PoissonKernelTest

class QuadratiK.poisson_kernel_test.PoissonKernelTest(rho: float, num_iter: int = 300, quantile: float = 0.95, random_state: int | None = None, n_jobs: int = 8)

Class for Poisson kernel-based quadratic distance tests of Uniformity on the Sphere. More details can be found in the User Guide.

Parameters

rhofloat

The value of concentration parameter used for the Poisson kernel function.

num_iterint, optional

Number of iterations for critical value estimation of Tn-statistic.

quantilefloat, optional

The quantile to use for critical value estimation.

random_stateint, None, optional.

Seed for random number generation. Defaults to None.

n_jobsint, optional.

n_jobs specifies the maximum number of concurrently running workers. If 1 is given, no joblib parallelism is used at all, which is useful for debugging. For more information on joblib n_jobs refer to - https://joblib.readthedocs.io/en/latest/generated/joblib.Parallel.html. Defaults to 8.

Attributes

test_type_str

The type of test performed on the data.

execution_timefloat

Time taken for the test method to execute.

u_statistic_h0_boolean

A logical value indicating whether or not the null hypothesis is rejected according to Tn.

u_statistic_tn_float

The value of the Tn-statistic.

u_statistic_cv_float

The empirical critical value for Tn.

v_statistic_h0_boolean

A logical value indicating whether or not the null hypothesis is rejected according to Sn.

v_statistic_sn_float

The value of the Sn-statistic.

v_statistic_cv_float

The critical value for Sn computed following the asymptotic distribution.

Note

A Tn-statistic is a type of U-statistic that is used to estimate a population parameter. It is based on the idea of averaging over all possible distinct combinations of a fixed size from a sample. A Sn-statistic is a type of V-statistic that considers all possible tuples of a certain size, not just distinct combinations and can be used in contexts where unbiasedness is not required.

References

Ding Y., Markatou M., Saraceno G. (2023). “Poisson Kernel-Based Tests for Uniformity on the d-Dimensional Sphere.” Statistica Sinica. doi: doi:10.5705/ss.202022.0347.

Examples

import numpy as np
np.random.seed(0)
from QuadratiK.poisson_kernel_test import PoissonKernelTest
# data generation
z = np.random.normal(size=(200, 3))
data_unif = z / np.sqrt(np.sum(z**2, axis=1, keepdims=True))
#performing the uniformity test
unif_test = PoissonKernelTest(rho = 0.7, random_state=42).test(data_unif)
print(unif_test)

PoissonKernelTest(
  Test Type=Poisson Kernel-based quadratic distance test of Uniformity on the Sphere,
  Execution Time=12.943207263946533 seconds,
  Tn-Statistic=0.5977824645431932,
  Tn-Statistic Critical Value=1.596531141687201,
  Tn-Statistic Null Hypothesis Rejected=False,
  Sn-Statistic=19.722614852087556,
  Sn-Statistic Critical Value=23.229486935225513,
  Sn-Statistic Null Hypothesis Rejected=False,
  Selected concentration parameter rho=0.7,
)

Methods

PoissonKernelTest.stats()	Function to generate descriptive statistics.
PoissonKernelTest.summary([print_fmt])	Summary function generates a table for the Poisson kernel test results and the summary statistics.
PoissonKernelTest.test(x)	Performs the Poisson kernel-based quadratic distance Goodness-of-fit tests for Uniformity for spherical data using the Poisson kernel with concentration parameter \(rho\).

---

PoissonKernelTest.stats() → DataFrame

Function to generate descriptive statistics.

Returns

summary_stats_dfpandas.DataFrame

Dataframe of descriptive statistics.

PoissonKernelTest.summary(print_fmt: str = 'simple_grid') → str

Summary function generates a table for the Poisson kernel test results and the summary statistics.

Parameters

print_fmtstr, optional.

Used for printing the output in the desired format. Supports all available options in tabulate, see here: https://pypi.org/project/tabulate/. Defaults to “simple_grid”.

Returns

summarystr

A string formatted in the desired output format with the kernel test results and summary statistics.

PoissonKernelTest.test(x: ndarray | DataFrame) → PoissonKernelTest

Performs the Poisson kernel-based quadratic distance Goodness-of-fit tests for Uniformity for spherical data using the Poisson kernel with concentration parameter \(rho\).

Parameters

xnumpy.ndarray, pandas.DataFrame

a numeric d-dim matrix of data points on the Sphere \(S^{(d-1)}\).

Returns

selfobject

Fitted estimator.