mixle.stats.univariate.continuous.rayleigh module

Create, estimate, and sample from a Rayleigh distribution.

Reference: Johnson, Kotz & Balakrishnan, Continuous Univariate Distributions (2nd ed., Wiley, 1994/95).

class RayleighDistribution(sigma, name=None, keys=None)

Bases: SequenceEncodableProbabilityDistribution

Rayleigh distribution with scale sigma > 0.

Parameters:

• sigma (float)

• name (str | None)

• keys (str | None)

classmethod compute_capabilities()

classmethod compute_declaration()

static exp_family_sufficient_statistics(x, engine)

Return Rayleigh sufficient statistics for generated scoring.

Parameters:

• x (tuple[Any, Any, Any])

• engine (Any)

Return type:

tuple[Any, …]

static exp_family_legacy_sufficient_statistics(x, params, engine)

Return per-row Rayleigh sufficient statistics in accumulator order.

Parameters:

• x (tuple[Any, Any, Any])

• params (dict[str, Any])

• engine (Any)

Return type:

tuple[Any, …]

static exp_family_natural_parameters(params, engine)

Return Rayleigh natural parameters for generated scoring.

Parameters:

• params (dict[str, Any])

• engine (Any)

Return type:

tuple[Any, …]

static exp_family_log_partition(params, engine)

Return Rayleigh log partition for generated scoring.

Parameters:

• params (dict[str, Any])

• engine (Any)

Return type:

Any

static exp_family_base_measure(x, engine)

Return Rayleigh support/base measure for generated scoring.

Parameters:

• x (tuple[Any, Any, Any])

• engine (Any)

Return type:

Any

density(x)

Return the probability density or mass at a single observation.

Parameters:

x (float)

Return type:

float

log_density(x)

Return the log-density or log-mass at a single observation.

Parameters:

x (float)

Return type:

float

seq_log_density(x)

Return vectorized log-density values for sequence-encoded observations.

Parameters:

x (tuple[ndarray, ndarray, ndarray])

Return type:

ndarray

static backend_log_density_from_params(vals, vals2, log_vals, sigma, engine)

Engine-neutral Rayleigh log-density from explicit parameters.

Parameters:

• vals (Any)

• vals2 (Any)

• log_vals (Any)

• sigma (Any)

• engine (Any)

Return type:

Any

backend_seq_log_density(x, engine)

Engine-neutral vectorized log-density for encoded data.

Parameters:

• x (tuple[Any, Any, Any])

• engine (Any)

Return type:

Any

classmethod backend_stacked_params(dists, engine)

Return stacked Rayleigh parameters for a homogeneous mixture kernel.

Parameters:

• dists (Sequence[RayleighDistribution])

• engine (Any)

Return type:

dict[str, Any]

classmethod backend_stacked_log_density(x, params, engine)

Return an (n, k) matrix of Rayleigh log densities.

Parameters:

• x (tuple[Any, Any, Any])

• params (dict[str, Any])

• engine (Any)

Return type:

Any

cdf(x)

Cumulative distribution function P(X <= x) (exact). The continuous ‘index of’ a value.

Parameters:

x (float)

Return type:

float

quantile(q)

Inverse CDF F^{-1}(q): the value at cumulative-probability index q (continuous unranking).

Parameters:

q (float)

Return type:

float

mean()

Mean E[X] of the distribution.

Return type:

float

variance()

Variance Var[X] of the distribution.

Return type:

float

entropy()

Differential entropy 1 + log(sigma/sqrt(2)) + gamma/2.

Return type:

float

skewness()

Skewness 2*sqrt(pi)(pi-3)/(4-pi)^1.5.

Return type:

float

kurtosis()

Excess kurtosis -(6pi^2-24pi+16)/(4-pi)^2.

Return type:

float

mode()

Mode (sigma).

Return type:

float

sampler(seed=None)

Return a sampler for drawing observations from this distribution.

