Solved – Which kernel function for Watson Nadaraya classifier

I am trying to implement a Watson Nadaraya classifier. There is one thing I didn't understand from the equation:

$${F}(x)=frac{sum_{i=1}^n K_h(x-X_i) Y_i}{sum_{i=1}^nK_h(x-X_i)}$$

What should I use for the kernel K?

I have a 2-dimensional dataset which has 1000 samples (each sample is like this: [-0.10984628, 5.53485135]).

What confuses me is, based on my data, the input of the kernel function will be something like this:

K([-0.62978309,  0.10464536]) 

And what I understand, it'll produce some number instead of an array, therefore I can go ahead and calculate F(x) which will also be a number. Then I'll check whether it is > or <= than zero. But I couldn't find any kernel that produces a number. So confused.

Edit: I tried to implement my classifier based on the comments, but I got a very low accuracy. I appreciate if someone notices what's wrong with it.

def gauss(x):         return (1.0 / np.sqrt(2 * np.pi)) * np.exp(- 0.5 * x**2)  def transform(X, h):         A = []         for i in X:                 A.append(stats.norm.pdf(i[0],0,h)*stats.norm.pdf(i[1],0,h))         return A       N = 100     # pre-assign some mean and variance     mean1 = (0,9)     mean2 = (0,5)     cov = [[0.3,0.7],[0.7,0.3]]      # generate a dataset     dataset1 = np.random.multivariate_normal(mean1,cov,N)     dataset2 = np.random.multivariate_normal(mean2,cov,N)     X = np.vstack((dataset1, dataset2))      # pre-assign labels     Y1 = [1]*N     Y2 = [-1]*N     Y = Y1 + Y2     # assing a width     h = 0.5      #now, transform the data     X2 = transform(X, h)      j = 0     predicted = []      for i in X2:             # apply the equation             fx = sum((gauss(i-X2))*Y)/float(np.sum(gauss(i-X2)))             # if fx>0, it belongs to class 1             if fx >0:                     predicted.append(1)             else:                     predicted.append(-1)             j = j+1 

You could take $K_h$ to be the density function for a bi-variate Gaussian distribution, with mean $x$, covariance matrix $hI$, and evaluated at $X_i$…

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