A case study on using generalized additive models to … · A case study on using generalized additive models to ﬁt credit rating ... German Credit Data ... capital requirements

A case study on using generalizedadditive models to fit credit ratingscores

Marlene Müller, [email protected]

This version: July 8, 2009, 14:32

Contents

Application: Credit Rating

Aim of this Talk

Case StudyGerman Credit DataAustralian Credit DataFrench Credit DataUC2005 Credit Data

Simulation Study

Conclusions

Appendix: Further PlotsAustralian Credit DataFrench Credit DataUC2005 Credit Data

1


- Basel II: capital requirements of a bank are adapted to the individual creditportfolio

- core terms: determine rating score and subsequently default probabilities(PDs) as a function of some explanatory variables

- further terms: loss given default, portfolio dependence structure

- in practice: often classical logit/probit-type models to estimate linearpredictors (scores) and probabilities (PDs)

- statistically: 2-group classification problem

risk management issues

- credit risk is ony one part of a bank’s total risk:; will be aggregated with other risks

- credit risk estimation from historical data:; stress-tests to simulate future extreme situations; need to easily adapt the rating system to possible future changes; possible need to extrapolate to segments without observations

2


- Basel II: capital requirements of a bank are adapted to the individual creditportfolio

- core terms: determine rating score and subsequently default probabilities(PDs) as a function of some explanatory variables

- further terms: loss given default, portfolio dependence structure

- in practice: often classical logit/probit-type models to estimate linearpredictors (scores) and probabilities (PDs)

- statistically: 2-group classification problem

risk management issues

- credit risk is ony one part of a bank’s total risk:; will be aggregated with other risks

- credit risk estimation from historical data:; stress-tests to simulate future extreme situations; need to easily adapt the rating system to possible future changes; possible need to extrapolate to segments without observations

2

(Simplified) Development of Rating Score and Default Probability

� raw data:

Xj measurements of several variables (“risk factors”)

� (nonlinear) transformation:

Xj → eXj = mj(Xj)

; handle outliers, allow for nonlinear dependence on raw risk factors

� rating score:S = w1

eX1 + . . . + wdeXd

� default probability:

PD = P(Y = 1|X) = G(w1eX1 + . . . + wd

eXd)

(where G is e.g. the logistic or gaussian cdf ; logit or probit)

3


� raw data:



Xj → eXj = mj(Xj)



eX1 + . . . + wdeXd


PD = P(Y = 1|X) = G(w1eX1 + . . . + wd

eXd)


3


� raw data:



Xj → eXj = mj(Xj)



eX1 + . . . + wdeXd


PD = P(Y = 1|X) = G(w1eX1 + . . . + wd

eXd)


3


� raw data:



Xj → eXj = mj(Xj)



eX1 + . . . + wdeXd


PD = P(Y = 1|X) = G(w1eX1 + . . . + wd

eXd)


3


� raw data:



Xj → eXj = mj(Xj)



eX1 + . . . + wdeXd


PD = P(Y = 1|X) = G(w1eX1 + . . . + wd

eXd)


3

Aim of this Talk

case study on (cross-sectional) rating data

- compare different approaches to generalized additive models (GAM)

- consider models that allow for additional categorical variables; partial linear terms (combination of GAM/GPLM)

� generalized additive models allow for a simultaneous fit of thetransformations from the raw data, the linear rating score and the defaultprobabilities

4

Outline of the Study

� credit data case study: 4 credit datasets

regressorsdataset sample defaults continuous discrete categoricalGerman Credit 1000 30.00% 3 – 17Australian Credit 678 55.90% 3 1 8French Credit 8178 5.86% 5 3 15UC2005 Credit 5058 23.92% 12 3 21

- differences between different approaches?- improvement of default predictions?

� simulation study: comparison of additive model (AM) and GAM fits

- differences between different approaches?- what if regressors are concurve? (nonlinear version of multicollinear)- do sample size and default rate matter?

