[RECALL, PRECISION] = VL_PR(LABELS, SCORES) computes the precision-recall (PR) curve. LABELS are the ground truth labels, greather than zero for a positive sample and smaller than zero for a negative one. SCORES are the scores of the samples obtained from a classifier, where lager scores should correspond to positive samples.

Samples are ranked by decreasing scores, starting from rank 1. PRECISION(K) and RECALL(K) are the precison and recall when samples of rank smaller or equal to K-1 are predicted to be positive and the remaining to be negative. So for example PRECISION(3) is the percentage of positive samples among the two samples with largest score. PRECISION(1) is the precision when no samples are predicted to be positive and is conventionally set to the value 1.

Set to zero the lables of samples that should be ignored in the evaluation. Set to -INF the scores of samples which are not retrieved. If there are samples with -INF score, then the PR curve may have maximum recall smaller than 1, unless the INCLUDEINF option is used (see below). The options NUMNEGATIVES and NUMPOSITIVES can be used to add additional surrogate samples with -INF score (see below).

[RECALL, PRECISION, INFO] = VL_PR(...) returns an additional structure INFO with the following fields:

info.auc

  The area under the precision-recall curve. If the INTERPOLATE option is set to FALSE, then trapezoidal interpolation is used to integrate the PR curve. If the INTERPOLATE option is set to TRUE, then the curve is piecewise constant and no other approximation is introduced in the calculation of the area. In the latter case, INFO.AUC is the same as INFO.AP.

info.ap

  Average precision as defined by TREC. This is the average of the precision observed each time a new positive sample is recalled. In this calculation, any sample with -INF score (unless INCLUDEINF is used) and any additional positive induced by NUMPOSITIVES has precision equal to zero. If the INTERPOLATE option is set to true, the AP is computed from the interpolated precision and the result is the same as INFO.AUC. Note that AP as defined by TREC normally does not use interpolation [1].

info.ap_interp_11

  11-points interpolated average precision as defined by TREC. This is the average of the maximum precision for recall levels greather than 0.0, 0.1, 0.2, ..., 1.0. This measure was used in the PASCAL VOC challenge up to the 2008 edition.

info.auc_pa08

  Deprecated. It is the same of INFO.AP_INTERP_11.

VL_PR(...) with no output arguments plots the PR curve in the current axis.

VL_PR() accepts the following options:

Interpolate false

  If set to true, use interpolated precision. The interpolated precision is defined as the maximum precision for a given recall level and onwards. Here it is implemented as the culumative maximum from low to high scores of the precision.

NumPositives []
NumNegatives []

  If set to a number, pretend that LABELS contains this may positive/negative labels. NUMPOSITIVES/NUMNEGATIVES cannot be smaller than the actual number of positive/negative entrires in LABELS. The additional positive/negative labels are appended to the end of the sequence, as if they had -INF scores (not retrieved). This is useful to evaluate large retrieval systems for which one stores ony a handful of top results for efficiency reasons.

IncludeInf false

  If set to true, data with -INF score SCORES is included in the evaluation and the maximum recall is 1 even if -INF scores are present. This option does not include any additional positive or negative data introduced by specifying NUMPOSITIVES and NUMNEGATIVES.

Stable false

  If set to true, RECALL and PRECISION are returned in the same order of LABELS and SCORES rather than being sorted by decreasing score (increasing recall). Samples with -INF scores are assigned RECALL and PRECISION equal to NaN.

NormalizePrior []

  If set to a scalar, reweights positive and negative labels so that the fraction of positive ones is equal to the specified value. This computes the normalised PR curves of [2]

About the PR curve

  This section uses the same symbols used in the documentation of the VL_ROC() function. In addition to those quantities, define:

        PRECISION(S) = TP(S) / (TP(S) + FP(S))
        RECALL(S) = TPR(S) = TP(S) / P

  The precision is the fraction of positivie predictions which are correct, and the recall is the fraction of positive labels that have been correctly classified (recalled). Notice that the recall is also equal to the true positive rate for the ROC curve (see VL_ROC()).

REFERENCES: [1] C. D. Manning, P. Raghavan, and H. Schutze. An Introduction to Information Retrieval. Cambridge University Press, 2008. [2] D. Hoiem, Y. Chodpathumwan, and Q. Dai. Diagnosing error in object detectors. In Proc. ECCV, 2012.

See also VL_ROC(), VL_HELP().