PySATL_NMVM_Module

Python package for statistical analysis of Normal Mean or/and Variance Mixtures.

Installation

Requirements:

pip install requirements.txt

For contributing:

pip install requirements.dev.txt

Requirements

• numpy~=1.26.4
• scipy~=1.13.1
• matplotlib~=3.8.4
• numba~=0.59.0
• mpmath~=1.3.0

Development:

• mypy~=1.10.0
• black~=24.4.2
• isort~=5.13.2
• mpmath~=1.3.0
• pytest~=7.4.4

Theoretical basis

Normal Mean Mixture

$Y_{NMM}$ is called Normal Mean Mixture (NMM) if it can be represented in classical form with parameters $\alpha$, $\beta$, $\gamma$ $\in \mathbf{R}$ and mixing density function $g(x)$ as:

$Y_{NMM} = \alpha + \beta \cdot \xi + \gamma \cdot N$, where $N \sim \mathcal{N}(0, 1), \xi \sim g(x)$ and $\xi \perp N$

Normal Variance Mixture

$Y_{NVM}$ is called Normal Variance Mixture (NVM) if it can be represented in classical form with parameters $\alpha$, $\beta$, $\gamma$ $\in \mathbf{R}$ and mixing density function $g(x)$ as:

$Y_{NVM} = \alpha + \gamma \cdot \sqrt{\xi} \cdot N$, where $N \sim \mathcal{N}(0, 1), \xi \sim g(x)$ and $\xi \perp N$

Normal Mean-Variance Mixture

$Y_{NMVM}$ is called Normal Mean-Variance Mixture (NMVM) if it can be represented in classical form with parameters $\alpha$, $\beta$, $\gamma$ $\in \mathbf{R}$ and mixing density function $g(x)$ as:

$Y_{NMVM}$ = $\alpha + \beta \cdot \xi + \gamma \cdot \sqrt{\xi} \cdot N$, where $N \sim \mathcal{N}(0, 1), \xi \sim g(x)$ and $\xi \perp N$

Problem with classical representation is that it is not unique. So usually it is more convenient to make some substitutions and get canonical representation of mixture.

Canonical representations

$Y_{NMM}(\xi, \sigma) = \xi + \sigma \cdot N$
$Y_{NVM}(\xi, \alpha) = \alpha + \sqrt{\xi} \cdot N$
$Y_{NMVM}(\xi, \alpha, \mu) = \alpha + \mu \cdot \xi + \sqrt{\xi} \cdot N$
where $\alpha, \mu, \sigma \in \mathbf{R}$; $N \sim \mathcal{N}(0, 1); \xi \sim g(x)$

Normal Mean or/and Variance Mixtures have useful applications in statistical analysis and financial math.

Mixture sample generation:

Classes are implemented in directory src.mixtures .

There are three classes of mixtures:

• NormalMeanMixtures
• NormalVarianceMixtures
• NormalMeanVarianceMixtures

One can select mixture representation (classical or canonical) by specifying mixture_form parameter.

Usage

Example of mixture object creation:

from src.mixtures.nm_mixture import *
from scipy.stats import norm

mixture = NormalMeanVarianceMixtures("classical", alpha=1.2, beta=2.2, gamma=1, distribution=norm)

Example of mixture sample generation:

from src.generators.nm_generator import NMGenerator
from src.mixtures.nm_mixture import *
from scipy.stats import expon

generator = NMGenerator()
mixture = NormalMeanMixtures("classical", alpha=1.6, beta=4, gamma=1, distribution=expon)
sample = generator.classical_generate(mixture, 5000)

Histogram of sample values:

Calculation of standard statistical characteristics of mixture:

• Calculation of pobability density function
  compute_pdf()
• Calculation of logarithm of pobability density function
  compute_logpdf()
• Cumulative distribution function
  compute_cdf()
• Moments calculation
  compute_moment()

Support algorithms

Randomized Quasi-Monte Carlo method (RQMC)

Computes given integral with the limits of integration being 0 and 1 using Randomized Quasi-Monte Carlo method.

Parameters:

func: integrated function
error_tolerance: pre-specified error tolerance
count: number of rows of random values matrix
base_n: number of columns of random values matrix
i_max: allowed number of cycles
a: parameter for quantile of normal distribution

Usage:

from src.algorithms.support_algorithms.rqmc import RQMC
rqmc = RQMC(lambda x: x**3 - x**2 + 1, error_tolerance=1e-5)

So rqmc()[0] is estimated integral value and rqmc()[1] is current error tolerance.

