Working with Intervals

pyjpt ships a first-class interval arithmetic library used internally for representing variable domains, evidence ranges, and query results. The same classes are fully public and useful on their own.

This tutorial covers:

1. ContinuousSet — a single real-valued interval with inclusive/exclusive bounds

2. IntSet — a contiguous range of integers

3. UnionSet — a disjoint union of any of the above

4. Set operations — intersection, union, difference, complement

5. Sampling, chopping, and transforming intervals

6. Using intervals in JPT queries

Imports

from jpt.base.intervals import (
    ContinuousSet, IntSet, UnionSet,
    INC, EXC,           # boundary type constants
    R, Z, EMPTY,        # predefined sets: all reals, all integers, empty set
)
import numpy as np

ContinuousSet

Represents a single real-valued interval. Bounds can be inclusive (INC, written [ / ]) or exclusive (EXC, written ( / )).

Creating intervals

# Constructor: ContinuousSet(lower, upper, left_bound, right_bound)
half_open = ContinuousSet(0, 1, INC, EXC)   # [0, 1)
closed    = ContinuousSet(2, 5, INC, INC)   # [2, 5]
open_     = ContinuousSet(0, 1, EXC, EXC)   # (0, 1)

print(half_open)   # [0.0, 1.0)
print(closed)      # [2.0, 5.0]
print(open_)       # (0.0, 1.0)

[0.0,1.0)
[2.0,5.0]
(0.0,1.0)

Parsing from strings

The most convenient way to build intervals is to parse a string:

i1 = ContinuousSet.parse('[0, 2]')
i2 = ContinuousSet.parse('(1.5, 4)')
i3 = ContinuousSet.parse('[-inf, 0)')
i4 = ContinuousSet.parse('[3, inf]')

for i in [i1, i2, i3, i4]:
    print(i)

[0.0,2.0]
(1.5,4.0)
[-∞,0.0)
[3.0,∞]

Basic properties

i = ContinuousSet.parse('[1, 5]')

print("min:   ", i.min)
print("max:   ", i.max)
print("width: ", i.width)
print("empty: ", i.isempty())
print("contains 3:", i.contains_value(3))
print("contains 6:", i.contains_value(6))

min:    1.0
max:    5.0
width:  4.0
empty:  0
contains 3: 1
contains 6: 0

Predefined constants

print("All reals R:", R)
print("Empty set:  ", EMPTY)
print("R is empty?", R.isempty())
print("EMPTY is empty?", EMPTY.isempty())

All reals R: (-∞,∞)
Empty set:   ∅
R is empty? 0
EMPTY is empty? 1

Set Operations

a = ContinuousSet.parse('[0, 3]')
b = ContinuousSet.parse('[2, 5]')

print("a:            ", a)
print("b:            ", b)
print("a ∩ b:        ", a.intersection(b))
print("a ∪ b:        ", a.union(b))
print("a \ b:        ", a.difference(b))
print("intersects?   ", a.intersects(b))
print("a ⊆ b?        ", a.issuperseteq(b))

a:             [0.0,3.0]
b:             [2.0,5.0]
a ∩ b:         [2.0,3.0]
a ∪ b:         [0.0,5.0]
a \ b:         [0.0,2.0)
intersects?    1
a ⊆ b?         0

Operator shortcuts &, |, - work too:

a = ContinuousSet.parse('[0, 4]')
b = ContinuousSet.parse('[1, 3]')

print(a & b)   # intersection
print(a | b)   # union
print(a - b)   # difference

[1.0,3.0]
[0.0,4.0]
[0.0,1.0) ∪ (3.0,4.0]

Complement

The complement of an interval with respect to ℝ:

i = ContinuousSet.parse('[1, 3]')
comp = i.complement()
print(f"complement of {i} = {comp}")

complement of [1.0,3.0] = (-∞,1.0) ∪ (3.0,∞)

Sampling

Draw uniform random samples from any interval:

i = ContinuousSet.parse('[2, 5]')
samples = i.sample(10)
print("10 samples from [2, 5]:", samples)
print("all in range:", all(2 <= x <= 5 for x in samples))

10 samples from [2, 5]: [2.22827182 3.55108613 2.26035635 2.79801308 3.46849101 4.29623478
 2.50564093 2.48938792 4.46170836 3.60877701]
all in range: True

IntSet — Integer Intervals

IntSet represents a contiguous range of integers. Boundaries are always inclusive.

from jpt.base.intervals import Z

z1 = IntSet(1, 10)          # {1, 2, ..., 10}
z2 = IntSet.parse('{5..15}')

print(z1)
print(z2)
print("intersection:", z1.intersection(z2))
print("union:       ", z1.union(z2))
print("size:        ", z1.size())
print("All integers:", Z)

{1..10}
{5..15}
intersection: {5..10}
union:        {1..15}
size:         10.0
All integers: ℤ

Iterate directly over an IntSet:

for n in IntSet(1, 5):
    print(n, end="  ")
print()

1  2  3  4  5

Sample integers:

z = IntSet(0, 100)
print("5 random integers from {0..100}:", z.sample(5))

5 random integers from {0..100}: [59. 65. 57.  6. 22.]

