# Implemented Performance Measures

This page shows the performance measures available for the different types of learning problems as well as general performance measures in alphabetical order. (See also the documentation about measures and makeMeasure for available measures and their properties.)

If you find that a measure is missing, you can either open an issue or try to implement a measure yourself.

Column Minim. indicates if the measure is minimized during, e.g., tuning or feature selection. Best and Worst show the best and worst values the performance measure can attain. For classification, column Multi indicates if a measure is suitable for multi-class problems. If not, the measure can only be used for binary classification problems.

The next six columns refer to information required to calculate the performance measure.

• Pred.: The Prediction object.
• Truth: The true values of the response variable(s) (for supervised learning).
• Probs: The predicted probabilities (might be needed for classification).
• Model: The WrappedModel (e.g., for calculating the training time).
• Feats: The predicted data (relevant for clustering).

Aggr. shows the default aggregation method tied to the measure.

### Classification

ID / Name Minim. Best Worst Multi Pred. Truth Probs Model Task Feats Aggr. Note
acc
Accuracy
1 0 X X X test.mean Defined as: mean(response == truth)
auc
Area under the curve
1 0 X X X test.mean Integral over the graph that results from computing fpr and tpr for many different thresholds.
bac
Balanced accuracy
1 0 X X test.mean Mean of true positive rate and true negative rate.
ber
Balanced error rate
X 0 1 X X X test.mean Mean of misclassification error rates on all individual classes.
brier
Brier score
X 0 1 X X X test.mean The Brier score is defined as the quadratic difference between the probability and the value (1,0) for the class. That means we use the numeric representation 1 and 0 for our target classes. It is similiar to the mean squared error in regression. multiclass.brier is the sum over all one vs. all comparisons and for a binary classifcation 2 * brier.
brier.scaled
Brier scaled
1 0 X X X test.mean Brier score scaled to [0,1], see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3575184/.
f1
F1 measure
1 0 X X test.mean Defined as: 2 * tp/ (sum(truth == positive) + sum(response == positive))
fdr
False discovery rate
X 0 1 X X test.mean Defined as: (fp) / (tn + fn).
fn
False negatives
X 0 Inf X X test.mean Sum of misclassified observations in the negative class. Also called misses.
fnr
False negative rate
X 0 1 X X test.mean Percentage of misclassified observations in the negative class.
fp
False positives
X 0 Inf X X test.mean Sum of misclassified observations in the positive class. Also called false alarms.
fpr
False positive rate
X 0 1 X X test.mean Percentage of misclassified observations in the positive class. Also called false alarm rate or fall-out.
gmean
G-mean
1 0 X X test.mean Geometric mean of recall and specificity.
gpr
Geometric mean of precision and recall.
1 0 X X test.mean Defined as: sqrt(ppv * tpr)
kappa
Cohen's kappa
1 -1 X X X test.mean Defined as: 1 - (1 - p0) / (1 - pe). With: p0 = 'observed frequency of agreement' and pe = 'expected agremeent frequency under independence
logloss
Logarithmic loss
X 0 Inf X X X test.mean Defined as: -mean(log(p_i)), where p_i is the predicted probability of the true class of observation i. Inspired by https://www.kaggle.com/wiki/MultiClassLogLoss.
lsr
Logarithmic Scoring Rule
0 -Inf X X X test.mean Defined as: mean(log(p_i)), where p_i is the predicted probability of the true class of observation i. This scoring rule is the same as the negative logloss, self-information or surprisal. See: Bickel, J. E. (2007). Some comparisons among quadratic, spherical, and logarithmic scoring rules. Decision Analysis, 4(2), 49-65.
mcc
Matthews correlation coefficient
1 -1 X X test.mean Defined as sqrt((tp + fp) * (tp + fn) * (tn + fp) * (tn + fn)), denominator set to 1 if 0
mmce
Mean misclassification error
X 0 1 X X X test.mean Defined as: mean(response != truth)
multiclass.au1p
Weighted average 1 vs. 1 multiclass AUC
1 0.5 X X X X test.mean Computes AUC of c(c - 1) binary classifiers while considering the a priori distribution of the classes. See Ferri et al.: https://www.math.ucdavis.edu/~saito/data/roc/ferri-class-perf-metrics.pdf.
multiclass.au1u
Average 1 vs. 1 multiclass AUC
1 0.5 X X X X test.mean Computes AUC of c(c - 1) binary classifiers (all possible pairwise combinations) while considering uniform distribution of the classes. See Ferri et al.: https://www.math.ucdavis.edu/~saito/data/roc/ferri-class-perf-metrics.pdf.
multiclass.aunp
Weighted average 1 vs. rest multiclass AUC
1 0.5 X X X X test.mean Computes the AUC treating a c-dimensional classifier as c two-dimensional classifiers, taking into account the prior probability of each class. See Ferri et al.: https://www.math.ucdavis.edu/~saito/data/roc/ferri-class-perf-metrics.pdf.
multiclass.aunu
Average 1 vs. rest multiclass AUC
1 0.5 X X X X test.mean Computes the AUC treating a c-dimensional classifier as c two-dimensional classifiers, where classes are assumed to have uniform distribution, in order to have a measure which is independent of class distribution change. See Ferri et al.: https://www.math.ucdavis.edu/~saito/data/roc/ferri-class-perf-metrics.pdf.
multiclass.brier
Multiclass Brier score
X 0 2 X X X X test.mean Defined as: (1/n) sum_i sum_j (y_ij - p_ij)^2, where y_ij = 1 if observation i has class j (else 0), and p_ij is the predicted probability of observation i for class j. From http://docs.lib.noaa.gov/rescue/mwr/078/mwr-078-01-0001.pdf.
npv
Negative predictive value
1 0 X X test.mean Defined as: (tn) / (tn + fn).
ppv
Positive predictive value
1 0 X X test.mean Defined as: tp / (tp + number of fp). Also called precision. If the denominator is 0, PPV is set to be either 1 or 0 depending on whether the highest probability prediction is positive (1) or negative (0).
qsr
1 -1 X X X test.mean Defined as: 1 - (1/n) sum_i sum_j (y_ij - p_ij)^2, where y_ij = 1 if observation i has class j (else 0), and p_ij is the predicted probablity of observation i for class j. This scoring rule is the same as 1 - multiclass.brier. See: Bickel, J. E. (2007). Some comparisons among quadratic, spherical, and logarithmic scoring rules. Decision Analysis, 4(2), 49-65.
ssr
Spherical Scoring Rule
1 0 X X X test.mean Defined as: mean(p_i(sum_j(p_ij))), where p_i is the predicted probability of the true class of observation i and p_ij is the predicted probablity of observation i for class j. See: Bickel, J. E. (2007). Some comparisons among quadratic, spherical, and logarithmic scoring rules. Decision Analysis, 4(2), 49-65.
tn
True negatives
Inf 0 X X test.mean Sum of correctly classified observations in the negative class. Also called correct rejections.
tnr
True negative rate
1 0 X X test.mean Percentage of correctly classified observations in the negative class. Also called specificity.
tp
True positives
Inf 0 X X test.mean Sum of all correctly classified observations in the positive class.
tpr
True positive rate
1 0 X X test.mean Percentage of correctly classified observations in the positive class. Also called hit rate or recall.
wkappa
1 -1 X X X test.mean Defined as: 1 - sum(weights * conf.mat) / sum(weights * expected.mat), the weight matrix measures seriousness of disagreement with the squared euclidean metric.

