Difference between revisions of "Logistic distribution"

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[[Category:Distribution Functions]]
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[[category:Continuous distributions]]
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[[category:Unbounded distributions]]
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[[category:Unimodal distributions]]
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[[category:Univariate distributions]]
  
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The logistic distribution describes a continuous, symmetric, smooth, uni-modal distribution with tails that are heavier than the [[Normal|normal distribution]].
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<center><code>Logistic(17, 10)</code> &rarr; [[Image:Logistic Distribution.jpg]]</center>
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== Functions ==
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=== Logistic( mean'', scale, over'' ) ===
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The distribution function. Use to define a quantity as being logistically-distributed.
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=== <div id="DensLogistic">Dens{{Release||5.1|_}}Logistic(x, mean'', scale'')</div>===
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{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}{{Release|5.2||''(New as a built-in function in [[Analytica 5.2]])''}}
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The probability density at «x» for a logistic distribution with «mean» and «scale». Equal to
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:<math>p(x) = {\eta \over {s ( 1 + \eta)^2} }</math>, where <math>\eta = \exp\left(-{ {x-mean}\over {scale}}\right)</math>
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=== <div id="CumLogistic">CumLogistic( x, mean'', scale'' )</div> ===
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{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}
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The cumulative density function, describing the probability of being less than or equal to «x». Given by
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:<math>
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F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)}
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</math>
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=== <div id="CumLogisticInv">CumLogisticInv( p, mean'', scale'' )</div> ===
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{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}
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The inverse cumulative probability function, also know as the quantile function. Returns the value for which has a «p» probability of being greater than or equal to the true value.
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:<math>
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F^{-1}(p) = «mean» - «scale» \ln\left({1-p}\over p\right)
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</math>
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=== Parameters ===
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* «mean»: The mean, which for the logistic distribution is also the mode and median. Any real number.
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* «scale»: optional, defaults to 1. Must be positive.
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* «over»: optional. A list of indexes to independently sample over.
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== Statistics ==
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Theoretical (i.e., in the absence of sampling error) for the logistic distribution are as follows.
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* [[Mean]] = «mean»
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* [[Variance]] = <math>{\pi^2}\over 3 «scale»</math>
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* [[Skewness]] = 0
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* [[Kurtosis]] = 6/5
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* [[Median]] = «mean»
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* [[Mode]] = «mean»
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== Parameter Estimation ==
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The parameters of the distribution can be estimated using:
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:<code>«mean» := [[Mean]](X, I) </code>
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:<code>«scale» := [[Sqrt]](3*[[Variance]](X, I))/[[Pi]]</code>
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== Applications ==
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=== [[LogisticRegression]] ===
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The logistic distribution is particularly convenient for determining dependent probabilities using [[Regression|linear regression]] techniques, where the probability of a [[binomial]] event depends monotonically on a continuous variable ''x''. For example, in a toxicology assay, ''x'' may be the dosage of a toxin, and ''p(x)'' the probability of death for an animal exposed to that dosage. Using p(x) = F(x), the [[logit]] of ''p'', given by
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:<code>[[Logit]](p(x)) = [[Ln]](p(x) / (1-p(x))) = x/s - m/s</code>
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has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data.
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==Examples==
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:<code>Logistic(17, 10)</code> &rarr; [[Image:Logistic Distribution.jpg]]
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== See Also ==
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* [[LogisticRegression]]
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* [[Mean]], [[Variance]], [[Skewness]], [[Kurtosis]]
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* A 2-term [[Keelin]] distribution is a [[Logistic]] distribution.
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* [[Distribution Densities Library]]

Latest revision as of 21:01, 9 October 2018




Release:

4.6  •  5.0  •  5.1  •  5.2  •  5.3  •  5.4  •  6.0  •  6.1  •  6.2  •  6.3  •  6.4  •  6.5


The logistic distribution describes a continuous, symmetric, smooth, uni-modal distribution with tails that are heavier than the normal distribution.

Logistic(17, 10) → Logistic Distribution.jpg

Functions

Logistic( mean, scale, over )

The distribution function. Use to define a quantity as being logistically-distributed.

DensLogistic(x, mean, scale)

(New as a built-in function in Analytica 5.2)

The probability density at «x» for a logistic distribution with «mean» and «scale». Equal to

[math]\displaystyle{ p(x) = {\eta \over {s ( 1 + \eta)^2} } }[/math], where [math]\displaystyle{ \eta = \exp\left(-{ {x-mean}\over {scale}}\right) }[/math]

CumLogistic( x, mean, scale )

The cumulative density function, describing the probability of being less than or equal to «x». Given by

[math]\displaystyle{ F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} }[/math]

CumLogisticInv( p, mean, scale )

The inverse cumulative probability function, also know as the quantile function. Returns the value for which has a «p» probability of being greater than or equal to the true value.

[math]\displaystyle{ F^{-1}(p) = «mean» - «scale» \ln\left({1-p}\over p\right) }[/math]

Parameters

  • «mean»: The mean, which for the logistic distribution is also the mode and median. Any real number.
  • «scale»: optional, defaults to 1. Must be positive.
  • «over»: optional. A list of indexes to independently sample over.

Statistics

Theoretical (i.e., in the absence of sampling error) for the logistic distribution are as follows.

Parameter Estimation

The parameters of the distribution can be estimated using:

«mean» := Mean(X, I)
«scale» := Sqrt(3*Variance(X, I))/Pi

Applications

LogisticRegression

The logistic distribution is particularly convenient for determining dependent probabilities using linear regression techniques, where the probability of a binomial event depends monotonically on a continuous variable x. For example, in a toxicology assay, x may be the dosage of a toxin, and p(x) the probability of death for an animal exposed to that dosage. Using p(x) = F(x), the logit of p, given by

Logit(p(x)) = Ln(p(x) / (1-p(x))) = x/s - m/s

has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data.

Examples

Logistic(17, 10) → Logistic Distribution.jpg

See Also

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