ROC Curve and Performance Metrics

This topic describes the performance metrics for classification, including the receiver operating characteristic (ROC) curve and the area under a ROC curve (AUC), and introduces the rocmetrics object, which you can use to compute performance metrics for binary and multiclass classification problems.

Introduction to ROC Curve

After training a classification model, you can examine the performance of the algorithm on a specific test data set. A common approach is to compute a gross measure of performance, such as quadratic loss or accuracy, averaged over the entire test data set. You can inspect the classifier performance more closely by plotting a ROC curve and computing performance metrics. For example, you can find the threshold that maximizes the classification accuracy, or assess how the classifier performs in the regions of high sensitivity and high specificity.

Receiver Operating Characteristic (ROC) Curve

A ROC curve shows the true positive rate (TPR, or sensitivity) versus the false positive rate (FPR, or 1-specificity) for different thresholds of classification scores.

Each point on a ROC curve corresponds to a pair of TPR and FPR values for a specific threshold value. You can find different pairs of TPR and FPR values by varying the threshold value, and then create a ROC curve using the pairs.

For a multiclass classification problem, you can use the one-versus-all coding design and find a ROC curve for each class. The one-versus-all coding design treats a multiclass classification problem as a set of binary classification problems, and assumes one class as positive and the rest as negative in each binary problem.

A binary classifier typically classifies an observation into a class that yields a larger score, which corresponds to a positive adjusted score for a one-versus-all binary classification problem. That is, a classifier typically uses 0 as a threshold and determines whether an observation is positive or negative. For example, if an adjusted score for an observation is 0.2, then the classifier with a threshold value of 0 assigns the observation to the positive class. You can find a pair of TPR and FPR values by applying the threshold value to all observations, and use the pair as a single point on a ROC curve. Now, assume you use a new threshold value of 0.25. Then, the classifier with a threshold value of 0.25 assigns the observation with an adjusted score of 0.2 to the negative class. By applying the new threshold to all observations, you can find a new pair of TPR and FPR values and have a new point on the a ROC curve. By repeating this process for various threshold values, you find pairs of TPR and FPR values and create a ROC curve using the pairs.

Area Under ROC Curve (AUC)

The area under a ROC curve (AUC) corresponds to the integral of a ROC curve (TPR values) with respect to FPR from FPR = 0 to FPR = 1.

The AUC provides an aggregate performance measure across all possible thresholds. The AUC values are in the range 0 to 1, and larger AUC values indicate better classifier performance.

• A perfect classifier always correctly assigns positive class observations to the positive class and has a true positive rate of 1 for any threshold values. Therefore, the line passing through [0,0], [0,1], and [1,1] represents the perfect classifier, and the AUC value is 1.

• A random classifier returns random score values and has the same values for the false positive rate and true positive rate for any threshold values. Therefore, the ROC curve for the random classifier lies on the diagonal line, and the AUC value is 0.5.

Performance Curve with MATLAB

You can compute a ROC curve and other performance curves by creating a rocmetrics object. The rocmetrics object supports both binary and multiclass classification problems and provides the following object functions:

• plot — Plot ROC or other classifier performance curves. plot returns a ROCCurve graphics object for each curve. You can modify the properties of the objects to control the appearance of each curve. For details, see ROCCurve Properties.

• average — Compute performance metrics for an average ROC curve for multiclass problems.

• addMetrics — Compute additional classification performance metrics.

You can also compute the confidence intervals of performance curves by providing cross-validated inputs or by bootstrapping the input data. Using confidence intervals requires Statistics and Machine Learning Toolbox™.

After training a classifier, use a performance curve to evaluate the classifier performance on test data. Various measures such as mean squared error, classification error, or exponential loss can summarize the predictive power of a classifier in a single number. However, a performance curve offers more information because it lets you explore the classifier performance across a range of thresholds on the classification scores.

ROC Curve for Multiclass Classification

For a multiclass classifier, the rocmetrics function computes the performance metrics of a one-versus-all ROC curve for each class, and the average function computes the metrics for an average of the ROC curves. You can use the plot function to plot a ROC curve for each class and the average ROC curve.

