Performance measurement *
Rootmeansquare deviation
The rootmeansquare deviation (RMSD) or rootmeansquare error (RMSE) is a frequently used measure of the differences between values (sample and population values) predicted by a model or an estimator and the values actually observed. Basically, the RMSD represents the sample standard deviation of the differences between predicted values and observed values. These individual differences are called residuals when the calculations are performed over the data sample that was used for estimation, and are called prediction errors when computed outofsample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a good measure of accuracy, but only to compare forecasting errors of different models for a particular variable and not between variables, as it is scaledependent.[1]
Formula
The RMSD of an estimator \hat{\theta} with respect to an estimated parameter
is defined as the square root of the mean square error:
For an unbiased estimator, the RMSD is the square root of the variance, known as the standard error.
The RMSD of predicted values
for times t of a regression's dependent variable
is computed for n different predictions as the square root of the mean of the squares of the deviations:
In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard". For example, when measuring the average difference between two time series
and
, the formula becomes
Normalized rootmeansquare deviation
Normalizing the RMSD facilitates the comparison between datasets or models with different scales. Though there is no consistent means of normalization in the literature, the range of the measured data defined as the maximum value minus the minimum value is a common choice:[2]
This value is commonly referred to as the normalized rootmeansquare deviation or error (NRMSD or NRMSE), and often expressed as a percentage, where lower values indicate less residual variance.
Another common choice is to normalize by the mean value of the measurements:[3]
When the RMSD is normalized by the mean measured value, is usually called coefficient of variation of the RMSE, CV(RMSE). It is analogous to the coefficient of variation with the RMSE taking the place of the standard deviation.
References

Hyndman, Rob J. Koehler, Anne B.; Koehler (2006). "Another look at measures of forecast accuracy". International Journal of Forecasting 22 (4): 679–688. doi:10.1016/j.ijforecast.2006.03.001.

"Coastal Inlets Research Program (CIRP) Wiki  Statistics". Retrieved 4 February 2015.

"FAQ: What is the coefficient of variation?". Retrieved 4 February 2015.

J. Scott Armstrong and Fred Collopy (1992). "Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons" (PDF). International Journal of Forecasting 8 (1): 69–80. doi:10.1016/01692070(92)90008w.

Anderson, M.P.; Woessner, W.W. (1992). Applied Groundwater Modeling: Simulation of Flow and Advective Transport (2nd ed.). Academic Press.

Ensemble Neural Network Model
Cohen's kappa
Cohen's kappa coefficient is a statistic which measures interrater agreement for qualitative (categorical) items. It is generally thought to be a more robust measure than simple percent agreement calculation, since κ takes into account the agreement occurring by chance. [1] Some researchers[2][citation needed] have expressed concern over κ's tendency to take the observed categories' frequencies as givens, which can have the effect of underestimating agreement for a category that is also commonly used; for this reason, κ is considered an overly conservative measure of agreement. Others[3][citation needed] contest the assertion that kappa "takes into account" chance agreement. To do this effectively would require an explicit model of how chance affects rater decisions. The socalled chance adjustment of kappa statistics supposes that, when not completely certain, raters simply guess—a very unrealistic scenario.
Calculation
Cohen's kappa measures the agreement between two raters who each classify N items into C mutually exclusive categories. The first mention of a kappalike statistic is attributed to Galton (1892),[4] see Smeeton (1985).[5]
The equation for κ is:
where Pr(a) is the relative observed agreement among raters, and Pr(e) is the hypothetical probability of chance agreement, using the observed data to calculate the probabilities of each observer randomly saying each category. If the raters are in complete agreement then κ = 1. If there is no agreement among the raters other than what would be expected by chance (as defined by Pr(e)), κ = 0.
The seminal paper introducing kappa as a new technique was published by Jacob Cohen in the journal Educational and Psychological Measurement in 1960.[6]
A similar statistic, called pi, was proposed by Scott (1955). Cohen's kappa and Scott's pi differ in terms of how Pr(e) is calculated.
Note that Cohen's kappa measures agreement between two raters only. For a similar measure of agreement (Fleiss' kappa) used when there are more than two raters, see Fleiss (1971). The Fleiss kappa, however, is a multirater generalization of Scott's pi statistic, not Cohen's kappa. Kappa is also used to compare performance in Machine Learning but the directional version known as Informedness or Youden's J statistic is argued to be more appropriate for supervised learning.[7]
Example
Suppose that you were analyzing data related to a group of 50 people applying for a grant. Each grant proposal was read by two readers and each reader either said "Yes" or "No" to the proposal. Suppose the dis/agreement count data were as follows, where A and B are readers, data on the diagonal slanting left shows the count of agreements and the data on the diagonal slanting right, disagreements:


B

B



yes

no

A

yes

20

5

A

no

10

15

Note that there were 20 proposals that were granted by both reader A and reader B, and 15 proposals that were rejected by both readers. Thus, the observed proportionate agreement is Pr(a) = (20 + 15) / 50 = 0.70
To calculate Pr(e) (the probability of random agreement) we note that:

Reader A said "Yes" to 25 applicants and "No" to 25 applicants. Thus reader A said "Yes" 50% of the time.

