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modelDiscrimination

Compute AUROC and ROC data

Description

example

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

example

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

Examples

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This example shows how to use fitLGDModel to fit data with a Regression model and then use modelDiscrimination to compute AUROC and ROC.

Load Data

Load the loss given default data.

load LGDData.mat
head(data)
ans=8×4 table
      LTV        Age         Type           LGD   
    _______    _______    ___________    _________

    0.89101    0.39716    residential     0.032659
    0.70176     2.0939    residential      0.43564
    0.72078     2.7948    residential    0.0064766
    0.37013      1.237    residential     0.007947
    0.36492     2.5818    residential            0
      0.796     1.5957    residential      0.14572
    0.60203     1.1599    residential     0.025688
    0.92005    0.50253    investment      0.063182

Partition Data

Separate the data into training and test partitions.

rng('default'); % for reproducibility
NumObs = height(data);

c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Create a Regression LGD Model

Use fitLGDModel to create a Regression model using training data. You can also use fitLGDModel to create a Tobit model by changing the lgdModel input argument to 'Tobit'.

lgdModel = fitLGDModel(data(TrainingInd,:),'Regression');
disp(lgdModel)    
  Regression with properties:

    ResponseTransform: "logit"
    BoundaryTolerance: 1.0000e-05
              ModelID: "Regression"
          Description: ""
      UnderlyingModel: [1x1 classreg.regr.CompactLinearModel]
        PredictorVars: ["LTV"    "Age"    "Type"]
          ResponseVar: "LGD"

Display the underlying model.

disp(lgdModel.UnderlyingModel)
Compact linear regression model:
    LGD_logit ~ 1 + LTV + Age + Type

Estimated Coefficients:
                       Estimate       SE        tStat       pValue  
                       ________    ________    _______    __________

    (Intercept)        -4.7549      0.36041    -13.193    3.0997e-38
    LTV                 2.8565      0.41777     6.8377    1.0531e-11
    Age                -1.5397     0.085716    -17.963    3.3172e-67
    Type_investment     1.4358       0.2475     5.8012     7.587e-09


Number of observations: 2093, Error degrees of freedom: 2089
Root Mean Squared Error: 4.24
R-squared: 0.206,  Adjusted R-Squared: 0.205
F-statistic vs. constant model: 181, p-value = 2.42e-104

Compute AUROC and ROC Data

Use modelDiscrimination to compute the AUROC and ROC for the test data set.

DiscMeasure = modelDiscrimination(lgdModel,data(TestInd,:))
DiscMeasure=table
                   AUROC 
                  _______

    Regression    0.67897

You can visualize the ROC data using modelDiscriminationPlot.

modelDiscriminationPlot(lgdModel,data(TestInd,:))

Figure contains an axes. The axes with title ROC Regression, AUROC = 0.67897 contains an object of type line. This object represents Regression.

This example shows how to use fitLGDModel to fit data with a Tobit model and then use modelDiscrimination to compute AUROC and ROC.

Load Data

Load the loss given default data.

load LGDData.mat
head(data)
ans=8×4 table
      LTV        Age         Type           LGD   
    _______    _______    ___________    _________

    0.89101    0.39716    residential     0.032659
    0.70176     2.0939    residential      0.43564
    0.72078     2.7948    residential    0.0064766
    0.37013      1.237    residential     0.007947
    0.36492     2.5818    residential            0
      0.796     1.5957    residential      0.14572
    0.60203     1.1599    residential     0.025688
    0.92005    0.50253    investment      0.063182

Partition Data

Separate the data into training and test partitions.

rng('default'); % for reproducibility
NumObs = height(data);

c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Create a Tobit LGD Model

Use fitLGDModel to create a Tobit model using training data.

lgdModel = fitLGDModel(data(TrainingInd,:),'tobit');
disp(lgdModel)    
  Tobit with properties:

      CensoringSide: "both"
          LeftLimit: 0
         RightLimit: 1
            ModelID: "Tobit"
        Description: ""
    UnderlyingModel: [1x1 risk.internal.credit.TobitModel]
      PredictorVars: ["LTV"    "Age"    "Type"]
        ResponseVar: "LGD"

Display the underlying model.

disp(lgdModel.UnderlyingModel)
Tobit regression model:
     LGD = max(0,min(Y*,1))
     Y* ~ 1 + LTV + Age + Type

Estimated coefficients:
                       Estimate        SE         tStat       pValue  
                       _________    _________    _______    __________

    (Intercept)         0.058257     0.027265     2.1367      0.032737
    LTV                  0.20126     0.031354     6.4189    1.6932e-10
    Age                -0.095407    0.0072653    -13.132             0
    Type_investment      0.10208     0.018058     5.6531    1.7915e-08
    (Sigma)              0.29288    0.0057036      51.35             0

Number of observations: 2093
Number of left-censored observations: 547
Number of uncensored observations: 1521
Number of right-censored observations: 25
Log-likelihood: -698.383

Compute AUROC and ROC Data

Use modelDiscrimination to compute the AUROC and ROC for the test data set.

DiscMeasure = modelDiscrimination(lgdModel,data(TestInd,:),'SegmentBy',"Type",'DiscretizeBy',"median")
DiscMeasure=2×1 table
                                AUROC 
                               _______

    Tobit, Type=residential    0.70101
    Tobit, Type=investment     0.73252

You can visualize the ROC using modelDiscriminationPlot.

modelDiscriminationPlot(lgdModel,data(TestInd,:),'SegmentBy',"Type",'DiscretizeBy',"median")

Figure contains an axes. The axes with title ROC Segmented by Type contains 2 objects of type line. These objects represent Tobit, residential, AUROC = 0.70101, Tobit, investment, AUROC = 0.73252.

Input Arguments

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Loss given default model, specified as a previously created Regression or Tobit object using fitLGDModel.

Data Types: object

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 Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

Data set identifier, specified as the comma-separated pair consisting of 'DataID' and a character vector or string. The DataID is included in the output for reporting purposes.

Data Types: char | string

Discretization method for LGD data, specified as the comma-separated pair consisting of 'DiscretizeBy' and a character vector or string.

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

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

  • 'positive' — Discretized response is 1 if observed LGD is positive, 0 otherwise (full recovery).

  • 'total' — Discretized response is 1 if observed LGD is greater than or equal to 1 (total loss), 0 otherwise.

Data Types: char | string

Name of a column in the data input, not necessarily a model variable, to be used to segment the data set, specified as the comma-separated pair consisting of '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

LGD values predicted for data by the reference model, specified as the comma-separated pair consisting of 'ReferenceID' and a NumRows-by-1 numeric vector. The modelDiscrimination output information is reported for both the lgdModel object and the reference model.

Data Types: double

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

Data Types: char | string

Output Arguments

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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.

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

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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 loss given default (LGD) models, the predicted LGD is used directly as the prediction. However, the observed LGD must be discretized into a binary variable. By default, observed LGD values greater than or equal to the mean observed LGD are assigned a value of 1, and values below the mean are assigned a value of 0. This discretized response is interpreted as “high LGD” vs. “low LGD.” Therefore, the modelDiscrimination function measures how well the predicted LGD separates the “high LGD” vs. the “low LGD” observations. You can change the discretization criterion with the DiscretizeBy name-value pair argument.

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

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

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

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

The parameter t sweeps through all the observed predicted LGD 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 Performance Curves.

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.

Introduced in R2021a