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# Generating Correlated Discrete Distributions OnThe Fly

IP.com Disclosure Number: IPCOM000240439D
Publication Date: 2015-Jan-30
Document File: 4 page(s) / 123K

## Publishing Venue

The IP.com Prior Art Database

## Abstract

A method is disclosed for generating correlated discrete distributions on the fly.

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This is the abbreviated version, containing approximately 41% of the total text.

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Generating Correlated Discrete Distributions OnThe Fly

Simulated data is often used for various kinds of prediction, in particular, risk analysis. To determine a desired distribution for variable(s), Distribution Fitting [1] may be used on a sample of real, historical data, where the amount of data in the sample may be relatively small. A huge amount of simulated data may be generated based on the relatively small amount of sample data.

It may be advantageous to generate the simulated data via a single, random variable. In order to simulate in this manner, one needs to know the probability distribution, including the distribution type and parameters. A native algorithm can be used for direct simulation of any particular distribution [2].

A set of variables can be simulated if all the variables are assumed to be independent of each other and each includes separate distribution. However, in reality random variables are often correlated, in particular, when found from Distribution Fitting [1].

In general, native simulation algorithms may not simulate variables in a way that provides proper distributions and correlations [3]. NORTA algorithms provide one way to simulate variables with different distributions and prescribed correlations.

In NORTA, several correlated variables with Normal Distribution are simulated and then each is transformed into a variable with prescribed distribution. The NORTA Transformation (NT) is performed by composition of Inverse of Cumulative Distribution Function (CDF) of the expected distribution and direct CDF for Normal Distribution [3].

Given a random variable x, the CDF of a given value x is the probability of the event that a given discrete variable is taking a value not exceeding the specified one,

The CDF can take values from 0.0 to 1.0. Therefore, if is a normally distributed

variable, can be transformed to a variable X by,

where is CDF for the normal distribution, while is CDF for the expected

distribution of .

The distributions may be continuous such as Triangular, Uniform, Normal, Lognormal,

1

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Exponential, Beta, Gamma, and Weibull. Alternatively, the distributions can be discrete such as Bernoulli, Binomial, Categorical, Poisson, and Negative Binomial [1].

For the continuous distributions, the CDF is continuously increasing, so inverse of CDF is uniquely defined. A reliable computational system performing NT can be used for such distributions.

Our subject is discrete distributions. Typically, a variable with discrete distribution takes consequent integer values 0, 1, 2… The Probability Mass Function (PMF) of a given value is the probability of the event that a given discrete variable is taking that value. A good example of a discrete distribution is Poisson Distribution with PMF,

where is the rate parameter of the Poisson Distribution [1]. The discrete variable x can take any non-negative integer value. The CDF for discrete distributions can be computed as,

For discrete distributi...