Polar Coordinate Features¶

These per-sample FORMAT fields transform Cartesian allele depth coordinates (AD_Ref, AD_Alt) into polar coordinates, separating biological identity from sequencing depth so that neither depends on the other.

Why Not DP and VAF?¶

Standard VCF metrics already encode depth (DP = AD_Ref + AD_Alt) and allele fraction (VAF = AD_Alt / DP). At first glance, these seem sufficient. The problem is geometric: DP and VAF have structural properties that degrade ML classification, and the polar transform fixes both simultaneously.

DP vs. PRAD: How We Measure "Weight of Evidence"¶

DP is a simple sum (the L₁ norm) of the allele depth vector. PRAD is the log-compressed Euclidean distance (the L₂ norm) — the magnitude of the evidence vector.

The difference matters because DP couples the two allele counts: for a fixed DP, increasing one allele's count forces the other's count down. This treats REF and ALT as interchangeable, which they are not — a somatic variant with AD=(95, 5) and a germline het with AD=(50, 50) both have DP=100, but represent fundamentally different signal geometries. PRAD treats REF and ALT as independent dimensions of a vector, so the distance from the origin (the "zero-information" state) reflects the true signal magnitude.

Crucially, sequencing noise scatters reads equally in all directions around the true allele depth point — a read is just as likely to land 2 counts above as 2 counts to the right. This means the zone of expected noise forms a roughly circular region, not a diagonal band. The Euclidean radius (PRAD) traces a circular arc that matches this noise shape naturally. A constant DP traces a straight diagonal that cuts through the noise at an angle, artificially coupling the two allele counts.

VAF vs. PANG: The Unstable Precision Problem¶

VAF is a linear ratio: Alt / (Ref + Alt). PANG is an angular ratio: atan2(Alt, Ref).

VAF has depth-dependent precision — its noise level changes with coverage. A single-read fluctuation at 10× depth swings VAF by 10%; the same fluctuation at 1000× barely moves it (0.1%). This means two VAF=0.5 data points at different depths carry radically different reliability, but the number itself gives no indication of this. An ML model using raw VAF must learn the "fan" (wedge) shape where the meaning of a VAF value depends on DP — a hidden dependency that forces the model to learn the interaction between VAF and DP rather than treating each independently.

PANG computes the angle of the allele depth vector, which is a pure ratio independent of magnitude. When paired with PRAD, the two make the noise spread uniform across the feature space: the flaring Cartesian wedge (where variance grows with depth) unrolls into a rectangular strip with constant-width noise at every depth. PANG answers "what is it?" (somatic vs. germline), PRAD answers "how confident are we?", and these two questions become independent — allowing ML models to learn each axis separately.

The Diagonal Collapse¶

A germline het at 20× sits at AD=(10, 10) in Cartesian space; the same variant at 2000× sits at AD=(1000, 1000). A distance-based classifier sees these as enormously far apart despite being biologically identical.

• With DP/VAF: the model sees VAF=0.5, DP=20 and VAF=0.5, DP=2000 — same VAF but wildly different DP. It must learn that "VAF=0.5 means germline" regardless of DP, which requires seeing examples at every depth.
• With PANG/PRAD: both points have PANG=0.785 (45°). The biological identity is a constant. Only the radius differs, and it does so on a bounded log scale (1.15 → 3.15). The entire diagonal collapses into a single PANG value, and the model only needs to learn that PANG ≈ π/4 means "germline."

This is the fundamental advantage: the polar transform algebraically separates what a variant is from how much evidence supports it, enabling ML models trained at one coverage to generalize to any other.

The polar transform "unrolls" the wedge and "collapses" the diagonal into a uniform rectangular strip where ML models can draw simple flat boundaries.

Polar Radius (PRAD)¶

In plain terms: PRAD answers "how much total evidence do I have?" on a compressed scale where 1.0 ≈ ~10 reads, 2.0 ≈ ~100 reads, 3.0 ≈ ~1000 reads.

Computation: PRAD = log10(1 + sqrt(AD_Ref² + AD_Alt²)), computed per-sample.

The log10-compressed Euclidean magnitude of the allele depth vector. Encodes total signal strength (sequencing depth) independent of allele fraction, with built-in normalization for cross-coverage ML generalization.

Why log10? The raw Euclidean radius sqrt(AD_Ref² + AD_Alt²) scales linearly with coverage, spanning 0 to >2800 at 2000× depth — a 100× dynamic range. An ML model trained at 30× WGS would see completely different PRAD distributions when applied to 100× WES data. The log10(1+r) compression reduces this to a compact [0, ~3.5] range while preserving monotonicity:

Value range: [0, ~3.5]

Interpretation: Higher PRAD means more sequencing evidence, regardless of whether it supports REF or ALT. Acts as a confidence tie-breaker for low-angle variants:

Coverage stability: PRAD still varies with coverage (1.15 at 20× to 3.15 at 2000×), but the log10 compression reduces the dynamic range from 100× to 3× and preserves monotonic ordering. For ML models that use PRAD as a confidence weight, this bounded range avoids feature-scale domination that occurs with raw allele counts. Tree-based models (XGBoost, Random Forest) are invariant to monotonic transforms and handle this naturally; linear models benefit from the compressed range.

Polar Angle (PANG)¶

In plain terms: PANG answers "what fraction of reads support the variant?" expressed as an angle. The key property: this angle is the same whether you have 10 reads or 10,000 — depth doesn't change it.

Computation: PANG = atan2(AD_Alt, AD_Ref), computed per-sample.

The arctangent of the allele depth ratio, in radians. Encodes the biological identity of the variant — the allele fraction — independent of total depth.

Value range: [0, π/2] radians (≈ [0, 1.571])

Interpretation:

Coverage stability: Perfectly coverage-invariant. PANG is a pure ratio (atan2(AD_Alt, AD_Ref)) — identical allele fractions produce identical angles at any depth. A 50% VAF het produces PANG ≈ 0.785 (45°) whether the sample is 20× or 2000×. This is the key advantage of the polar transform: allele identity is algebraically separated from sequencing depth.

Case-Control "Four-Feature" Paradigm¶

Each sample independently computes its own PRAD and PANG from its own allele depths. The downstream ML model receives a four-element feature vector:

[PRAD_Control, PANG_Control, PRAD_Case, PANG_Case]

Both PRAD (log10-compressed, range [0, ~3.5]) and PANG (ratio-based, range [0, π/2]) are coverage-invariant, ensuring the four-feature paradigm produces comparable values regardless of whether the sample is 20× or 2000×.

This avoids cross-sample relational metrics that would violate VCF FORMAT semantics and eliminates sentinel/NaN imputation issues. The model learns the relational delta itself: