Credit Scorecards with Constrained Logistic Regression Coefficients

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To compute scores for a creditscorecard object with constraints for equality, inequality, or bounds on the coefficients of the logistic regression model, use fitConstrainedModel. Unlike fitmodel, fitConstrainedModel solves for both the unconstrained and constrained problem. The current solver used to minimize an objective function for fitConstrainedModel is fmincon, from the Optimization Toolbox™.

This example has three main sections. First, fitConstrainedModel is used to solve for the coefficients in the unconstrained model. Then, fitConstrainedModel demonstrates how to use several types of constraints. Finally, fitConstrainedModel uses bootstrapping for the significance analysis to determine which predictors to reject from the model.

Create the `creditscorecard` Object and Bin data

load CreditCardData.mat
sc = creditscorecard(data,'IDVar','CustID');
sc = autobinning(sc);

Unconstrained Model Using `fitConstrainedModel`

Solve for the unconstrained coefficients using fitConstrainedModel with default values for the input parameters. fitConstrainedModel uses the internal optimization solver fmincon from the Optimization Toolbox™. If you do not set any constraints, fmincon treats the model as an unconstrained optimization problem. The default parameters for the LowerBound and UpperBound are -Inf and +Inf, respectively. For the equality and inequality constraints, the default is an empty numeric array.

[sc1,mdl1] = fitConstrainedModel(sc);
coeff1 = mdl1.Coefficients.Estimate;
disp(mdl1.Coefficients);

                   Estimate 
                   _________

    (Intercept)      0.70246
    CustAge           0.6057
    TmAtAddress       1.0381
    ResStatus         1.3794
    EmpStatus        0.89648
    CustIncome       0.70179
    TmWBank           1.1132
    OtherCC           1.0598
    AMBalance         1.0572
    UtilRate       -0.047597

Unlike fitmodel which gives p-values, when using fitConstrainedModel, you must use bootstrapping to find out which predictors are rejected from the model, when subject to constraints. This is illustrated in the "Significance Bootstrapping" section.

Using `fitmodel` to Compare the Results and Calibrate the Model

fitmodel fits a logistic regression model to the Weight-of-Evidence (WOE) data and there are no constraints. You can compare the results from the "Unconstrained Model Using fitConstrainedModel" section with those of fitmodel to verify that the model is well calibrated.

Now, solve the unconstrained problem by using fitmodel. Note that fitmodel and fitConstrainedModel use different solvers. While fitConstrainedModel uses fmincon, fitmodel uses stepwiseglm by default. To include all predictors from the start, set the 'VariableSelection' name-value pair argument of fitmodel to 'fullmodel'.

[sc2,mdl2] = fitmodel(sc,'VariableSelection','fullmodel','Display','off');
coeff2 = mdl2.Coefficients.Estimate;
disp(mdl2.Coefficients);

                   Estimate        SE         tStat        pValue  
                   _________    ________    _________    __________

    (Intercept)      0.70246    0.064039       10.969    5.3719e-28
    CustAge           0.6057     0.24934       2.4292      0.015131
    TmAtAddress       1.0381     0.94042       1.1039       0.26963
    ResStatus         1.3794      0.6526       2.1137      0.034538
    EmpStatus        0.89648     0.29339       3.0556     0.0022458
    CustIncome       0.70179     0.21866       3.2095     0.0013295
    TmWBank           1.1132     0.23346       4.7683    1.8579e-06
    OtherCC           1.0598     0.53005       1.9994      0.045568
    AMBalance         1.0572     0.36601       2.8884     0.0038718
    UtilRate       -0.047597     0.61133    -0.077858       0.93794

figure
plot(coeff1,'*')
hold on
plot(coeff2,'s')
xticklabels(mdl1.Coefficients.Properties.RowNames)
ylabel('Model Coefficients')
title('Unconstrained Model Coefficients')
legend({'Calculated by fitConstrainedModel with defaults','Calculated by fimodel'},'Location','best')
grid on

Figure contains an axes object. The axes object with title Unconstrained Model Coefficients, ylabel Model Coefficients contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Calculated by fitConstrainedModel with defaults, Calculated by fimodel.

