# plot

Plot receiver operating characteristic (ROC) curves and other performance curves

## Syntax

## Description

`plot(`

creates
a receiver operating characteristic (ROC) curve, which is a plot of the true positive
rate (TPR) versus the false positive rate (FPR), for each class in the `rocObj`

)`ClassNames`

property of the
`rocmetrics`

object
`rocObj`

. The function marks the model operating point for each
curve, and displays the value of the area under the ROC curve (AUC) and the class name for the curve in the legend.

`plot(___,`

specifies additional options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, `Name=Value`

)`AverageROCType="macro",ClassNames=[]`

computes the average performance metrics using the macro-averaging method and plots the average ROC curve only.

`[`

also returns graphics objects for the model operating points and diagonal line.`curveObj`

,`graphicsObjs`

] = plot(___)

## Examples

### Plot ROC Curve

Load a sample of predicted classification scores and true labels for a classification problem.

`load('flowersDataResponses.mat')`

`trueLabels`

is the true labels for an image classification problem and `scores`

is the softmax prediction scores. `scores`

is an N-by-K array where N is the number of observations and K is the number of classes.

trueLabels = flowersData.trueLabels; scores = flowersData.scores;

Load the class names. The column order of `scores`

follows the class order stored in c`lassNames`

.

classNames = flowersData.classNames;

Create a `rocmetrics`

object by using the true labels in `trueLabels`

and the classification scores in `scores`

. Specify the column order of `scores`

using `classNames`

.

rocObj = rocmetrics(trueLabels,scores,classNames);

`rocObj`

is a `rocmetrics`

object that stores the AUC values and performance metrics for each class in the `AUC`

and `Metrics`

properties. Display the `AUC`

property.

rocObj.AUC

`ans = `*1x5 single row vector*
0.9781 0.9889 0.9728 0.9809 0.9732

Plot the ROC curve for each class.

plot(rocObj)

The filled circle markers indicate the model operating points. The legend displays the class name and AUC value for each curve.

Plot the macro average ROC curve.

`plot(rocObj,AverageROCType=["macro"],ClassNames=[])`

### Plot Precision-Recall Curve and Detection Error Tradeoff (DET) Graph

Create a `rocmetrics`

object and plot performance curves by using the `plot`

function. Specify the `XAxisMetric`

and `YAxisMetric`

name-value arguments of the `plot`

function to plot different types of performance curves other than the ROC curve. If you specify new metrics when you call the `plot`

function, the function computes the new metrics and then uses them to plot the curve.

Load a sample of true labels and the prediction scores for a classification problem. For this example, there are five classes: daisy, dandelion, roses, sunflowers, and tulips. The class names are stored in `classNames`

. The scores are the softmax prediction scores generated using the `predict`

function. `scores`

is an N-by-K array where N is the number of observations and K is the number of classes. The column order of `scores`

follows the class order stored in c`lassNames`

.

```
load('flowersDataResponses.mat')
scores = flowersData.scores;
trueLabels = flowersData.trueLabels;
classNames = flowersData.classNames;
```

Create a `rocmetrics`

object. The `rocmetrics`

function computes the FPR and TPR at different thresholds.

rocObj = rocmetrics(trueLabels,scores,classNames);

Plot the precision-recall curve for the first class. Specify the *y*-axis metric as precision (or positive predictive value) and the *x*-axis metric as recall (or true positive rate). The `plot`

function computes the new metric values and plots the curve. Compute the area under the precision-recall curve using the trapezoidal method of the `trapz`

function, and display the value in the legend.

curveObj = plot(rocObj,ClassNames=classNames(1), ... YAxisMetric="PositivePredictiveValue",XAxisMetric="TruePositiveRate"); xyData = rmmissing([curveObj.XData curveObj.YData]); auc = trapz(xyData(:,1),xyData(:,2)); legend(join([string(classNames(1)) " (AUC = " string(auc) ")"],""), ... Location="southwest") title("Precision-Recall Curve")

Plot the detection error tradeoff (DET) graph for the first class. Specify the *y*-axis metric as the false negative rate and the *x*-axis metric as the false positive rate. Use a log scale for the *x*-axis and *y*-axis.

f = figure; plot(rocObj,ClassNames=classNames(1), ... YAxisMetric="FalseNegativeRate",XAxisMetric="FalsePositiveRate") f.CurrentAxes.XScale = "log"; f.CurrentAxes.YScale = "log"; title("DET Graph")

### Plot Confidence Intervals

Compute the confidence intervals for FPR and TPR for fixed threshold values by using bootstrap samples, and plot the confidence intervals for TPR on the ROC curve by using the `plot`

function. This examples requires Statistics and Machine Learning Toolbox™.

