
= TPR + TNR − 1 		Prevalence threshold (PT)
= ⁠√TPR × FPR − FPR/TPR − FPR⁠
Actual condition
 Positive (P) [a] True positive (TP),
hit[b] 		False negative (FN),
miss, underestimation 		True positive rate (TPR), recall, sensitivity (SEN), probability of detection, hit rate, power
= ⁠TP/P⁠ = 1 − FNR 		False negative rate (FNR),
miss rate
type II error [c]
= ⁠FN/P⁠ = 1 − TPR
Negative (N)[d] False positive (FP),
false alarm, overestimation 		True negative (TN),
correct rejection[e] 		False positive rate (FPR),
probability of false alarm, fall-out
type I error [f]
= ⁠FP/N⁠ = 1 − TNR 		True negative rate (TNR),
specificity (SPC), selectivity
= ⁠TN/N⁠ = 1 − FPR
Prevalence
= ⁠P/P + N⁠ 		Positive predictive value (PPV), precision
= ⁠TP/TP + FP⁠ = 1 − FDR 		Negative predictive value (NPV)
= ⁠TN/TN + FN⁠ = 1 − FOR 		Positive likelihood ratio (LR+)
= ⁠TPR/FPR⁠ 		Negative likelihood ratio (LR−)
= ⁠FNR/TNR⁠
Accuracy (ACC)
= ⁠TP + TN/P + N⁠ 		False discovery rate (FDR)
= ⁠FP/TP + FP⁠ = 1 − PPV 		False omission rate (FOR)
= ⁠FN/TN + FN⁠ = 1 − NPV 		Markedness (MK), deltaP (Δp)
= PPV + NPV − 1 		Diagnostic odds ratio (DOR)
= ⁠LR+/LR−⁠
Balanced accuracy (BA)
= ⁠TPR + TNR/2⁠ 		F1 score
= ⁠2 PPV × TPR/PPV + TPR⁠ = ⁠2 TP/2 TP + FP + FN⁠ 		Fowlkes–Mallows index (FM)
= √PPV × TPR 		phi or Matthews correlation coefficient (MCC)
= √TPR × TNR × PPV × NPV - √FNR × FPR × FOR × FDR 		Threat score (TS), critical success index (CSI), Jaccard index
= ⁠TP/TP + FN + FP⁠
  1. ^ the number of real positive cases in the data
  2. ^ A test result that correctly indicates the presence of a condition or characteristic
  3. ^ Type II error: A test result which wrongly indicates that a particular condition or attribute is absent
  4. ^ the number of real negative cases in the data
  5. ^ A test result that correctly indicates the absence of a condition or characteristic
  6. ^ Type I error: A test result which wrongly indicates that a particular condition or attribute is present


See also

References

  1. ^ Fawcett, Tom (2006). "An Introduction to ROC Analysis" (PDF). Pattern Recognition Letters. 27 (8): 861–874. doi:10.1016/j.patrec.2005.10.010. S2CID 2027090.
  2. ^ Provost, Foster; Tom Fawcett (2013-08-01). "Data Science for Business: What You Need to Know about Data Mining and Data-Analytic Thinking". O'Reilly Media, Inc.
  3. ^ Powers, David M. W. (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies. 2 (1): 37–63.
  4. ^ Ting, Kai Ming (2011). Sammut, Claude; Webb, Geoffrey I. (eds.). Encyclopedia of machine learning. Springer. doi:10.1007/978-0-387-30164-8. ISBN 978-0-387-30164-8.
  5. ^ Brooks, Harold; Brown, Barb; Ebert, Beth; Ferro, Chris; Jolliffe, Ian; Koh, Tieh-Yong; Roebber, Paul; Stephenson, David (2015-01-26). "WWRP/WGNE Joint Working Group on Forecast Verification Research". Collaboration for Australian Weather and Climate Research. World Meteorological Organisation. Retrieved 2019-07-17.
  6. ^ Chicco D, Jurman G (January 2020). "The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation". BMC Genomics. 21 (1): 6-1–6-13. doi:10.1186/s12864-019-6413-7. PMC 6941312. PMID 31898477.
  7. ^ Chicco D, Toetsch N, Jurman G (February 2021). "The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two-class confusion matrix evaluation". BioData Mining. 14 (13): 13. doi:10.1186/s13040-021-00244-z. PMC 7863449. PMID 33541410.
  8. ^ Tharwat A. (August 2018). "Classification assessment methods". Applied Computing and Informatics. 17: 168–192. doi:10.1016/j.aci.2018.08.003.
  9. ^ Stehman, Stephen V. (1997). "Selecting and interpreting measures of thematic classification accuracy". Remote Sensing of Environment. 62 (1): 77–89. Bibcode:1997RSEnv..62...77S. doi:10.1016/S0034-4257(97)00083-7.
  10. ^ Powers, David M. W. (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies. 2 (1): 37–63. S2CID 55767944.
  11. ^ a b Piryonesi S. Madeh; El-Diraby Tamer E. (2020-03-01). "Data Analytics in Asset Management: Cost-Effective Prediction of the Pavement Condition Index". Journal of Infrastructure Systems. 26 (1): 04019036. doi:10.1061/(ASCE)IS.1943-555X.0000512.
  12. ^ "The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation". BMC Genomics. 21 (1): 6-1–6-13. January 2020. doi:10.1186/s12864-019-6413-7. PMC 6941312. PMID 31898477. {{cite journal}}: Unknown parameter |authors= ignored (help)CS1 maint: unflagged free DOI (link)
  13. ^ Fawcett, Tom (2006). "An Introduction to ROC Analysis" (PDF). Pattern Recognition Letters. 27 (8): 861–874. doi:10.1016/j.patrec.2005.10.010. S2CID 2027090.
  14. ^ Provost, Foster; Tom Fawcett (2013-08-01). "Data Science for Business: What You Need to Know about Data Mining and Data-Analytic Thinking". O'Reilly Media, Inc.
  15. ^ Powers, David M. W. (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies. 2 (1): 37–63.
  16. ^ Ting, Kai Ming (2011). Sammut, Claude; Webb, Geoffrey I. (eds.). Encyclopedia of machine learning. Springer. doi:10.1007/978-0-387-30164-8. ISBN 978-0-387-30164-8.
  17. ^ Brooks, Harold; Brown, Barb; Ebert, Beth; Ferro, Chris; Jolliffe, Ian; Koh, Tieh-Yong; Roebber, Paul; Stephenson, David (2015-01-26). "WWRP/WGNE Joint Working Group on Forecast Verification Research". Collaboration for Australian Weather and Climate Research. World Meteorological Organisation. Retrieved 2019-07-17.
  18. ^ Chicco D, Jurman G (January 2020). "The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation". BMC Genomics. 21 (1): 6-1–6-13. doi:10.1186/s12864-019-6413-7. PMC 6941312. PMID 31898477.
  19. ^ Chicco D, Toetsch N, Jurman G (February 2021). "The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two-class confusion matrix evaluation". BioData Mining. 14 (13): 13. doi:10.1186/s13040-021-00244-z. PMC 7863449. PMID 33541410.
  20. ^ Tharwat A. (August 2018). "Classification assessment methods". Applied Computing and Informatics. 17: 168–192. doi:10.1016/j.aci.2018.08.003.


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