Parameters:

seed (int | None)

Return type:

RayleighSampler

estimator(pseudo_count=None)

Return an estimator for fitting this distribution from data.

Parameters:

pseudo_count (float | None)

Return type:

RayleighEstimator

dist_to_encoder()

Return the data encoder used by this distribution for vectorized methods.

Return type:

RayleighDataEncoder

class RayleighSampler(dist, seed=None)

Bases: DistributionSampler

Draw iid Rayleigh observations.

Parameters:

• dist (RayleighDistribution)

• seed (int | None)

sample(size=None)

Draw observations.

Combinator samplers (mixture/sequence/…) accept batched. With batched=True (the default) each child stream is drawn in one vectorized call instead of a per-draw Python loop – far faster. Because every child sampler owns an independent RandomState, batching consumes each stream in the same order as the loop, so the draws are identical to the legacy path. batched=False forces that legacy per-draw loop as a guaranteed- stable reference. Leaf samplers are already vectorized and ignore the flag.

Parameters:

size (int | None)

Return type:

float | ndarray

class RayleighAccumulator(name=None, keys=None)

Bases: SequenceEncodableStatisticAccumulator

Accumulate weighted squared observations.

Parameters:

• name (str | None)

• keys (str | None)

update(x, weight, estimate)

Parameters:

• x (float)

• weight (float)

• estimate (RayleighDistribution | None)

Return type:

None

initialize(x, weight, rng)

Parameters:

• x (float)

• weight (float)

• rng (RandomState | None)

Return type:

None

seq_update(x, weights, estimate)

Parameters:

• x (tuple[ndarray, ndarray, ndarray])

• weights (ndarray)

• estimate (RayleighDistribution | None)

Return type:

None

seq_initialize(x, weights, rng)

Parameters:

• x (tuple[ndarray, ndarray, ndarray])

• weights (ndarray)

• rng (RandomState | None)

Return type:

None

combine(suff_stat)

Parameters:

suff_stat (tuple[float, float])

Return type:

RayleighAccumulator

value()

Return type:

tuple[float, float]

from_value(x)

Parameters:

x (tuple[float, float])

Return type:

RayleighAccumulator

key_merge(stats_dict)

Pool this accumulator’s statistics into stats_dict under its merge key.

The structural default implements the common single-key pattern: store the accumulator under self.keys the first time the key is seen, else combine into the one already there. Accumulators with several named keys (e.g. an HMM’s init/trans/state keys) or a non-accumulator stats payload override this. A keys of None (the default) is a no-op.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

key_replace(stats_dict)

Replace this accumulator’s statistics from the pooled stats_dict entry (see key_merge).

Parameters:

stats_dict (dict[str, Any])

Return type:

None

acc_to_encoder()

Return type:

RayleighDataEncoder

class RayleighAccumulatorFactory(name=None, keys=None)

Bases: StatisticAccumulatorFactory

Factory for RayleighAccumulator.

Parameters:

• name (str | None)

• keys (str | None)

make()

Return type:

RayleighAccumulator

class RayleighEstimator(pseudo_count=None, suff_stat=None, min_sigma=1.0e-8, name=None, keys=None)

Bases: ParameterEstimator

Closed-form MLE estimator for Rayleigh scale.

Parameters:

• pseudo_count (float | None)

• suff_stat (float | None)

• min_sigma (float)

• name (str | None)

• keys (str | None)

accumulator_factory()

Return type:

RayleighAccumulatorFactory

estimate(nobs, suff_stat)

Parameters:

• nobs (float | None)

• suff_stat (tuple[float, float])

Return type:

RayleighDistribution

class RayleighDataEncoder

Bases: DataSequenceEncoder

Encode Rayleigh observations with x, x**2, and log(x).

seq_encode(x)

Encode the iid observation sequence x for vectorized evaluation.

Parameters:

x (Sequence[float])

Return type:

tuple[ndarray, ndarray, ndarray]