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Mathematics in machine learning -- common probability distribution (XIII): Logistic Distribution
2022-07-06 18:39:00 【von Neumann】
Catalogues :《 Mathematics in machine learning 》 General catalogue
Related articles :
· Common probability distribution ( One ): Bernoulli distribution (Bernoulli Distribution )
· Common probability distribution ( Two ): Category distribution (Multinoulli Distribution )
· Common probability distribution ( 3、 ... and ): The binomial distribution (Binomial Distribution )
· Common probability distribution ( Four ): Uniform distribution (Uniform Distribution )
· Common probability distribution ( 5、 ... and ): Gaussian distribution (Gaussian Distribution )/ Normal distribution (Normal Distribution )
· Common probability distribution ( 6、 ... and ): An index distribution (Exponential Distribution )
· Common probability distribution ( 7、 ... and ): Laplacian distribution (Laplace Distribution )
· Common probability distribution ( 8、 ... and ): Dirac distribution (Dirac Distribution )
· Common probability distribution ( Nine ): Empirical distribution (Empirical Distribution )
· Common probability distribution ( Ten ): Beta distribution (Beta Distribution )
· Common probability distribution ( 11、 ... and ): Dirichlet distribution (Dirichlet Distribution )
· Common probability distribution ( Twelve ): Structured probability model / Graph model
· Common probability distribution ( 13、 ... and ): Logistic distribution (Logistic Distribution )
set up X X X It's a continuous random variable , X X X Obey the logistic distribution (Logistic Distribution ) Refer to X X X It has the following distribution function and density function :
F ( x ) = P ( X ≤ x ) = 1 1 + e − ( x − μ ) γ F(x)=P(X\leq x)=\frac{1}{1+e^{-\frac{(x-\mu)}{\gamma}}} F(x)=P(X≤x)=1+e−γ(x−μ)1
f ( x ) = F ′ ( x ) = e − ( x − μ ) γ γ ( 1 + e − ( x − μ ) γ ) 2 f(x)=F'(x)=\frac{e^{-\frac{(x-\mu)}{\gamma}}}{\gamma(1+e^{-\frac{(x-\mu)}{\gamma}})^2} f(x)=F′(x)=γ(1+e−γ(x−μ))2e−γ(x−μ)
among , μ \mu μ For position parameters , γ > 0 \gamma>0 γ>0 For shape parameters .
Density function of logistic distribution f ( x ) f(x) f(x) And distribution function F ( x ) F(x) F(x) The figure of is shown in the figure below . The distribution function belongs to the logical tearing function , The graph is a S Shape curve (sigmoid curve). The curve is in point ( μ , 1 2 ) (\mu,\frac{1}{2}) (μ,21) It's central symmetry , The meet
F ( − x + μ ) − 1 2 = − F ( x − μ ) + 1 2 F(-x+\mu)-\frac{1}{2}=-F(x-\mu)+\frac{1}{2} F(−x+μ)−21=−F(x−μ)+21
The curve grows faster near the center , Slow growth at both ends . shape parameter γ \gamma γ The smaller the value of , The faster the curve grows near the center .
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