5

Generalized Additive Model

� logit/probit are special cases of the generalized linear model (GLM)

E(Y |X) = G“

X⊤

β”

� “classic” generalized additive model

E(Y |X) = G

8

<

:

c +

pX

j=1

mj(Xj)

9

=

;

mj nonparametric

� generalized additive partial linear model (semiparametric GAM)

E(Y |X1, X2) = G

8

<

:

c + X⊤

1 β +

pX

j=1

mj(X2j)

9

=

;

mj nonparametric

linear part

- allows for known transformation functions

- allows to add / control for categorical regressors

6



E(Y |X) = G“

X⊤

β”


E(Y |X) = G

8

<

:

c +

pX

j=1

mj(Xj)

9

=

;

mj nonparametric


E(Y |X1, X2) = G

8

<

:

c + X⊤

1 β +

pX

j=1

mj(X2j)

9

=

;

mj nonparametric

linear part



6



E(Y |X) = G“

X⊤

β”


E(Y |X) = G

8

<

:

c +

pX

j=1

mj(Xj)

9

=

;

mj nonparametric


E(Y |X1, X2) = G

8

<

:

c + X⊤

1 β +

pX

j=1

mj(X2j)

9

=

;

mj nonparametric

linear part



6

R “Standard” Tools

two main approaches for GAM in

- gam::gam ; backfitting with local scoring (Hastie and Tibshirani; 1990)

- mgcv::gam ; penalized regression splines (Wood; 2006)

; compare these procedures under the default settings of gam::gam andmgcv::gam

competing estimators:

- logit binary GLM with G(u) = 1/{1 + exp(−u)} (logistic cdf as link)

- logit2, logit3 binary GLM with 2nd / 3rd order polynomial terms for thecontinuous regressors

- logitc binary GLM with continuous regressors categorized (4–5 levels)

- gam binary GAM using gam::gam with s() terms for continuous

- mgcv binary GAM using mgcv::gam

7

German Credit Data

� from http://www.stat.uni-muenchen.de/service/datenarchiv/kredit/kredit_e.html

regressorsdataset name sample defaults continuous discrete categoricalGerman 1000 30.00% 3 – 17

� 3 continuous regressors: age, amount, duration (time to maturity)

� use 10 CV subsamples for validation

� stratified data (true default rate ≈ 5%)

� important findings:- some observation(s) that seem to confuse mgcv::gam in one CV subsample

(→ see following slides)- however, mgcv::gam seems to improve deviance and discriminatory power

w.r.t. gam::gam- estimation times of mgcv::gam are between 4 to 7 times higher than for

gam::gam (not more than around a second, though)- if we only use the continuous regressors: both GAM estimators are comparable

to logit cubic additive functions

8

http://www.stat.uni-muenchen.de/service/datenarchiv/kredit/kredit_e.html

German Credit Data

� from http://www.stat.uni-muenchen.de/service/datenarchiv/kredit/kredit_e.html

regressorsdataset name sample defaults continuous discrete categoricalGerman 1000 30.00% 3 – 17

� 3 continuous regressors: age, amount, duration (time to maturity)

� use 10 CV subsamples for validation


� important findings:- some observation(s) that seem to confuse mgcv::gam in one CV subsample

(→ see following slides)- however, mgcv::gam seems to improve deviance and discriminatory power

w.r.t. gam::gam- estimation times of mgcv::gam are between 4 to 7 times higher than for

gam::gam (not more than around a second, though)- if we only use the continuous regressors: both GAM estimators are comparable

to logit cubic additive functions

8

http://www.stat.uni-muenchen.de/service/datenarchiv/kredit/kredit_e.html

German Credit Data: Additive Functions

3.0 3.2 3.4 3.6 3.8 4.0 4.2

−0.

50.

00.

