Difference between revisions of "Probabilistic calculation"

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Analytica performs probabilistic evaluation of probability distributions through simulation — by computing a random sample of values from the actual probability distribution for each uncertain quantity. The result of evaluating a distribution is represented internally as an array of the sample values, indexed by Run. Run is an index variable that identifies each sample iteration by an integer from 1 to '''''Samplesize'''''.
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[[Category:Analytica User Guide]]
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<breadcrumbs>Analytica User Guide > Expressing Uncertainty > {{PAGENAME}}</breadcrumbs>
  
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You can display a probabilistic value using a variety of '''''uncertainty view options''''' (page 29) — the mean, statistics, probability bands, probability density (or mass function), and cumulative distribution function. All these views are derived or estimated from the underlying sample array, which you can inspect using the last uncertainty view, Sample.
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Analytica can calculate the value of each variable deterministically -- generating its [[Mid]] value -- or probabilistically, generating a probability distribution -- its [[Evaluation Modes|Prob value]] -- if the definition of the variable or any of its predecessors contains a probability distribution.  It does probabilistic calculation as a random sample of values from the actual probability distribution for each uncertain quantity. Each probabilistic value is represented internally as a sample, an array of the sample values indexed by [[Run]]. '''Run''' is a system index variable that identifies each sample iteration by an integer from 1 to [[SampleSize|Samplesize]].
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You can display a probabilistic value using a variety of [[uncertainty views]] — including mean, statistics, probability bands, probability density (or mass function), cumulative distribution function, and Sample view. All these views are derived or estimated from the underlying sample array, which you can see directly in the Sample view. For example, with a Samplesize = 6, here is the probabilistic value of A:
  
 
'''Example'''
 
'''Example'''
  
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''A:= Normal(10, 2) →''
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:<code>A := Normal(10, 2) &rarr;</code>
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:{| class="wikitable"
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''Iteration (Run)''
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!! colspan="6" style="text-align: left;" | Iteration (Run) &#9654;
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{| class="wikitable"
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|-
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|10.74
 
|10.74
 
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'''Tip: '''''The values in a sample are generated at random from the distribution; if you try this example and display the result as a table, you might see values different from those shown here. To reproduce this example, reset the random number seed to 99 and use the default sampling method and random number method (see “Uncertainty Setup dialog” on page 257)''.
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<tip title="Tip"> The values in a sample are generated at random from the distribution; if you try this example and display the result as a table, you might see values different from those shown here. To reproduce this example, reset the random number seed to 99 and use the default sampling method and random number method (see [[Uncertainty Setup dialog]].</tip>
  
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For each sample run, a random value is generated from each probability distribution in the model. Output variables of uncertain variables are calculated by calculating a value for each value of ''Run''.
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For each sample run, a random value is generated from each probability distribution in the model. Output variables of uncertain variables are calculated by calculating a value for each value of [[Run]].
  
 
'''Example'''
 
'''Example'''
  
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''B:= Normal(5, 1) →''
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:<code>B := Normal(5, 1) &rarr;</code>
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:{| class="wikitable"
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''Iteration (Run)''
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!! colspan="6" style="text-align: left;" | Iteration (Run) &#9654;
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|5.09
 
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''C:= A + B →
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:<code>C := A + B &rarr;</code>
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Iteration (Run)''
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:{| class="wikitable"
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!! colspan="6" style="text-align: left;" | Iteration (Run) &#9654;
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''Iteration (Run)''
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Notice that each sample value of C is equal to the sum of the corresponding values of A and B.
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Notice that each sample value of '''C''' is equal to the sum of the corresponding values of '''A''' and '''B'''.
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To control the probabilistic simulation, as well as views of probabilistic results, use the Uncertainty Setup dialog (page 257).
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To control the probabilistic simulation, as well as views of probabilistic results, use the [[Uncertainty Setup dialog]].
  
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'''Tip: '''''If you try to apply an '''array-reducing function''' (page 194) to a probability distribution across Run,
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<tip title="Tip">If you try to apply an [[Array-reducing_functions|array-reducing function]] to a probability distribution across [[Run]], Analytica returns the distribution’s mid value.''</tip>
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Analytica returns the distribution’s mid value.''
 
  
 
'''Example:'''
 
'''Example:'''
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:<code>X := Beta(2, 3)</code>
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:<code>Mid(X) &arr; 0.3857</code>
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:<code>Max(X, Run) &rarr; 0.3857</code>
  
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''X:= Beta(2, 3)''
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To evaluate the input parameters probabilistically and reduce across [[Run]], use [[Sample]]().
  
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''Mid(X) → 0.3857 and Max(X, Run) → 0.3857''
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'''Example:'''
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:<code>Max(Sample(X), Run) &arr; 0.8892</code>
  
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To evaluate the input parameters probabilistically and reduce across Run, use Sample() (page 301).
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== See Also ==
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*[[Uncertainty view of a result]]
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*[[Evaluation Modes]]
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*[[Mid]]
  
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'''Example:'''
 
  
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''Max(Sample(X), Run) → 0.8892''
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<footer>Including a distribution in a definition/ {{PAGENAME}} / Monte Carlo and probabilistic simulation </footer>

Latest revision as of 17:04, 2 June 2016

Analytica User GuideExpressing UncertaintyProbabilistic calculation


Analytica can calculate the value of each variable deterministically -- generating its Mid value -- or probabilistically, generating a probability distribution -- its Prob value -- if the definition of the variable or any of its predecessors contains a probability distribution. It does probabilistic calculation as a random sample of values from the actual probability distribution for each uncertain quantity. Each probabilistic value is represented internally as a sample, an array of the sample values indexed by Run. Run is a system index variable that identifies each sample iteration by an integer from 1 to Samplesize.

You can display a probabilistic value using a variety of uncertainty views — including mean, statistics, probability bands, probability density (or mass function), cumulative distribution function, and Sample view. All these views are derived or estimated from the underlying sample array, which you can see directly in the Sample view. For example, with a Samplesize = 6, here is the probabilistic value of A:

Example

A := Normal(10, 2) →
Iteration (Run) ▶
1 2 3 4 5 6
10.74 13.2 9.092 11.44 9.519 13.03
Tip
The values in a sample are generated at random from the distribution; if you try this example and display the result as a table, you might see values different from those shown here. To reproduce this example, reset the random number seed to 99 and use the default sampling method and random number method (see Uncertainty Setup dialog.

For each sample run, a random value is generated from each probability distribution in the model. Output variables of uncertain variables are calculated by calculating a value for each value of Run.

Example

B := Normal(5, 1) →
Iteration (Run) ▶
1 2 3 4 5 6
5.09 4.94 4.65 6.60 5.24 6.96
C := A + B →
Iteration (Run) ▶
1 2 3 4 5 6
15.83 18.13 13.75 18.04 14.76 19.99

Notice that each sample value of C is equal to the sum of the corresponding values of A and B. To control the probabilistic simulation, as well as views of probabilistic results, use the Uncertainty Setup dialog.

Tip
If you try to apply an array-reducing function to a probability distribution across Run, Analytica returns the distribution’s mid value.

Example:

X := Beta(2, 3)
Mid(X) &arr; 0.3857
Max(X, Run) → 0.3857

To evaluate the input parameters probabilistically and reduce across Run, use Sample().

Example:

Max(Sample(X), Run) &arr; 0.8892

See Also


Including a distribution in a definitionProbabilistic calculationMonte Carlo and probabilistic simulation
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