Difference between revisions of "Random"

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[[Category:Doc Status C]] <!-- For Lumina use, do not change -->
 
[[Category:Doc Status C]] <!-- For Lumina use, do not change -->
 
   
 
   
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= Function Random =
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__TOC__
  
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Random generates a single random value from a probability distribution. It is not a probability distribution ''per se'', such as Normal() or Uniform(). It generates a single value, not a sample indexed by Run, and it does so whether evaluated in a deterministic (Mid) or probabilistic context. For example.
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==Random(''dist, method, over'')==
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Random(Normal(10, 2))
 
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generates a single value generated at random from the specified Normal distribution.
 
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Random()
 
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with no parameters, it returns a single uniformly-distributed random number between 0 and 1.
 
  
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== Declaration ==
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Function [[Random]] generates a single random value from a probability distribution. It is not a probability distribution ''per se'', such as [[Normal]]() or [[Uniform]](). It generates a single value, not a sample indexed by [[Run]], and it does so whether evaluated in a deterministic ([[Mid]]) or probabilistic context. For example.
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:<code>Random(Normal(10, 2))</code>
  
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Random(dist: Optional Unevaluated; method: Optional Scalar; over: ... Optional Index)
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generates a single value generated at random from the specified [[Normal]] distribution. [[Random]]() with no parameters returns a single uniformly-distributed random number between 0 and 1.
  
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Parameters:
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Because there is often a need to access a random number generator stream, such as for rejection sampling, Metropolis-Hastings simulation, etc,  [[Random]]() makes it possible to get such values, even if the global sampling method is Latin hypercube, and efficiently since it isn't necessary to generate an entire sample.  [[Random]] can return variates from a wide variety of distributions. It is even possible to write user-defined distribution functions for custom distributions that work with random.
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* '''''dist''''' : If specified, must be an explicit call to a distribution function that supports single-sample generation (see below).  If you specify no distribution, it defaults to Uniform(0,1). If dist is a multivariate distribution, indexed by I, it truents
 
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* '''''method''''' Selects the algorithm used to generate the random number.  Possible value are: 0=use system default, 1=Minimal standard, 2=L'Ecuyer, 3=Knuth.
 
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* '''''over:''''' A convenient way to list indexes that independent random numbers will be generated over.  (This will also occur if the index(es) occur in any of the other parameters).
 
  
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== Description ==
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The full declaration of  the function is
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:[[Random]](''dist'': Optional Unevaluated; ''method'': Optional Scalar; ''over'': ... Optional Index)
  
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Random is not a distribution-function per-se, as Uniform(0,1) is.  However, one often needs access to a random number generator stream, such as for rejection sampling, Metropolis-Hastings simulation, etc.  Random() makes it possible to get such values, even if the global sampling method is Latin hypercube, and efficiently since it isn't necessary to generate an entire sample.  Random can return variates from a wide variety of distributions.  It is even possible to write user-defined distribution functions for custom distributions that work with random.
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==Optional parameters==
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===Dist===
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If specified, «dist» must be a call to a distribution function that supports single-sample generation (most do, but see below).  If you specify no distribution, it defaults to [[Uniform]](0,1). If «dist» is a multivariate distribution, indexed by <code>I</code>, it returns an array of random samples indexed by <code>I</code>.  
  
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== Examples ==
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===Method===
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Selects the algorithm used to generate the random number.  Possible value are:
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:<code>0</code>: use system default
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:<code>1</code>: Minimal standard
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:<code>2</code>: L'Ecuyer
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:<code>3</code>: Knuth
  
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Random(Uniform(-100, 100)):  Returns a single real-valued random number uniformly selected between -100 and 100.
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===Over===
  
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Random(Uniform(1, 100, integer: True)): Returns a random integer between 1 and 100 inclusive.
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Give it an index or list of indexes. Random will choose an independent random number for each index value, or combination of index values.  It also does this if any of the other parameters is an array with one or more indexes.
  