Reader B said "Yes" to 30 applicants and "No" to 20 applicants. Thus reader B said "Yes" 60% of the time.
Therefore the probability that both of them would say "Yes" randomly is 0.50 · 0.60 = 0.30 and the probability that both of them would say "No" is 0.50 · 0.40 = 0.20. Thus the overall probability of random agreement is Pr(e) = 0.3 + 0.2 = 0.5.
So now applying our formula for Cohen's Kappa we get:
Same percentages but different numbers
A case sometimes considered to be a problem with Cohen's Kappa occurs when comparing the Kappa calculated for two pairs of raters with the two raters in each pair having the same percentage agreement but one pair give a similar number of ratings while the other pair give a very different number of ratings.[8] For instance, in the following two cases there is equal agreement between A and B (60 out of 100 in both cases) so we would expect the relative values of Cohen's Kappa to reflect this. However, calculating Cohen's Kappa for each:


B

B



yes

no

A

yes

45

15

A

no

25

15



B

B



yes

no

A

yes

25

35

A

no

5

35

we find that it shows greater similarity between A and B in the second case, compared to the first. This is because while the percentage agreement is the same, the percentage agreement that would occur 'by chance' is significantly higher in the first case (0.54 compared to 0.46).
References

Carletta, Jean. (1996) Assessing agreement on classification tasks: The kappa statistic. Computational Linguistics, 22(2), pp. 249–254.

Strijbos, J.; Martens, R.; Prins, F.; Jochems, W. (2006). "Content analysis: What are they talking about?". Computers & Education 46: 29–48. doi:10.1016/j.compedu.2005.04.002.

Uebersax, JS. (1987). "Diversity of decisionmaking models and the measurement of interrater agreement" . Psychological Bulletin 101: 140–146. doi:10.1037/00332909.101.1.140.

Galton, F. (1892). Finger Prints Macmillan, London.

Smeeton, N.C. (1985). "Early History of the Kappa Statistic". Biometrics 41: 795.

Cohen, Jacob (1960). "A coefficient of agreement for nominal scales". Educational and Psychological Measurement 20 (1): 37–46. doi:10.1177/001316446002000104

Powers, David M. W. (2012). "The Problem with Kappa" . Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUSUNSUP Workshop.

Kilem Gwet (May 2002). "InterRater Reliability: Dependency on Trait Prevalence and Marginal Homogeneity" . Statistical Methods for InterRater Reliability Assessment 2: 1–10.
Terminology and derivations from a confusion matrix
true positive (TP)
eqv. with hit
true negative (TN)
eqv. with correct rejection
false positive (FP)
eqv. with false alarm, Type I error
false negative (FN)
eqv. with miss, Type II error
sensitivity or true positive rate (TPR)
eqv. with hit rate, recall
specificity (SPC) or true negative rate
precision or positive predictive value (PPV)
negative predictive value (NPV)
fallout or false positive rate (FPR)
false discovery rate (FDR)
accuracy (ACC)
F1 score
is the harmonic mean of precision and sensitivity
Matthews correlation coefficient (MCC)
Informedness
Markedness
Sources: Fawcett (2006) and Powers (2011).[1][2]
Definitions
Imagine a study evaluating a new test that screens people for a disease. Each person taking the test either has or does not have the disease. The test outcome can be positive (predicting that the person has the disease) or negative (predicting that the person does not have the disease). The test results for each subject may or may not match the subject's actual status. In that setting:

True positive: Sick people correctly diagnosed as sick

False positive: Healthy people incorrectly identified as sick

True negative: Healthy people correctly identified as healthy

False negative: Sick people incorrectly identified as healthy
In general, Positive = identified and negative = rejected. Therefore:

True positive = correctly identified

False positive = incorrectly identified

True negative = correctly rejected

False negative = incorrectly rejected
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:
Sensitivity and specificity
Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function:

Sensitivity (also called the true positive rate, or the recall in some fields) measures the proportion of actual positives which are correctly identified as such (e.g., the percentage of sick people who are correctly identified as having the condition), and is complementary to the false negative rate.