5

age

s(ag

e,1)

Variable age (mgcv and blue: gam)

6 7 8 9

−4

−2

02

amount

s(am

ount

,4.4

9)

Variable amount (mgcv and blue: gam)

1.5 2.0 2.5 3.0 3.5 4.0

−2

−1

01

2

duration

s(du

ratio

n,1)

Variable duration (mgcv and blue: gam)

9

How to Compare Binary GLM Fits?

� preferably by out-of-sample validation ; block cross-validation approach:leave out subsamples of x% from the fitting procedure, estimate from theremaining (100-x)% and calculate validation criteria from the x% left-out

� two criteria for comparison: deviance (→ goodness of fit) and accuracyratios AR from CAP curves (→ discriminatory power)

� CAP curve (Lorenz curve) and the accuracy ratio AR:

- plot the empirical cdf of the fitted scoresagainst the empirical cdf of the fitted

default sample scores (precisely 1 − bF vs.

1 − bF(.|Y = 1))

- AR is the area between CAP curve anddiagonal in relation to the correspondingarea for the best possible CAP curve (bestpossible ∼= perfect separation)

- relation to ROC: compares bF(.|Y = 0) andbF(.|Y = 1) and it holds

AR = 2 AUC−1

PD

1−F(s)

best possible CAP curve

Percentage

100%

100%

1−F (s)1

CAP curve

2

1_G

Percentage of applicants

of defaults

10

How to Compare Binary GLM Fits?

� preferably by out-of-sample validation ; block cross-validation approach:leave out subsamples of x% from the fitting procedure, estimate from theremaining (100-x)% and calculate validation criteria from the x% left-out

� two criteria for comparison: deviance (→ goodness of fit) and accuracyratios AR from CAP curves (→ discriminatory power)

� CAP curve (Lorenz curve) and the accuracy ratio AR:

- plot the empirical cdf of the fitted scoresagainst the empirical cdf of the fitted

default sample scores (precisely 1 − bF vs.

1 − bF(.|Y = 1))

- AR is the area between CAP curve anddiagonal in relation to the correspondingarea for the best possible CAP curve (bestpossible ∼= perfect separation)

- relation to ROC: compares bF(.|Y = 0) andbF(.|Y = 1) and it holds

AR = 2 AUC−1

PD

1−F(s)

best possible CAP curve

Percentage

100%

100%

1−F (s)1

CAP curve

2

1_G

Percentage of applicants

of defaults

10

German Credit Data: Comparison

logit1 logit2 logit3 logitc gam mgcv

0.45

0.50

0.55

0.60

0.65

German: Accuracy Ratios (AR)

2 4 6 8 10

0.45

0.50

0.55

0.60

0.65

German: Accuracy Ratios (AR)

sample

2 4 6 8 10

0.45

0.50

0.55

0.60

0.65

gam vs. mgcv (AR)

sample


100

150

200

250

300

350

German: Deviances

2 4 6 8 10

100

150

200

250

300

350

German: Deviances

sample


02

46

810

German: Estimation Times

Figure: Out of sample comparison (blockwise CV with 10 blocks) for various estimators,accuracy ratios from CAP curves (upper panels), deviance values and estimation times(lower panels)

11

German Credit Data: Models with only Continuous Regressors


0.0

0.1

0.2

0.3

0.4

0.5

German−Metric: Accuracy Ratios (AR)

2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

German−Metric: Accuracy Ratios (AR)

sample

2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

gam vs. mgcv (AR)

sample


110

115

120

125

130

German−Metric: Deviances

2 4 6 8 10

110

115

120

125

130

German−Metric: Deviances

sample


0.0

0.1

0.2

0.3

0.4

0.5

German−Metric: Estimation Times


12

Australian Credit Data

� from http://archive.ics.uci.edu/ml/datasets/Statlog+(Australian+Credit+Approval)

� used for estimation:

regressorsdataset name sample defaults continuous discrete categoricalAustralian 678 55.90% 3 1 8

� use only 7 CV subsamples for validation

� original A13 and A14 were dropped since actually multicollinear with A10, someobservations were dropped because of very few categories

� A10 was transformed to log(1 + A10), nevertheless used only as a linear predictor(as half of the observations have the same value)

� important findings:

- essentially, the estimated additive function for A2 differs between mgcv::gamand gam::gam

- gam::gam mostly outperforms than all other estimates (recall, that however thenumber of CV subsamples is rather small!)