Load a sample of true labels and the prediction scores for a classification problem. For this example, there are five classes: daisy, dandelion, roses, sunflowers, and tulips. The class names are stored in `classNames`

. The scores are the softmax prediction scores generated using the `predict`

function. `scores`

is an N-by-K array where N is the number of observations and K is the number of classes. The column order of `scores`

follows the class order stored in c`lassNames`

.

```
load('flowersDataResponses.mat')
scores = flowersData.scores;
trueLabels = flowersData.trueLabels;
classNames = flowersData.classNames;
```

Create a `rocmetrics`

object by using the true labels in `trueLabels`

and the classification scores in `scores`

. Specify the column order of `scores`

using `classNames`

. Specify `NumBootstraps`

as 100 to use 100 bootstrap samples to compute the confidence intervals.

rocObj = rocmetrics(trueLabels,scores,classNames,NumBootstraps=100);

Plot the ROC curve and the confidence intervals for TPR. Specify `ShowConfidenceIntervals=true`

to show the confidence intervals.

plot(rocObj,ShowConfidenceIntervals=true)

The shaded area around each curve indicates the confidence intervals. `rocmetrics`

computes the ROC curves using the scores. The confidence intervals represent the estimates of uncertainty for the curve.

Specify one class to plot by using the `ClassNames`

name-value argument.

`plot(rocObj,ShowConfidenceIntervals=true,ClassNames="daisy")`

## Input Arguments

`rocObj`

— Object evaluating classification performance

`rocmetrics`

object

Object evaluating classification performance, specified as a `rocmetrics`

object.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

**Example: **`YAxisMetric="PositivePredictiveValue",XAxisMetric="TruePositiveRate"`

plots the precision (positive predictive value) versus the recall (true positive rate),
which represents a precision-recall curve.

`AverageROCType`

— Method for averaging ROC curves

`"none"`

(default) | `"micro"`

| `"macro"`

| `"weighted"`

| string array | cell array of character vectors

Method for averaging ROC curves, specified as `"none"`

, `"micro"`

, `"macro"`

, `"weighted"`

, a string array of method names, or a cell array of method names.

If you specify

`"none"`

(default), the`plot`

function does not create the average ROC curve. The`AverageROCType`

value must be`"none"`

if`plot`

creates performance curves other than a ROC curve.If you specify multiple methods as a string array or a cell array of character vectors, then the

`plot`

function plots multiple average ROC curves using the specified methods.If you specify one or more averaging methods and specify

`ClassNames=[]`

, then the`plot`

function plots only the average ROC curves.

`plot`

computes the averages of performance metrics for a multiclass classification problem, and plots the average ROC curves using these methods:

`"micro"`

(micro-averaging) —`plot`

finds the average performance metrics by treating all one-versus-all binary classification problems as one binary classification problem. The function computes the confusion matrix components for the combined binary classification problem, and then computes the average FPR and TPR using the values of the confusion matrix.`"macro"`

(macro-averaging) —`plot`

computes the average values for FPR and TPR by averaging the values of all one-versus-all binary classification problems.`"weighted"`

(weighted macro-averaging) —`plot`

computes the weighted average values for FPR and TPR using the macro-averaging method and using the prior class probabilities (the`Prior`

property of`rocObj`

) as weights.

The algorithm type determines the length of the vectors in the `XData`

, `YData`

, and
`Thresholds`

properties of a `ROCCurve`

object, returned by `plot`

,
for the average ROC curve. For more details, see Average of Performance Metrics.

**Example: **`AverageROCType="macro"`

**Example: **`AverageROCType=["micro","macro"]`

**Data Types: **`char`

| `string`

`ClassNames`

— Class labels to plot

`rocObj.ClassNames`

(default) | categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels to plot, specified as a categorical, character, or string array, logical or
numeric vector, or cell array of character vectors. The values and data types in
`ClassNames`

must match those of the class names in the `ClassNames`

property
of `rocObj`

. (The software treats character or string arrays as cell arrays of character vectors.)

If you specify multiple class labels, the

`plot`

function plots a ROC curve for each class.If you specify

`ClassNames=[]`

and specify one or more averaging methods using`AverageROCType`

, then the`plot`

function plots only the average ROC curves.

**Example: **`ClassNames=["red","blue"]`

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

| `categorical`

`ShowConfidenceIntervals`

— Flag to show confidence intervals of *y*-axis metric

`false`

or `0`

(default) | `true`

or `1`

Flag to show the confidence intervals of the *y*-axis metric (`YAxisMetric`

), specified as logical `0`

(`false`

) or `1`

(`true`

).

The `ShowConfidenceIntervals`

value can be `true`

only if
the `Metrics`

property of
`rocObj`

contains the confidence intervals for the
*y*-axis metric.

**Example: **`ShowConfidenceIntervals=true`

Using confidence intervals requires Statistics and Machine Learning Toolbox™.