Specificity (also called the true negative rate) measures the proportion of negatives which are correctly identified as such (e.g., the percentage of healthy people who are correctly identified as not having the condition), and is complementary to the false positive rate.
For any test, there is usually a tradeoff between the measures. For instance, in an airport security setting in which one is testing for potential threats to safety, scanners may be set to trigger on lowrisk items like belt buckles and keys (low specificity), in order to reduce the risk of missing objects that do pose a threat to the aircraft and those aboard (high sensitivity). This tradeoff can be represented graphically as a receiver operating characteristic curve.
A perfect predictor would be described as 100% sensitive (e.g., all sick are identified as sick) and 100% specific (e.g., all healthy are not identified as sick); however, theoretically any predictor will possess a minimum error bound known as the Bayes error rate.
Sensitivity
Sensitivity relates to the test's ability to identify a condition correctly. Consider the example of a medical test used to identify a disease. Sensitivity of the test is the proportion of people known to have the disease, who test positive for it. Mathematically, this can be expressed as:
A negative result in a test with high sensitivity is useful for ruling out disease. A high sensitivity test is reliable when its result is negative, since it rarely misdiagnoses those who have the disease. A test with 100% sensitivity will recognize all patients with the disease by testing positive. A negative test result would definitively rule out presence of the disease in a patient.
A positive result in a test with high sensitivity is not useful for ruling in disease. Suppose a 'bogus' test kit is designed to show only one reading, positive. When used on diseased patients, all patients test positive, giving the test 100% sensitivity. However, sensitivity by definition does not take into account false positives. The bogus test also returns positive on all healthy patients, giving it a false positive rate of 100%, rendering it useless for diagnosing or "ruling in" the disease.
Sensitivity is not the same as the precision or positive predictive value (ratio of true positives to combined true and false positives), which is as much a statement about the proportion of actual positives in the population being tested as it is about the test.
The calculation of sensitivity does not take into account indeterminate test results. If a test cannot be repeated, indeterminate samples either should be excluded from the analysis (the number of exclusions should be stated when quoting sensitivity) or can be treated as false negatives (which gives the worstcase value for sensitivity and may therefore underestimate it).
A test with high sensitivity has a low type II error rate. In nonmedical contexts, sensitivity is sometimes called recall.
Specificity
Specificity relates to the test's ability to exclude a condition correctly. Consider the example of a medical test for diagnosing a disease. Specificity of a test is the proportion of healthy patients known not to have the disease, who will test negative for it. Mathematically, this can also be written as:
Positive result in a test with high specificity is useful for ruling in disease. The test rarely gives positive results in healthy patients. A test with 100% specificity will read negative, and accurately exclude disease from all healthy patients. A positive result will highlight a high probability of the presence of disease.[3]
Negative result in a test with high specificity is not useful for ruling out disease. Assume a 'bogus' test is designed to read only negative. This is administered to healthy patients, and reads negative on all of them. This will give the test a specificity of 100%. Specificity by definition does not take into account false negatives. The same test will also read negative on diseased patients, therefore it has a false negative rate of 100%, and will be useless for ruling out disease.
A test with a high specificity has a low type I error rate.
Estimation of errors in quoted sensitivity or specificity
Sensitivity and specificity values alone may be highly misleading. The 'worstcase' sensitivity or specificity must be calculated in order to avoid reliance on experiments with few results. For example, a particular test may easily show 100% sensitivity if tested against the gold standard four times, but a single additional test against the gold standard that gave a poor result would imply a sensitivity of only 80%. A common way to do this is to state the binomial proportion confidence interval, often calculated using a Wilson score interval.
Confidence intervals for sensitivity and specificity can be calculated, giving the range of values within which the correct value lies at a given confidence level (e.g., 95%).[12]
Terminology in information retrieval
In information retrieval, the positive predictive value is called precision, and sensitivity is called recall. Unlike the Specificity vs Sensitivity tradeoff, these measures are both independent of the number of true negatives, which is generally unknown and much larger than the actual numbers of relevant and retrieved documents. This assumption of very large numbers of true negatives versus positives is rare in other applications.[2]
The Fscore can be used as a single measure of performance of the test for the positive class. The Fscore is the harmonic mean of precision and recall:
In the traditional language of statistical hypothesis testing, the sensitivity of a test is called the statistical power of the test, although the word power in that context has a more general usage that is not applicable in the present context. A sensitive test will have fewer Type II errors.
References

Fawcett, Tom (2006). "An Introduction to ROC Analysis". Pattern Recognition Letters 27 (8): 861–874. doi:10.1016/j.patrec.2005.10.010.

Powers, David M W (2011). "Evaluation: From Precision, Recall and FMeasure to ROC, Informedness, Markedness & Correlation" (PDF). Journal of Machine Learning Technologies 2 (1): 37–63.

"SpPins and SnNouts". Centre for Evidence Based Medicine (CEBM). Retrieved 26 December 2013.

Mangrulkar, Rajesh. "Diagnostic Reasoning I and II". Retrieved 24 January 2012.

"EvidenceBased Diagnosis". Michigan State University.

