diagnostics

Diagnostics — Outlier Detection

Overview

The diagnostics module provides functions for detecting outliers in fitted regression models. The outlier_dummy() function identifies observations that lie outside the expected distribution bounds and creates dummy variables that can be used to re-estimate the model with outlier effects removed.

Import

# R — function is in the greybox namespace
library(greybox)

# Python
from greybox.diagnostics import outlier_dummy, OutlierResult

Algorithm

1. Extract residuals from the fitted model
2. Compute standardised (rstandard) or studentised (rstudent) residuals using leverage (hat values)
3. Determine critical bounds based on the model's distribution family and the specified confidence level
4. Flag observations whose standardised residuals fall outside the bounds
5. Return a matrix of dummy variables (one column per outlier)

outlier_dummy — Detect Outliers and Create Dummy Variables

Parameters

R: outlierdummy()

Parameter	Type	Default	Description
object	alm	—	Fitted alm model
level	numeric	0.999	Confidence level for outlier detection
type	character	"rstandard"	Residual type: "rstandard" or "rstudent"

Python: outlier_dummy()

Parameter	Type	Default	Description
model	ALM	—	Fitted ALM model
level	float	0.999	Confidence level for outlier detection
type	str	"rstandard"	Residual type: "rstandard" or "rstudent"

Return Value

R: outlierdummy object

Field	Description
$outliers	Matrix of dummy variables (or NULL if no outliers)
$statistic	Critical values used for detection
$id	Indices of outlier observations
$level	Confidence level used
$type	Residual type used
$errors	Standardised/studentised residuals

Python: OutlierResult

Attribute	Type	Description
.outliers	np.ndarray or None	Matrix of dummy variables (n_obs x n_outliers), or None if no outliers
.statistic	np.ndarray	Lower and upper critical bounds
.id	np.ndarray	Indices of outlier observations
.level	float	Confidence level used
.type	str	Residual type used ("rstandard" or "rstudent")
.errors	np.ndarray	The standardised/studentised residuals

Distribution-Specific Bounds

The critical bounds are computed using the quantile function of the distribution family:

Distribution	Quantile Distribution
dnorm, dlnorm, ds, dls, dbcnorm, dfnorm, drectnorm	Normal
dlaplace, dllaplace	Laplace
dlogis	Logistic
dt	Student's t (df = residual df)
dgnorm, dlgnorm	Generalised Normal
dgamma	Gamma
dexp	Exponential

Examples

Basic Outlier Detection

# R
library(greybox)

x <- rnorm(100)
y <- 2 * x + rnorm(100)
y[50] <- 100  # inject outlier

model <- alm(y ~ x, distribution="dnorm")
result <- outlierdummy(model, level=0.999)
print(result$id)       # Which observations are outliers
print(result$outliers)  # Dummy variable matrix

# Python
import numpy as np
from greybox.alm import ALM
from greybox.formula import formula
from greybox.diagnostics import outlier_dummy

np.random.seed(42)
x = np.random.randn(100)
y = 2 * x + np.random.randn(100)
y[50] = 100  # inject outlier

data = {"y": y, "x": x}
y_vec, X = formula("y ~ x", data)

model = ALM(distribution="dnorm")
model.fit(X, y_vec)

result = outlier_dummy(model, level=0.999)
print(f"Outlier indices: {result.id}")
print(f"Critical bounds: {result.statistic}")

Re-estimating with Outlier Dummies

# Python — add outlier dummies to the model
if result.outliers is not None:
    X_with_dummies = np.column_stack([X, result.outliers])
    model2 = ALM(distribution="dnorm")
    model2.fit(X_with_dummies, y_vec)
    print(f"Original scale: {model.scale:.4f}")
    print(f"With dummies:   {model2.scale:.4f}")

Using Studentised Residuals

# Python — rstudent is more sensitive to single outliers
result_student = outlier_dummy(model, level=0.999, type="rstudent")
print(f"Outliers (rstudent): {result_student.id}")

R vs Python Function Names

R Function	Python Function
outlierdummy()	outlier_dummy()

References

• Cook, R.D. and Weisberg, S. (1982). Residuals and Influence in Regression. Chapman and Hall.
• Svetunkov, I. (2023). Statistics for Business Analytics. https://openforecast.org/sba/