Python code for computing fused density estimators (FDEs). Fused density estimation is a computationally tractable method for nonparametric density estimation from univariate and geometric network data. Further details on fused density estimation can be found here. This package also contains a number of auxiliary functions for importing and building geometric networks from OpenStreetMaps (OSM) XML files.
| FDE on Geometric Network | Univariate FDE |
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Update 10/08/22: Big thanks to Yu Miao, a statistics graduate student at UCLA, for updating the original Python 2 code to Python 3!
- Clone this repository to a local directory.
- Install OSQP, and (optionally, for interior-point capabilities) CVXOPT
pip install OSQP CVXOPT
Univariate FDEs can be declared with the syntax
fde = UnivarFDE((a,b), P)
where P is a collection of 1-D observations between a and b. For a penalty parameter lambda, the fde can be generated and solved via
fde.GenerateProblem()
fde.SolveProblem(lam)
Lastly, one can plot the fde with
fde.Plot()
plt.show()
Lastly, one can also perform cross-validation on the penalty parameter with
fde.CrossValidate()
which returns a float.
Geometric network FDEs can be declared with the syntax
fde = FDE(L,P)
L is an array with elements as the segments of the geometric network. A segment is an array of 2D points. By default, these are assumed to be of the form (longitude, latitude) and segment distances are calculated from these coordinates. As an example of this syntax,
L = [[[x1, y1], [x2, y2], [x3, y3]], [[x4, y4], [x2, y2], [x5, y5]]]
This example is a geometric network with two segments. The segments intersect at [x2, y2]. P is an array of points from L, so that
P = [[x1, y1], [x3, y3], [x4, y4], [x5, y5]]
places observations at the endpoints of the above network. Generating, solving, plotting, and finding a lambda-parameter via cross-validation is all done as in the univariate case.
fde.GenerateProblem()
lam = fde.CrossValidate()
fde.SolveProblem(lam)
fde.Plot()
plt.show()
In this example, we will estimate a normal density with 100 data points.
import sys
sys.path.append("../../FDE-Tools") #put package in path
from FDE import *
import scipy.stats as stats
P = stats.norm.rvs(size = 100) #generate data
(a,b) = (-4,4)
fde = UnivarFDE((a,b), P) #Declare and solve FDE
fde.GenerateProblem()
fde.SolveProblem(.030)
fde.Plot()
x = np.linspace(-4,4,100)
plt.plot(x, stats.norm.pdf(x), color = 'darkorange')
plt.show()
#Perform cross-validation
print("Performing Cross Validation...")
lam = fde.CrossValidate()
fde.SolveProblem(lam)
print("Optimal lambda parameter: " + str(lam))
plt.plot(x, stats.norm.pdf(x), color = 'darkorange')
fde.Plot()
plt.show()
In this example we will load a geometric network from an OSM XML file ("SanDiego.xml"). We will also extract observations from the XML file as the locations of eateries in the downloaded region.
import sys
sys.path.append("../../FDE-Tools") #put package in path
from FDE import *
from HelperFunctions import * #HelperFunctions includes functions for importing OSM data
#A window to focus on
lon = (-117.7, -117.149)
lat = (32.7071, 32.7216)
#Load data from xml, filter it to desired window
print("Loading data from file")
L = LoadMapXML("SanDiego.xml")
L = [FilterData(l, lon, lat) for l in L]
P = GetEateries("SanDiego.xml")
P = FilterData(P, lon, lat)
#This will likely take a while to load. Save L & P with numpy.save for quick reloading.
#Declare and solve the FDE
print("Declaring fused density estimator")
fde = FDE(L,P)
print("Generating Problem")
fde.GenerateProblem()
print("Solving Problem")
fde.SolveProblem(.022)
fde.Plot()
plt.show()