"Sensitivity and Specificity". Emory University Medical School Evidence Based Medicine course.

Baron, JA (Apr–Jun 1994). "Too bad it isn't true.....". Medical decision making : an international journal of the Society for Medical Decision Making 14 (2): 107. doi:10.1177/0272989X9401400202. PMID 8028462.

Boyko, EJ (Apr–Jun 1994). "Ruling out or ruling in disease with the most sensitive or specific diagnostic test: short cut or wrong turn?". Medical decision making : an international journal of the Society for Medical Decision Making 14 (2): 175–179. doi:10.1177/0272989X9401400210. PMID 8028470.

Pewsner, D; Battaglia, M; Minder, C; Marx, A; Bucher, HC; Egger, M (Jul 24, 2004). "Ruling a diagnosis in or out with "SpPIn" and "SnNOut": a note of caution". BMJ (Clinical research ed.) 329 (7459): 209–13. doi:10.1136/bmj.329.7459.209. PMC 487735. PMID 15271832.

Gale, SD; Perkel, DJ (Jan 20, 2010). "A basal ganglia pathway drives selective auditory responses in songbird dopaminergic neurons via disinhibition". The Journal of neuroscience : the official journal of the Society for Neuroscience 30 (3): 1027–1037. doi:10.1523/JNEUROSCI.358509.2010. PMC 2824341. PMID 20089911.

Macmillan, Neil A.; Creelman, C. Douglas (15 September 2004). Detection Theory: A User's Guide. Psychology Press. p. 7. ISBN 9781410611147.

"Diagnostic test online calculator calculates sensitivity, specificity, likelihood ratios and predictive values from a 2x2 table  calculator of confidence intervals for predictive parameters". medcalc.org.
Accuracy and precision
Accuracy and precision are defined in terms of systematic and random errors. The more common definition associates accuracy with systematic errors and precision with random errors. Another definition, advanced by ISO, associates trueness with systematic errors and precision with random errors, and defines accuracy as the combination of both trueness and precision.
Common definition
In the fields of science, engineering, industry, and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's true value.[1] The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.[1][2] Although the two words precision and accuracy can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.
A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. The result would be a consistent yet inaccurate string of results from the flawed experiment. Eliminating the systematic error improves accuracy but does not change precision.
A measurement system is considered valid if it is both accurate and precise. Related terms include bias (nonrandom or directed effects caused by a factor or factors unrelated to the independent variable) and error (random variability).
The terminology is also applied to indirect measurements—that is, values obtained by a computational procedure from observed data.
In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement.
In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number of decimal or binary digits.
Statistical literature prefers to use the terms bias and variability instead of accuracy and precision. Bias is the amount of inaccuracy and variability is the amount of imprecision.
Accuracy is the proximity of measurement results to the true value; precision, the repeatability, or reproducibility of the measurement
Quantification
In industrial instrumentation, accuracy is the measurement tolerance, or transmission of the instrument and defines the limits of the errors made when the instrument is used in normal operating conditions.[3]
Ideally a measurement device is both accurate and precise, with measurements all close to and tightly clustered around the true value. The accuracy and precision of a measurement process is usually established by repeatedly measuring some traceable reference standard. Such standards are defined in the International System of Units (abbreviated SI from French: Système international d'unités) and maintained by national standards organizations such as the National Institute of Standards and Technology in the United States.
This also applies when measurements are repeated and averaged. In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of measurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged measurements will be closer to a normal distribution than that of individual measurements.
With regard to accuracy we can distinguish:

the difference between the mean of the measurements and the reference value, the bias. Establishing and correcting for bias is necessary for calibration.

the combined effect of that and precision.
A common convention in science and engineering is to express accuracy and/or precision implicitly by means of significant figures. Here, when not explicitly stated, the margin of error is understood to be onehalf the value of the last significant place. For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 8,436 m would imply a margin of error of 0.5 m (the last significant digits are the units).
A reading of 8,000 m, with trailing zeroes and no decimal point, is ambiguous; the trailing zeroes may or may not be intended as significant figures. To avoid this ambiguity, the number could be represented in scientific notation: 8.0 × 103 m indicates that the first zero is significant (hence a margin of 50 m) while 8.000 × 103 m indicates that all three zeroes are significant, giving a margin of 0.5 m. Similarly, it is possible to use a multiple of the basic measurement unit: 8.0 km is equivalent to 8.0 × 103 m. In fact, it indicates a margin of 0.05 km (50 m). However, reliance on this convention can lead to false precision errors when accepting data from sources that do not obey it.
Precision is sometimes stratified into:

Repeatability — the variation arising when all efforts are made to keep conditions constant by using the same instrument and operator, and repeating during a short time period; and

