modelDiscrimination

Compute AUROC and ROC data

Since R2021b

Syntax

DiscMeasure = modelDiscrimination(eadModel,data)

[DiscMeasure,DiscData] = modelDiscrimination(___,Name=Value)

Description

DiscMeasure = modelDiscrimination(eadModel,data) computes the area under the receiver operating characteristic curve (AUROC). modelDiscrimination supports segmentation and comparison against a reference model and alternative methods to discretize the EAD response into a binary variable.

[DiscMeasure,DiscData] = modelDiscrimination(___,Name=Value) specifies options using one or more name-value arguments in addition to the input arguments in the previous syntax.

Examples

Compute AUROC and ROC Using Tobit EAD Model

This example shows how to use fitEADModel to create a Tobit model and then use modelDiscrimination to compute AUROC and ROC.

Load EAD Data

Load the EAD data.

load EADData.mat
head(EADData)

    UtilizationRate    Age     Marriage        Limit         Drawn          EAD    
    _______________    ___    ___________    __________    __________    __________

        0.24359        25     not married         44776         10907         44740
        0.96946        44     not married    2.1405e+05    2.0751e+05         40678
              0        40     married        1.6581e+05             0    1.6567e+05
        0.53242        38     not married    1.7375e+05         92506        1593.5
         0.2583        30     not married         26258        6782.5        54.175
        0.17039        54     married        1.7357e+05         29575        576.69
        0.18586        27     not married         19590          3641        998.49
        0.85372        42     not married    2.0712e+05    1.7682e+05    1.6454e+05

rng('default');
NumObs = height(EADData);
c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Select Model Type

Select a model type for Tobit or Regression.

ModelType = "Tobit";

Select Conversion Measure

Select a conversion measure for the EAD response values.

ConversionMeasure = "LCF";

Create Tobit EAD Model

Use fitEADModel to create a Tobit model using the TrainingInd data.

eadModel = fitEADModel(EADData(TrainingInd,:),ModelType,PredictorVars={'UtilizationRate','Age','Marriage'}, ...
    ConversionMeasure=ConversionMeasure,DrawnVar="Drawn",LimitVar="Limit",ResponseVar="EAD");
disp(eadModel);

  Tobit with properties:

        CensoringSide: "both"
            LeftLimit: 0
           RightLimit: 1
              ModelID: "Tobit"
          Description: ""
      UnderlyingModel: [1x1 risk.internal.credit.TobitModel]
        PredictorVars: ["UtilizationRate"    "Age"    "Marriage"]
          ResponseVar: "EAD"
             LimitVar: "Limit"
             DrawnVar: "Drawn"
    ConversionMeasure: "lcf"

Display the underlying model. The underlying model's response variable is the transformation of the EAD response data. Use the 'LiimitVar' and 'DrwanVar' name-value arguments to modify the transformation.

disp(eadModel.UnderlyingModel);

Tobit regression model:
     EAD_lcf = max(0,min(Y*,1))
     Y* ~ 1 + UtilizationRate + Age + Marriage

Estimated coefficients:
                             Estimate         SE         tStat        pValue  
                            __________    __________    ________    __________

    (Intercept)                0.22467      0.031085      7.2276    6.4149e-13
    UtilizationRate             0.4714      0.020682      22.793             0
    Age                     -0.0014209    0.00075844     -1.8735      0.061111
    Marriage_not married     -0.010543      0.015817    -0.66654       0.50512
    (Sigma)                     0.3618     0.0049991      72.374             0

Number of observations: 2627
Number of left-censored observations: 0
Number of uncensored observations: 2626
Number of right-censored observations: 1
Log-likelihood: -1057.9

Predict EAD

EAD prediction operates on the underlying compact statistical model and then transforms the predicted values back to the EAD scale. You can specify the predict function with different options for the 'ModelLevel' name-value argument.

predictedEAD = predict(eadModel,EADData(TestInd,:),ModelLevel="ead");
predictedConversion = predict(eadModel,EADData(TestInd,:),ModelLevel="ConversionMeasure");

Validate EAD Model

For model validation, use modelDiscrimination, modelDiscriminationPlot, modelCalibration, and modelCalibrationPlot.

Use modelDiscrimination and then modelDiscriminationPlot to plot the ROC curve.