- estimation times of mgcv::gam are around 3 to 5 times higher than forgam::gam (less than a second, though)

13

http://archive.ics.uci.edu/ml/datasets/Statlog+(Australian+Credit+Approval)

French Credit Data

� data were already analyzed with GPLMs in Müller and Härdle (2003), here used forestimation:

regressorsdataset name sample defaults continuous discrete categoricalFrench 8178 5.86% 5 3 15

� use the same preprocessing as in as in Müller and Härdle (2003)

� the original estimation + validation samples were merged, use 20 CV subsamples forvalidation instead

� continuous variables are X1, X2, X3, X4 and X6, in particular X3, X4 and X6 areknown to have nonlinear form in a GAM


- it is confirmed that additive functions for X3, X4 and X6 should be modelled bya nonlinear function be nonlinear

- again observation(s) "confusing" mgcv::gam in one of the subsamples- all estimates show similar discriminatory power, though with a slightly better

performance for both mgcv::gam and gam::gam- estimation times of mgcv::gam are around 15 to 24 times higher than for

gam::gam (for the largest model: 20-40 sec. on a 3Ghz Intel CPU for thesubsamples of about 7800 observations)

14

UC2005 Credit Data

� data from the 2005 UC data mining competitionwere already analyzed with GPLMsin Müller and Härdle (2003), here used for estimation:

regressorsdataset name sample defaults continuous discrete categoricalUC2005 5058 23.92% 12 3 21

� the original estimation + validation + quiz samples were merged, use again 20 CVsubsamples for validation


� several of the variables have been preprocessed with a log-transform or to binary

� in general, the data haven’t been very carefully analysed, it’s use is rather meant a“proof-of concept”


- there are again observations "confusing" mgcv::gam in one of the subsamples- performance of mgcv::gam and gam::gam w.r.t. is very similar and

outperforms the other approaches (closest to them is the logit fit with cubicadditive functions)

- estimation times of mgcv::gam are around 8 to 40 times higher than forgam::gam (for the largest model: 5-8 min on a 3Ghz Intel CPU for up to 400seconds for the subsamples of about 4800 observations)

15

Simulation Study for (G)PLM

E(Y |X, T) = β1X1 + β2X2 + m(T)

which of the (G)AM estimators is preferable ...?

� ... to fit the additive component functions and/or the regression function?

� ... w.r.t. discriminatory power in the GPLM/GAM cases?

� ... from a practical point of view (comp. speed, numerical stability etc.)?

simulation setup:

β1 = 1, β2 = −1, m(t) = 1.5 cos(πt) + c

X1, U, T ∼ Uniform[-1,1], X2 ∼ m (ρT + (1 − ρ)U) (centered)

nsim = 1000, n ∈ {100, 1000, 10000}, ρ ∈ {0.0, 0.7}, c ∈ {0,−1,−2}

� X2 and T are nonlinearly dependent (if ρ = 0.7) or independent otherwise

� sample size n up to 10000 which is a possible size for credit data

� the intercept c controls for the default rate (15%–50%) in the GPLM

16


E(Y |X, T) = β1X1 + β2X2 + m(T)





simulation setup:

β1 = 1, β2 = −1, m(t) = 1.5 cos(πt) + c


nsim = 1000, n ∈ {100, 1000, 10000}, ρ ∈ {0.0, 0.7}, c ∈ {0,−1,−2}




16


E(Y |X, T) = β1X1 + β2X2 + m(T)





simulation setup:

β1 = 1, β2 = −1, m(t) = 1.5 cos(πt) + c


nsim = 1000, n ∈ {100, 1000, 10000}, ρ ∈ {0.0, 0.7}, c ∈ {0,−1,−2}




16

Simulation Study: Additive Components for GPLM

gammgcv

ρ = 0.7c = −2

n = 1000

gam mgcv

0.00

000

0.00

010

0.00

020

0.00

030

MSE m(t)

gam mgcv

0.00

000

0.00

005

0.00

010

0.00

015

0.00

020

0.00

025

0.00

030

MSE beta1

gam mgcv

0.00

000

0.00

005

0.00

010

0.00

015

0.00

020

0.00

025

MSE beta2

gammgcv

ρ = 0.7c = −2

n = 10000

gam mgcv

0e+

002e

−05

4e−

056e

−05

MSE m(t)

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

2.5e

−05

MSE beta1

gam mgcv

0e+

001e

−05

2e−

053e

−05

4e−

05

MSE beta2

17

Simulation Study: Independent Components vs. Dependent

gammgcv

ρ = 0.7

c = 0

n = 10000

gam mgcv

0e+

001e

−05

2e−

053e

−05

4e−

05

MSE m(t)

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

2.5e

−05

3.0e

−05

MSE beta1

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

MSE beta2

gammgcv

ρ = 0.7

c = −2

n = 10000

gam mgcv

0e+

002e

−05

4e−

056e

−05

MSE m(t)

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

2.5e

−05

MSE beta1

gam mgcv

0e+

001e

−05

2e−

053e

−05

4e−

05

MSE beta2

18

Simulation Study: Comparison with Components for PLM

gammgcv

ρ = 0.7c = 0

n = 1000

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

MSE m(t)

gam mgcv

0e+

001e

−07

2e−

073e

−07

4e−

07

MSE beta1

gam mgcv

0e+

001e

−06

2e−

063e

−06

4e−

065e

−06

MSE beta2

gammgcv

ρ = 0.7c = 0

n = 10000

gam mgcv

0.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

MSE m(t)

gam mgcv

0e+

001e

−08

2e−

083e

−08

4e−

085e

−08

6e−

08

MSE beta1

gam mgcv

0.0e

+00

5.0e

−07

1.0e

−06

1.5e

−06

2.0e

−06

2.5e

−06

3.0e

−06

MSE beta2

19

Simulation Study: Deviance and Discriminatory Power for GPLM

gammgcv

ρ = 0.7c = −2

n = 1000

gam mgcv

650

700

750

800

850

Deviances

gam mgcv

0.35

0.40

0.45

0.50

0.55

0.60

Accuracy Ratios (AR)

gam mgcv

0.30

0.35

0.40

0.45

0.50

Kolmogorov Stats (T)

gammgcv

ρ = 0.7c = −2

n = 10000

gam mgcv

7200

7300

7400

7500

7600

7700

7800

Deviances

gam mgcv

0.44

0.46

0.48

0.50

Accuracy Ratios (AR)

gam mgcv

0.32

0.34

0.36

0.38

Kolmogorov Stats (T)

in fact, most of the gam::gam deviances are larger here than the mgcv::gam deviances

and gam::gam fits have smaller discriminatory power

20

Simulation Study: Estimation Times for GPLM

gammgcv

ρ = 0.7c = −2

gam mgcv

0.1

0.2

0.3

0.4

0.5

0.6

Estimation Time

gam mgcv

12

34

56

Estimation Time

n = 1000 n = 10000

(estimation times in sec. on a Xeon 2.50GHz)

21

Conclusions

� typically, categorical regressors improve fit significantly, thereforeestimation methods for credit data should adequately use these

� backfitting + local scoring (gam::gam) provides fast and numerically stableresults

� there is however clear indication, that penalized regression splines(mgcv::gam) may provide more precise estimates of the additivecomponent functions; its current drawbacks are:

- estimation time (increasing with model complexity, categorical variables)- effects are to be seen only in large samples

� thus: no clear recommendation, no “ultimate method”; clearly topics for more research

22

References

Härdle, W., Müller, M., Sperlich, S. and Werwatz, A. (2004). Nonparametric and SemiparametricModeling: An Introduction, Springer, New York.

Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models, Vol. 43 of Monographs onStatistics and Applied Probability, Chapman and Hall, London.

Müller, M. (2001). Estimation and testing in generalized partial linear models — a comparative study,Statistics and Computing 11: 299–309.

Müller, M. and Härdle, W. (2003). Exploring credit data, in G. Bol, G. Nakhaeizadeh, S. Rachev, T. Ridderand K.-H. Vollmer (eds), Credit Risk - Measurement, Evaluation and Management, Physica-Verlag.

Speckman, P. E. (1988). Regression analysis for partially linear models, Journal of the Royal StatisticalSociety, Series B 50: 413–436.

Wood, S. N. (2006). Generalized Additive Models: An Introduction with R, Texts in Statistical Science,Chapman and Hall, London.

23

Australian Credit Data: Additive Functions

3.0 3.5 4.0

−1.

0−

0.5

0.0

0.5

A2

s(A

2,1)

Variable A2 (mgcv and blue: gam)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

−3

−2

−1

01

23

A3

s(A

3,4.

35)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

−10

−5

0

A7

s(A

7,6.

27)


24

Australian Credit Data: Comparison


0.75

0.80

0.85

0.90

Australian: Accuracy Ratios (AR)

1 2 3 4 5 6 7

0.75

0.80

0.85

0.90

Australian: Accuracy Ratios (AR)

sample

1 2 3 4 5 6 7

0.75

0.80

0.85

0.90

gam vs. mgcv (AR)

sample


5060

7080

90

Australian: Deviances

1 2 3 4 5 6 7

5060

7080

90

Australian: Deviances

sample


0.0

0.1

0.2

0.3

0.4

0.5

0.6

Australian: Estimation Times


25

Australian Credit Data: Models with only Continuous Regressors


0.30

0.35

0.40

0.45

0.50

0.55

0.60

Australian−Metric: Accuracy Ratios (AR)

1 2 3 4 5 6 7

0.30

0.35

0.40

0.45

0.50

0.55

0.60

Australian−Metric: Accuracy Ratios (AR)

sample

1 2 3 4 5 6 7

0.30

0.35

0.40

0.45

0.50

0.55

0.60

gam vs. mgcv (AR)

sample


105

110

115

120

125

Australian−Metric: Deviances

1 2 3 4 5 6 7

105

110

115

120

125

Australian−Metric: Deviances

sample


0.00

0.05

0.10

0.15

0.20

0.25

Australian−Metric: Estimation Times


26

French Credit Data: Additive Functions

−1 0 1 2 3

−0.

50.

00.

51.

0

X1

s(X

1,1)

Variable X1 (mgcv and blue: gam)

−1 0 1 2 3

−1.

5−

1.0

−0.

50.

00.

5

X2

s(X

2,1)


0 1 2 3

−0.

4−

0.3

−0.

2−

0.1

0.0

0.1

X3

s(X

3,1)


0 1 2 3

−0.

50.

00.

51.

0

X4

s(X

4,5.

19)


0 1 2 3−

1.5

−1.

0−

0.5

0.0

0.5

X6

s(X

6,3.