Reproducibility — the variation arising using the same measurement process among different instruments and operators, and over longer time periods.
In binary classification
Accuracy is also used as a statistical measure of how well a binary classification test correctly identifies or excludes a condition.
That is, the accuracy is the proportion of true results (both true positives and true negatives) among the total number of cases examined.[6] To make the context clear by the semantics, it is often referred to as the "rand accuracy".[citation needed] It is a parameter of the test.
On the other hand, precision or positive predictive value is defined as the proportion of the true positives against all the positive results (both true positives and false positives)
An accuracy of 100% means that the measured values are exactly the same as the given values.
Also see Sensitivity and specificity.
Accuracy may be determined from sensitivity and specificity, provided prevalence is known, using the equation:
The accuracy paradox for predictive analytics states that predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy. It may be better to avoid the accuracy metric in favor of other metrics such as precision and recall.[citation needed] In situations where the minority class is more important, Fmeasure may be more appropriate, especially in situations with very skewed class imbalance.
Another useful performance measure is the balanced accuracy[citation needed] which avoids inflated performance estimates on imbalanced datasets. It is defined as the arithmetic mean of sensitivity and specificity, or the average accuracy obtained on either class:
If the classifier performs equally well on either class, this term reduces to the conventional accuracy (i.e., the number of correct predictions divided by the total number of predictions). In contrast, if the conventional accuracy is above chance only because the classifier takes advantage of an imbalanced test set, then the balanced accuracy, as appropriate, will drop to chance.[7] A closely related chance corrected measure is:
A direct approach to debiasing and renormalizing Accuracy is Cohen's kappa, whilst Informedness has been shown to be a Kappafamily debiased renormalization of Recall.[9] Informedness and Kappa have the advantage that chance level is defined to be 0, and they have the form of a probability. Informedness has the stronger property that it is the probability that an informed decision is made (rather than a guess), when positive. When negative this is still true for the absolutely value of Informedness, but the information has been used to force an incorrect response.[8]
References

JCGM 200:2008 International vocabulary of metrology — Basic and general concepts and associated terms (VIM)

Taylor, John Robert (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. pp. 128–129. ISBN 093570275X.

Creus, Antonio. Instrumentación Industrial[citation needed]

BS ISO 57251: "Accuracy (trueness and precision) of measurement methods and results  Part 1: General principles and definitions.", p.1 (1994)

BS 54971: "Precision of test methods. Guide for the determination of repeatability and reproducibility for a standard test method." (1979)

Metz, CE (October 1978). "Basic principles of ROC analysis" . Semin Nucl Med. 8 (4): 283–98.

Brodersen, K.H.; Ong, C.S.; Stephan, K.E.; Buhmann, J.M. (2010). "The balanced accuracy and its posterior distribution". Proceedings of the 20th International Conference on Pattern Recognition: 3121–24.

Powers, David M W (2011). "Evaluation: From Precision, Recall and FMeasure to ROC, Informedness, Markedness & Correlation" . Journal of Machine Learning Technologies 2 (1): 37–63.

Powers, David M. W. (2012). The Problem with Kappa. Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUSUNSUP Workshop.
F1 score
In statistical analysis of binary classification, the F1 score (also Fscore or Fmeasure) is a measure of a test's accuracy. It considers both the precision p and the recall r of the test to compute the score: p is the number of correct positive results divided by the number of all positive results, and r is the number of correct positive results divided by the number of positive results that should have been returned. The F1 score can be interpreted as a weighted average of the precision and recall, where an F1 score reaches its best value at 1 and worst score at 0.
The traditional Fmeasure or balanced Fscore (F1 score) is the harmonic mean of precision and recall:
The general formula for positive real β is:
The formula in terms of Type I and type II errors:
Two other commonly used F measures are the
measure, which weights recall higher than precision, and the
measure, which puts more emphasis on precision than recall.
The Fmeasure was derived so that
"measures the effectiveness of retrieval with respect to a user who attaches β times as much importance to recall as precision".[1] It is based on Van Rijsbergen's effectiveness measure
Their relationship is
where
.
Gmeasure
While the Fmeasure is the harmonic mean of Recall and Precision, the Gmeasure is the geometric mean.[2]
References

Van Rijsbergen, C. J. (1979). Information Retrieval (2nd ed.). Butterworth.

