Using scipy.stats.gaussian_kde with 2 dimensional data

Question

I'm trying to use the scipy.stats.gaussian_kde class to smooth out some discrete data collected with latitude and longitude information, so it shows up as somewhat similar to a contour map in the end, where the high densities are the peak and low densities are the valley.

I'm having a hard time putting a two-dimensional dataset into the gaussian_kde class. I've played around to figure out how it works with 1 dimensional data, so I thought 2 dimensional would be something along the lines of:

from scipy import stats
from numpy import array
data = array([[1.1, 1.1],
              [1.2, 1.2],
              [1.3, 1.3]])
kde = stats.gaussian_kde(data)
kde.evaluate([1,2,3],[1,2,3])

which is saying that I have 3 points at [1.1, 1.1], [1.2, 1.2], [1.3, 1.3]. and I want to have the kernel density estimation using from 1 to 3 using width of 1 on x and y axis.

When creating the gaussian_kde, it keeps giving me this error:

raise LinAlgError("singular matrix")
numpy.linalg.linalg.LinAlgError: singular matrix

Looking into the source code of gaussian_kde, I realize that the way I'm thinking about what dataset means is completely different from how the dimensionality is calculate, but I could not find any sample code showing how multi-dimension data works with the module. Could someone help me with some sample ways to use gaussian_kde with multi-dimensional data?

Answer 1

This example seems to be what you're looking for:

import numpy as np
import scipy.stats as stats
from matplotlib.pyplot import imshow

# Create some dummy data
rvs = np.append(stats.norm.rvs(loc=2,scale=1,size=(2000,1)),
                stats.norm.rvs(loc=0,scale=3,size=(2000,1)),
                axis=1)

kde = stats.kde.gaussian_kde(rvs.T)

# Regular grid to evaluate kde upon
x_flat = np.r_[rvs[:,0].min():rvs[:,0].max():128j]
y_flat = np.r_[rvs[:,1].min():rvs[:,1].max():128j]
x,y = np.meshgrid(x_flat,y_flat)
grid_coords = np.append(x.reshape(-1,1),y.reshape(-1,1),axis=1)

z = kde(grid_coords.T)
z = z.reshape(128,128)

imshow(z,aspect=x_flat.ptp()/y_flat.ptp())

Axes need fixing, obviously.

You can also do a scatter plot of the data with

scatter(rvs[:,0],rvs[:,1])

Answer 2

I think you are mixing up kernel density estimation with interpolation or maybe kernel regression. KDE estimates the distribution of points if you have a larger sample of points.

I'm not sure which interpolation you want, but either the splines or rbf in scipy.interpolate will be more appropriate.

If you want one-dimensional kernel regression, then you can find a version in scikits.statsmodels with several different kernels.

update: here is an example (if this is what you want)

>>> data = 2 + 2*np.random.randn(2, 100)
>>> kde = stats.gaussian_kde(data)
>>> kde.evaluate(np.array([[1,2,3],[1,2,3]]))
array([ 0.02573917,  0.02470436,  0.03084282])

gaussian_kde has variables in rows and observations in columns, so reversed orientation from the usual in stats. In your example, all three points are on a line, so it has perfect correlation. That is, I guess, the reason for the singular matrix.

Adjusting the array orientation and adding a small noise, the example works, but still looks very concentrated, for example you don't have any sample point near (3,3):

>>> data = np.array([[1.1, 1.1],
              [1.2, 1.2],
              [1.3, 1.3]]).T
>>> data = data + 0.01*np.random.randn(2,3)
>>> kde = stats.gaussian_kde(data)
>>> kde.evaluate(np.array([[1,2,3],[1,2,3]]))
array([  7.70204299e+000,   1.96813149e-044,   1.45796523e-251])

Answer 3

I found it difficult to understand the SciPy manual's description of how gaussian_kde works with 2D data. Here is an explanation which is intended to complement @endolith 's example. I divided the code into several steps with comments to explain the less intuitive bits.

First, the imports:

import numpy as np
import scipy.stats as st
from matplotlib.pyplot import imshow, show

Create some dummy data: these are 1-D arrays of the "X" and "Y" point coordinates.

np.random.seed(142)  # for reproducibility
x = st.norm.rvs(loc=2, scale=1, size=2000)
y = st.norm.rvs(loc=0, scale=3, size=2000)

For 2-D density estimation the gaussian_kde object has to be initialised with an array with two rows containing the "X" and "Y" datasets. In NumPy terminology, we "stack them vertically":

xy = np.vstack((x, y))

so the "X" data is in the first row xy[0,:] and the "Y" data are in the second row xy[1,:] and xy.shape is (2, 2000). Now create the gaussian_kde object:

dens = st.gaussian_kde(xy)

We will evaluate the estimated 2-D density PDF on a 2-D grid. There is more than one way of creating such a grid in NumPy. I show here an approach which is different from (but functionally equivalent to) @endolith 's method:

gx, gy = np.mgrid[x.min():x.max():128j, y.min():y.max():128j]
gxy = np.dstack((gx, gy)) # shape is (128, 128, 2)

gxy is a 3-D array, the [i,j]-th element of gxy contains a 2-element list of the corresponding "X" and "Y" values: gxy[i, j] 's value is [ gx[i], gy[j] ].

We have to invoke dens() (or dens.pdf() which is the same thing) on each of the 2-D grid points. NumPy has a very elegant function for this purpose:

z = np.apply_along_axis(dens, 2, gxy)

In words, the callable dens (could have been dens.pdf as well) is invoked along axis=2 (the third axis) in the 3-D array gxy and the values should be returned as a 2-D array. The only glitch is that the shape of z will be (128,128,1) and not (128,128) what I expected. Note that the documentation says that:

The shape of out [the return value, L.D.] is identical to the shape of arr, except along the axis dimension. This axis is removed, and replaced with new dimensions equal to the shape of the return value of func1d. So if func1d returns a scalar out will have one fewer dimensions than arr.

Most likely dens() returned a 1-long tuple and not a scalar which I was hoping for. I didn't investigate the issue any further, because this is easy to fix:

z = z.reshape(128, 128)

after which we can generate the image:

imshow(z, aspect=gx.ptp() / gy.ptp())
show()  # needed if you try this in PyCharm

Here is the image. (Note that I have implemented @endolith 's version as well and got an image indistinguishable from this one.)

Answer 4

The example posted in the top answer didn't work for me. I had to tweak it little bit and it works now:

import numpy as np
import scipy.stats as stats
from matplotlib import pyplot as plt

# Create some dummy data
rvs = np.append(stats.norm.rvs(loc=2,scale=1,size=(2000,1)),
                stats.norm.rvs(loc=0,scale=3,size=(2000,1)),
                axis=1)

kde = stats.kde.gaussian_kde(rvs.T)

# Regular grid to evaluate kde upon
x_flat = np.r_[rvs[:,0].min():rvs[:,0].max():128j]
y_flat = np.r_[rvs[:,1].min():rvs[:,1].max():128j]
x,y = np.meshgrid(x_flat,y_flat)
grid_coords = np.append(x.reshape(-1,1),y.reshape(-1,1),axis=1)

z = kde(grid_coords.T)
z = z.reshape(128,128)

plt.imshow(z,aspect=x_flat.ptp()/y_flat.ptp())
plt.show()