ModelLevel = "ConversionMeasure";

[DiscMeasure1,DiscData1] = modelDiscrimination(eadModel,EADData(TestInd,:),ShowDetails=true,ModelLevel=ModelLevel)

DiscMeasure1=1×3 table
              AUROC      Segment      SegmentCount
             _______    __________    ____________

    Tobit    0.70893    "all_data"        1751    

DiscData1=1534×3 table
        X             Y           T   
    __________    _________    _______

             0            0    0.63602
             0    0.0027778    0.63602
             0    0.0041667     0.6349
    0.00096993    0.0055556    0.63377
    0.00096993    0.0069444    0.63265
     0.0019399    0.0083333    0.63152
     0.0029098    0.0097222     0.6304
     0.0029098     0.015278    0.62927
     0.0029098     0.016667    0.62922
     0.0029098     0.018056     0.6288
     0.0029098     0.019444    0.62864
     0.0038797     0.022222    0.62814
     0.0038797        0.025    0.62767
     0.0048497     0.026389    0.62701
     0.0048497     0.033333    0.62654
     0.0058196     0.033333    0.62618
      ⋮

modelDiscriminationPlot(eadModel,EADData(TestInd,:),ModelLevel=ModelLevel,SegmentBy="Marriage");

Compute AUROC and ROC Using Beta EAD Model

This example shows how to use fitEADModel to create a Beta model and then use modelDiscrimination to compute AUROC and ROC.

Load EAD Data

Load the EAD data.

load EADData.mat
head(EADData)

    UtilizationRate    Age     Marriage        Limit         Drawn          EAD    
    _______________    ___    ___________    __________    __________    __________

        0.24359        25     not married         44776         10907         44740
        0.96946        44     not married    2.1405e+05    2.0751e+05         40678
              0        40     married        1.6581e+05             0    1.6567e+05
        0.53242        38     not married    1.7375e+05         92506        1593.5
         0.2583        30     not married         26258        6782.5        54.175
        0.17039        54     married        1.7357e+05         29575        576.69
        0.18586        27     not married         19590          3641        998.49
        0.85372        42     not married    2.0712e+05    1.7682e+05    1.6454e+05

rng('default');
NumObs = height(EADData);
c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Select Model Type

Select a model type for a Beta model.

ModelType = "Beta";

Select Conversion Measure

Select a conversion measure for the EAD response values.

ConversionMeasure = "LCF";

Create Beta EAD Model

Use fitEADModel to create a Beta model using the TrainingInd data.

eadModel = fitEADModel(EADData(TrainingInd,:),ModelType,PredictorVars={'UtilizationRate','Age','Marriage'}, ...
    ConversionMeasure=ConversionMeasure,DrawnVar="Drawn",LimitVar="Limit",ResponseVar="EAD");
disp(eadModel);

  Beta with properties:

    BoundaryTolerance: 1.0000e-07
              ModelID: "Beta"
          Description: ""
      UnderlyingModel: [1x1 risk.internal.credit.BetaModel]
        PredictorVars: ["UtilizationRate"    "Age"    "Marriage"]
          ResponseVar: "EAD"
             LimitVar: "Limit"
             DrawnVar: "Drawn"
    ConversionMeasure: "lcf"

Display the underlying model. The underlying model's response variable is the transformation of the EAD response data. Use the 'LiimitVar' and 'DrwanVar' name-value arguments to modify the transformation.

disp(eadModel.UnderlyingModel);

Beta regression model:
     logit(EAD_lcf) ~ 1_mu + UtilizationRate_mu + Age_mu + Marriage_mu
     log(EAD_lcf) ~ 1_phi + UtilizationRate_phi + Age_phi + Marriage_phi

Estimated coefficients:
                                Estimate        SE         tStat        pValue  
                                _________    _________    ________    __________

    (Intercept)_mu               -0.65566      0.11484     -5.7093    1.2614e-08
    UtilizationRate_mu             1.7014     0.078094      21.787             0
    Age_mu                       -0.00559    0.0027603     -2.0252      0.042952
    Marriage_not married_mu     -0.012576     0.052098     -0.2414       0.80926
    (Intercept)_phi              -0.50132     0.094625     -5.2979    1.2685e-07
    UtilizationRate_phi           0.39731     0.066707       5.956    2.9304e-09
    Age_phi                     -0.001167    0.0023161    -0.50386       0.61441
    Marriage_not married_phi    -0.013275     0.042627    -0.31143        0.7555

