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Calibrate the alignment to a central tendency

Source: R/pd.R
scr_calibrate.Rd

Re-anchors the probability of default of a scorecard to a long-run average default rate (the central tendency, CT) without touching the points: the result is a new alignment (I*, S*) such that predict(alignment, raw, type = "prob") is the calibrated PD, while the scorecard keeps its own alignment for the score. Four methods:

Usage

scr_calibrate(
  x,
  target,
  sample_rate = NULL,
  method = NULL,
  ar_target = NULL,
  segment = NULL,
  raw = NULL,
  y = NULL,
  sample = "holdout"
)

Arguments

x

An scr_scorecard() (uses the ln(odds) and outcome of sample), an scr_align() (pass raw and, for the two-parameter methods, y) or a numeric vector of event ln(odds) (aligned directly to the default 600/50/20 scale).

target

The central tendency: a number in (0, 1) or an scr_dr from scr_default_rate() (its lra$mean is used). With segment, a named vector with one CT per segment.

sample_rate

Event rate of the calibration sample; NULL uses the mean of y.

method

"intercept", "logodds_ab", "scaling" or "qmm"; NULL uses config$pd_calibration.

ar_target

Target accuracy ratio for "logodds_ab" and "qmm".

segment

Optional vector of segment labels, one per calibration row: one alignment per segment is fitted as well.

raw, y

Raw ln(odds) and 0/1 outcome when x is not a scorecard.

sample

Sample of the scorecard used for the calibration.

Value

An object of class scr_pd_calibration: alignment (the new scr_align), alignment_before, method, ct, target_source, sample_rate, shift (change of the event intercept), shift_prior (the closed-form King-Zeng shift), slope_ratio (S* / S), mean_pd_before, mean_pd_after, ar_before, ar_after (observed), ar_implied_before, ar_implied_after, n, segments (table and alignments when segment is given), ledger. Also ar_target, note and sample.

Details

"intercept"

The prior-correction shift of King and Zeng (2001), \(\delta = \ln[\tau(1-\bar y) / ((1-\tau)\bar y)]\), added to the event ln(odds); S unchanged, so the rank order and every discrimination statistic are untouched. The closed form is exact on the odds; when the calibration sample is available the shift is refined by a one-dimensional root so that the mean PD equals the CT exactly (the closed form is reported as shift_prior).

"logodds_ab"

Tasche (2013): ln(odds*) = a + b ln(odds), with (a, b) solving mean(PD*) = CT and implied accuracy ratio equal to ar_target (default: the accuracy ratio observed on the sample). With the observed accuracy ratio this is Tasche's quasi-moment matching (QMM) proper. The implied AUC is the probability that a default has a higher PD than a non-default when the PDs are true: each score carries weight PD* among the defaults and 1 - PD* among the non-defaults, ties counted one half.

"qmm"

The outcome-free variant of the same two equations: the target accuracy ratio is the implied AR of the current PDs, so no outcome is needed and the implied discriminatory power of the uncalibrated curve is carried over to the new level.

"scaling"

PD* = PD * CT / ybar. The proportional rescaling is not a logit map, so the slope is the least-squares projection of logit(PD*) on the ln(odds) and the intercept is solved to the CT.

References

King, G. and Zeng, L. (2001). Logistic regression in rare events data. Political Analysis, 9(2), 137-163.

Tasche, D. (2013). The art of probability-of-default curve calibration. Journal of Credit Risk, 9(4), 63-103.

See also

Other irb-pd: predict.scr_grades(), predict.scr_pd(), scr_grades(), scr_master_scale(), scr_migration(), scr_moc(), scr_pd(), scr_pd_pit_ttc(), scr_pd_validate()

Examples

set.seed(1)
l <- stats::qlogis(0.12) + stats::rnorm(2000)
y <- stats::rbinom(2000, 1, stats::plogis(l))
cal <- scr_calibrate(l, target = 0.04, y = y)
cal
#> <scr_pd_calibration> method intercept | CT 4.000% (numeric) | sample rate 16.450% | n 2,000
#>   event ln(odds)* = -1.642243 +1.000000 * ln(odds)   [prior shift -1.552934]
#>   alignment: I 0.000000 -> 1.642243 | S -1.000000 -> -1.000000
#>   mean PD 15.709% -> 4.000% | AR observed 0.5532 -> 0.5532 | AR implied 0.5061 -> 0.5237
mean(predict(cal$alignment, l, type = "prob"))
#> [1] 0.04
scr_calibrate(l, target = 0.04, y = y, method = "logodds_ab", ar_target = 0.55)
#> <scr_pd_calibration> method logodds_ab | CT 4.000% (numeric) | sample rate 16.450% | n 2,000
#>   event ln(odds)* = -1.570900 +1.065298 * ln(odds)   [prior shift -1.552934]
#>   alignment: I 0.000000 -> 1.570900 | S -1.000000 -> -1.065298
#>   mean PD 15.709% -> 4.000% | AR observed 0.5532 -> 0.5532 | AR implied 0.5061 -> 0.5500