**Data Types: **`logical`

`ShowDiagonalLine`

— Flag to show diagonal line

`true`

or `1`

| `false`

or `0`

Flag to show the diagonal line that extends from `[0,0]`

to `[1,1]`

, specified as logical `1`

(`true`

) or `0`

(`false`

).

The default value is `true`

if you plot a ROC curve or an average ROC curve, and `false`

otherwise.

In the ROC curve plot, the diagonal line represents a random classifier, and the line passing through `[0,0]`

, `[0,1]`

, and `[1,1]`

represents a perfect classifier.

**Example: **`ShowDiagonalLine=false`

**Data Types: **`logical`

`ShowModelOperatingPoint`

— Flag to show model operating point

`true`

or `1`

| `false`

or `0`

Flag to show the model operating point, specified as logical `1`

(`true`

) or `0`

(`false`

).

The default value is `true`

for a ROC curve, and `false`

for an average ROC curve. The `ShowModelOperatingPoint`

value must be false for performance curves other than ROC.

**Example: **`ShowModelOperatingPoint=false`

**Data Types: **`logical`

`XAxisMetric`

— Metric for *x*-axis

`"FalsePositiveRate"`

(default) | name of performance metric | function handle

Metric for the *x*-axis, specified as a character vector or string scalar of the built-in metric name or a custom metric name, or a function handle (`@metricName`

).

Built-in metrics — Specify one of the following built-in metric names by using a character vector or string scalar.

Name Description `"TruePositives"`

or`"tp"`

Number of true positives (TP) `"FalseNegatives"`

or`"fn"`

Number of false negatives (FN) `"FalsePositives"`

or`"fp"`

Number of false positives (FP) `"TrueNegatives"`

or`"tn"`

Number of true negatives (TN) `"SumOfTrueAndFalsePositives"`

or`"tp+fp"`

Sum of TP and FP `"RateOfPositivePredictions"`

or`"rpp"`

Rate of positive predictions (RPP), `(TP+FP)/(TP+FN+FP+TN)`

`"RateOfNegativePredictions"`

or`"rnp"`

Rate of negative predictions (RNP), `(TN+FN)/(TP+FN+FP+TN)`

`"Accuracy"`

or`"accu"`

Accuracy, `(TP+TN)/(TP+FN+FP+TN)`

`"TruePositiveRate"`

or`"tpr"`

True positive rate (TPR), also known as recall or sensitivity, `TP/(TP+FN)`

`"FalseNegativeRate"`

,`"fnr"`

, or`"miss"`

False negative rate (FNR), or miss rate, `FN/(TP+FN)`

`"FalsePositiveRate"`

or`"fpr"`

False positive rate (FPR), also known as fallout or 1-specificity, `FP/(TN+FP)`

`"TrueNegativeRate"`

,`"tnr"`

, or`"spec"`

True negative rate (TNR), or specificity, `TN/(TN+FP)`

`"PositivePredictiveValue"`

,`"ppv"`

, or`"prec"`

Positive predictive value (PPV), or precision, `TP/(TP+FP)`

`"NegativePredictiveValue"`

or`"npv"`

Negative predictive value (NPV), `TN/(TN+FN)`

`"ExpectedCost"`

or`"ecost"`

Expected cost,

`(TP*cost(P|P)+FN*cost(N|P)+FP*cost(P|N)+TN*cost(N|N))/(TP+FN+FP+TN)`

, where`cost`

is a 2-by-2 misclassification cost matrix containing`[0,cost(N|P);cost(P|N),0]`

.`cost(N|P)`

is the cost of misclassifying a positive class (`P`

) as a negative class (`N`

), and`cost(P|N)`

is the cost of misclassifying a negative class as a positive class.The software converts the

`K`

-by-`K`

matrix specified by the`Cost`

name-value argument of`rocmetrics`

to a 2-by-2 matrix for each one-versus-all binary problem. For details, see Misclassification Cost Matrix.The software computes the scale vector using the prior class probabilities (

`Prior`

) and the number of classes in`Labels`

, and then scales the performance metrics according to this scale vector. For details, see Performance Metrics.Custom metric stored in the

`Metrics`

property — Specify the name of a custom metric stored in the`Metrics`

property of the input object`rocObj`

. The`rocmetrics`

function names a custom metric`"CustomMetricN"`

, where`N`

is the number that refers to the custom metric. For example, specify`XAxisMetric="CustomMetric1"`

to use the first custom metric in`Metrics`

as a metric for the*x*-axis.Custom metric — Specify a new custom metric by using a function handle. A custom function that returns a performance metric must have this form:

metric = customMetric(C,scale,cost)

The output argument

`metric`

is a scalar value.A custom metric is a function of the confusion matrix (

`C`

), scale vector (`scale`

), and cost matrix (`cost`

). The software finds these input values for each one-versus-all binary problem. For details, see Performance Metrics.`C`

is a`2`

-by-`2`

confusion matrix consisting of`[TP,FN;FP,TN]`

.`scale`

is a`2`

-by-`1`

scale vector.`cost`

is a`2`

-by-`2`

misclassification cost matrix.