One-Versus-All (OVA) Coding Design

The one-versus-all (OVA) coding design reduces a multiclass classification problem to a set of binary classification problems. In this coding design, each binary classification treats one class as positive and the rest of the classes as negative. rocmetrics uses the OVA coding design for multiclass classification and evaluates the performance on each class by using the binary classification that the class is positive.

For example, the OVA coding design for three classes formulates three binary classifications:

$$\begin{array}{cccc}& \text{Binary 1}& \text{Binary }2& \text{Binary 3}\\ \text{Class 1}& 1& -1& -1\\ \text{Class 2}& -1& 1& -1\\ \text{Class 3}& -1& -1& 1\end{array}$$

Each row corresponds to a class, and each column corresponds to a binary classification problem. The first binary classification assumes that class 1 is a positive class and the rest of the classes are negative. rocmetrics evaluates the performance on the first class by using the first binary classification problem.

rocmetrics applies the OVA coding design to a binary classification problem as well if you specify classification scores as a two-column matrix. rocmetrics formulates two one-versus-all binary classification problems each of which treats one class as a positive class and the other class as a negative class, and rocmetrics finds two ROC curves. You can use one of them to evaluate the binary classification problem.

Average of Performance Metrics

You can compute metrics for an average ROC curve by using the average function. Alternatively, you can use the plot function to compute the metrics and plot the average ROC curve. For examples, see Find Average ROC Curve (example for average).

average and plot support three algorithms for computing the average false positive rate (FPR) and average true positive rate (TPR) to find the average ROC curve:

• Micro-averaging — The software combines all one-versus-all binary classification problems into one binary classification problem and computes the average performance metrics as follows:

  1. Convert the values in the Labels property of a rocmetrics object to logical values where logical 1 (true) indicates a positive class for each binary problem.

  2. Stack the converted vectors of labels, one vector from each binary problem, into a single vector.

  3. Convert the matrix that contains the adjusted values of the classification scores (the Scores property) into a vector by stacking the columns of the matrix.

  4. Compute the components of the confusion matrix for the combined binary problem for each threshold (each distinct value of adjusted scores). A confusion matrix contains the number of instances for true positive (TP), false negative (FN), false positive (FP), and true negative (TN).

  5. Compute the average FPR and TPR based on the components of the confusion matrix.

• Macro-averaging — The software computes the average values for FPR and TPR by averaging the values of all one-versus-all binary classification problems.

  The software uses three metrics—threshold, FPR, and TPR—to compute the average values as follows:

  1. Determine a fixed metric. If you specify FixedMetric of rocmetrics as "FalsePositiveRate" or "TruePositiveRate", then the function holds the specified metric fixed. Otherwise, the function holds the threshold values fixed.

  2. Find all distinct values in the Metrics property for the fixed metric.

  3. Find the corresponding values for the other two metrics for each binary problem.

  4. Average the FPR and TPR values of all binary problems.

• Weighted macro-averaging — The software computes the weighted average values for FPR and TPR using the macro-averaging algorithm and using the prior class probabilities (the Prior property) as weights.

Performance Metrics

The rocmetrics object supports these built-in performance metrics:

• Number of true positives (TP)

• Number of false negatives (FN)

• Number of false positives (FP)

• Number of true negatives (TN)

• Sum of TP and FP

• Rate of positive predictions (RPP)

• Rate of negative predictions (RNP)

• Accuracy

• True positive rate (TPR), recall, or sensitivity

• False negative rate (FNR), or miss rate

• False positive rate (FPR), fallout, or 1-specificity

• True negative rate (TNR), or specificity

• Positive predictive value (PPV), or precision

• Negative predictive value (NPR)

• Expected cost

rocmetrics also supports a custom metric specified as a function handle. For details, see the AdditionalMetrics name-value argument of the rocmetrics function.

rocmetrics computes performance metric values for various thresholds for each one-versus-all binary classification problem using a confusion matrix, scale vector, and misclassification cost matrix. Each performance metric is a function of a confusion matrix and scale vector. The expected cost is also a function of the misclassification cost matrix, as is a custom metric.

