Generate correlated random variables python cholesky

Table of Contents Show

How to Generate Correlated Random Numbers
Precise Problem Definition
Special Cases
Box–Muller transform
The Student-t Copula
Archimedean Copulas
Empirical Copula
Applications
Implementation
How do you generate a correlated normal random variable in Python?
When can we use Cholesky factorization?
Is Cholesky factorization unique?
What is correlation between two random variables?

1 How to Generate Correlated Random Numbers
- 1.1 Precise Problem Definition
- 1.2 Special Cases
  - 1.2.1 Cholesky Decomposition
  - 1.2.2 Singular Value Decomposition
  - 1.2.3 Prespecified Linear Models
- 1.3 Box–Muller transform
2 Copulas
- 2.1 The Student-t Copula
3 Archimedean Copulas
4 Empirical Copula
5 Applications
6 Implementation

How to Generate Correlated Random Numbers

Generation of correlated random numbers is of wide applicability in many domains of quantitative analysis and risk modelling. This article is a review of approaches.

Precise Problem Definition

The more precisely defined question is how to generate random numbers

according to a defined multivariate probability distribution.

Special Cases

The most commonly encountered special case is that of multivariate normal distribution given by the density:

For the purposes of this article will ignore non-zero means and non-unit (unscaled) variances as those aspects can be handled on a univariate basis. We will focus on the Correlation Matrix.

In general the methodologies involve generating realizations of the random (vector)

on the basis of a random vector

of uncorrelated normal variables, which in turn are (typically) produced by a random vector

of uncorrelated uniform variables.

Cholesky Decomposition

Given the variance-covariance matrix

(that is positive definite), the Cholesky decomposition is

Upon simulation of random vectors

the correlated realisations are provided by:

where L is a lower triangular matrix that is effectively the "square-root" of the correlation matrix

Singular Value Decomposition

When the correlation matrix is estimated empirically it may be the case that it fails to be positive semi-definite, in which case the Cholesky decomposition may fail. One option is to adjust the correlation matrix. Another option is to pursue a singular value decomposition

, where U, V are orthogonal matrices and D is diagonal (with possibly reduced rank, ie. some eigenvalues are zero).

Then

Prespecified Linear Models

It is not always the case that our input data or modelling framework is based on covariance matrix. If the dependency between dependent variables is explicitly derived in terms of a multi-factor model wher the factors

are uncorrelated then the problem is reduced to generating independent random numbers and subsequently constructing suitable sums:

The simplest example being the case

Box–Muller transform

The wikipedia:Box–Muller transform is an interesting special case, primarily of interest for educational purposes. In the first step uncorrelated normal numbers are generated via

Those can then be combined as per the formulas of the previous section on linear models

Copulas

In various applications the multi-variate dependency cannot be assumed to be a Gaussian. In this case we require to have a specified copula which provides for more general dependency structures.

The copula of

is defined as the cumulative distribution function of

via

The Student-t Copula

This is a special copula as it is linked to the Gaussian distribution

Archimedean Copulas

Clayton
Gumbel
Frank

Empirical Copula

Applications

Monte Carlo Simulation of Credit Portfolios

Implementation

A list of open source implementations

Library / Package	Language	Characteristics
XXX	Python	TD
XXX	R	TD
XXX	C++	TD
XXX	Julia	TD

How do you generate a correlated normal random variable in Python?

To generate correlated normally distributed random samples, one can first generate uncorrelated samples, and then multiply them by a matrix C such that CCT=R, where R is the desired covariance matrix. C can be created, for example, by using the Cholesky decomposition of R, or from the eigenvalues and eigenvectors of R.

When can we use Cholesky factorization?

Cholesky factorization can be applied to a symmetric matrix which is not positive definite but the process does not possess the numerical stability of the positive definite case. Furthermore, one or more rows in P may be purely imaginary. For example, This is not implemented in Matlab.

Is Cholesky factorization unique?

The Cholesky factorization is a particular form of this factorization in which X is upper triangular with positive diagonal elements; it is usually written as A = RTR or A = LLT and it is unique.

What is correlation between two random variables?

Covariance is the measure of the joint variability of two random variables [5]. It shows the degree of linear dependence between two random variables. Positive covariance implies that there is a direct linear relationship i.e. increase in one variable corresponds with greater values in the other.