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  Random(Over: I): Returns an array of independent uniform random numbers between 0 and 1 indexed by I.  The numbers are independent (i.e., Monte Carlo sampled, never Latin Hypercube).
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== Examples ==
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:<code>Random(Uniform(-100, 100))</code>  
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::Returns a single real-valued random number uniformly selected between -100 and 100.
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:<code>Random(Uniform(1, 100, integer: True))</code>
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::Returns a random integer between 1 and 100 inclusive.
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:<code>Random(Over: I)</code>
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::Returns an array of independent uniform random numbers between 0 and 1 indexed by <code>I</code>.  The numbers are independent (i.e., Monte Carlo sampled, never Latin Hypercube).
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:<code>Random(Over: I, J)</code>
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::Returns a 2-D array of independent uniform random numbers between 0 and 1, indexed by <code>I</code> and <code>J</code>.  All numbers in the array are sampled independently.
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:<code>Random(Uniform(min: Array(I, J, 0), max: 1))</code>
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::This is functionally equivalent to the preceding example. It demonstrates how the «over» parameter is only a convenience, but results in an easier to interpret syntax.
  
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Random(Over: I, J): Returns a 2-D array of independent uniform random numbers between 0 and 1, indexed by I and J.  All numbers in the array are sampled independently.
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==Details and more examples==
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=== Distribution function support for single samples ===
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Random supports only those distribution functions with parameter «singleMethod», usually declared as:
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:<code>singleSampleMethod: Optional Atomic Numeric</code>
  
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  Random(Uniform(min: Array(I, J, 0), max: 1)): This is functionally equivalent to the preceding example. It demonstrates how the Over parameter is only a convenience, but results in an easier to interpret syntax.[[Category:Distribution Functions]]
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When the parameter is provided, the distribution function must return a single random variate from the distribution indicated by the other parameters. Random will fill in this parameter with one of the following values, indicating which sampling method should be used:
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[[Category:Doc Status C]]
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Possible values for «singleMethod»:
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:<code>0</code>: use default method
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:<code>1</code>: use Minimal standard
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:<code>2</code>: use L'Ecuyer
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:<code>3</code>: use Knuth
  
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=== Distribution Function Support for Single Samples ===
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As an example, consider what happens when <code>Random(Normal(2, 3))</code> is evaluated.  The [[Random]] function checks that its parameter is an acceptable distribution function, and then it evaluates:
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:<code>Normal(2, 3, singleSampleMethod: 0)</code>
  
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Random supports only those distribution functions with parameter singleMethod, usually declared as:
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User-defined functions can support single-variate generation, and therefore can be used as a parameter to [[Random]], if they have a parameter named «singleMethod».
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singleSampleMethod: Optional Atomic Numeric
 
  
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When the parameter is provided, the distribution function must return a single random variate from the distribution indicated by the other parameters.  Random will fill in this parameter with one of the following values, indicating which sampling method should be used:
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===Functions supported===
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Possible values for singleMethod:
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[[Random]](dist) supports any of these built-in probability distributions functions as the distribution:
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0 = use default method
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<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
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1 = use Minimal standard
 
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2 = use L'Ecuyer
 
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3 = use Knuth
 
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As an example, consider what happens when Random(Normal(2,3)) is evaluated.  The Random function checks that its parameter is an acceptable distribution function, and then it evaluates:
 
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  Normal(2, 3, singleSampleMethod: 0)
 
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Random(dist) supports any of these built-in probability distributions functions as the distribution:
 
 
* [[Bernoulli]]
 
* [[Bernoulli]]
 
* [[Beta]]
 
* [[Beta]]
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* [[Uniform]]
 
* [[Uniform]]
 
* [[Weibull]]
 
* [[Weibull]]
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</div>
  
 
It also works for these distributions from the ''Distribution variations'' library:
 
It also works for these distributions from the ''Distribution variations'' library:
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<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
 
* [[Beta_m_sd]]
 
* [[Beta_m_sd]]
 
* [[Erlang]]
 
* [[Erlang]]
 
* [[Gamma_m_sd]]
 
* [[Gamma_m_sd]]
 
* [[InverseGaussian]]
 
* [[InverseGaussian]]
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* LogNormal_m_sd (but note that this one is superceded by LogNormal)
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* [[Lognormal_m_sd]] (but note that this one is superseded by [[LogNormal]])
 
* [[Lorenzian]]
 
* [[Lorenzian]]
 
* [[NegBinomial]]
 
* [[NegBinomial]]
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* [[Smooth_Fractile]]
 
* [[Smooth_Fractile]]
 
* [[Wald]]
 
* [[Wald]]
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</div>
  
 
It also works for these distributions from the ''Multivariate Distributions'' library:
 
It also works for these distributions from the ''Multivariate Distributions'' library:
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<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
 
* [[BiNormal]]
 
