modelAccuracy

Compute R-square, RMSE, correlation, and sample mean error of predicted and observed LGDs

Syntax

AccMeasure = modelAccuracy(lgdModel,data)

[AccMeasure,AccData] = modelAccuracy(___,Name,Value)

Description

AccMeasure = modelAccuracy(lgdModel,data) computes the R-square, root mean square error (RMSE), correlation, and sample mean error of observed vs. predicted loss given default (LGD) data. modelAccuracy supports comparison against a reference model and also supports different correlation types. By default, modelAccuracy computes the metrics in the LGD scale. You can use the ModelLevel name-value pair argument to compute metrics using the underlying model's transformed scale.

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

Examples

Compute R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs Using Regression LGD Model

This example shows how to use fitLGDModel to fit data with a Regression model and then use modelAccuracy to compute the R-Square, RMSE, correlation, and sample mean error of predicted and observed LGDs.

Load Data

Load the loss given default data.

load LGDData.mat
head(data)

      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 Regression LGD Model

Use fitLGDModel to create a Regression model using training data.

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 R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs

Use modelAccuracy to compute the RSquared, RMSE, Correlation, and SampleMeanError of the predicted and observed LGDs for the test data set.

[AccMeasure,AccData] = modelAccuracy(lgdModel,data(TestInd,:))

AccMeasure=1×4 table
                  RSquared     RMSE      Correlation    SampleMeanError
                  ________    _______    ___________    _______________

    Regression    0.070867    0.25988      0.26621          0.10759    

AccData=1394×3 table
    Observed     Predicted_Regression    Residuals_Regression
    _________    ____________________    ____________________

    0.0064766         0.00091169               0.0055649     
     0.007947          0.0036758               0.0042713     
     0.063182            0.18774                -0.12456     
            0          0.0010877              -0.0010877     
      0.10904           0.011213                0.097823     
            0           0.041992               -0.041992     
      0.89463           0.052947                 0.84168     
            0         3.7188e-06             -3.7188e-06     
     0.072437          0.0090124                0.063425     
     0.036006           0.023928                0.012078     
            0          0.0034833              -0.0034833     
      0.39549          0.0065253                 0.38896     
     0.057675           0.071956               -0.014281     
     0.014439          0.0061499                0.008289     
            0          0.0012183              -0.0012183     
            0          0.0019828              -0.0019828     
      ⋮

Generate a scatter plot of predicted and observed LGDs using modelAccuracyPlot.

modelAccuracyPlot(lgdModel,data(TestInd,:),'ModelLevel',"underlying")

Compute R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs Using Tobit LGD Model

This example shows how to use fitLGDModel to fit data with a Tobit model and then use modelAccuracy to compute R-Square, RMSE, correlation, and sample mean error of predicted and observed LGDs.

Load Data

Load the loss given default data.

load LGDData.mat
head(data)

      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 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     0.027279     2.1356      0.032828
    LTV                  0.20126     0.031383     6.4129    1.7592e-10
    Age                -0.095407    0.0072435    -13.171             0
    Type_investment      0.10208     0.018054     5.6544    1.7785e-08
    (Sigma)              0.29288    0.0057071     51.318             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 R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs

Use modelAccuracy to compute RSquared, RMSE, Correlation, and SampleMeanError of predicted and observed LGDs for the test data set.

[AccMeasure,AccData] = modelAccuracy(lgdModel,data(TestInd,:),'CorrelationType',"kendall")

AccMeasure=1×4 table
             RSquared     RMSE      Correlation    SampleMeanError
             ________    _______    ___________    _______________

    Tobit    0.08527     0.23712      0.29964         -0.034412   

AccData=1394×3 table
    Observed     Predicted_Tobit    Residuals_Tobit
    _________    _______________    _______________

    0.0064766       0.087889           -0.081412   
     0.007947        0.12432            -0.11638   
     0.063182        0.32043            -0.25724   
            0       0.093354           -0.093354   
      0.10904        0.16718           -0.058144   
            0        0.22382            -0.22382   
      0.89463        0.23695             0.65768   
            0       0.010234           -0.010234   
     0.072437         0.1592           -0.086761   
     0.036006        0.19893            -0.16292   
            0        0.12764            -0.12764   
      0.39549        0.14568              0.2498   
     0.057675        0.26181            -0.20413   
     0.014439        0.14483            -0.13039   
            0       0.094123           -0.094123   
            0        0.10944            -0.10944   
      ⋮

Generate a scatter plot of the predicted and observed LGDs using modelAccuracyPlot.

modelAccuracyPlot(lgdModel,data(TestInd,:))

Compute R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs Using Beta LGD Model

This example shows how to use fitLGDModel to fit data with a Beta model and then use modelAccuracy to compute R-Square, RMSE, correlation, and sample mean error of predicted and observed LGDs.

Load Data

Load the loss given default data.

load LGDData.mat
head(data)

      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 Beta LGD Model

Use fitLGDModel to create a Beta model using training data.

lgdModel = fitLGDModel(data(TrainingInd,:),'Beta');
disp(lgdModel)    

  Beta with properties:

    BoundaryTolerance: 1.0000e-05
              ModelID: "Beta"
          Description: ""
      UnderlyingModel: [1x1 risk.internal.credit.BetaModel]
        PredictorVars: ["LTV"    "Age"    "Type"]
          ResponseVar: "LGD"

Display the underlying model.

disp(lgdModel.UnderlyingModel)

Beta regression model:
     logit(LGD) ~ 1_mu + LTV_mu + Age_mu + Type_mu
     log(LGD) ~ 1_phi + LTV_phi + Age_phi + Type_phi