• Confusion matrix — A confusion matrix contains the number of instances for true positive (TP), false negative (FN), false positive (FP), and true negative (TN). rocmetrics computes confusion matrices for various threshold values for each binary problem.

• Scale vector — A scale vector is defined by the prior class probabilities and the number of classes in true labels. rocmetrics finds the probabilities and number of classes for each binary problem from the prior class probabilities specified by the Prior name-value argument and the true labels specified by the Labels input argument.

• Misclassification cost matrix — rocmetrics converts the misclassification cost matrix specified by the Cost name-value argument to the values for each binary problem.

By default, rocmetrics uses all distinct adjusted score values as threshold values for each binary problem. For more details on threshold values, see Thresholds, Fixed Metric, and Fixed Metric Values.

Confusion Matrix

A confusion matrix is defined as

$$\left[\begin{array}{cc}TP& FN\\ FP& TN\end{array}\right],$$

where

• P stands for "positive".

• N stands for "negative".

• T stands for "true".

• F stands for "false".

For example, the first row of the confusion matrix defines how the classifier identifies instances of the positive class: TP is the count of correctly identified positive instances, and FN is the count of positive instances misidentified as negative.

rocmetrics computes confusion matrices for various threshold values for each one-versus-all binary classification. The one-versus-all binary classification model classifies an observation into a positive class if the score for the observation is greater than or equal to the threshold value.

Prior Class Probabilities

By default, rocmetrics uses empirical probabilities, which are class frequencies in the true labels.

rocmetrics normalizes the 1-by-K prior probability vector π to a 1-by-2 vector for each one-versus-all binary classification, where K is the number of classes.

The prior probabilities for the kth binary classification in which the positive class is the kth class is $$\left[{\pi }_k,1-{\pi }_k\right]$$, where π_k is the prior probability for class k in the multiclass problem.

Scale Vector

rocmetrics defines a scale vector s_k of size 2-by-1 for each one-versus-all binary classification problem:

$${\text{s}}_k=\frac{1}{{\pi }_{k}N+\left(1-{\pi }_k\right)P}\left[\begin{array}{c}{\pi }_{k}N\\ \left(1-{\pi }_k\right)P\end{array}\right],$$

where P and N represent the total instances of positive class and negative class, respectively. That is, P is the sum of TP and FN, and N is the sum of FP and TN. s_k(1) (first element of s_k) and s_k(2) (second element of s_k) are the scales for the positive class (kth class) and negative class (the rest), respectively.

rocmetrics applies the scale values as multiplicative factors to the counts from the corresponding class. That is, the function multiplies counts from the positive class by s_k(1) and counts from the negative class by s_k(2). For example, to compute the positive predictive value (PPV = TP/(TP+FP)) for the kth binary problem, rocmetrics scales PPV as follows:

$$PPV=\frac{s_k(1)\cdot TP}{s_k(1)\cdot TP+s_k(2)\cdot FP}.$$

Misclassification Cost Matrix

By default, rocmetrics uses a K-by-K cost matrix C, where C(i,j) = 1 if i ~= j, and C(i,j) = 0 if i = j. C(i,j) is the cost of classifying a point into class j if its true class is i (that is, the rows correspond to the true class and the columns correspond to the predicted class).

rocmetrics normalizes the K-by-K cost matrix C to a 2-by-2 matrix for each one-versus-all binary classification:

$$C_k=\left[\begin{array}{cc}0& {\text{cost}}_k(N|P)\\ {\text{cost}}_k(P|N)& 0\end{array}\right].$$

C_k is the cost matrix for the kth binary classification in which the positive class is the kth class, where costk(N|P) is the cost of misclassifying a positive class as a negative class, and costk(P|N) is the cost of misclassifying a negative class as a positive class.

For class k, let π_k⁺ and π_k^- be K-by-1 vectors with the following values:

$${\pi }_{ki}^{+}=\{\begin{array}{ll}{\pi }_i\hfill & \text{if }k=i,\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}{\pi }_{ki}^{-}=\{\begin{array}{ll}0\hfill & \text{if }k=i,\hfill \\ {\pi }_i\hfill & \text{otherwise}.\hfill \end{array}$$

π_ki⁺ and π_ki^- are the ith elements of π_k⁺ and π_k^-, respectively.

