Compute moments with pymot: tutorial¶

This tutorial aims at demonstrating how to use the moments module of pymot. A Jupyter notebook version of this tutorial will be available soon.

Load `pymot` and necessary modules¶

import torch
import numpy as np
import matplotlib.pyplot as plt

import pymot.moments
import pymot.quantile

dtype=torch.float64 # double float precision
device = torch.device('cpu') # or cuda if available

Compute moments with `pymot`¶

Four families of (generalized) moments are avaible in pymot (see below). Let \(\mu\) be a probability measure on \(\mathbb{R}\).

Moments can be computed with estimate_moments from the moments module. More particulary, it computes the \(k\) first moments \((m_\ell)_{\ell=0}^{k-1}\)

# Parameters
type_moments = 'om' # ['om', 'laplace', 'feller', 'bins']
num_moments = 20

# Data
X_samples = torch.randn([1, 500], dtype=dtype)

# Compute the num_moments first om moments
my_moments = pymot.moments.estimate_moments(X_samples,
                                            num_moments,
                                            type_moments,
                                            dtype=dtype)

Family 1: Ordinary moments¶

The ordinary moment (OM) of order \(\ell\) is defined as

(1)¶\[m_\ell \triangleq \mathbb{E}_{X\sim\mu} [X^\ell]\]

Family 2: Laplace moments¶

Let \(\lambda\) be a positive scalar and assume that \(\mu\) is supported in \(\mathbb{R}_0\). The laplace moment (laplace) of order \(\ell\) is defined as

(2)¶\[m_\ell \triangleq \mathbb{E}_{X\sim\mu} [e^{-\lambda\ell X}]\]

Family 3: Feller moments¶

Let \(\lambda\) be a positive scalar and assume that \(\mu\) is supported in \(\mathbb{R}_0\). The feller moment (feller) of order \(\ell\) is defined as

(3)¶\[m_\ell \triangleq \mathbb{E}_{X\sim\mu} [e^{-\lambda X} X^\ell]\]

Family 4: Discretization of the quantile function¶

The last family of generalized moments consists in a discretization of the quantile function / pseudo inverse of the CDF. Let \(k\) be a positive integer. The \(\ell\) bins is defined as

(4)¶\[m_\ell \triangleq \frac{1}{k}\int_{\frac{\ell}{k}}^{\frac{\ell+1}{k}} Q_{\mu}(q)\,\mathrm{d}q\]

where \(Q_{\mu}\) denotes the quantile function / inverse CDF associated to \(X\sim\mu\).

Approximate quantile function with `pymot`¶

The quantile module provides methods to approximate a quantile function based on its moments.

# Parameters
qfunc = pymot.quantile.get_func_quantile(my_moments,
                                         dtype=dtype,
                                         device=device)

# plots
f, ax = plt.subplots(1, 1, figsize=(6, 6))

rangeq = = np.linspace(0, 1, 201)
ax.plt(rangeq, np.array([qfunc(q) for q in rangeq]))

`pymot` and multidimensional random variable¶

When \(X\sim\mu\) is a random variable defined in \(\mathbb{R}^d\) with \(d>1\), pymot computes the moments along each one-dimensional direction.

# Parameters
type_moments = 'om' # ['om', 'laplace', 'feller', 'bins']
num_dim = 5
num_moments = 20

# Data
X_samples = torch.randn([num_dim, 500], dtype=dtype)

# Compute the num_moments first om moments
my_moments = pymot.moments.estimate_moments(X_samples,
                                            num_moments,
                                            type_moments,
                                            dtype=dtype)

# my_moments contains (among other) a num_dim x num_moments tensor.

Compute moments with pymot: tutorial¶

Load `pymot` and necessary modules¶