As both the tables and the plot show, the model coefficients match. You can be confident that this implementation of fitConstrainedModel is well calibrated.

Constrained Model

In the constrained model approach, you solve for the values of the coefficients $b_{i}$ of the logistic model, subject to constraints. The supported constraints are bound, equality, or inequality. The coefficients maximize the likelihood-of-default function defined, for observation $i$ , as:

$L_{i} = {p ({Default}_{i})}^{y_{i}} \times {(1 - p ({Default}_{i}))}^{1 - y_{i}}$

where:

${p ({Default}_{i}) = \frac{1}{1 + e^{- b x_{i}}}}^{}$
$b = [b_{1} b_{2} . . . b_{K}]$ is an unknown model coefficient
$x_{i} = [x_{i1} x_{2} . . . x_{iK}]$ is the predictor values at observation $i$
$y_{i}$ is the response value; a value of 1 represents default and a value of 0 represents non-default

This formula is for non-weighted data. When observation i has weight $w_{i}$ , it means that there are $w_{i}$ as many observations i. Therefore, the probability that default occurs at observation i is the product of the probabilities of default:

$p_{i} = \underset{w_{i} times}{{p ({Default}_{i})}^{y_{i}} * {p ({Default}_{i})}^{y_{i}} * . . . * {p ({Default}_{i})}^{y_{i}}} = {p ({Default}_{i})}^{{w_{i} * y}_{i}}$

Likewise, the probability of non-default for weighted observation i is:

${\hat{p}}_{i} = \underset{w_{i} times}{{p ({~ Default}_{i})}^{{1 - y}_{i}} * {p (~ {Default}_{i})}^{1 - y_{i}} * . . . * {p (~ {Default}_{i})}^{1 - y_{i}}} = {(1 - p ({Default}_{i}))}^{w_{i} * (1 - y_{i})}$

For weighted data, if there is default at a given observation i whose weight is $w_{i}$ , it is as if there was a $w_{i}$ count of that one observation, and all of them either all default, or all non-default. $w_{i}$ may or may not be an integer.

Therefore, for the weighted data, the likelihood-of-default function for observation i in the first equation becomes

$L_{i} = {p ({Default}_{i})}^{{w_{i} * y}_{i}} \times {(1 - p ({Default}_{i}))}^{w_{i} * (1 - y_{i})}$

By assumption, all defaults are independent events, so the objective function is

$L = L_{1} \times L_{2} \times . . . \times L_{N}$

or, in more convenient logarithmic terms:

$\log (L) = \sum_{i = 1}^{N} w_{i} * [y_{i} \log (p ({Default}_{i})) + (1 - y_{i}) \log (1 - p ({Default}_{i}))]$

Apply Constraints on the Coefficients

After calibrating the unconstrained model as described in the "Unconstrained Model Using fitConstrainedModel" section, you can solve for the model coefficients subject to constraints. You can choose lower and upper bounds such that $0 \leq b_{i} \leq 1, \forall i = 1 . . . K$ , except for the intercept. Also, since the customer age and customer income are somewhat correlated, you can also use additional constraints on their coefficients, for example, $| b_{C u s A g e} - b_{C u s t I n c o m e} | < 0.1$ . The coefficients corresponding to the predictors 'CustAge' and 'CustIncome' in this example are $b_{2}$ and $b_{6}$ , respectively.