34)


27

French Credit Data: Comparison


0.3

0.4

0.5

0.6

French: Accuracy Ratios (AR)

5 10 15 20

0.3

0.4

0.5

0.6

French: Accuracy Ratios (AR)

sample

5 10 15 20

0.3

0.4

0.5

0.6

gam vs. mgcv (AR)

sample


150

200

250

300

350

400

French: Deviances

5 10 15 20

150

200

250

300

350

400

French: Deviances

sample


050

100

150

200

French: Estimation Times


28

French Credit Data: Models with only Significant Regressors


0.30

0.35

0.40

0.45

0.50

0.55

French−Signif: Accuracy Ratios (AR)

5 10 15 20

0.30

0.35

0.40

0.45

0.50

0.55

French−Signif: Accuracy Ratios (AR)

sample

5 10 15 20

0.30

0.35

0.40

0.45

0.50

0.55

gam vs. mgcv (AR)

sample


160

170

180

190

French−Signif: Deviances

5 10 15 20

160

170

180

190

French−Signif: Deviances

sample


05

1015

French−Signif: Estimation Times


29

French Credit Data: Models with only Metric Regressors


0.1

0.2

0.3

0.4

0.5

French−Metric: Accuracy Ratios (AR)

5 10 15 20

0.1

0.2

0.3

0.4

0.5

French−Metric: Accuracy Ratios (AR)

sample

5 10 15 20

0.1

0.2

0.3

0.4

0.5

gam vs. mgcv (AR)

sample


170

175

180

185

190

195

200

French−Metric: Deviances

5 10 15 20

170

175

180

185

190

195

200

French−Metric: Deviances

sample


02

46

810

12

French−Metric: Estimation Times


30

UC2005 Credit Data: Additive Functions

−2 0 2 4

−1.

5−

1.0

−0.

50.

00.

51.

0

X1

s(X

1,2.

13)


−4 −2 0 2

−0.

50.

00.

51.

0

X4

s(X

4,1.

84)


−3 −2 −1 0 1 2 3

−1

01

23

X5

s(X

5,5.

24)


−1 0 1 2

−2

02

46

8

X13

s(X

13,4

.43)


0 2 4

−2

−1

01

X14

s(X

14,2

.11)


−2 −1 0 1

−0.

40.

00.

20.

40.

60.

81.

0

X15

s(X

15,2

.77)


−1 0 1 2 3

−3

−2

−1

01

X26

s(X

26,1

.64)


0 1 2 3

01

23

4

X28

s(X

28,4

.43)


−1 0 1 2 3

−2

02

4

X29

s(X

29,2

.6)


−1 0 1 2 3 4 5

−2.

5−

1.5

−0.

50.

00.

5

X30

s(X

30,4

.43)


−1 0 1 2 3 4 5

−2

02

46

X33

s(X

33,7

.99)


0 2 4 6 8

−6

−4

−2

02

X37

s(X

37,2

.1)


31

UC2005 Credit Data: Comparison


0.92

0.94

0.96

0.98

UC2005: Accuracy Ratios (AR)

5 10 15 20

0.92

0.94

0.96

0.98

UC2005: Accuracy Ratios (AR)

sample

5 10 15 20

0.92

0.94

0.96

0.98

gam vs. mgcv (AR)

sample


5010

015

020

025

030

0

UC2005: Deviances

5 10 15 20

5010

015

020

025

030

0

UC2005: Deviances

sample


010

020

030

040

050

0

UC2005: Estimation Times


32

UC2005 Credit Data: Models with only Metric Regressors


0.85

0.90

0.95

UC2005−Metric: Accuracy Ratios (AR)

5 10 15 20

0.85

0.90

0.95

UC2005−Metric: Accuracy Ratios (AR)

sample

5 10 15 20

0.85

0.90

0.95

gam vs. mgcv (AR)

sample


6080

100

120

140

160

UC2005−Metric: Deviances

5 10 15 20

6080

100

120

140

160

UC2005−Metric: Deviances

sample


020

4060

80

UC2005−Metric: Estimation Times


33

Documents

A case study on using generalized additive models to … · A case study on using generalized additive models to ﬁt credit rating ... German Credit Data ... capital requirements