# 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,AccData] = modelAccuracy(pdModel,data(Ind,:),GroupingVar,'DataID',DataSetChoice)

`AccMeasure=`*table*
RMSE
_________
Logistic, grouped by YOB, Training 0.0004142

`AccData=`*16×4 table*
ModelID YOB PD GroupCount
__________ ___ _________ __________
"Observed" 1 0.017421 58092
"Observed" 2 0.012305 56723
"Observed" 3 0.011382 55524
"Observed" 4 0.010741 54650
"Observed" 5 0.00809 53770
"Observed" 6 0.0066747 53186
"Observed" 7 0.0032198 36959
"Observed" 8 0.0018757 19193
"Logistic" 1 0.017185 58092
"Logistic" 2 0.012791 56723
"Logistic" 3 0.01131 55524
"Logistic" 4 0.010615 54650
"Logistic" 5 0.0083982 53770
"Logistic" 6 0.0058744 53186
"Logistic" 7 0.0035872 36959
"Logistic" 8 0.0023689 19193

`%disp(AccMeasure) `

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 | `customLifetimePDModel`

object

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

, `Probit`

, or `Cox`

object using
`fitLifetimePDModel`

. Alternatively, you can create a custom
probability of default model using `customLifetimePDModel`

.

**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 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(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 `'PD'`

column of
`AccData`

is a the PD data. The last column of
`AccData`

is a `'GroupCount'`

column
with the group counts 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.

## Version History

**Introduced in R2020b**

### R2022b: Support for `customLifetimePDModel`

model

The `pdModel`

input supports an option for a
`customLifetimePDModel`

model object that you can create using
`customLifetimePDModel`

.

### R2022a: Additional column for `AccData`

for `GroupCount`

There is an additional column for `AccData`

for
`GroupCount`

for PD models.

### R2022a: `GroupCount`

column automatically included in `AccData`

outputs

Starting in R2022a, the `AccData`

output of
`modelAccuracy`

contains an additional column for
`GroupCount`

with the group counts data.

If you extract the end column from the `AccData`

output using
`AccData{:,end}`

, the `end`

column is
different than previous releases of `modelAccuracy`

.

## See Also

`modelDiscrimination`

| `modelDiscriminationPlot`

| `modelAccuracyPlot`

| `predictLifetime`

| `predict`

| `fitLifetimePDModel`

| `Logistic`

| `Probit`

| `Cox`

| `customLifetimePDModel`

### 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

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