Neat-o Mathematics

kde.py

import numpy as np

# Example kernels
def uniform_kernel(x):
    norm_coeff = .5
    return .5*np.bitwise_and(-1<x,1>x)
  
def normal_kernel(x):
    norm_coeff = (2*np.pi)**-.5
    return .5*np.exp(-.5*x**2)
  

def KDE(input_samples,kernel,bandwidth=1.0):
    """
    Takes in an array of samples, a kernel func (e.g. unigorm_kernel,normal_kernel),
    and an optional "bandwidth" smoothing parameter
    
    """
    def f(x):
        return 1./(bandwidth*len(input_samples))*kernel(1./bandwidth*(x-input_samples)).sum()
    return np.vectorize(f)
  
###
# Example:
###
# import matplotlib.pyplot as plt
# samples = np.hstack([np.random.normal(0,1,100),np.random.normal(3,.5,100)])
# domain = np.linspace(-5,10,1500)
# f,ax = plt.subplots(2)
# ax[0].hist(samples,normed=True)
# ax[0].plot(domain,KDE(samples,normal_kernel)(domain))
# ax[0].set_title('normal kernel')
# ax[1].hist(samples,normed=True)
# ax[1].plot(domain,KDE(samples,uniform_kernel)(domain))
# ax[1].set_title('uniform kernel')

logmap.py

def logmap(n,x0,r=4):
    _logmapcache={}
    def _logmap(n):
        if n==0:
            return x0
        xprev=_logmapcache[n] if n in _logmapcache else _logmap(n-1)
        return r*xprev*(1-xprev)
    return _logmap(n)