### Regression

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
arsq
1 0 X X test.mean Defined as: 1 - (1 - rsq) * (p / (n - p - 1L)). Adjusted R-squared is only defined for normal linear regression.
expvar
Explained variance
1 0 X X test.mean Similar to measure rsq (R-squared). Defined as explained_sum_of_squares / total_sum_of_squares.
kendalltau
Kendall's tau
1 -1 X X test.mean Defined as: Kendall's tau correlation between truth and response. Only looks at the order. See Rosset et al.: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.1398&rep=rep1&type=pdf.
mae
Mean of absolute errors
X 0 Inf X X test.mean Defined as: mean(abs(response - truth))
mape
Mean absolute percentage error
X 0 Inf X X test.mean Defined as the abs(truth_i - response_i) / truth_i. Won't work if any truth value is equal to zero. In this case the output will be NA.
medae
Median of absolute errors
X 0 Inf X X test.mean Defined as: median(abs(response - truth)).
medse
Median of squared errors
X 0 Inf X X test.mean Defined as: median((response - truth)^2).
mse
Mean of squared errors
X 0 Inf X X test.mean Defined as: mean((response - truth)^2)
msle
Mean squared logarithmic error
X 0 Inf X X test.mean Defined as: mean((log(response + 1, exp(1)) - log(truth + 1, exp(1)))^2). This measure is mostly used for count data, note that all predicted and actual target values must be greater or equal '-1' to compute the measure.
rae
Relative absolute error
X 0 Inf X X test.mean Defined as sum_of_absolute_errors / mean_absolute_deviation. Undefined for single instances and when every truth value is identical. In this case the output will be NA.
rmse
Root mean squared error
X 0 Inf X X test.rmse The RMSE is aggregated as sqrt(mean(rmse.vals.on.test.sets^2)). If you don't want that, you could also use test.mean.
rmsle
Root mean squared logarithmic error
X 0 Inf X X test.mean Defined as: sqrt(msle). Definition taken from: Definition taken from: https: / /www.kaggle.com / wiki / RootMeanSquaredLogarithmicError. This measure is mostly used for count data, note that all predicted and actual target values must be greater or equal '-1' to compute the measure.
rrse
Root relative squared error
X 0 Inf X X test.mean Defined as sqrt (sum_of_squared_errors / total_sum_of_squares). Undefined for single instances and when every truth value is identical. In this case the output will be NA.
rsq
Coefficient of determination
1 -Inf X X test.mean Also called R-squared, which is 1 - residual_sum_of_squares / total_sum_of_squares.
sae
Sum of absolute errors
X 0 Inf X X test.mean Defined as: sum(abs(response - truth))
spearmanrho
Spearman's rho
1 -1 X X test.mean Defined as: Spearman's rho correlation between truth and response. Only looks at the order. See Rosset et al.: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.1398&rep=rep1&type=pdf.
sse
Sum of squared errors
X 0 Inf X X test.mean Defined as: sum((response - truth)^2)

