mixle.stats.univariate.continuous.nakagami module¶

Nakagami distribution – the amplitude/envelope law of Nakagami-m fading.

A positive-support family for signal envelopes (wireless/radar/sonar fading, also reliability and hydrology). With shape m >= 1/2 and spread omega = E[X^2] > 0,

f(x; m, omega) = 2 m^m / (Gamma(m) omega^m) * x^(2m-1) * exp(-m x^2 / omega), x > 0,

so X^2 ~ Gamma(m, omega/m); m = 1/2 is the half-normal and m = 1 the Rayleigh. The CDF is the regularized lower incomplete gamma, it samples exactly via a Gamma draw, and it has a clean closed-form method-of-moments fit: omega = E[X^2] and m = E[X^2]^2 / Var[X^2].

Reference: Nakagami, “The m-distribution – a general formula of intensity distribution of rapid fading”, in Statistical Methods in Radio Wave Propagation (1960).

class NakagamiDistribution(m, omega, name=None, keys=None)[source]

Bases: SequenceEncodableProbabilityDistribution

Nakagami distribution with shape m >= 1/2 and spread omega = E[X^2] > 0.

Parameters:

m (float)
omega (float)
name (str | None)
keys (str | None)

density(x)[source]

Return the probability density at x.

Parameters:: x (float)
Return type:: float

log_density(x)[source]

Return the log-density at x (-inf for x <= 0).

Parameters:: x (float)
Return type:: float

seq_log_density(x)[source]

Return vectorized log-density for a sequence-encoded array of observations.

Parameters:: x (ndarray)
Return type:: ndarray

classmethod compute_capabilities()[source]

classmethod compute_declaration()[source]

static backend_legacy_sufficient_statistics(x, params, engine)[source]

Per-row Nakagami power sums in accumulator order (count, sum x^2, sum x^4).

Parameters:

x (Any)
params (dict[str, Any])
engine (Any)

Return type:

tuple[Any, …]

static backend_log_density_from_params(x, m, omega, engine)[source]

Engine-neutral Nakagami log-density from explicit parameters (-inf for x <= 0).

Parameters:

x (Any)
m (Any)
omega (Any)
engine (Any)

Return type:

Any

backend_seq_log_density(x, engine)[source]

Engine-neutral vectorized log-density for encoded data.

Parameters:

x (Any)
engine (Any)

Return type:

Any

classmethod backend_stacked_params(dists, engine)[source]

Stacked Nakagami parameters for a homogeneous mixture kernel.

Parameters:

dists (Sequence[NakagamiDistribution])
engine (Any)

Return type:

dict[str, Any]

classmethod backend_stacked_log_density(x, params, engine)[source]

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

Parameters:

x (ndarray)
params (dict[str, Any])
engine (Any)

Return type:

Any

classmethod backend_stacked_sufficient_statistics(x, weights, params, engine)[source]

Stacked Nakagami power sums (count, sum x^2, sum x^4) using engine-resident arrays.

Parameters:

x (ndarray)
weights (Any)
params (dict[str, Any])
engine (Any)

Return type:

tuple[Any, Any, Any]

cdf(x)[source]

Cumulative distribution function P(X <= x) = P(m, m x^2 / omega) (0 for x <= 0).

Parameters:: x (float)
Return type:: float

quantile(q)[source]

Inverse CDF F^{-1}(q).

Parameters:: q (float)
Return type:: float

mean()[source]

Mean (Gamma(m+1/2)/Gamma(m)) sqrt(omega/m).

Return type:: float

variance()[source]

Variance omega - mean^2 (since E[X^2] = omega).

Return type:: float

sampler(seed=None)[source]

Return a sampler (X = sqrt(Gamma(m, omega/m))).

Parameters:: seed (int | None)
Return type:: NakagamiSampler

estimator(pseudo_count=None)[source]

Return a closed-form method-of-moments estimator.

Parameters:: pseudo_count (float | None)
Return type:: NakagamiEstimator

dist_to_encoder()[source]

Return the data encoder used by this distribution (the raw value).

Return type:: NakagamiDataEncoder

class NakagamiSampler(dist, seed=None)[source]

Bases: DistributionSampler

Draw X = sqrt(G) with G ~ Gamma(shape=m, scale=omega/m) (so X^2 ~ Gamma(m, omega/m)).

Parameters:

dist (NakagamiDistribution)
seed (int | None)

sample(size=None)[source]

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 NakagamiAccumulator(name=None, keys=None)[source]

Bases: SequenceEncodableStatisticAccumulator

Accumulate the weighted power sums (count, sum x^2, sum x^4) for the moment fit.

Parameters:

name (str | None)
keys (str | None)

update(x, weight, estimate)[source]

Parameters:

x (float)
weight (float)
estimate (NakagamiDistribution | None)

Return type:

None

initialize(x, weight, rng)[source]

Parameters:

x (float)
weight (float)
rng (RandomState | None)

Return type:

None

seq_update(x, weights, estimate)[source]

Parameters:

x (ndarray)
weights (ndarray)
estimate (Any)

Return type:

None

seq_initialize(x, weights, rng)[source]

Parameters:

x (ndarray)
weights (ndarray)
rng (RandomState | None)

Return type:

None

combine(suff_stat)[source]

Parameters:: suff_stat (tuple[float, float, float])
Return type:: NakagamiAccumulator

value()[source]

Return type:: tuple[float, float, float]

from_value(x)[source]

Parameters:: x (tuple[float, float, float])
Return type:: NakagamiAccumulator

key_merge(stats_dict)[source]

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)[source]

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()[source]

Return type:: NakagamiDataEncoder

class NakagamiAccumulatorFactory(name=None, keys=None)[source]

Bases: StatisticAccumulatorFactory

Factory for NakagamiAccumulator.

Parameters:

name (str | None)
keys (str | None)

make()[source]

Return type:: NakagamiAccumulator

class NakagamiEstimator(m_min=0.5, name=None, keys=None)[source]

Bases: ParameterEstimator

Method-of-moments estimator: omega = E[X^2], m = E[X^2]^2 / Var[X^2] (clamped m >= 1/2).

Parameters:

m_min (float)
name (str | None)
keys (str | None)

accumulator_factory()[source]

Return type:: NakagamiAccumulatorFactory

estimate(nobs, suff_stat)[source]

Parameters:

nobs (float | None)
suff_stat (tuple[float, float, float])

Return type:

NakagamiDistribution

class NakagamiDataEncoder[source]

Bases: DataSequenceEncoder

Encode observations as a float array.

seq_encode(x)[source]

Encode the iid observation sequence x for vectorized evaluation.

Parameters:: x (Sequence[float])
Return type:: ndarray