Estimated coefficients:
                           Estimate       SE        tStat       pValue  
                           ________    ________    _______    __________

    (Intercept)_mu          -1.3772     0.13201    -10.433             0
    LTV_mu                  0.60269     0.15087     3.9947    6.7023e-05
    Age_mu                 -0.47464    0.040264    -11.788             0
    Type_investment_mu      0.45372    0.085143     5.3289     1.094e-07
    (Intercept)_phi        -0.16337     0.12591    -1.2975       0.19462
    LTV_phi                0.055892     0.14719    0.37973       0.70419
    Age_phi                 0.22887    0.040335     5.6743    1.5863e-08
    Type_investment_phi    -0.14102    0.078155    -1.8044      0.071311

Number of observations: 2093
Log-likelihood: -5291.04

Compute R-Square, RMSE, Correlation, and Sample Mean Error of Predicted and Observed LGDs

Use modelAccuracy to compute RSquared, RMSE, Correlation, and SampleMeanError of predicted and observed LGDs for the test data set.

[AccMeasure,AccData] = modelAccuracy(lgdModel,data(TestInd,:),'CorrelationType',"kendall")

AccMeasure=1×4 table
            RSquared     RMSE      Correlation    SampleMeanError
            ________    _______    ___________    _______________

    Beta    0.080804    0.24112      0.29448         -0.052396   

AccData=1394×3 table
    Observed     Predicted_Beta    Residuals_Beta
    _________    ______________    ______________

    0.0064766       0.093695         -0.087218   
     0.007947        0.14915           -0.1412   
     0.063182        0.35263          -0.28944   
            0       0.096434         -0.096434   
      0.10904        0.18858         -0.079542   
            0         0.2595           -0.2595   
      0.89463        0.26767           0.62696   
            0       0.021315         -0.021315   
     0.072437        0.17736          -0.10492   
     0.036006        0.22556          -0.18955   
            0        0.13369          -0.13369   
      0.39549        0.16768            0.2278   
     0.057675        0.29159          -0.23392   
     0.014439         0.1617          -0.14726   
            0        0.10506          -0.10506   
            0         0.1161           -0.1161   
      ⋮

Generate a scatter plot of the predicted and observed LGDs using modelAccuracyPlot.

modelAccuracyPlot(lgdModel,data(TestInd,:))

Input Arguments

lgdModel — Loss given default model

Regression object | Tobit object | Beta object

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

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.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [AccMeasure,AccData] = modelAccuracy(lgdModel,data(TestInd,:),'DataID','Testing','CorrelationType','spearman')

CorrelationType — Correlation type

"pearson" (default) | character vector with value of 'pearson', 'spearman', or 'kendall' | string with value of "pearson", "spearman", or "kendall'"

Correlation type, specified as the comma-separated pair consisting of 'CorrelationType' and a character vector or string.

Data Types: char | string

DataID — Data set identifier

"" (default) | character vector | string

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

ModelLevel — Model level

'top' (default) | character vector with value 'top' or 'underlying' | string with value "top" or "underlying"

Model level, specified as the comma-separated pair consisting of 'ModelLevel' and a character vector or string.

• 'top' — The accuracy metrics are computed in the LGD scale at the top model level.

• 'underlying' — For a Regression model only, the metrics are computed in the underlying model's transformed scale. The metrics are computed on the transformed LGD data.

Note

ModelLevel has no effect for a Tobit or Beta model because there is no response transformation.

Data Types: char | string

ReferenceLGD — LGD values predicted for data by reference model

[ ] (default) | numeric vector

LGD values predicted for data by the reference model, specified as the comma-separated pair consisting of 'ReferenceLGD' and a NumRows-by-1 numeric vector. The modelAccuracy output information is reported for both the lgdModel 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 the comma-separated pair consisting of 'ReferenceID' and a character vector or string. 'ReferenceID' is used in the modelAccuracy output for reporting purposes.

Data Types: char | string

Output Arguments

AccMeasure — Accuracy measure

table

Accuracy measure, returned as a table with columns 'RSquared', 'RMSE', 'Correlation', and 'SampleMeanError'. AccMeasure has one row if only the lgdModel accuracy is measured and it has two rows if reference model information is given. The row names of AccMeasure report the model ID and data ID (if provided).

AccData — Accuracy data

table

Accuracy data, returned as a table with observed LGD values, predicted LGD values, and residuals (observed minus predicted). Additional columns for predicted and residual values are included for the reference model, if provided. The ModelID and ReferenceID labels are appended in the column names.

More About

Model Accuracy

Model accuracy measures the accuracy of the predicted probability of LGD values using different metrics.

• R-squared — To compute the R-squared metric, modelAccuracy fits a linear regression of the observed LGD values against the predicted LGD values

  $$LG{D}_{obs}=a+b\ast LG{D}_{pred}+\epsilon$$

  The R-square of this regression is reported. For more information, see Coefficient of Determination (R-Squared).

• RMSE — To compute the root mean square error (RMSE), modelAccuracy uses the following formula where N is the number of observations:

  $$RMSE=\sqrt{\frac{1}{N}{\displaystyle {\sum }_{i=1}^N(LG{D}_i^{obs}-LG{D}_i^{pred}}{)}^2}$$

• Correlation — This is the correlation between the observed and predicted LGD:

  $$corr(LG{D}_{obs},LG{D}_{pred})$$

  For more information and details about the different correlation types, see corr.

• Sample mean error — This is the difference between the mean observed LGD and the mean predicted LGD or, equivalently, the mean of the residuals:

  $$SampleMeanError=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N(LG{D}_i^{obs}-LG{D}_i^{pred})}$$

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.

Version History

Introduced in R2021a

R2022b: Support for Beta model

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