Parameter estimation algorithms for Normal Mean or/and Variance Mixtures:

• Semiparametric estimation of parameter $\mu$ in Normal Mean-Variance mixture.

  Parameter $\mu$ estimation for $Y_{NMVM}(\xi, \alpha, \mu) = \alpha + \mu \cdot \xi + \sqrt{\xi} \cdot N$
  where $\alpha = 0; N \sim \mathcal{N}(0, 1); \xi \sim g(x)$

  Algorithm was created by Belomestny & Panov(2017)

  Parameters

  m - Search area radius
  tolerance - Defines error tolerance to stop bisection algorithm
  max_iterations - Maximum allowed iterations for bisection algorithm
  omega - Lipschitz continuous odd function on $\mathbf{R}$ with compact support\

  Algorithm returns estimated $\mu$ value and estimation successfulness.

  Usage

  from scipy.stats import halfnorm
  from src.estimators.semiparametric.nmv_semiparametric_estimator import NMVSemiParametricEstimator
  from src.generators.nmv_generator import NMVGenerator
  from src.mixtures.nmv_mixture import NormalMeanVarianceMixtures
  
  # Generate sample of NMV-mixture with parameter mu = 2 to test the algorithm
  mixture = NormalMeanVarianceMixtures("canonical", alpha=0, mu=2, distribution=halfnorm)
  sample = NMVGenerator().canonical_generate(mixture, 1000)
  
  # Create new estimator for NMV-mixture and select an algorithm by its name
  estimator = NMVSemiParametricEstimator("mu_estimation", {"m": 10, "tolerance": 10 ** -5})
  estimate_result = estimator.estimate(sample)
  print("Estimated mu value: ", estimate_result.value, "\nEstimation is successful: ", estimate_result.success)

  Output:

  Estimated mu value:  2.0798295736312866 
  Estimation is successful:  True
  

• Estimation of mixing density for given $\mu$ in Normal Mean-Variance mixture.

  Mixing density function $g(x)$ estimation for $Y_{NMVM}(\xi, \alpha, \mu) = \alpha + \mu \cdot \xi + \sqrt{\xi} \cdot N$
  where $\alpha = 0, \mu \in \mathbf{R}; N \sim \mathcal{N}(0, 1); \xi \sim g(x)$

  Algorithm was created by Belomestny & Panov(2017)

  Parameters

  mu - Parameter $\mu$ of NMV-mixture
  gmm - Parameter $\gamma$ for Belomestny & Panov(2017) algorithm
  u_value - Parameter $U_n$ for Belomestny & Panov(2017) algorithm
  v_value - Parameter $V_n$ for Belomestny & Panov(2017) algorithm
  x_data - Points to estimate function $g(x)$

  Algorithm returns list of estimated g(x) values and estimation successfulness.

  Usage:

  For demonstration purposes to use the algorithm we can generate sample from NMV mixture with parameter $\mu = 1$, size 25 values and mixing density function $g(x) = exp(-x)$

  from scipy.stats import expon
  from src.generators.nmv_generator import NMVGenerator
  from src.mixtures.nmv_mixture import NormalMeanVarianceMixtures
  
  real_g = expon.pdf # Real g(x)
  
  mixture = NormalMeanVarianceMixtures("canonical", alpha=0, mu=1, distribution=expon)
  sample = NMVGenerator().canonical_generate(mixture, 25)

  Now we assume that we know nothing about $g(x)$, and try to estimate it in points 0.5, 1 and 2.

  from src.estimators.semiparametric.nmv_semiparametric_estimator import NMVSemiParametricEstimator
  
  x_data = [0.5, 1, 2] # x points, where to estimate g(x)
  estimator = NMVSemiParametricEstimator(
      "g_estimation_given_mu", {"x_data": x_data, "u_value": 7.6, "v_value": 0.9}
  )
  est = estimator.estimate(sample) # est is EstimateResult object

  Result of the estimation can be accesses by est.list_value

  x = 0.5, Real g(x) value:  0.6065 Estimated:  0.9727
  x = 1,   Real g(x) value:  0.3678 Estimated:  0.5227
  x = 2,   Real g(x) value:  0.1353 Estimated:  0.1866
  

Reference list

• Hintz, E., Hofert, M., & Lemieux, C. (2019). Normal variance mixtures: Distribution, density and parameter estimation. Comput. Stat. Data Anal., 157, 107175.
• Belomestny, D., & Panov, V. (2017). Semiparametric estimation in the normal variance-mean mixture model. Statistics, 52, 571 - 589.

Contributors:

• Andreev Sergey - andreev-sergej
• Knyazev Dmitrii - Engelsgeduld