Difference between revisions of "Logistic distribution"
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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]]. | 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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| + | :<code>Logistic(17, 10)</code> → [[Image:Logistic Distribution.jpg]] | ||
== Distribution functions == | == Distribution functions == | ||
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The distribution function. Use to define a quantity as being logistically-distributed. | The distribution function. Use to define a quantity as being logistically-distributed. | ||
=== Dens{{Release||5.1|_}}Logistic(x, mean'', scale'')=== | === Dens{{Release||5.1|_}}Logistic(x, mean'', scale'')=== | ||
| + | {{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 | The probability density at «x» for a logistic distribution with «mean» and «scale». Equal to | ||
:<math>p(x) = {\eta \over {s ( 1 + \eta)^2} }</math>, where <math>\eta = \exp\left(-{ {x-mean}\over {scale}}\right)</math> | :<math>p(x) = {\eta \over {s ( 1 + \eta)^2} }</math>, where <math>\eta = \exp\left(-{ {x-mean}\over {scale}}\right)</math> | ||
=== CumLogistic(x, mean'', scale'') === | === CumLogistic(x, mean'', scale'') === | ||
| + | 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 | The cumulative density function, describing the probability of being less than or equal to «x». Given by | ||
:<math> | :<math> | ||
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=== CumLogisticInv(p, mean'', scale'') === | === CumLogisticInv(p, mean'', scale'') === | ||
| + | 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. | 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> | ||
| + | F^{-1}(p) = «mean» - «scale» \ln\left({1-p}\over p\right) | ||
| + | </math> | ||
== Parameters == | == Parameters == | ||
Revision as of 00:05, 29 September 2018
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4.6 • 5.0 • 5.1 • 5.2 • 5.3 • 5.4 • 6.0 • 6.1 • 6.2 • 6.3 • 6.4 • 6.5 |
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The logistic distribution describes a continuous, symmetric, smooth, uni-modal distribution with tails that are heavier than the normal distribution.
Logistic(17, 10)→
Distribution 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)
To use, add the Distribution Densities Library to your model.
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)
To use, add the Distribution Densities Library to your model.
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.
- Mean = «mean»
- Variance = [math]\displaystyle{ {\pi^2}\over 3 «scale» }[/math]
- Skewness = 0
- Kurtosis = 6/5
- Median = «mean»
- Mode = «mean»
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
has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data.
Parameter Estimation
The parameters of the distribution can be estimated using:
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