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Breast Cancer Wisconsin

This page compares the rules that HGP, a decision tree (DT), RIPPER, and BoolXAI learn on the Breast Cancer Wisconsin (Diagnostic) dataset.1 The dataset ships with scikit-learn and has 569 tumors described by 30 numeric features: 357 benign and 212 malignant.

All methods use one stratified 80/20 train/test split with seed 42, and the positive class is benign. Each method runs with its default parameters:

  • HGP: BooleanRuleClassifier with 1000 epochs, on the bins of the default StandardBinarizer.
  • DT: the scikit-learn decision tree.
  • RIPPER: wittgenstein on the raw features, which it splits into its own bins.
  • BoolXAI: on 5 quantile bins per feature.

The test set has 114 tumors, so a single split says little about accuracy. See PMLB for the comparison over 70 datasets.

Rules

Each rule below is true for the tumors it predicts benign. The rules use the hgp-lib notation: , separates the operands of an operator, and ~ negates the literal or operator that follows. The DT and RIPPER models are written as hgp-lib rules. DT thresholds and BoolXAI bin edges are shown with 4 significant digits.

HGP

Rule size: 21 nodes. Literals: 15. Test F1: 0.9437.

HGP evolves one rule of nested And and Or operators over the binarizer's bins.

And(
  ~115.35 <= worst perimeter,
  And(
    ~0.125 <= mean compactness < 0.1293,
    ~1.807 <= texture error < 1.9,
    Or(
      Or(
        ~Or(
          16.8 <= worst radius < 17.51,
          696.25 <= mean area
        ),
        mean texture < 16.395
      ),
      29.3 <= worst texture < 30.91
    ),
    ~739.2 <= worst area < 743.45,
    ~0.176 <= worst concave points
  ),
  ~17.51 <= worst radius,
  ~17.51 <= worst radius,
  ~1.807 <= texture error < 1.9,
  ~739.2 <= worst area < 743.45,
  Or(
    ~mean texture < 16.395,
    ~1.807 <= texture error < 1.9
  )
)

A tumor is benign if its worst perimeter, worst radius, and worst concave points are below a threshold, its mean compactness, texture error, and worst area lie outside a bin, and one operand of the first Or holds. hgp-lib does not simplify rules: three literals appear more than once, and the last Or always holds, because the rule already requires ~1.807 <= texture error < 1.9. Without them, the rule has 10 literals and 13 nodes and makes the same predictions.

DT

Rule size: 56 nodes. Literals: 46. Test F1: 0.9286.

The tree becomes the disjunction of its 9 paths to benign leaves.

Or(
  And(
    ~worst radius <= 16.8,
    ~texture error <= 0.4731,
    worst concavity <= 0.1907,
    worst texture <= 30.98
  ),
  And(
    ~worst radius <= 16.8,
    texture error <= 0.4731
  ),
  And(
    worst radius <= 16.8,
    ~worst concave points <= 0.1359,
    ~worst texture <= 25.62,
    worst symmetry <= 0.2682,
    mean concave points <= 0.06432
  ),
  And(
    worst radius <= 16.8,
    ~worst concave points <= 0.1359,
    worst texture <= 25.62,
    worst area <= 817.1,
    worst smoothness <= 0.1786
  ),
  And(
    worst radius <= 16.8,
    worst concave points <= 0.1359,
    area error <= 91.56,
    ~area error <= 38.6,
    ~texture error <= 2.007,
    concave points error <= 0.01308
  ),
  And(
    worst radius <= 16.8,
    worst concave points <= 0.1359,
    area error <= 91.56,
    ~area error <= 38.6,
    texture error <= 2.007
  ),
  And(
    worst radius <= 16.8,
    worst concave points <= 0.1359,
    area error <= 91.56,
    area error <= 38.6,
    ~smoothness error <= 0.003294,
    ~worst texture <= 33.27,
    ~worst texture <= 33.8
  ),
  And(
    worst radius <= 16.8,
    worst concave points <= 0.1359,
    area error <= 91.56,
    area error <= 38.6,
    ~smoothness error <= 0.003294,
    worst texture <= 33.27
  ),
  And(
    worst radius <= 16.8,
    worst concave points <= 0.1359,
    area error <= 91.56,
    area error <= 38.6,
    smoothness error <= 0.003294,
    worst texture <= 26.64
  )
)

Each path repeats the tests it shares with other paths, so the root test on worst radius appears in all 9 conjunctions.

RIPPER

Rule size: 19 nodes. Literals: 15. Test F1: 0.8702.

RIPPER learns a list of rules for the positive class and predicts it when any rule fires. As a single rule, this is a disjunction of conjunctions, here of 12 conjunctions with one or two literals each. The bins are RIPPER's own, as wittgenstein prints them.

Or(
  perimeter error = <1.28,
  mean concave points = 0.011 - 0.018,
  worst concavity = 0.089 - 0.14,
  concavity error = <0.007,
  worst perimeter = 86.17 - 90.53,
  worst radius = 11.16 - 12.47,
  worst radius = 13.85 - 14.91,
  And(
    worst concave points = 0.083 - 0.097,
    mean concave points = 0.033 - 0.046
  ),
  worst radius = <11.16,
  worst radius = 12.47 - 13.3,
  And(
    worst concavity = 0.14 - 0.18,
    mean texture = <14.0
  ),
  And(
    mean texture = 15.49 - 16.76,
    mean fractal dimension = 0.068 - 0.072
  )
)

Each literal tests one bin, so a range that spans several bins needs one conjunction per bin. For example, three conjunctions cover worst radius below 13.3.

BoolXAI

Rule size: 6 nodes. Literals: 5. Test F1: 0.8955.

BoolXAI refines one rule by simulated annealing. Besides And and Or, it uses the higher-order Boolean operators AtLeast, AtMost, and Choose. AtMost1 is true when at most one of its operands is true.

AtMost1(
  ~worst concavity < 0.08854,
  0.1524 <= mean concavity,
  15.85 <= worst radius < 20.13,
  1243 <= worst area,
  900.3 <= mean area
)

A tumor is benign if at most one of the five conditions holds, such as a worst area of at least 1243.

  1. Breast Cancer Wisconsin (Diagnostic), https://archive.ics.uci.edu/dataset/17/breast+cancer+wisconsin+diagnostic, loaded with sklearn.datasets.load_breast_cancer. ↩