Logistic Function 逻辑函数 （sigmoid）

Logistic Function to Represent Transition

A logistic function (also called a sigmoid) is a non-linear function bounded between 0 and 1. As $t\to -\infty ,p(s|t)\to 0$ and, as $t\to +\infty ,p(s|t)\to 1$. The expression for a logistic probability distribution for sleep as a function of time is:

$$p(s|t)=\frac{1}{1+e^{\beta t}}$$

Several logistic functions with various $\beta$ parameters are shown below:

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from IPython.core.pylabtools import figsize

figsize(12, 6)

# Logistic function with only beta
def logistic(x, beta):
    return 1. / (1. + np.exp(beta * x))

# Plot examples with different betas 
x = np.linspace(-5, 5, 1000)
for beta in [-5, -1, 0.5, 1, 5]:
    plt.plot(x, logistic(x, beta), label = r"$\beta$ = %.1f" % beta)

plt.legend();
plt.title(r'Logistic Function with Different $\beta$ values');