### Survival analysis

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
cindex
Harrell's Concordance index
1 0 X X test.mean Fraction of all pairs of subjects whose predicted survival times are correctly ordered among all subjects that can actually be ordered. In other words, it is the probability of concordance between the predicted and the observed survival.
cindex.uno
Uno's Concordance index
1 0 X X X test.mean Fraction of all pairs of subjects whose predicted survival times are correctly ordered among all subjects that can actually be ordered. In other words, it is the probability of concordance between the predicted and the observed survival. Corrected by weighting with IPCW as suggested by Uno. Implemented in survAUC::UnoC.
iauc.uno
Uno's estimator of cumulative AUC for right censored time-to-event data
1 0 X X X X test.mean To set an upper time limit, set argument max.time (defaults to max time in complete task). Implemented in survAUC::AUC.uno.

### Cluster analysis

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
db
Davies-Bouldin cluster separation measure
X 0 Inf X X test.mean Ratio of the within cluster scatter, to the between cluster separation, averaged over the clusters. See ?clusterSim::index.DB.
dunn
Dunn index
Inf 0 X X test.mean Defined as the ratio of the smallest distance between observations not in the same cluster to the largest intra-cluster distance. See ?clValid::dunn.
G1
Calinski-Harabasz pseudo F statistic
Inf 0 X X test.mean Defined as ratio of between-cluster variance to within cluster variance. See ?clusterSim::index.G1.
G2
Baker and Hubert adaptation of Goodman-Kruskal's gamma statistic
Inf 0 X X test.mean Defined as: (number of concordant comparisons - number of discordant comparisons) / (number of concordant comparisons + number of discordant comparisons). See ?clusterSim::index.G2.
silhouette
Rousseeuw's silhouette internal cluster quality index
Inf 0 X X test.mean Silhouette value of an observation is a measure of how similar an object is to its own cluster compared to other clusters. The measure is calculated as the average of all silhouette values. See ?clusterSim::index.S.

### Cost-sensitive classification

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
mcp
Misclassification penalty
X 0 Inf X X test.mean Average difference between costs of oracle and model prediction.
meancosts
Mean costs of the predicted choices
X 0 Inf X X test.mean Defined as: mean(y), where y is the vector of costs for the predicted classes.

Note that in case of ordinary misclassification costs you can also generate performance measures from cost matrices by function makeCostMeasure. For details see the tutorial page on cost-sensitive classification and also the page on custom performance measures.

### Multilabel classification

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
multilabel.acc
Accuracy (multilabel)
1 0 X X test.mean Averaged proportion of correctly predicted labels with respect to the total number of labels for each instance, following the definition by Charte and Charte: https: / /journal.r-project.org / archive / 2015 - 2 / charte-charte.pdf. Fractions where the denominator becomes 0 are replaced with 1 before computing the average across all instances.
multilabel.f1
F1 measure (multilabel)
1 0 X X test.mean Harmonic mean of precision and recall on a per instance basis (Micro-F1), following the definition by Montanes et al.: http: / /www.sciencedirect.com / science / article / pii / S0031320313004019. Fractions where the denominator becomes 0 are replaced with 1 before computing the average across all instances.
multilabel.hamloss
Hamming loss
X 0 1 X X test.mean Proportion of labels that are predicted incorrectly, following the definition by Charte and Charte: https://journal.r-project.org/archive/2015-2/charte-charte.pdf.
multilabel.ppv
Positive predictive value (multilabel)
1 0 X X test.mean Also called precision. Averaged ratio of correctly predicted labels for each instance, following the definition by Charte and Charte: https: / /journal.r-project.org / archive / 2015 - 2 / charte-charte.pdf. Fractions where the denominator becomes 0 are ignored in the average calculation.
multilabel.subset01
Subset-0-1 loss
X 0 1 X X test.mean Proportion of observations where the complete multilabel set (all 0-1-labels) is predicted incorrectly, following the definition by Charte and Charte: https://journal.r-project.org/archive/2015-2/charte-charte.pdf.
multilabel.tpr
TPR (multilabel)
1 0 X X test.mean Also called recall. Averaged proportion of predicted labels which are relevant for each instance, following the definition by Charte and Charte: https: / /journal.r-project.org / archive / 2015 - 2 / charte-charte.pdf. Fractions where the denominator becomes 0 are ignored in the average calculation.

### General performance measures

ID / Name Minim. Best Worst Pred. Truth Probs Model Task Feats Aggr. Note
featperc
Percentage of original features used for model
X 0 1 X X test.mean Useful for feature selection.
timeboth
timetrain + timepredict
X 0 Inf X X test.mean
timepredict
Time of predicting test set
X 0 Inf X test.mean
timetrain
Time of fitting the model
X 0 Inf X test.mean