K  = length(sc.PredictorVars);
lb = [-Inf;zeros(K,1)];
ub = [Inf;ones(K,1)];
AIneq = [0 -1 0 0 0 1 0 0 0 0;0 -1 0 0 0 -1 0 0 0 0];
bIneq = [0.05;0.05];
Options = optimoptions('fmincon','SpecifyObjectiveGradient',true,'Display','off');
[sc3,mdl] = fitConstrainedModel(sc,'AInequality',AIneq,'bInequality',bIneq,...
    'LowerBound',lb,'UpperBound',ub,'Options',Options);

figure
plot(coeff1,'*','MarkerSize',8)
hold on
plot(mdl.Coefficients.Estimate,'.','MarkerSize',12)
line(xlim,[0 0],'color','k','linestyle',':')
line(xlim,[1 1],'color','k','linestyle',':')
text(1.1,0.1,'Lower bound')
text(1.1,1.1,'Upper bound')
grid on

xticklabels(mdl.Coefficients.Properties.RowNames)
ylabel('Model Coefficients')
title('Comparison Between Unconstrained and Constrained Solutions')
legend({'Unconstrained','Constrained'},'Location','best')

Figure contains an axes object. The axes object with title Comparison Between Unconstrained and Constrained Solutions, ylabel Model Coefficients contains 6 objects of type line, text. One or more of the lines displays its values using only markers These objects represent Unconstrained, Constrained.

Significance Bootstrapping

For the unconstrained problem, standard formulas are available for computing p-values, which you use to evaluate which coefficients are significant and which are to be rejected. However, for the constrained problem, standard formulas are not available, and the derivation of formulas for significance analysis is complicated. A practical alternative is to perform significance analysis through bootstrapping.

In the bootstrapping approach, when using fitConstrainedModel, you set the name-value argument 'Bootstrap' to true and chose a value for the name-value argument 'BootstrapIter'. Bootstrapping means that $NIter$ samples (with replacement) from the original observations are selected. In each iteration, fitConstrainedModel solves for the same constrained problem as the "Constrained Model" section. fitConstrainedModel obtains several values (solutions) for each coefficient $b_{i}$ and you can plot these as a boxplot or histogram. Using the boxplot or histogram, you can examine the median values to evaluate whether the coefficients are away from zero and how much the coefficients deviate from their means.

lb = [-Inf;zeros(K,1)];
ub = [Inf;ones(K,1)];
AIneq = [0 -1 0 0 0 1 0 0 0 0;0 1 0 0 0 -1 0 0 0 0];
bIneq = [0.05;0.05];
c0 = zeros(K,1);
NIter = 100;
Options = optimoptions('fmincon','SpecifyObjectiveGradient',true,'Display','off');
rng('default')

[sc,mdl] = fitConstrainedModel(sc,'AInequality',AIneq,'bInequality',bIneq,...
    'LowerBound',lb,'UpperBound',ub,'Bootstrap',true,'BootstrapIter',NIter,'Options',Options);

figure
boxplot(mdl.Bootstrap.Matrix,mdl.Coefficients.Properties.RowNames)
hold on
line(xlim,[0 0],'color','k','linestyle',':')
line(xlim,[1 1],'color','k','linestyle',':')
title('Bootstrapping with N = 100 Iterations')
ylabel('Model Coefficients')

Figure contains an axes object. The axes object with title Bootstrapping with N = 100 Iterations, ylabel Model Coefficients contains 72 objects of type line. One or more of the lines displays its values using only markers

The solid red lines in the boxplot indicate that the median values and the bottom and top edges are for the $25^{th}$ and $75^{th}$ percentiles. The "whiskers" are the minimum and maximum values, not including outliers. The dotted lines represent the lower and upper bound constraints on the coefficients. In this example, the coefficients cannot be negative, by construction.

To help decide which predictors to keep in the model, assess the proportion of times each coefficient is zero.

Tol = 1e-6;
figure
bar(100*sum(mdl.Bootstrap.Matrix<= Tol)/NIter)
ylabel('% of Zeros')
title('Percentage of Zeros Over Bootstrap Iterations')
xticklabels(mdl.Coefficients.Properties.RowNames)
grid on

Figure contains an axes object. The axes object with title Percentage of Zeros Over Bootstrap Iterations, ylabel % of Zeros contains an object of type bar.

Based on the plot, you can reject 'UtilRate' since it has the highest number of zero values. You can also decide to reject 'TmAtAddress' since it shows a peak, albeit small.

