Difference between revisions of "Logistic distribution"
m (Lchrisman moved page Logistic to Logistic distribution: Working on a new format with all distribution functions on the same wiki page.) |
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[[Category:Distribution Functions]] | [[Category:Distribution Functions]] | ||
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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]]. | ||
| − | == Logistic(mean, scale) == | + | == Distribution functions == |
| − | + | === Logistic(mean'', scale, over'') === | |
| − | The logistic | + | The distribution function. Use to define a quantity as being logistically-distributed. |
| − | + | === Dens{{Release||5.1|_}}Logistic(x, mean'', scale'')=== | |
| + | 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> | ||
| + | === CumLogistic(x, mean'', scale'') === | ||
| + | The cumulative density function, describing the probability of being less than or equal to «x». Given by | ||
:<math> | :<math> | ||
F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} | F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} | ||
</math> | </math> | ||
| − | The distribution is | + | === 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. | ||
| + | |||
| + | == 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>{\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 [[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 | 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 | ||
| − | :<code>Logit(p(x)) = Ln(p(x) / (1-p(x))) = x/s - m/s</code> | + | :<code>[[Logit]](p(x)) = [[Ln]](p(x) / (1-p(x))) = x/s - m/s</code> |
has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data. | 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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== Parameter Estimation == | == Parameter Estimation == | ||
The parameters of the distribution can be estimated using: | The parameters of the distribution can be estimated using: | ||
| − | :<code>«mean» := Mean(X, I) </code> | + | :<code>«mean» := [[Mean]](X, I) </code> |
| − | :<code>«scale» := Sqrt(3*Variance(X, I))/Pi</code> | + | :<code>«scale» := [[Sqrt]](3*[[Variance]](X, I))/[[Pi]]</code> |
==Example== | ==Example== | ||
Revision as of 23:56, 28 September 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 |
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The logistic distribution describes a continuous, symmetric, smooth, uni-modal distribution with tails that are heavier than the normal distribution.
Distribution functions
Logistic(mean, scale, over)
The distribution function. Use to define a quantity as being logistically-distributed.
DensLogistic(x, mean, scale)
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.
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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