Number of observations: 2627
Log-likelihood: -3140.21

Predict EAD

EAD prediction operates on the underlying compact statistical model and then transforms the predicted values back to the EAD scale. You can specify the predict function with different options for the 'ModelLevel' name-value argument.

predictedEAD = predict(eadModel,EADData(TestInd,:),ModelLevel="ead");
predictedConversion = predict(eadModel,EADData(TestInd,:),ModelLevel="ConversionMeasure");

Validate EAD Model

For model validation, use modelDiscrimination, modelDiscriminationPlot, modelCalibration, and modelCalibrationPlot.

Use modelDiscrimination and then modelDiscriminationPlot to plot the ROC curve.

ModelLevel = "ConversionMeasure";

[DiscMeasure1,DiscData1] = modelDiscrimination(eadModel,EADData(TestInd,:),ShowDetails=true,ModelLevel=ModelLevel)

DiscMeasure1=1×3 table
             AUROC      Segment      SegmentCount
            _______    __________    ____________

    Beta    0.70895    "all_data"        1751    

DiscData1=1534×3 table
        X             Y           T   
    __________    _________    _______

             0            0    0.71675
             0    0.0027778    0.71675
             0    0.0041667    0.71561
             0    0.0055556    0.71533
    0.00096993    0.0069444    0.71447
    0.00096993    0.0097222    0.71419
    0.00096993     0.011111    0.71333
    0.00096993     0.018056    0.71304
     0.0019399     0.018056     0.7128
     0.0029098     0.019444    0.71218
     0.0048497     0.019444     0.7119
     0.0058196     0.020833    0.71104
     0.0067895     0.020833    0.71075
     0.0067895     0.022222    0.71022
     0.0067895     0.027778    0.70989
     0.0067895     0.029167    0.70968
      ⋮

modelDiscriminationPlot(eadModel,EADData(TestInd, :),ModelLevel=ModelLevel,SegmentBy="Marriage");

Input Arguments

eadModel — Exposure at default model

Regression object | Tobit object | Beta object

Exposure at default model, specified as a previously created Regression, Tobit, or Beta object using fitEADModel.

Data Types: object

data — Data

table

Data, specified as a NumRows-by-NumCols table with predictor and response values. The variable names and data types must be consistent with the underlying model.

Data Types: table

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: [DiscMeasure,DiscData] = modelDiscrimination(eadModel,data(TestInd,:),DataID='Testing',DiscretizeBy='median')

DataID — Data set identifier

"" (default) | character vector | string

Data set identifier, specified as DataID and a character vector or string. The DataID is included in the output for reporting purposes.

Data Types: char | string

DiscretizeBy — Discretization method for EAD data at defined ModelLevel

'mean' (default) | character vector with value 'mean' or 'median' | string with value "mean" or "median"

Discretization method for EAD data at the defined ModelLevel, specified as DiscretizeBy and a character vector or string.

• 'mean' — Discretized response is 1 if observed EAD is greater than or equal to the mean EAD, 0 otherwise.

• 'median' — Discretized response is 1 if observed EAD is greater than or equal to the median EAD, 0 otherwise.

Data Types: char | string

SegmentBy — Name of column in data input used to segment data set

"" (default) | character vector | string

Name of a column in the data input, not necessarily a model variable, to be used to segment the data set, specified as SegmentBy and a character vector or string. One AUROC is reported for each segment, and the corresponding ROC data for each segment is returned in the optional output.

Data Types: char | string

ShowDetails — Indicates if output includes columns showing segment value and segment count

false (default) | logical

Since R2022a

Indicates if the output includes columns showing segment value and segment count, specified as the comma-separated pair consisting of 'ShowDetails' and a scalar logical.

Data Types: logical

ModelLevel — Model level

"ead" (default) | character vector with value 'ead', 'conversionMeasure', or 'conversionTransform' | string with value "ead", "conversionMeasure", or "conversionTransform"

Model level, specified as ModelLevel and a character vector or string.

Note

Regression models support all three model levels, but a Tobit or Beta model supports only a ModelLevel for "ead" and "conversionMeasure".