Set the Corresponding Coefficients to Zero

To set the corresponding coefficients to zero, set their upper bound to zero and solve the model again using the original data set.

ub(3) = 0;
ub(end) = 0;
[sc,mdl] = fitConstrainedModel(sc,'AInequality',AIneq,'bInequality',bIneq,'LowerBound',lb,'UpperBound',ub,'Options',Options);
Ind = (abs(mdl.Coefficients.Estimate) <= Tol);
ModelCoeff = mdl.Coefficients.Estimate(~Ind);
ModelPreds = mdl.Coefficients.Properties.RowNames(~Ind)';

figure
hold on
plot(ModelCoeff,'.','MarkerSize',12)
ylim([0.2 1.2])
ylabel('Model Coefficients')
xticklabels(ModelPreds)
title('Selected Model Coefficients After Bootstrapping')
grid on

Figure contains an axes object. The axes object with title Selected Model Coefficients After Bootstrapping, ylabel Model Coefficients contains a line object which displays its values using only markers.

Set Constrained Coefficients Back Into the `creditscorecard`

Now that you have solved for the constrained coefficients, use setmodel to set the model's coefficients and predictors. Then you can compute the (unscaled) points.

ModelPreds = ModelPreds(2:end);
sc = setmodel(sc,ModelPreds,ModelCoeff);
p = displaypoints(sc);

disp(p)

      Predictors               Bin              Points  
    ______________    _____________________    _________

    {'CustAge'   }    {'[-Inf,33)'        }     -0.16725
    {'CustAge'   }    {'[33,37)'          }     -0.14811
    {'CustAge'   }    {'[37,40)'          }    -0.065607
    {'CustAge'   }    {'[40,46)'          }     0.044404
    {'CustAge'   }    {'[46,48)'          }      0.21761
    {'CustAge'   }    {'[48,58)'          }      0.23404
    {'CustAge'   }    {'[58,Inf]'         }      0.49029
    {'CustAge'   }    {'<missing>'        }          NaN
    {'ResStatus' }    {'Tenant'           }    0.0044307
    {'ResStatus' }    {'Home Owner'       }      0.11932
    {'ResStatus' }    {'Other'            }      0.30048
    {'ResStatus' }    {'<missing>'        }          NaN
    {'EmpStatus' }    {'Unknown'          }    -0.077028
    {'EmpStatus' }    {'Employed'         }      0.31459
    {'EmpStatus' }    {'<missing>'        }          NaN
    {'CustIncome'}    {'[-Inf,29000)'     }     -0.43795
    {'CustIncome'}    {'[29000,33000)'    }    -0.097814
    {'CustIncome'}    {'[33000,35000)'    }     0.053667
    {'CustIncome'}    {'[35000,40000)'    }     0.081921
    {'CustIncome'}    {'[40000,42000)'    }     0.092364
    {'CustIncome'}    {'[42000,47000)'    }      0.23932
    {'CustIncome'}    {'[47000,Inf]'      }      0.42477
    {'CustIncome'}    {'<missing>'        }          NaN
    {'TmWBank'   }    {'[-Inf,12)'        }     -0.15547
    {'TmWBank'   }    {'[12,23)'          }    -0.031077
    {'TmWBank'   }    {'[23,45)'          }    -0.021091
    {'TmWBank'   }    {'[45,71)'          }      0.36703
    {'TmWBank'   }    {'[71,Inf]'         }      0.86888
    {'TmWBank'   }    {'<missing>'        }          NaN
    {'OtherCC'   }    {'No'               }     -0.16832
    {'OtherCC'   }    {'Yes'              }      0.15336
    {'OtherCC'   }    {'<missing>'        }          NaN
    {'AMBalance' }    {'[-Inf,558.88)'    }      0.34418
    {'AMBalance' }    {'[558.88,1254.28)' }    -0.012745
    {'AMBalance' }    {'[1254.28,1597.44)'}    -0.057879
    {'AMBalance' }    {'[1597.44,Inf]'    }     -0.19896
    {'AMBalance' }    {'<missing>'        }          NaN

Using the unscaled points, you can follow the remainder of the Credit Scorecard Modeling Workflow to compute scores and probabilities of default and to validate the model.

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