Powers, David M W (2011). "Evaluation: From Precision, Recall and FMeasure to ROC, Informedness, Markedness & Correlation" . Journal of Machine Learning Technologies 2 (1): 37–63.
Receiver operating characteristic
In statistics, a receiver operating characteristic (ROC), or ROC curve, is a graphical plot that illustrates the performance of a binary classifier system as its discrimination threshold is varied. The curve is created by plotting the true positive rate against the false positive rate at various threshold settings. (The truepositive rate is also known as sensitivity in biomedical informatics, or recall in machine learning. The falsepositive rate is also known as the fallout and can be calculated as 1  specificity). The ROC curve is thus the sensitivity as a function of fallout. In general, if the probability distributions for both detection and false alarm are known, the ROC curve can be generated by plotting the cumulative distribution function (area under the probability distribution from
to
) of the detection probability in the yaxis versus the cumulative distribution function of the falsealarm probability in xaxis.
ROC analysis provides tools to select possibly optimal models and to discard suboptimal ones independently from (and prior to specifying) the cost context or the class distribution. ROC analysis is related in a direct and natural way to cost/benefit analysis of diagnostic decision making.
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was soon introduced to psychology to account for perceptual detection of stimuli. ROC analysis since then has been used in medicine, radiology, biometrics, and other areas for many decades and is increasingly used in machine learning and data mining research.
The ROC is also known as a relative operating characteristic curve, because it is a comparison of two operating characteristics (TPR and FPR) as the criterion changes.[1]
Basic concept
A classification model (classifier or diagnosis) is a mapping of instances between certain classes/groups. The classifier or diagnosis result can be a real value (continuous output), in which case the classifier boundary between classes must be determined by a threshold value (for instance, to determine whether a person has hypertension based on a blood pressure measure). Or it can be a discrete class label, indicating one of the classes.
Let us consider a twoclass prediction problem (binary classification), in which the outcomes are labeled either as positive (p) or negative (n). There are four possible outcomes from a binary classifier. If the outcome from a prediction is p and the actual value is also p, then it is called a true positive (TP); however if the actual value is n then it is said to be a false positive (FP). Conversely, a true negative (TN) has occurred when both the prediction outcome and the actual value are n, and false negative (FN) is when the prediction outcome is n while the actual value is p.
To get an appropriate example in a realworld problem, consider a diagnostic test that seeks to determine whether a person has a certain disease. A false positive in this case occurs when the person tests positive, but actually does not have the disease. A false negative, on the other hand, occurs when the person tests negative, suggesting they are healthy, when they actually do have the disease.
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:
ROC space
The contingency table can derive several evaluation "metrics" (see infobox). To draw a ROC curve, only the true positive rate (TPR) and false positive rate (FPR) are needed (as functions of some classifier parameter). The TPR defines how many correct positive results occur among all positive samples available during the test. FPR, on the other hand, defines how many incorrect positive results occur among all negative samples available during the test.
A ROC space is defined by FPR and TPR as x and y axes respectively, which depicts relative tradeoffs between true positive (benefits) and false positive (costs). Since TPR is equivalent to sensitivity and FPR is equal to 1 − specificity, the ROC graph is sometimes called the sensitivity vs (1 − specificity) plot. Each prediction result or instance of a confusion matrix represents one point in the ROC space.
The best possible prediction method would yield a point in the upper left corner or coordinate (0,1) of the ROC space, representing 100% sensitivity (no false negatives) and 100% specificity (no false positives). The (0,1) point is also called a perfect classification. A completely random guess would give a point along a diagonal line (the socalled line of nodiscrimination) from the left bottom to the top right corners (regardless of the positive and negative base rates). An intuitive example of random guessing is a decision by flipping coins (heads or tails). As the size of the sample increases, a random classifier's ROC point migrates towards (0.5,0.5).
The diagonal divides the ROC space. Points above the diagonal represent good classification results (better than random), points below the line poor results (worse than random). Note that the output of a consistently poor predictor could simply be inverted to obtain a good predictor.
Let us look into four prediction results from 100 positive and 100 negative instances:
Plots of the four results above in the ROC space are given in the figure. The result of method A clearly shows the best predictive power among A, B, and C. The result of B lies on the random guess line (the diagonal line), and it can be seen in the table that the accuracy of B is 50%. However, when C is mirrored across the center point (0.5,0.5), the resulting method C′ is even better than A. This mirrored method simply reverses the predictions of whatever method or test produced the C contingency table. Although the original C method has negative predictive power, simply reversing its decisions leads to a new predictive method C′ which has positive predictive power. When the C method predicts p or n, the C′ method would predict n or p, respectively. In this manner, the C′ test would perform the best. The closer a result from a contingency table is to the upper left corner, the better it predicts, but the distance from the random guess line in either direction is the best indicator of how much predictive power a method has. If the result is below the line (i.e. the method is worse than a random guess), all of the method's predictions must be reversed in order to utilize its power, thereby moving the result above the random guess line.
Curves in ROC space
Classifications are often based on a continuous random variable. Write the probability for belonging in the class as a function of a decision/threshold parameter
as
and the probability of not belonging to the class as
. The false positive rate FPR is given by
, dT and the true positive rate is
, dT . The ROC curve plots parametrically TPR(T) versus FPR(T) with T as the varying parameter.
Further interpretations
Sometimes, the ROC is used to generate a summary statistic. Common versions are:

