如何用高斯 kernel 构造一个 kernel 密度估计？

Question

I want to construct an estimator of the unknown density f using the gaussian kernel here:

I've found that the kernel density estimator of f is:

I've tried to implement a code using these two formula but it's leading me nowhere:

n = 100
X = np.random.normal(0,1,n)
a = min(X)
b = max(X)
sigma = 1

Xplot = np.linspace(a,b, num = len(X))
ftrue = np.zeros((100,1))
ftrue = (np.exp(-(Xplot**2))/(2*(sigma**2)))/(np.sqrt(2*np.pi)*sigma)
#np.exp(-0.5*Xplot**2)/np.sqrt(2*np.pi)

def K (x,y,sigma) :
    return((np.exp(-((x-y)**2))/(2*(sigma**2)))/(np.sqrt(2*np.pi)*sigma))

fest = np.zeros((100,1))
fest2 = np.zeros((100,1))
mat = np.zeros((100,n))
KER = np.zeros((100,n))

for i in range(100):
    for j in range(n):
        U = (Xplot[i]-X[j])/sigma 
        mat[i,j] = np.exp(-0.5*U**2)/np.sqrt(2*np.pi)
        # KER[i,j] = KernelDensity(U, kernel="gaussian")
        KER[i,j] = (np.mean(K(Xplot[i],X[j],sigma))) / sigma # I'm not sure about that 

fest = mat.mean(axis=1) 
fest2 = KER.mean(axis=1)

plt.plot(Xplot,ftrue,'r')
plt.plot(Xplot,fest,'b')
plt.plot(Xplot,fest2,'g')

Do you have an idea about how to construct this kernel density estimator?

Thank you !

Answer 1

Shameless plug for my own library .

There are utility functions in here for kernel density estimation.

from local_models.local_models import GaussianKernel
from local_models.utils import kernel_density
import numpy as np
import matplotlib.pyplot as plt

X_train = np.random.normal(size=100).reshape(-1,1)
y_train = np.ones_like(X_train).flatten()
X_test = np.linspace(-4,4,1000).reshape(-1,1)

kernel = GaussianKernel(bandwidth = 0.1)
KDE = kernel_density(X_test, X_train, kernel)

plt.plot(X_test, KDE)
plt.scatter(X_train, y_train, c='r')
plt.show()

Answer 2

I tried this, is this correct? :

def kernel(x, y, sigma):
    return (1 / np.sqrt(2 * np.pi * sigma)) * np.exp(-(x - y)**2 / (2 * sigma**2))
 
n=100
X=np.random.normal(0,1,n)
a = min(X)
b = max(X)
Xplot = np.linspace(a,b, num = len(X))
sigma=[0.1,1,4]
t = len(X)
print(t)

ftrue = np.zeros((100,1))
ftrue = np.exp(-0.5*Xplot**2)/np.sqrt(2*np.pi)
plt.plot(Xplot,ftrue,'r')
plt.legend(loc='upper left')

y=[np.mean(kernel(X, Xplot[i], sigma[0]))/sigma[0] for i in range(t)]
plt.plot(Xplot,y,'b')
plt.legend(loc='upper left')

y=[np.mean(kernel(X, Xplot[i], sigma[1]))/sigma[1] for i in range(t)]
plt.plot(Xplot,y, 'orange')
plt.legend(loc='upper left')


y=[np.mean(kernel(X, Xplot[i], sigma[2]))/sigma[2] for i in range(t)]
plt.plot(Xplot,y, 'green' )
plt.legend(loc='upper left')