Data Types: char | string

ReferenceEAD — EAD values predicted for data by reference model

[] (default) | numeric vector

EAD values predicted for data by the reference model, specified as ReferenceEAD and a NumRows-by-1 numeric vector. The modelDiscrimination output information is reported for both the eadModel object and the reference model.

Data Types: double

ReferenceID — Identifier for the reference model

'Reference' (default) | character vector | string

Identifier for the reference model, specified as ReferenceID and a character vector or string. 'ReferenceID' is used in the modelDiscrimination output for reporting purposes.

Data Types: char | string

Output Arguments

DiscMeasure — AUROC information for each model and each segment

table

AUROC information for each model and each segment, returned as a table. DiscMeasure has a single column named 'AUROC' and the number of rows depends on the number of segments and whether you use a ReferenceID for a reference model. The row names of DiscMeasure report the model IDs, segment, and data ID. If the optional ShowDetails name-value argument is true, the DiscMeasure output displays Segment and SegmentCount columns.

Note

If you do not specify SegmentBy and use ShowDetails to request the segment details, the two columns are added and show the Segment column as "all_data" and the sample size (minus missing values) for the SegmentCount column.

DiscData — ROC data for each model and each segment

table

ROC data for each model and each segment, returned as a table. There are three columns for the ROC data, with column names 'X', 'Y', and 'T', where the first two are the X and Y coordinates of the ROC curve, and T contains the corresponding thresholds. For more information, see Model Discrimination or perfcurve.

If you use SegmentBy, the function stacks the ROC data for all segments and DiscData has a column with the segmentation values to indicate where each segment starts and ends.

If reference model data is given, the DiscData outputs for the main and reference models are stacked, with an extra column 'ModelID' indicating where each model starts and ends.

More About

Model Discrimination

Model discrimination measures the risk ranking.

The modelDiscrimination function computes the area under the receiver operator characteristic (AUROC) curve, sometimes called simply the area under the curve (AUC). This metric is between 0 and 1 and higher values indicate better discrimination.

To compute the AUROC, you need a numeric prediction and a binary response. For EAD models, the predicted EAD is used directly as the prediction. However, the observed EAD must be discretized into a binary variable. By default, observed EAD values greater than or equal to the mean observed EAD are assigned a value of 1, and values below the mean are assigned a value of 0. This discretized response is interpreted as "high EAD" vs. "low EAD." Therefore, the modelDiscrimination function measures how well the predicted EAD separates the "high EAD" vs. the "low EAD" observations. You can change the level to compute the model discrimination with the ModelLevel name-value pair argument and the discretization criterion with the DiscretizeBy name-value pair argument.

To plot the receiver operator characteristic (ROC) curve, use the modelDiscriminationPlot function. However, if you need the ROC curve data, use the optional DiscData output argument from the modelDiscrimination function.

The ROC curve is a parametric curve that plots the proportion of

• High EAD cases with predicted EAD greater than or equal to a parameter t, or true positive rate (TPR)

• Low EAD cases with predicted EAD greater than or equal to the same parameter t, or false positive rate (FPR)

The parameter t sweeps through all the observed predicted EAD values for the given data. The DiscData optional output contains the TPR in the 'X' column, the FPR in the 'Y' column, and the corresponding parameters t in the 'T' column. For more information about ROC curves, see ROC Curve and Performance Metrics.

References

[1] Baesens, Bart, Daniel Roesch, and Harald Scheule. Credit Risk Analytics: Measurement Techniques, Applications, and Examples in SAS. Wiley, 2016.

[2] Bellini, Tiziano. IFRS 9 and CECL Credit Risk Modelling and Validation: A Practical Guide with Examples Worked in R and SAS. San Diego, CA: Elsevier, 2019.

[3] Brown, Iain. Developing Credit Risk Models Using SAS Enterprise Miner and SAS/STAT: Theory and Applications. SAS Institute, 2014.

[4] Roesch, Daniel and Harald Scheule. Deep Credit Risk. Independently published, 2020.

Version History

Introduced in R2021b

R2022b: Support for Beta model

The eadModel input supports an option for a Beta model object that you can create using fitEADModel.

R2022a: Additional option for ShowDetails

There is an additional name-value pair for ShowDetails to indicate if the DiscMeasure output includes columns for Segment value and the SegmentCount.