3.4 Variable Transformation & Scaling

1.3 Feature Transformations & Scaling

Exam Focus: Actuarial rating variables rarely enter predictive models in raw linear format. Skewed exposures, high-cardinality predictors, non-linear age curves, and regulatory mandates require sophisticated feature engineering. The CAS PCPA exam requires complete mastery of power transformations (Box-Cox vs. Yeo-Johnson), mathematical sensitivity to feature scaling across distinct algorithms (LASSO vs. tree ensembles vs. GLMs), optimal supervised binning, and spline basis functions that enforce monotonic risk relativities.

In property and casualty insurance, raw features exhibit non-linearities, heavy right-skewness, and severe exposure clustering. For instance, commercial property replacement values span several orders of magnitude, driver crash risk surges for teenagers before leveling off in adulthood, and commercial liability financial metrics can be zero or negative. Preparing these features for a Generalized Linear Model demands deliberate mathematical transformation of the predictors.

---

1. Power Transformations for Skewed Actuarial Features

Many statistical modeling techniques assume that continuous predictors relate linearly to the link-transformed response and possess homoscedastic variance. When raw features violate these assumptions, power transformations stabilize variance and improve model fit.

                          ┌────────────────────────────────────────────────────────┐
                          │                 Power Transformations                  │
                          └────────────────────────────────────────────────────────┘
                                                       │
         ┌─────────────────────────────────────────────┼─────────────────────────────────────────────┐
         ▼                                             ▼                                             ▼
  ┌──────────────┐                              ┌──────────────┐                              ┌──────────────┐
  │   Natural    │                              │   Box-Cox    │                              │ Yeo-Johnson  │
  │  Logarithm   │                              │ (Strictly >0)│                              │ (All Reals)  │
  └──────┬───────┘                              └──────┬───────┘                              └──────┬───────┘
         │                                             │                                             │
         ▼                                             ▼                                             ▼
Compresses monetary values:                    Parametric family with                        Accommodates zero and
y = ln(x) for x > 0.                           optimal λ via MLE:                            negative insurance
Linearizes multiplicative                      y^(λ) = (y^λ - 1) / λ.                        covariates (e.g. net
relationships in log-link GLMs.                Fails if y ≤ 0.                               profit, experience mod).

The Natural Logarithm

The natural log transformation y = ln(x) (for x > 0) is the most ubiquitous actuarial transformation. It is standard for monetary variables (Total Insured Value, annual payroll, vehicle replacement cost) and exposure durations.

• Multiplicative Interpretation: In a log-link GLM (ln(μ) = β₀ + β₁ · ln(X)), exponentiating yields μ = exp(β₀) · X^(β₁). The coefficient β₁ represents the elasticity of the expected loss cost with respect to X. If β₁ = 1, the loss cost scales in direct, 1-to-1 proportion with exposure (which is why exposure enters frequency GLMs with an offset constraint β = 1).
• Handling Zeroes: When features contain zeroes (e.g., prior claim count, number of minor speeding convictions), analysts sometimes apply ln(x + c), where c is a small arbitrary constant (e.g., c = 1). However, the choice of c artificially warps the shape of the curve near zero, distorting rating relativities for preferred zero-violation risks.

Box-Cox Transformation

George Box and David Cox (1964) introduced a generalized parametric family of power transformations designed to transform continuous data into approximately normal distributions:

y^(λ) = (y^λ - 1) / λ if λ ≠ 0, and ln(y) if λ = 0

Mathematical Constraints & Maximum Likelihood Estimation:

• Positivity Requirement: The classical Box-Cox transformation is strictly defined only for strictly positive variables (y > 0). It cannot accommodate zero or negative values.
• Optimal Parameter Estimation: The transformation parameter λ is estimated simultaneously via Maximum Likelihood Estimation (MLE) on the model residuals. The profile log-likelihood function maximized over λ is: L(λ) = -(n / 2) · ln(σ̂²(λ)) + (λ - 1) ∑ ln(y_i) Where σ̂²(λ) = (1 / n) ∑ [y_i^(λ) - ȳ^(λ)]².

