SciPy KDE渐变 [英] SciPy KDE gradient

问题描述

我正在使用内核密度估计(KDE)的SciPy实现( http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gaussian_kde.html )，到目前为止，它的运行情况还不错.但是，我现在想获取特定点的KDE梯度.

I am using the SciPy implementation of the kernel density estimate (KDE) (http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gaussian_kde.html), which is working fine so far. However, I would now like to obtain the gradient of the KDE at a particular point.

我看过了该库的Python源代码，但是还没有弄清楚如何轻松实现此功能.有人知道这样做的方法吗?

I have looked at the Python source for the library, but haven't been able to figure out how to easily implement this functionality. Is anybody aware of a method to do this?

推荐答案

如果查看所引用的源，您会发现密度估计是根据数据集中所有点的贡献构建的.假设只有一个点points[:,i]您现在要评估(第219-222行):

If you look at the source you referenced, you'll see that the density estimation is constructed from contributions from all points in the dataset Assuming there's only one point points[:,i] you want to evaluate for the moment (lines 219–222):

diff = self.dataset - points[:, i, newaxis]
tdiff = dot(self.inv_cov, diff)
energy = sum(diff * tdiff, axis=0) / 2.0
result[i] = sum(exp(-energy), axis=0)

对于矩阵表示法(没有可用的LaTeX?)，将对数据集中的单个点D和要评估为的点p进行写

In matrix notation (no LaTeX available?), this would be written, for a single point D from the dataset and point p to be evaluated as

d = D - p
t = Cov^-1 d
e = 1/2 d^T t
r = exp(-e)

您要寻找的渐变是grad(r) = (dr/dx, dr/dy):

dr/dx = d(exp(-e))/dx 
      = -de/dx exp(-e)
      = -d(1/2 d^T Cov^-1 d)/dx exp(-e)
      = -(Cov^-1 d) exp(-e)

与dr/dy类似.因此，您要做的就是计算项Cov^-1 d并将其与您已经获得的结果相乘.

Likewise for dr/dy. Hence all you need to do is calculate the term Cov^-1 d and multiply it with the result you already obtained.

result = zeros((self.d,m), dtype=float)    
[...]
diff = self.dataset - points[:, i, newaxis]
tdiff = dot(self.inv_cov, diff)
energy = sum(diff * tdiff, axis=0) / 2.0
grad = dot(self.inv_cov, diff)
result[:,i] = sum(grad * exp(-energy), axis=1)

出于某种原因，我需要在计算grad时放下-1以获得与评估在所有四个方向上在p和p+delta处的密度估计值一致的结果，这当然是我可能会想到的迹象离开这里.

For some reason I needed to drop the -1 when calculating grad to obtain results coherent with a evaluating the density estimation at p and p+delta in all four directions, which is a sign I might of course be way off here.

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