The

`plot`

function names a custom metric`"Custom Metric"`

for the axis label.The software does not support cross-validation for a custom metric. Instead, you can specify to use bootstrap when you create a

`rocmetrics`

object.

If you specify a new metric instead of one in the `Metrics`

property of
the input object `rocObj`

, the `plot`

function
computes and plots the metric values. If you compute confidence intervals when you
create `rocObj`

, the `plot`

function also computes
confidence intervals for the new metric.

The `plot`

function ignores `NaN`

s in the performance metric values. Note that the positive predictive value (PPV) is
`NaN`

for the reject-all threshold for which `TP`

= `FP`

= `0`

, and the negative predictive value (NPV) is `NaN`

for the
accept-all threshold for which `TN`

= `FN`

= `0`

. For more details, see Thresholds, Fixed Metric, and Fixed Metric Values.

**Example: **`XAxisMetric="FalseNegativeRate"`

**Data Types: **`char`

| `string`

| `function_handle`

`YAxisMetric`

— Metric for *y*-axis

`"TruePositiveRate"`

(default) | name of performance metric | function handle

Metric for the *y*-axis, specified as a character vector or string scalar of
the built-in metric name or custom metric name, or a function handle
(`@metricName`

). For details, see `XAxisMetric`

.

**Example: **`YAxisMetric="FalseNegativeRate"`

**Data Types: **`char`

| `string`

| `function_handle`

## Output Arguments

`curveObj`

— Object for performance curve

`ROCCurve`

object | array of `ROCCurve`

objects

Object for the performance curve, returned as a `ROCCurve`

object or an array of `ROCCurve`

objects. `plot`

returns a `ROCCurve`

object for each performance curve.

Use `curveObj`

to query and modify properties of the plot after creating
it. For a list of properties, see ROCCurve Properties.

`graphicsObjs`

— Graphics objects

graphics array

Graphics objects for the model operating points and diagonal line, returned as a graphics array containing `Scatter`

and `Line`

objects.

`graphicsObjs`

contains a `Scatter`

object for each model operating point (if

) and a `ShowModelOperatingPoint`

=true`Line`

object for the diagonal line (if

). Use `ShowDiagonalLine`

=true`graphicsObjs`

to query and modify properties of the model operating points and diagonal line after creating the plot. For a list of properties, see Scatter Properties and Line Properties.

## More About

### Receiver Operating Characteristic (ROC) Curve

A ROC curve shows the true positive rate versus the false positive rate for different thresholds of classification scores.

The true positive rate and the false positive rate are defined as follows:

True positive rate (TPR), also known as recall or sensitivity —

`TP/(TP+FN)`

, where TP is the number of true positives and FN is the number of false negativesFalse positive rate (FPR), also known as fallout or 1-specificity —

`FP/(TN+FP)`

, where FP is the number of false positives and TN is the number of true negatives

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 each class,
`rocmetrics`

uses all distinct adjusted score values
as threshold values to create a ROC curve.

For a multiclass classification problem, `rocmetrics`

formulates a set
of one-versus-all binary
classification problems to have one binary problem for each class, and finds a ROC
curve for each class using the corresponding binary problem. Each binary problem
assumes one class as positive and the rest as negative.

For a binary classification problem, if you specify the classification scores as a
matrix, `rocmetrics`

formulates two one-versus-all binary
classification problems. Each of these problems treats one class as a positive class
and the other class as a negative class, and `rocmetrics`

finds two
ROC curves. Use one of the curves to evaluate the binary classification
problem.

For more details, see ROC Curve and Performance Metrics.

### 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.

### 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:

$$\begin{array}{cccc}& \text{Binary1}& \text{Binary}2& \text{Binary3}\\ \text{Class1}& 1& -1& -1\\ \text{Class2}& -1& 1& -1\\ \text{Class3}& -1& -1& 1\end{array}$$

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.

## Algorithms

### 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 [*s*_{1},*s*_{2},*s*_{3}] in a row of `Scores`

for a classification problem with
three classes, the adjusted score values are [*s*_{1}-`max`

(*s*_{2},*s*_{3}),*s*_{2}-`max`

(*s*_{1},*s*_{3}),*s*_{3}-`max`

(*s*_{1},*s*_{2})].

`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.

## References

[1] Sebastiani, Fabrizio. "Machine Learning in Automated Text Categorization." *ACM Computing Surveys* 34, no. 1 (March 2002): 1–47.

## Version History

**Introduced in R2022b**

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