# modelAccuracy

Compute RMSE of predicted and observed PDs on grouped data

## Syntax

## Description

computes the root mean squared error (RMSE) of the observed compared to the
predicted probabilities of default (PD). `AccMeasure`

= modelAccuracy(`pdModel`

,`data`

,`GroupBy`

)`GroupBy`

is
required and can be any column in the `data`

input (not
necessarily a model variable). The `modelAccuracy`

function
computes the observed PD as the default rate of each group and the predicted PD
as the average PD for each group. `modelAccuracy`

supports
comparison against a reference model.

`[`

specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.`AccMeasure`

,`AccData`

] = modelAccuracy(___,`Name,Value`

)

## Examples

### Compute Model Accuracy for Logistic Lifetime PD Model

This example shows how to use `fitLifetimePDModel`

to fit data with a `Logistic`

model and then use `modelAccuracy`

to compute the root mean squared error (RMSE) of the observed probabilities of default (PDs) with respect to the predicted PDs.

**Load Data**

Load the credit portfolio data.

```
load RetailCreditPanelData.mat
disp(head(data))
```

ID ScoreGroup YOB Default Year __ __________ ___ _______ ____ 1 Low Risk 1 0 1997 1 Low Risk 2 0 1998 1 Low Risk 3 0 1999 1 Low Risk 4 0 2000 1 Low Risk 5 0 2001 1 Low Risk 6 0 2002 1 Low Risk 7 0 2003 1 Low Risk 8 0 2004

disp(head(dataMacro))

Year GDP Market ____ _____ ______ 1997 2.72 7.61 1998 3.57 26.24 1999 2.86 18.1 2000 2.43 3.19 2001 1.26 -10.51 2002 -0.59 -22.95 2003 0.63 2.78 2004 1.85 9.48

Join the two data components into a single data set.

data = join(data,dataMacro); disp(head(data))

ID ScoreGroup YOB Default Year GDP Market __ __________ ___ _______ ____ _____ ______ 1 Low Risk 1 0 1997 2.72 7.61 1 Low Risk 2 0 1998 3.57 26.24 1 Low Risk 3 0 1999 2.86 18.1 1 Low Risk 4 0 2000 2.43 3.19 1 Low Risk 5 0 2001 1.26 -10.51 1 Low Risk 6 0 2002 -0.59 -22.95 1 Low Risk 7 0 2003 0.63 2.78 1 Low Risk 8 0 2004 1.85 9.48

**Partition Data**

Separate the data into training and test partitions.

nIDs = max(data.ID); uniqueIDs = unique(data.ID); rng('default'); % For reproducibility c = cvpartition(nIDs,'HoldOut',0.4); TrainIDInd = training(c); TestIDInd = test(c); TrainDataInd = ismember(data.ID,uniqueIDs(TrainIDInd)); TestDataInd = ismember(data.ID,uniqueIDs(TestIDInd));

**Create Logistic Lifetime PD Model**

Use `fitLifetimePDModel`

to create a `Logistic`

model using the training data.

pdModel = fitLifetimePDModel(data(TrainDataInd,:),"Logistic",... 'AgeVar','YOB',... 'IDVar','ID',... 'LoanVars','ScoreGroup',... 'MacroVars',{'GDP','Market'},... 'ResponseVar','Default'); disp(pdModel)

Logistic with properties: ModelID: "Logistic" Description: "" Model: [1x1 classreg.regr.CompactGeneralizedLinearModel] IDVar: "ID" AgeVar: "YOB" LoanVars: "ScoreGroup" MacroVars: ["GDP" "Market"] ResponseVar: "Default"

Display the underlying model.

disp(pdModel.Model)