UnionSet — Disjoint Unions

When two intervals do not overlap, their union is a UnionSet:

a = ContinuousSet.parse('[0, 1]')
b = ContinuousSet.parse('[3, 5]')
u = a.union(b)
print(type(u).__name__, u)
print("contains 0.5:", u.contains_value(0.5))
print("contains 2.0:", u.contains_value(2.0))
print("contains 4.0:", u.contains_value(4.0))

UnionSet [0.0,1.0] ∪ [3.0,5.0]
contains 0.5: 1
contains 2.0: 0
contains 4.0: 1

Build a UnionSet directly from a list of intervals:

pieces = [
    ContinuousSet.parse('[-2, -1]'),
    ContinuousSet.parse('[0, 1]'),
    ContinuousSet.parse('[2, 3]'),
]
u = UnionSet(pieces)
print(u)
print("min:", u.min, "  max:", u.max)
print("sample 6 points:", u.sample(6))

[-2.0,-1.0] ∪ [0.0,1.0] ∪ [2.0,3.0]
min: -2.0   max: 3.0
sample 6 points: [0.23038297 0.46269014 2.43488776 2.30432159 0.00416605 0.46396241]

Simplifying a UnionSet

If contiguous intervals were added separately, simplify() merges them:

u = UnionSet([
    ContinuousSet.parse('[0, 1]'),
    ContinuousSet.parse('[1, 2]'),   # contiguous — will be merged
    ContinuousSet.parse('[5, 6]'),
])
print("before simplify:", u)
print("after  simplify:", u.simplify())

before simplify: [0.0,1.0] ∪ [1.0,2.0] ∪ [5.0,6.0]
after  simplify: [0.0,2.0] ∪ [5.0,6.0]

Chopping Intervals

Split an interval at a list of points — useful for discretisation:

i = ContinuousSet.parse('[0, 10]')
chops = list(i.chop([2, 4, 7]))
for c in chops:
    print(c)

[0.0,2.0)
[2.0,4.0)
[4.0,7.0)
[7.0,10.0]

Transforming Boundaries

Apply an arbitrary function to both boundaries while keeping the bound types:

i = ContinuousSet.parse('[1, 3]')
squared = i.transform(lambda x: x**2)
print(f"transform {i} by x² → {squared}")

transform [1.0,3.0] by x² → [1.0,9.0]

Using Intervals in JPT Queries

Intervals are the native language for specifying evidence ranges and reading back posterior domains in pyjpt.

Evidence as intervals

import pandas as pd, sklearn.datasets
from jpt.variables import infer_from_dataframe
from jpt.trees import JPT

iris = sklearn.datasets.load_iris()
df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['species'] = [iris.target_names[t] for t in iris.target]

variables = infer_from_dataframe(df)
vnames = {v.name: v for v in variables}
model = JPT(variables, min_samples_leaf=0.1)
model.fit(df)
print(model)

JPT#innernodes = 6, #leaves = 7 (13 total)

Pass a [lo, hi] list or a ContinuousSet directly as evidence:

# P(species=virginica | petal length ∈ [5, 7])
p = model.infer(
    query={'species': 'virginica'},
    evidence={'petal length (cm)': ContinuousSet.parse('[5, 7]')}
)
print(f"P(virginica | petal length ∈ [5,7]) = {p:.4f}")

# Equivalent shorthand — list is auto-converted
p2 = model.infer(
    query={'species': 'virginica'},
    evidence={'petal length (cm)': [5, 7]}
)
print(f"Same via list shorthand:              {p2:.4f}")

P(virginica | petal length ∈ [5,7]) = 0.9756
Same via list shorthand:              0.9756

Posterior domains are intervals

The posterior distribution over a numeric variable carries its support as a ContinuousSet (or UnionSet for multi-modal posteriors):

post = model.posterior(
    variables=[vnames['petal length (cm)'], vnames['petal width (cm)']],
    evidence={'species': 'virginica'},
)

for vname, dist in post.items():
    lo, hi = dist.ppf(.01), dist.ppf(.99)
    print(f"{vname.name}: 99% range ≈ [{lo:.2f}, {hi:.2f}], mean ≈ {dist.expectation():.2f}")

petal length (cm): 99% range ≈ [0.46, 1.79], mean ≈ 5.17
petal width (cm): 99% range ≈ [0.34, 1.71], mean ≈ 1.95

Building multi-range evidence with UnionSet

Exclude a band from the evidence by passing a UnionSet:

# Sepal length outside [5, 6] — i.e., either < 5 or > 6
exclude_mid = ContinuousSet.parse('[4, 5)').union(ContinuousSet.parse('(6, 8]'))
print("evidence range:", exclude_mid)

p = model.infer(
    query={'species': 'setosa'},
    evidence={'sepal length (cm)': exclude_mid}
)
print(f"P(setosa | sepal length ∉ [5,6]) = {p:.4f}")

evidence range: [4.0,5.0) ∪ (6.0,8.0]
P(setosa | sepal length ∉ [5,6]) = 0.3041