Yeo-Johnson Transformation

To resolve Box-Cox's inability to handle non-positive values, In-Kwon Yeo and Richard Johnson (2000) formulated an extended power transformation supporting the entire real line (y ∈ ℝ):

• If λ ≠ 0 and y ≥ 0: ψ(λ, y) = [(y + 1)^λ - 1] / λ
• If λ = 0 and y ≥ 0: ψ(λ, y) = ln(y + 1)
• If λ ≠ 2 and y < 0: ψ(λ, y) = -[(-y + 1)^(2 - λ) - 1] / (2 - λ)
• If λ = 2 and y < 0: ψ(λ, y) = -ln(-y + 1)

Why Yeo-Johnson is Essential for P&C Commercial Lines:

Commercial underwriting datasets frequently contain legitimate zero and negative financial metrics:

1. Net Annual Operating Profit / Cash Flow: Used as an underwriting solvency predictor in commercial surety and directors & officers (D&O) liability.
2. Prior Year Experience Rating Modification Change: ΔMod = Mod_t - Mod_(t-1), where negative values indicate improving safety records.
3. Fleet Size Net Growth: Commercial auto fleets expanding (positive) or contracting (negative). Yeo-Johnson seamlessly normalizes these mixed-sign features without requiring artificial offset constants.

---

2. Feature Scaling & Algorithmic Sensitivity

Feature scaling rescales independent variables to a common numerical range. While critical for some algorithms, it is entirely irrelevant for others.

                                  ┌───────────────────────────────────┐
                                  │    Scaling Methods Comparison     │
                                  └───────────────────────────────────┘
                                                    │
         ┌──────────────────────────────────────────┼──────────────────────────────────────────┐
         ▼                                          ▼                                          ▼
  ┌──────────────┐                           ┌──────────────┐                           ┌──────────────┐
  │   Min-Max    │                           │Standardizati.│                           │Robust Scaler │
  │ Normalizati. │                           │  (Z-Score)   │                           │    (IQR)     │
  └──────┬───────┘                           └──────┬───────┘                           └──────┬───────┘
         │                                          │                                          │
         ▼                                          ▼                                          ▼
  x' = (x - xmin) /                          x' = (x - μ) / σ                            x' = (x - x̃) / IQR
     (xmax - xmin)                                  │                                          │
         │                                          ▼                                          ▼
Maps to [0, 1]. Highly                     Mean = 0, StDev = 1.                        Resistant to extreme
sensitive to extreme                       Standard choice for                         P&C loss and exposure
outliers (tail squishing).                 regularized models.                         outliers in predictors.

Scaling Methods Formulation:

1. Min-Max Normalization: x' = (x - x_min) / (x_max - x_min) ∈ [0, 1] Vulnerability: Highly sensitive to outliers. A single massive commercial payroll will compress 99% of policyholders into a tight range between 0.00 and 0.02.
2. Standardization (Z-Score Scaling): x' = (x - μ) / σ (with mean 0, unit variance)
3. Robust Scaling: x' = (x - median(x)) / IQR(x) Centers at median and scales by IQR, remaining completely impervious to outlier corruption.

Algorithmic Sensitivity Matrix

---

3. Discretization & Binning of Continuous Actuarial Predictors

Actuaries frequently discretize continuous variables into discrete intervals (bins) to reflect rating territories, driver age brackets, or vehicle symbol groupings.

                                 ┌─────────────────────────────────────┐
                                 │         Binning Methodologies       │
                                 └─────────────────────────────────────┘
                                                    │
         ┌──────────────────────────────────────────┼──────────────────────────────────────────┐
         ▼                                          ▼                                          ▼
  ┌──────────────┐                           ┌──────────────┐                           ┌──────────────┐
  │ Equal-Width  │                           │Equal-Freq.   │                           │ Supervised   │
  │   Binning    │                           │ (Quantiles)  │                           │ (Tree-Based) │
  └──────┬───────┘                           └──────┬───────┘                           └──────┬───────┘
         │                                          │                                          │
         ▼                                          ▼                                          ▼
Interval width Δ = (xmax-xmin)/k.          Each bin contains N/k records.              Decision tree optimizes
Creates sparse or empty                    Ensures equal credibility,                  splits based on Poisson/
tail buckets in skewed P&C                 but clumps distinct high-                   Gamma deviance while enforcing
exposure distributions.                    hazard risks into single bin.               minimum exposure constraints.

1. Equal-Width Binning

Divides the range [x_min, x_max] into k intervals of identical width Δ = (x_max - x_min) / k.

• Actuarial Flaw: Because insurance data is skewed, the vast majority of policyholders fall into the first two bins, while upper bins have near-zero exposure, creating credibility deficiencies and volatile rating relativities.

2. Equal-Frequency (Quantile) Binning

Sorts data and establishes cutoffs such that each bin contains exactly 1/k of the total exposure or policy count (e.g., deciles or ventiles).

• Advantage: Guarantees sufficient earned car-years or payroll in every bucket, ensuring that credibility criteria (e.g., Classical or Bühlmann credibility) are satisfied.
• Disadvantage: Bins in the extreme tails span massive numerical ranges, masking distinct high-hazard risk profiles.

3. Supervised Binning via Decision Trees

Uses a shallow single-variable decision tree (e.g., CART) with a Poisson or Gamma deviance splitting criterion to identify optimal cutoffs.

• Algorithm: The tree iteratively splits the feature to maximize the reduction in Poisson deviance: ΔD = 2 ∑ [y_i · ln(y_i / μ̂_parent) - (y_i - μ̂_parent)] - ∑{k ∈ {L, R}} 2 ∑{i ∈ k} [y_i · ln(y_i / μ̂_k) - (y_i - μ̂_k)]
• Constraints: The actuary enforces a strict minimum exposure constraint per leaf node (e.g., minimum 1,000 earned car-years), ensuring every discovered bin is statistically credible and actuarially defensible.

---

4. Rating Step Functions, Splines & Monotonic Regulatory Curves

Traditional insurance rating manuals employ discrete step functions for continuous rating variables. For instance, driver age is traditionally banded into rating classes (16–20, 21–24, 25–29, 30–64, 65+).

          Rating Relativities: Step Functions vs. Smooth Monotonic Splines
       Relativity
        │
    2.5 ┼───█████████
        │   (Age 16-20)
    2.0 ┼               █████████ ◄── Cliff-Edge Discontinuity
        │               (Age 21-24)   (Rate drops 20% on 21st birthday)
    1.5 ┼                               █████████
        │                               (Age 25-29)   • • • • • • • Smooth Spline Curve
    1.0 ┼───────────────────────────────────────────────────────── (Monotonic GAM)
        └──────┬────────────┬──────────────┬──────────────┬──────► Driver Age
              16           21             25             30

The Problem of "Cliff-Edge" Step Functions

• Notch Effects: A step function creates artificial rate discontinuities. An insured turning 25 experiences an immediate 20% premium reduction on their birthday, even though their underlying driving maturity evolved continuously.
• Adverse Selection and Gaming: Policyholders and independent agents exploit arbitrary boundaries (e.g., misreporting driving experience or garaging zip codes to fall just inside a cheaper rating bucket).

Natural Cubic Splines and B-Splines in Actuarial GLMs

To eliminate cliff edges, modern actuarial ratemaking replaces step functions with continuous spline basis functions within Generalized Additive Models (GAMs):

ln(μ) = β₀ + f(Age) + ∑ β_j · Z_j

Where f(X) is represented as a linear combination of spline basis functions b_k(X):

f(X) = ∑ γ_k · b_k(X)

• Natural Cubic Splines: Cubic polynomials between internal knots k₁, k₂, ..., k_m, constrained to be strictly linear beyond the boundary knots (k₁ and k_m). This linearity constraint prevents erratic polynomial oscillations (Runge's phenomenon) in the sparse data tails.
• Continuous Derivatives: Natural cubic splines possess continuous first and second derivatives (C² continuity), guaranteeing completely smooth rate transitions across all ages.

Regulatory Defensibility and Monotonicity Constraints

State insurance regulators (such as California CDI, Texas TDI, and Florida OIR) review rating algorithms under statutory standards requiring that rates be neither excessive, inadequate, nor unfairly discriminatory. A common cause of regulatory rate filing objection is non-monotonicity caused by data noise.

The Regulatory Objection:

Suppose an unconstrained spline or unregularized high-degree polynomial fits the raw empirical loss cost for driver age. Due to local sample noise, the model might estimate that 23-year-olds have higher claim frequency than 22-year-olds, but lower than 24-year-olds.

Regulators will reject this filing as actuarially unsound and unfairly discriminatory because there is no causal or actuarial justification for 23-year-olds being riskier than 22-year-olds.

The Solution: Monotonic P-Splines and Shape-Constrained GAMs (SC-GAMs)

Actuaries enforce mathematical shape constraints directly into the spline optimization:

d f(X) / dX ≤ 0 for all X ∈ [16, 65]

By constraining the derivative of the driver age spline to be strictly non-positive (f'(x) ≤ 0), the model guarantees a monotonically decreasing risk curve from teenage years through maturity. This simultaneously satisfies actuarial soundness, eliminates cliff edges, and complies with state insurance department filing standards.