* [[BiNormal]]
 
* [[Dirichlet]]
 
* [[Dirichlet]]
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* [[Normal_serial_correl]]
 
* [[Normal_serial_correl]]
 
* [[UniformSpherical]]
 
* [[UniformSpherical]]
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</div>
  
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Random does not support these built-in distribution functions:
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===Functions not supported===
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[[Random]] does not support these built-in distribution functions:
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<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
 
* [[ChanceDist]]
 
* [[ChanceDist]]
 
* [[Fractiles]]
 
* [[Fractiles]]
 
* [[ProbDist]]
 
* [[ProbDist]]
 
* [[Truncate]]
 
* [[Truncate]]
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</div>
  
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User-defined functions can support single-variate generation, and therefore can be used as a parameter to Random, if they have a parameter named singleMethod.
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== See Also ==
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* [[RandomType]]
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= See Also =
 
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* [[Shuffle]]
 
* [[Shuffle]]
 
* [[Sample]]
 
* [[Sample]]
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* [[Distribution Functions]]
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* [[Distribution Densities Library]]

Latest revision as of 20:01, 13 April 2016


Random(dist, method, over)

Function Random generates a single random value from a probability distribution. It is not a probability distribution per se, such as Normal() or Uniform(). It generates a single value, not a sample indexed by Run, and it does so whether evaluated in a deterministic (Mid) or probabilistic context. For example.

Random(Normal(10, 2))

generates a single value generated at random from the specified Normal distribution. Random() with no parameters returns a single uniformly-distributed random number between 0 and 1.

Because there is often a need to access a random number generator stream, such as for rejection sampling, Metropolis-Hastings simulation, etc, Random() makes it possible to get such values, even if the global sampling method is Latin hypercube, and efficiently since it isn't necessary to generate an entire sample. Random can return variates from a wide variety of distributions. It is even possible to write user-defined distribution functions for custom distributions that work with random.

The full declaration of the function is

Random(dist: Optional Unevaluated; method: Optional Scalar; over: ... Optional Index)

Optional parameters

Dist

If specified, «dist» must be a call to a distribution function that supports single-sample generation (most do, but see below). If you specify no distribution, it defaults to Uniform(0,1). If «dist» is a multivariate distribution, indexed by I, it returns an array of random samples indexed by I.

Method

Selects the algorithm used to generate the random number. Possible value are:

0: use system default
1: Minimal standard
2: L'Ecuyer
3: Knuth

Over

Give it an index or list of indexes. Random will choose an independent random number for each index value, or combination of index values. It also does this if any of the other parameters is an array with one or more indexes.

Examples

Random(Uniform(-100, 100))
Returns a single real-valued random number uniformly selected between -100 and 100.
Random(Uniform(1, 100, integer: True))
Returns a random integer between 1 and 100 inclusive.
Random(Over: I)
Returns an array of independent uniform random numbers between 0 and 1 indexed by I. The numbers are independent (i.e., Monte Carlo sampled, never Latin Hypercube).
Random(Over: I, J)
Returns a 2-D array of independent uniform random numbers between 0 and 1, indexed by I and J. All numbers in the array are sampled independently.
Random(Uniform(min: Array(I, J, 0), max: 1))
This is functionally equivalent to the preceding example. It demonstrates how the «over» parameter is only a convenience, but results in an easier to interpret syntax.

Details and more examples

Distribution function support for single samples

Random supports only those distribution functions with parameter «singleMethod», usually declared as:

singleSampleMethod: Optional Atomic Numeric

When the parameter is provided, the distribution function must return a single random variate from the distribution indicated by the other parameters. Random will fill in this parameter with one of the following values, indicating which sampling method should be used: Possible values for «singleMethod»:

0: use default method
1: use Minimal standard
2: use L'Ecuyer
3: use Knuth

As an example, consider what happens when Random(Normal(2, 3)) is evaluated. The Random function checks that its parameter is an acceptable distribution function, and then it evaluates:

Normal(2, 3, singleSampleMethod: 0)

User-defined functions can support single-variate generation, and therefore can be used as a parameter to Random, if they have a parameter named «singleMethod».

Functions supported

Random(dist) supports any of these built-in probability distributions functions as the distribution:

It also works for these distributions from the Distribution variations library:

It also works for these distributions from the Multivariate Distributions library:

Functions not supported

Random does not support these built-in distribution functions:

See Also

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