The cost of classifying a positive-class (class k) observation into the negative class (the rest) is

$${\text{cost}}_k(N|P)={\left({\pi }_k^{+}\right)}^{\prime }C{\pi }_k^{-}.$$

Similarly, the cost of classifying a negative-class observation into the positive class is

$${\text{cost}}_k(P|N)={\left({\pi }_k^{-}\right)}^{\prime }C{\pi }_k^{+}.$$

Classification Scores and Thresholds

The rocmetrics function determines threshold values from the input classification scores or the FixedMetricValues name-value argument.

Classification Score Input for rocmetrics

rocmetrics accepts classification scores (Scores) in a matrix of size n-by-K or a vector of length n, where n is the number of observations and K is the number classes. For cross-validated data, Scores can be a cell array of vectors or a cell array of matrices.

• Matrix of size n-by-K — Specify Scores using the output argument of the predict function for a deep neural network for classification. Each row of the output contains classification scores for an observation for all classes. The order of the classes matches the order of the classes in your network. You can specify Scores as a matrix for both binary classification and multiclass classification problems.

  If you use a matrix format, rocmetrics adjusts the classification scores for each class relative to the scores for the rest of the classes. Specifically, the adjusted score for a class given an observation is the difference between the score for the class and the maximum value of the scores for the rest of the classes. For more details, see Adjusted Scores for Multiclass Classification Problem.

• Vector of length n — Specify Scores using a vector when you have classification scores for one class only. A vector input is also suitable when you want to use a different type of adjusted scores for a multiclass problem. As an example, consider a problem with three classes, A, B, and C. If you want to compute a performance curve for separating classes A and B, with C ignored, you need to address the ambiguity in selecting A over B. You can use the score ratio s(A)/s(B) or score difference s(A)–s(B) and pass the vector to rocmetrics; this approach can depend on the nature of the scores and their normalization.

You can use rocmetrics with any classifier or any function that returns a numeric score for an instance of input data.

• A high score returned by a classifier for a given instance and class signifies that the instance is likely from the respective class.

• A low score signifies that the instance is not likely from the respective class.

rocmetrics does not impose any requirements on the input score range. Because of this lack of normalization, you can use rocmetrics to process scores returned by any classification, regression, or fit functions. rocmetrics does not make any assumptions about the nature of input scores.

rocmetrics is intended for use with classifiers that return scores, not those that return only predicted classes. Consider a classifier that returns only classification labels, 0 or 1, for data with two classes. In this case, the performance curve reduces to a single point because the software can split classified instances into positive and negative categories in one way only.

Adjusted Scores for Multiclass Classification Problem

For each class, rocmetrics adjusts the classification scores (input argument Scores of rocmetrics) relative to the scores for the rest of the classes if you specify Scores as a matrix. Specifically, the adjusted score for a class given an observation is the difference between the score for the class and the maximum value of the scores for the rest of the classes.

For example, if you have [s₁,s₂,s₃] in a row of Scores for a classification problem with three classes, the adjusted score values are [s₁-max(s₂,s₃),s₂-max(s₁,s₃),s₃-max(s₁,s₂)].

rocmetrics computes the performance metrics using the adjusted score values for each class.

For a binary classification problem, you can specify Scores as a two-column matrix or a column vector. Using a two-column matrix is a simpler option because the predict function of a classification object returns classification scores as a matrix, which you can pass to rocmetrics. If you pass scores in a two-column matrix, rocmetrics adjusts scores in the same way that it adjusts scores for multiclass classification, and it computes performance metrics for both classes. You can use the metric values for one of the two classes to evaluate the binary classification problem. The metric values for a class returned by rocmetrics when you pass a two-column matrix are equivalent to the metric values returned by rocmetrics when you specify classification scores for the class as a column vector.

Model Operating Point

The model operating point represents the FPR and TPR corresponding to the typical threshold value.

