Terminology and derivationsconfusion matrix
condition positive (P)
the number of real positive cases in the data
condition negative (N)
the number of real negative cases in the data true positive (TP)
A test result that correctly indicates the presence of a condition or characteristic
true negative (TN)
A test result that correctly indicates the absence of a condition or characteristic
false positive (FP)
A test result which wrongly indicates that a particular condition or attribute is present
false negative (FN)
A test result which wrongly indicates that a particular condition or attribute is absent sensitivity , recall , hit rate , or true positive rate (TPR)T P R = T P P = T P T P + F N = 1 − F N R {\displaystyle \mathrm {TPR} ={\frac {\mathrm {TP} }{\mathrm {P} }}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}=1-\mathrm {FNR} } specificity , selectivity or true negative rate (TNR)T N R = T N N = T N T N + F P = 1 − F P R {\displaystyle \mathrm {TNR} ={\frac {\mathrm {TN} }{\mathrm {N} }}={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FP} }}=1-\mathrm {FPR} } precision or positive predictive value (PPV)P P V = T P T P + F P = 1 − F D R {\displaystyle \mathrm {PPV} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FP} }}=1-\mathrm {FDR} } negative predictive value (NPV)N P V = T N T N + F N = 1 − F O R {\displaystyle \mathrm {NPV} ={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FN} }}=1-\mathrm {FOR} } miss rate or false negative rate (FNR)
F N R = F N P = F N F N + T P = 1 − T P R {\displaystyle \mathrm {FNR} ={\frac {\mathrm {FN} }{\mathrm {P} }}={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TP} }}=1-\mathrm {TPR} } fall-out or false positive rate (FPR)F P R = F P N = F P F P + T N = 1 − T N R {\displaystyle \mathrm {FPR} ={\frac {\mathrm {FP} }{\mathrm {N} }}={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TN} }}=1-\mathrm {TNR} } false discovery rate (FDR)F D R = F P F P + T P = 1 − P P V {\displaystyle \mathrm {FDR} ={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TP} }}=1-\mathrm {PPV} } false omission rate (FOR)F O R = F N F N + T N = 1 − N P V {\displaystyle \mathrm {FOR} ={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TN} }}=1-\mathrm {NPV} } Positive likelihood ratio (LR+)L R + = T P R F P R {\displaystyle \mathrm {LR+} ={\frac {\mathrm {TPR} }{\mathrm {FPR} }}} Negative likelihood ratio (LR-)L R − = F N R T N R {\displaystyle \mathrm {LR-} ={\frac {\mathrm {FNR} }{\mathrm {TNR} }}} prevalence threshold (PT)P T = F P R T P R + F P R {\displaystyle \mathrm {PT} ={\frac {\sqrt {\mathrm {FPR} }}{{\sqrt {\mathrm {TPR} }}+{\sqrt {\mathrm {FPR} }}}}} threat score (TS) or critical success index (CSI)
T S = T P T P + F N + F P {\displaystyle \mathrm {TS} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} +\mathrm {FP} }}} Prevalence P P + N {\displaystyle {\frac {\mathrm {P} }{\mathrm {P} +\mathrm {N} }}} accuracy (ACC)A C C = T P + T N P + N = T P + T N T P + T N + F P + F N {\displaystyle \mathrm {ACC} ={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {P} +\mathrm {N} }}={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {TP} +\mathrm {TN} +\mathrm {FP} +\mathrm {FN} }}} balanced accuracy (BA)
B A = T P R + T N R 2 {\displaystyle \mathrm {BA} ={\frac {TPR+TNR}{2}}} F1 score is the harmonic mean of precision and sensitivity : F 1 = 2 × P P V × T P R P P V + T P R = 2 T P 2 T P + F P + F N {\displaystyle \mathrm {F} _{1}=2\times {\frac {\mathrm {PPV} \times \mathrm {TPR} }{\mathrm {PPV} +\mathrm {TPR} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}
phi coefficient (φ or rφ ) or Matthews correlation coefficient (MCC)M C C = T P × T N − F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) {\displaystyle \mathrm {MCC} ={\frac {\mathrm {TP} \times \mathrm {TN} -\mathrm {FP} \times \mathrm {FN} }{\sqrt {(\mathrm {TP} +\mathrm {FP} )(\mathrm {TP} +\mathrm {FN} )(\mathrm {TN} +\mathrm {FP} )(\mathrm {TN} +\mathrm {FN} )}}}} Fowlkes–Mallows index (FM)F M = T P T P + F P × T P T P + F N = P P V × T P R {\displaystyle \mathrm {FM} ={\sqrt {{\frac {TP}{TP+FP}}\times {\frac {TP}{TP+FN}}}}={\sqrt {PPV\times TPR}}} informedness or bookmaker informedness (BM)B M = T P R + T N R − 1 {\displaystyle \mathrm {BM} =\mathrm {TPR} +\mathrm {TNR} -1} markedness (MK) or deltaP (Δp)M K = P P V + N P V − 1 {\displaystyle \mathrm {MK} =\mathrm {PPV} +\mathrm {NPV} -1} Diagnostic odds ratio (DOR)D O R = L R + L R − {\displaystyle \mathrm {DOR} ={\frac {\mathrm {LR+} }{\mathrm {LR-} }}} Sources: Fawcett (2006),[1] [2] [3] [4] [5] D. Chicco & G. Jurman (2020, 2021, 2023) ,[6] [7] [8] [9] [10]
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↑ Fawcett, Tom (2006). "An Introduction to ROC Analysis" (PDF) . Pattern Recognition Letters . 27 (8): 861–874. doi :10.1016/j.patrec.2005.10.010 . ↑ 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 . ↑ 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. ↑ 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 ↑ 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 . ↑ 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 . ↑ 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): 1-22. doi :10.1186/s13040-021-00244-z . PMC 7863449 PMID 33541410 . ↑ Chicco D.; Jurman G. (2023). "The Matthews correlation coefficient (MCC) should replace the ROC AUC as the standard metric for assessing binary classification" . BioData Mining . 16 (1). doi :10.1186/s13040-023-00322-4 PMC 9938573 ↑ Tharwat A. (August 2018). "Classification assessment methods" . Applied Computing and Informatics . doi :10.1016/j.aci.2018.08.003 ↑ Balayla, Jacques (2020). "Prevalence threshold (ϕe) and the geometry of screening curves" . PLoS One . 15 (10). doi :10.1371/journal.pone.0240215 . 