the intercept of the ROC curve with the line at 90 degrees to the nodiscrimination line (also called Youden's J statistic)

the area between the ROC curve and the nodiscrimination line[citation needed]

the area under the ROC curve, or "AUC" ("Area Under Curve"), or A' (pronounced "aprime"),[3] or "cstatistic".[4]

d' (pronounced "dprime"), the distance between the mean of the distribution of activity in the system under noisealone conditions and its distribution under signalalone conditions, divided by their standard deviation, under the assumption that both these distributions are normal with the same standard deviation. Under these assumptions, it can be proved that the shape of the ROC depends only on d'.
However, any attempt to summarize the ROC curve into a single number loses information about the pattern of tradeoffs of the particular discriminator algorithm.
Area under the curve
When using normalized units, the area under the curve (often referred to as simply the AUC, or AUROC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative').[5] This can be seen as follows: the area under the curve is given by (the integral boundaries are reversed as large T has a lower value on the xaxis)
.The angular brackets denote average from the distribution of negative samples.
It can further be shown that the AUC is closely related to the Mann–Whitney U,[6][7] which tests whether positives are ranked higher than negatives. It is also equivalent to the Wilcoxon test of ranks.[7] The AUC is related to the Gini coefficient
by the formula
, where:
In this way, it is possible to calculate the AUC by using an average of a number of trapezoidal approximations.
It is also common to calculate the Area Under the ROC Convex Hull (ROC AUCH = ROCH AUC) as any point on the line segment between two prediction results can be achieved by randomly using one or other system with probabilities proportional to the relative length of the opposite component of the segment.[9] Interestingly, it is also possible to invert concavities – just as in the figure the worse solution can be reflected to become a better solution; concavities can be reflected in any line segment, but this more extreme form of fusion is much more likely to overfit the data.[10]
The machine learning community most often uses the ROC AUC statistic for model comparison.[11] However, this practice has recently been questioned based upon new machine learning research that shows that the AUC is quite noisy as a classification measure[12] and has some other significant problems in model comparison.[13][14] A reliable and valid AUC estimate can be interpreted as the probability that the classifier will assign a higher score to a randomly chosen positive example than to a randomly chosen negative example. However, the critical research[12][13] suggests frequent failures in obtaining reliable and valid AUC estimates. Thus, the practical value of the AUC measure has been called into question,[14] raising the possibility that the AUC may actually introduce more uncertainty into machine learning classification accuracy comparisons than resolution. Nonetheless, the coherence of AUC as a measure of aggregated classification performance has been vindicated, in terms of a uniform rate distribution,[15] and AUC has been linked to a number of other performance metrics such as the Brier score.[16]
One recent explanation of the problem with ROC AUC is that reducing the ROC Curve to a single number ignores the fact that it is about the tradeoffs between the different systems or performance points plotted and not the performance of an individual system, as well as ignoring the possibility of concavity repair, so that related alternative measures such as Informedness[17] or DeltaP are recommended.[18] These measures are essentially equivalent to the Gini for a single prediction point with DeltaP' = Informedness = 2AUC1, whilst DeltaP = Markedness represents the dual (viz. predicting the prediction from the real class) and their geometric mean is the Matthews correlation coefficient.[17]
Other measures
In engineering, the area between the ROC curve and the nodiscrimination line is sometimes preferred (equivalent to subtracting 0.5 from the AUC), and referred to as the discrimination.[citation needed] In psychophysics, the Sensitivity Index d' (dprime), ΔP' or DeltaP' is the most commonly used measure[19] and is equivalent to twice the discrimination, being equal also to Informedness, deskewed WRAcc and Gini Coefficient in the single point case (single parameterization or single system).[17] These measures all have the advantage that 0 represents chance performance whilst 1 represents perfect performance, and 1 represents the "perverse" case of full informedness used to always give the wrong response.[20]
These varying choices of scale are fairly arbitrary since chance performance always has a fixed value: for AUC it is 0.5, but these alternative scales bring chance performance to 0 and allow them to be interpreted as Kappa statistics. Informedness has been shown to have desirable characteristics for Machine Learning versus other common definitions of Kappa such as Cohen Kappa and Fleiss Kappa.[17][21]
Sometimes it can be more useful to look at a specific region of the ROC Curve rather than at the whole curve. It is possible to compute partial AUC.[22] For example, one could focus on the region of the curve with low false positive rate, which is often of prime interest for population screening tests.[23] Another common approach for classification problems in which P ≪ N (common in bioinformatics applications) is to use a logarithmic scale for the xaxis.[24]
Detection error tradeoff graph