The typical threshold value depends on the input format of the Scores argument (classification scores) specified when you create a rocmetrics object:

• If you specify Scores as a matrix, rocmetrics assumes that the values in Scores are the scores for a multiclass classification problem and uses adjusted score values. A multiclass classification model classifies an observation into a class that yields the largest score, which corresponds to a nonnegative score in the adjusted scores. Therefore, the threshold value is 0.

• If you specify Scores as a column vector, rocmetrics assumes that the values in Scores are posterior probabilities of the class specified in ClassNames. A binary classification model classifies an observation into a class that yields a higher posterior probability, that is, a posterior probability greater than 0.5. Therefore, the threshold value is 0.5.

For a binary classification problem, you can specify Scores as a two-column matrix or a column vector. However, if the classification scores are not posterior probabilities, you must specify Scores as a matrix. A binary classifier classifies an observation into a class that yields a larger score, which is equivalent to a class that yields a nonnegative adjusted score. Therefore, if you specify Scores as a matrix for a binary classifier, rocmetrics can find a correct model operating point using the same scheme that it applies to a multiclass classifier. If you specify classification scores that are not posterior probabilities as a vector, rocmetrics cannot identify a correct model operating point because it always uses 0.5 as a threshold for the model operating point.

The plot function displays a filled circle marker at the model operating point for each ROC curve (see ShowModelOperatingPoint). The function chooses a point corresponding to the typical threshold value. If the curve does not have a data point for the typical threshold value, the function finds a point that has the smallest threshold value greater than the typical threshold. The point on the curve indicates identical performance to the performance of the typical threshold value.

Thresholds, Fixed Metric, and Fixed Metric Values

rocmetrics finds the ROC curves and other metric values that correspond to the fixed values (FixedMetricValues name-value argument) of the fixed metric (FixedMetric name-value argument), and stores the values in the Metrics property as a table.

The default FixedMetric value is "Thresholds", and the default FixedMetricValues value is "all". For each class, rocmetrics uses all distinct adjusted score values as threshold values, computes the components of the confusion matrix for each threshold value, and then computes performance metrics using the confusion matrix components.

If you use the default FixedMetricValues value ("all"), specifying a nondefault FixedMetric value does not change the software behavior unless you specify to compute confidence intervals. If rocmetrics computes confidence intervals, then it holds FixedMetric fixed at FixedMetricValues and computes confidence intervals for other metrics. For more details, see Pointwise Confidence Intervals.

If you specify a nondefault value for FixedMetricValues, rocmetrics finds the threshold values corresponding to the specified fixed metric values (FixedMetricValues for FixedMetric) and computes other performance metric values using the threshold values.

• If you set the UseNearestNeighbor name-value argument to false, then rocmetrics uses the exact threshold values corresponding to the specified fixed metric values.

• If you set UseNearestNeighbor to true, then among the adjusted scores, rocmetrics finds a value that is the nearest to the threshold value corresponding to each specified fixed metric value.

The Metrics property includes an additional threshold value that replicates the largest threshold value for each class so that a ROC curve starts from the origin (0,0). The additional threshold value represents the reject-all threshold, for which TP = FP = 0 (no positive instances, that is, zero true positive instances and zero false positive instances).

Another special threshold in Metrics is the accept-all threshold, which is the smallest threshold value for which TN = FN = 0 (no negative instances, that is, zero true negative instances and zero false negative instances).

Note that the positive predictive value (PPV = TP/(TP+FP)) is NaN for the reject-all threshold, and the negative predictive value (NPV = TN/(TN+FN)) is NaN for the accept-all threshold.

NaN Score Values

rocmetrics processes NaN values in the classification score input (Scores) in one of two ways:

• If you specify NaNFlag="omitnan" (default), then rocmetrics discards rows with NaN scores.

• If you specify NaNFlag="includenan", then rocmetrics adds the instances of NaN scores to false classification counts in the respective class for each one-versus-all binary classification. That is, for any threshold, the software counts instances with NaN scores from the positive class as false negative (FN), and counts instances with NaN scores from the negative class as false positive (FP). The software computes the metrics corresponding to a threshold of 1 by setting the number of true positive (TP) instances to zero and setting the number of true negative (TN) instances to the total count minus the NaN count in the negative class.

