I’m working on kernel density estimation (KDE) using python for road accidents data, but here I will use 1d data just for illustration. When fitting the KDE model, I’ve noticed that I get different results when considering all data points versus restricting to those within the bandwidth. Specifically,when using only points within the bandwidth I get almost the same values of density both in areas with few points and areas with many point which to me seems counterintuitive. I don’t know how to interpret this or if it is some code problem.

def KDE(x_list,radius):
    return (1/(len(x_list)*radius))*np.sum([K(x/radius) for x in x_list])

def kde_val(x,dati):
    return K((x-xi))

dataset = np.array([10,11,10,55,56,57,58,59])

x_range = np.linspace(dataset.min()-0.3, dataset.max()+0.3, num=600)

# bandwith values for experimentation
H = [30, 40, 50,30, 40, 50]
n_samples = dataset.size

# line properties for different bandwith values
color_list = ["brown","black","yellow","blue","red","green"]
alpha_list = [0.8, 1, 0.8,0.8, 1, 0.8]
width_list = [1.7,2.5,1.7,1.7,2.5,1.7]

plt.figure(figsize=(10,4))
# iterate over bandwith values
i=0
for h, color, alpha, width in zip(H, color_list, alpha_list, width_list):
    i+=1
    
    # iterate over datapoints
    y_range=[]
    for x in x_range:
        a=x-dataset
        b=a[abs(a) <= h] #consider only datapoints within the bandwidth h
        if i>3: #when i>3 I consider all datapoints
            b=x-dataset
        y_range.append(KDE(b,h))
    y_range=np.array(y_range)
    
    plt.plot(x_range, y_range, 
             color=color, alpha=alpha, linewidth=width, 
             label=f'{h}')

    plt.plot(dataset, np.zeros_like(dataset) , 's', 
             markersize=8, color='black')
    

KDE

I couldn’t find any libraries that ignote datapoints outside the bandwidth to compare the results.