An alternative to the ROC curve is the detection error tradeoff (DET) graph, which plots the false negative rate (missed detections) vs. the false positive rate (false alarms) on nonlinearly transformed x and yaxes. The transformation function is the quantile function of the normal distribution, i.e., the inverse of the cumulative normal distribution. It is, in fact, the same transformation as zROC, below, except that the complement of the hit rate, the miss rate or false negative rate, is used. This alternative spends more graph area on the region of interest. Most of the ROC area is of little interest; one primarily cares about the region tight against the yaxis and the top left corner – which, because of using miss rate instead of its complement, the hit rate, is the lower left corner in a DET plot. The DET plot is used extensively in the automatic speaker recognition community, where the name DET was first used. The analysis of the ROC performance in graphs with this warping of the axes was used by psychologists in perception studies halfway the 20th century, where this was dubbed "double probability paper".[citation needed]
Zscore
If a standard score is applied to the ROC curve, the curve will be transformed into a straight line.[25] This zscore is based on a normal distribution with a mean of zero and a standard deviation of one. In memory strength theory, one must assume that the zROC is not only linear, but has a slope of 1.0. The normal distributions of targets (studied objects that the subjects need to recall) and lures (non studied objects that the subjects attempt to recall) is the factor causing the zROC to be linear.
The linearity of the zROC curve depends on the standard deviations of the target and lure strength distributions. If the standard deviations are equal, the slope will be 1.0. If the standard deviation of the target strength distribution is larger than the standard deviation of the lure strength distribution, then the slope will be smaller than 1.0. In most studies, it has been found that the zROC curve slopes constantly fall below 1, usually between 0.5 and 0.9.[26] Many experiments yielded a zROC slope of 0.8. A slope of 0.8 implies that the variability of the target strength distribution is 25% larger than the variability of the lure strength distribution.[27]
Another variable used is d' (d prime) (discussed above in "Other measures"), which can easily be expressed in terms of zvalues. Although d' is a commonly used parameter, it must be recognized that it is only relevant when strictly adhering to the very strong assumptions of strength theory made above.[28]
The zscore of an ROC curve is always linear, as assumed, except in special situations. The Yonelinas familiarityrecollection model is a twodimensional account of recognition memory. Instead of the subject simply answering yes or no to a specific input, the subject gives the input a feeling of familiarity, which operates like the original ROC curve. What changes, though, is a parameter for Recollection (R). Recollection is assumed to be allornone, and it trumps familiarity. If there were no recollection component, zROC would have a predicted slope of 1. However, when adding the recollection component, the zROC curve will be concave up, with a decreased slope. This difference in shape and slope result from an added element of variability due to some items being recollected. Patients with anterograde amnesia are unable to recollect, so their Yonelinas zROC curve would have a slope close to 1.0.[29]
ROC curves beyond binary classification
The extension of ROC curves for classification problems with more than two classes has always been cumbersome, as the degrees of freedom increase quadratically with the number of classes, and the ROC space has c(c1) dimensions, where c is the number of classes.[35] Some approaches have been made for the particular case with three classes (threeway ROC).[36] The calculation of the volume under the ROC surface (VUS) has been analyzed and studied as a performance metric for multiclass problems.[37] However, because of the complexity of approximating the true VUS, some other approaches [38] based on an extension of AUC are more popular as an evaluation metric.
Given the success of ROC curves for the assessment of classification models, the extension of ROC curves for other supervised tasks has also been investigated. Notable proposals for regression problems are the socalled regression error characteristic (REC) Curves [39] and the Regression ROC (RROC) curves.[40] In the latter, RROC curves become extremely similar to ROC curves for classification, with the notions of asymmetry, dominance and convex hull. Also, the area under RROC curves is proportional to the error variance of the regression model.
ROC curve is related to the lift and uplift curves,[41][42] which are used in uplift modelling. The ROC curve itself has also been used as the optimization metric in uplift modeling.[43][44]
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Matthews correlation coefficient
The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (twoclass) classifications, introduced by biochemist Brian W. Matthews in 1975.[1] 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 chisquare statistic for a 2×2 contingency table
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[citation needed]. 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:
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:[1]
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 (deltap) and informedness (deltap').[2][3]
References

Matthews, B. W. (1975). "Comparison of the predicted and observed secondary structure of T4 phage lysozyme". Biochimica et Biophysica Acta (BBA)  Protein Structure 405 (2): 442–451. doi:10.1016/00052795(75)901099.

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Powers, David M W (2011). "Evaluation: From Precision, Recall and FMeasure to ROC, Informedness, Markedness & Correlation" . Journal of Machine Learning Technologies 2 (1): 37–63.

Fawcelt, Tom (2006). "An Introduction to ROC Analysis". Pattern Recognition Letters 27 (8): 861 – 874. doi:10.1016/j.patrec.2005.10.010.
* https://en.wikipedia.org/