Consider an example with two rows in the positive class and two rows in the negative class, each pair having a NaN score:

True Class Label	Classification Score
Negative	0.2
Negative	NaN
Positive	0.7
Positive	NaN

If you discard rows with NaN scores (NaNFlag="omitnan"), then as the score threshold varies, rocmetrics computes performance metrics as shown in the following table. For example, a threshold of 0.5 corresponds to the middle row where rocmetrics classifies rows 1 and 3 correctly and omits rows 2 and 4.

Threshold	TP	FN	FP	TN
1	0	1	0	1
0.5	1	0	0	1
0	1	0	1	0

If you add rows with NaN scores to the false category in their respective classes (NaNFlag="includenan"), rocmetrics computes performance metrics as shown in the following table. For example, a threshold of 0.5 corresponds to the middle row where rocmetrics counts rows 2 and 4 as incorrectly classified. Notice that only the FN and FP columns differ between these two tables.

Threshold	TP	FN	FP	TN
1	0	2	1	1
0.5	1	1	1	1
0	1	1	2	0

Pointwise Confidence Intervals

rocmetrics computes pointwise confidence intervals for the performance metrics, including the score thresholds, by using either bootstrap samples or cross-validated data. The object stores the values in the Metrics property.

Note

Using confidence intervals requires Statistics and Machine Learning Toolbox.

• Bootstrap — To compute confidence intervals using bootstrapping, set the NumBootstraps name-value argument to a positive integer. rocmetrics generates NumBootstraps bootstrap samples. The function creates each bootstrap sample by randomly selecting n out of the n rows of input data with replacement. For an example, see Compute Confidence Intervals Using Bootstrapping.

• Cross-validation — To compute confidence intervals using cross-validation, specify cross-validated data for true class labels (Labels), classification scores (Scores), and observation weights (Weights) using cell arrays. rocmetrics treats elements in the cell arrays as cross-validation folds.

You cannot specify both options. If you specify a custom metric in AdditionalMetrics, you must use bootstrap to compute confidence intervals. rocmetrics does not support cross-validation for a custom metric.

rocmetrics holds FixedMetric (threshold, FPR, TPR, or a metric specified in AdditionalMetrics) fixed at FixedMetricValues and computes the confidence intervals on AUC and other metrics for the points corresponding to the values in FixedMetricValues.

• Threshold averaging (TA) (when FixedMetric is "Thresholds" (default)) — rocmetrics estimates confidence intervals for performance metrics at fixed threshold values. The function takes samples at the fixed thresholds and averages the corresponding metric values.

• Vertical averaging (VA) (when FixedMetric is a performance metric) — rocmetrics estimates confidence intervals for thresholds and other performance metrics at the fixed metric values. The function takes samples at the fixed metric values and averages the corresponding threshold and metric values.

The function estimates confidence intervals for the AUC value only when FixedMetric is "Thresholds", "FalsePositiveRate", or "TruePositiveRate".

References

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[3] Davis, J., and M. Goadrich. “The Relationship Between Precision-Recall and ROC Curves.” Proceedings of ICML ’06, 2006, pp. 233–240.

[4] Moskowitz, C. S., and M. S. Pepe. “Quantifying and Comparing the Predictive Accuracy of Continuous Prognostic Factors for Binary Outcomes.” Biostatistics 5, no. 1 (2004): 113–27.

[5] Huang, Y., M. S. Pepe, and Z. Feng. “Evaluating the Predictiveness of a Continuous Marker.” U. Washington Biostatistics Paper Series, 2006, 250–61.

[6] Briggs, W. M., and R. Zaretzki. “The Skill Plot: A Graphical Technique for Evaluating Continuous Diagnostic Tests.” Biometrics 64, no. 1 (2008): 250–256.

[7] Bettinger, R. “Cost-Sensitive Classifier Selection Using the ROC Convex Hull Method.” SAS Institute, 2003.