Compact generalized linear regression model: logit(Default) ~ 1 + ScoreGroup + YOB + GDP + Market Distribution = Binomial Estimated Coefficients: Estimate SE tStat pValue __________ _________ _______ ___________ (Intercept) -2.7422 0.10136 -27.054 3.408e-161 ScoreGroup_Medium Risk -0.68968 0.037286 -18.497 2.1894e-76 ScoreGroup_Low Risk -1.2587 0.045451 -27.693 8.4736e-169 YOB -0.30894 0.013587 -22.738 1.8738e-114 GDP -0.11111 0.039673 -2.8006 0.0051008 Market -0.0083659 0.0028358 -2.9502 0.0031761 388097 observations, 388091 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 1.85e+03, p-value = 0

**Compute Model Accuracy**

Model accuracy measures how accurate the predicted probabilities of default are. For example, if the model predicts a 10% PD for a group, does the group end up showing an approximate 10% default rate, or is the eventual rate much higher or lower? While model discrimination measures the risk ranking only, model accuracy measures the accuracy of the predicted risk levels.

`modelAccuracy`

computes the root mean squared error (RMSE) of the observed PDs with respect to the predicted PDs. A grouping variable is required and it can be any column in the data input (not necessarily a model variable). The `modelAccuracy`

function computes the observed PD as the default rate of each group and the predicted PD as the average PD for each group.

DataSetChoice = "Training"; if DataSetChoice=="Training" Ind = TrainDataInd; else Ind = TestDataInd; end GroupingVar = "YOB"; AccMeasure = modelAccuracy(pdModel,data(Ind,:),GroupingVar,'DataID',DataSetChoice); disp(AccMeasure)

RMSE _________ Logistic, grouped by YOB, Training 0.0004142

Visualize the model accuracy using `modelAccuracyPlot`

.

`modelAccuracyPlot(pdModel,data(Ind,:),GroupingVar,'DataID',DataSetChoice);`

You can use more than one variable for grouping. For this example, group by the variables `YOB`

and `ScoreGroup`

.

AccMeasure = modelAccuracy(pdModel,data(Ind,:),["YOB","ScoreGroup"],'DataID',DataSetChoice); disp(AccMeasure)

RMSE __________ Logistic, grouped by YOB, ScoreGroup, Training 0.00066239

Now visualize the two grouping variables using `modelAccuracyPlot`

.

modelAccuracyPlot(pdModel,data(Ind,:),["YOB","ScoreGroup"],'DataID',DataSetChoice);

## Input Arguments

`pdModel`

— Probability of default model

`Logistic`

object | `Probit`

object | `Cox`

object

Probability of default model, specified as a previously created
`Logistic`

,
`Probit`

, or `Cox`

object using
`fitLifetimePDModel`

.

**Data Types: **`object`

`data`

— Data

table

Data, specified as a
`NumRows`

-by-`NumCols`

table with
projected predictor values to make lifetime predictions. The predictor
names and data types must be consistent with the underlying
model.

**Data Types: **`table`

`GroupBy`

— Name of column in `data`

input used to group the data

string | character vector

Name of column in the `data`

input used to group the
data, specified as a string or character vector.
`GroupBy`

does not have to be a model variable
name. For each group designated by `GroupBy`

, the
`modelAccuracy`

function computes the observed
default rates and average predicted PDs are computed to measure the
RMSE.

**Data Types: **`string`

| `char`

### Name-Value 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:**

```
[AccMeasure,AccData] =
modelAccuracy(pdModel,data(Ind,:),'GroupBy',["YOB","ScoreGroup"],'DataID',"DataSetChoice")
```

`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. `DataID`

is included in the
`modelAccuracy`

output for reporting
purposes.

**Data Types: **`char`

| `string`

`ReferencePD`

— Conditional PD values predicted for `data`

by reference model

`[]`

(default) | numeric vector

`ReferenceID`

— Identifier for 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`

— RMSE values

table

Accuracy measure, returned as a table.

RMSE values, returned as a single-column `'RMSE'`

table. The table has one row if only the `pdModel`

accuracy is measured and it has two rows if reference model information
is given. The row names of `AccMeasure`

report the
model IDs, grouping variables, and data ID.

