PSD

"plan de sondage déterminantal" in French , "determinal sampling design" in English.

Introduction

The goal of sampling theory is to acquire knowledge of a parameter of interest using only partial information. This is done by means of a sampling design, through which a random subset is observed, and the construction of an estimator. The properties of the sampling design are thus of crucial importance to get “good” estimators.

Determinantal Sampling Designs

Determinantal Sampling Designs are a family of sampling designs that adresses all theses issues. They are indexed by Hermitian contracting matrices, called kernel, and have known inclusion probabilities for any order, among many others properties.

Definition

A sampling design $\mathbb{P}$ on a finite set $U$ is a determinantal sampling design if there exists a Hermitian matrix $K$ indexed by $U$, called kernel, such that for all $s \in 2^U \quad \sum_{s' \sqsupseteq s}\mathbb{P}( s' ) = \mathrm{det} ( K_{|s} )$. This sampling design is denoted by $\mathrm{DSD}(K)$.

Notation

• $N$ population size
• $n$ number of individual select
• $Q$ number of auxiliary variables
• $\Pi$ a probability vector
• $X$ a auxilary matrix
• $r$ a real number, penality factor
• $K$ a hermitian projection matrix, kernel
• $D_\pi$ a diagonal matrix

subject to :

• $N$ is a interger such that $N > 1$
• $n$ is a interger such that $1 < n < N$
• $Q$ is a interger such that $Q > 1$
• $\Pi$ a vector such that $\Pi = (\Pi_1,\cdots,\Pi_N)$ and $0 < \Pi_i < 1 , i=1\cdots=N$ and $\mathrm{sum}(\Pi)= n$
• $X$ is a matrix such that $\mathrm{size}(X) = (N,Q)$
• $r$ is a number such that $r \in \mathbb{R}$
• $K$ is a matrix such that $\mathrm{size}(K) = (N,N)$, $\mathrm{diag}(K) = \Pi$, $K = v \cdot v^\top$ with $v$ a matrix $v^\top \cdot v = \mathbb{I}_n$ $\mathrm{size}(v) = (N,n)$
• $D_\pi = \mathrm{diag}(\Pi)$

For more information there is a documentation.

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descriptive on the directorys

data

data is a directory, where there are directory to use for the programs. Each directory is composed of a descriptive file and a csv file.

name of the sub-directory

• fickus
• Kr
• Ppi
• train

docs

docs is a directory where the images and documentation are stored.

python_

python_ is a directory, where thre are python programs.

matlab_

matlab_ is directory, where thre are matalab programs.

to get started

PYTHON

To use the program with the manopt method, it is neccessary in progpymanopt.py to change the variable Pi and X and r in the

if __name__ == '__main__'

Pi must be of type numpy dim-1 and X must be of type numpy dim-2 and r must be of type int or float.

MATLAB

manopt

To use the program with the manopt method, it is neccessary in progmanopt.m to change the variable Pi and X and r. Pi must be of real column vector and X must be of real matrice and r must be of type int or float.

penlab

disclamers the programs penlab runs but they do not give satisfactory results.

To use the program with the penlab method, it is neccessary in progpenlab.m to change the variable $K$ and X. K,Pi. K must be of real matrix and and Pi real column vector and X must be of real matrice.

rotation

To use the program with the rotation method, it is neccessary in progrotation.m to change the variable K and X. K must be of real or hermitian matrix and X must be of real matrice.

pre-requisites

PYTHON

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the use of program progmanopt requires :

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if the packages are not installed. you can install them the command line in directory python_:

pip install -r requirements.txt

if the command did not work, you can install the packages one by one with the command

pip install [package]

MATLAB

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the use of program progpenlab requires the installation of the package .
the use of program progmanopt requires the installation of the package .
the use of program progrotation in matlab requires no packages.

Contibutor

• Vincent LOONIS
• Fabrice Nathan TCHAZOU KAMWA
• Loik johan ACAKPO-ADDRA alias @loik77360

Links

packages

• penlab
• manopt
• pandas
• autograd
• pytorch

Documentation

• rotation
• PSD and fickus