**Note**

The reported RMSE values depend on the grouping variable for
the required `GroupBy`

argument.

`AccData`

— Observed and predicted PD values for each group

table

Accuracy data, returned as a table.

Observed and predicted PD values for each group, returned as a table.
The reported observed PD values correspond to the observed default rate
for each group. The reported predicted PD values are the average PD
values predicted by the `pdModel`

object for each
group, and similarly for the reference model. The
`modelAccuracy`

function stacks the PD data,
placing the observed values for all groups first, then the predicted PDs
for the `pdModel`

, and then the predicted PDs for the
reference model, if given.

The column `'ModelID'`

identifies which rows
correspond to the observed PD, `pdModel`

, or
reference model. The table also has one column for each grouping
variable showing the unique combinations of grouping values. The last
column of `AccData`

is a `'PD'`

column
with the PD data.

## More About

### Model Accuracy

*Model accuracy* measures the accuracy
of the predicted probability of default (PD) values.

To measure model accuracy, also called model calibration, you must compare the predicted PD values to the observed default rates. For example, if a group of customers is predicted to have an average PD of 5%, then is the observed default rate for that group close to 5%?

The `modelAccuracy`

function requires a grouping variable to
compute average predicted PD values within each group and the average observed
default rate also within each group. `modelAccuracy`

uses the
root mean squared error (RMSE) to measure the deviations between the observed
and predicted values across groups. For example, the grouping variable could be
the calendar year, so that rows corresponding to the same calendar year are
grouped together. Then, for each year the software computes the observed default
rate and the average predicted PD. The `modelAccuracy`

function
then applies the RMSE formula to obtain a single measure of the prediction error
across all years in the sample.

Suppose there are N observations in the data set, and there are
*M* groups
*G*_{1},…,*G*_{M}.
The default rate for group *G*_{i} is

$$D{R}_{i}=\frac{{D}_{i}}{{N}_{i}}$$

where:

*D*_{i} is the number of
defaults observed in group
*G*_{i}.

*N*_{i} is the number
of observations in group
*G*_{i}.

The average predicted probability of default
*PD*_{i} for
group *G*_{i} is

$$P{D}_{i}=\frac{1}{{N}_{i}}{\displaystyle {\sum}_{j\in {G}_{i}}PD(j)}$$

where *PD*(*j*) is the probability of
default for observation *j*. In other words, this is the
average of the predicted PDs within group
*G*_{i}.

Therefore, the RMSE is computed as

$$RMSE\text{}=\sqrt{{\displaystyle {\sum}_{i=1}^{M}\left(\frac{{N}_{i}}{N}\right){(D{R}_{i}-P{D}_{i})}^{2}}}$$

The RMSE, as defined, depends on the selected grouping variable. For example, grouping by calendar year and grouping by years-on-books might result in different RSME values.

Use `modelAccuracyPlot`

to
visualize observed default rates and predicted PD values on grouped data.

## 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] Breeden, Joseph.
*Living with CECL: The Modeling Dictionary.* Santa Fe, NM:
Prescient Models LLC, 2018.

[4] Roesch, Daniel and Harald
Scheule. *Deep Credit Risk: Machine Learning with Python.*
Independently published, 2020.

## See Also

`modelDiscrimination`

| `modelDiscriminationPlot`

| `modelAccuracyPlot`

| `predictLifetime`

| `predict`

| `fitLifetimePDModel`

| `Logistic`

| `Probit`

| `Cox`

### Topics

- Basic Lifetime PD Model Validation
- Compare Logistic Model for Lifetime PD to Champion Model
- Compare Lifetime PD Models Using Cross-Validation
- Expected Credit Loss Computation
- Compare Model Discrimination and Accuracy to Validate of Probability of Default
- Compare Probability of Default Using Through-the-Cycle and Point-in-Time Models
- Overview of Lifetime Probability of Default Models

**Introduced in R2020b**

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