"""
Univariate Kernel Density Estimators
References
----------
Racine, Jeff. (2008) "Nonparametric Econometrics: A Primer," Foundation and
Trends in Econometrics: Vol 3: No 1, pp1-88.
http://dx.doi.org/10.1561/0800000009
https://en.wikipedia.org/wiki/Kernel_%28statistics%29
Silverman, B.W. Density Estimation for Statistics and Data Analysis.
"""
import numpy as np
from scipy import integrate, stats
from statsmodels.sandbox.nonparametric import kernels
from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.validation import array_like, float_like
from . import bandwidths
from .kdetools import forrt, revrt, silverman_transform
from .linbin import fast_linbin
# Kernels Switch for estimators
kernel_switch = dict(
gau=kernels.Gaussian,
epa=kernels.Epanechnikov,
uni=kernels.Uniform,
tri=kernels.Triangular,
biw=kernels.Biweight,
triw=kernels.Triweight,
cos=kernels.Cosine,
cos2=kernels.Cosine2,
tric=kernels.Tricube
)
def _checkisfit(self):
try:
self.density
except Exception:
raise ValueError("Call fit to fit the density first")
# Kernel Density Estimator Class
[docs]class KDEUnivariate(object):
"""
Univariate Kernel Density Estimator.
Parameters
----------
endog : array_like
The variable for which the density estimate is desired.
Notes
-----
If cdf, sf, cumhazard, or entropy are computed, they are computed based on
the definition of the kernel rather than the FFT approximation, even if
the density is fit with FFT = True.
`KDEUnivariate` is much faster than `KDEMultivariate`, due to its FFT-based
implementation. It should be preferred for univariate, continuous data.
`KDEMultivariate` also supports mixed data.
See Also
--------
KDEMultivariate
kdensity, kdensityfft
Examples
--------
>>> import statsmodels.api as sm
>>> import matplotlib.pyplot as plt
>>> nobs = 300
>>> np.random.seed(1234) # Seed random generator
>>> dens = sm.nonparametric.KDEUnivariate(np.random.normal(size=nobs))
>>> dens.fit()
>>> plt.plot(dens.cdf)
>>> plt.show()
"""
def __init__(self, endog):
self.endog = array_like(endog, "endog", ndim=1, contiguous=True)
[docs] def fit(
self,
kernel="gau",
bw="normal_reference",
fft=True,
weights=None,
gridsize=None,
adjust=1,
cut=3,
clip=(-np.inf, np.inf),
):
"""
Attach the density estimate to the KDEUnivariate class.
Parameters
----------
kernel : str
The Kernel to be used. Choices are:
- "biw" for biweight
- "cos" for cosine
- "epa" for Epanechnikov
- "gau" for Gaussian.
- "tri" for triangular
- "triw" for triweight
- "uni" for uniform
bw : str, float, callable
The bandwidth to use. Choices are:
- "scott" - 1.059 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "silverman" - .9 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "normal_reference" - C * A * nobs ** (-1/5.), where C is
calculated from the kernel. Equivalent (up to 2 dp) to the
"scott" bandwidth for gaussian kernels. See bandwidths.py
- If a float is given, its value is used as the bandwidth.
- If a callable is given, it's return value is used.
The callable should take exactly two parameters, i.e.,
fn(x, kern), and return a float, where:
* x - the clipped input data
* kern - the kernel instance used
fft : bool
Whether or not to use FFT. FFT implementation is more
computationally efficient. However, only the Gaussian kernel
is implemented. If FFT is False, then a 'nobs' x 'gridsize'
intermediate array is created.
gridsize : int
If gridsize is None, max(len(x), 50) is used.
cut : float
Defines the length of the grid past the lowest and highest values
of x so that the kernel goes to zero. The end points are
-/+ cut*bw*{min(x) or max(x)}
adjust : float
An adjustment factor for the bw. Bandwidth becomes bw * adjust.
Returns
-------
KDEUnivariate
The instance fit,
"""
if isinstance(bw, str):
self.bw_method = bw
else:
self.bw_method = "user-given"
if not callable(bw):
bw = float_like(bw, "bw")
endog = self.endog
if fft:
if kernel != "gau":
msg = "Only gaussian kernel is available for fft"
raise NotImplementedError(msg)
if weights is not None:
msg = "Weights are not implemented for fft"
raise NotImplementedError(msg)
density, grid, bw = kdensityfft(
endog,
kernel=kernel,
bw=bw,
adjust=adjust,
weights=weights,
gridsize=gridsize,
clip=clip,
cut=cut,
)
else:
density, grid, bw = kdensity(
endog,
kernel=kernel,
bw=bw,
adjust=adjust,
weights=weights,
gridsize=gridsize,
clip=clip,
cut=cut,
)
self.density = density
self.support = grid
self.bw = bw
self.kernel = kernel_switch[kernel](h=bw) # we instantiate twice,
# should this passed to funcs?
# put here to ensure empty cache after re-fit with new options
self.kernel.weights = weights
if weights is not None:
self.kernel.weights /= weights.sum()
self._cache = {}
return self
@cache_readonly
def cdf(self):
"""
Returns the cumulative distribution function evaluated at the support.
Notes
-----
Will not work if fit has not been called.
"""
_checkisfit(self)
kern = self.kernel
if kern.domain is None: # TODO: test for grid point at domain bound
a, b = -np.inf, np.inf
else:
a, b = kern.domain
def func(x, s):
return kern.density(s, x)
support = self.support
support = np.r_[a, support]
gridsize = len(support)
endog = self.endog
probs = [
integrate.quad(func, support[i - 1], support[i], args=endog)[0]
for i in range(1, gridsize)
]
return np.cumsum(probs)
@cache_readonly
def cumhazard(self):
"""
Returns the hazard function evaluated at the support.
Notes
-----
Will not work if fit has not been called.
"""
_checkisfit(self)
return -np.log(self.sf)
@cache_readonly
def sf(self):
"""
Returns the survival function evaluated at the support.
Notes
-----
Will not work if fit has not been called.
"""
_checkisfit(self)
return 1 - self.cdf
@cache_readonly
def entropy(self):
"""
Returns the differential entropy evaluated at the support
Notes
-----
Will not work if fit has not been called. 1e-12 is added to each
probability to ensure that log(0) is not called.
"""
_checkisfit(self)
def entr(x, s):
pdf = kern.density(s, x)
return pdf * np.log(pdf + 1e-12)
kern = self.kernel
if kern.domain is not None:
a, b = self.domain
else:
a, b = -np.inf, np.inf
endog = self.endog
# TODO: below could run into integr problems, cf. stats.dist._entropy
return -integrate.quad(entr, a, b, args=(endog,))[0]
@cache_readonly
def icdf(self):
"""
Inverse Cumulative Distribution (Quantile) Function
Notes
-----
Will not work if fit has not been called. Uses
`scipy.stats.mstats.mquantiles`.
"""
_checkisfit(self)
gridsize = len(self.density)
return stats.mstats.mquantiles(self.endog, np.linspace(0, 1, gridsize))
[docs] def evaluate(self, point):
"""
Evaluate density at a point or points.
Parameters
----------
point : {float, ndarray}
Point(s) at which to evaluate the density.
"""
_checkisfit(self)
return self.kernel.density(self.endog, point)
# Kernel Density Estimator Functions
def kdensity(
x,
kernel="gau",
bw="normal_reference",
weights=None,
gridsize=None,
adjust=1,
clip=(-np.inf, np.inf),
cut=3,
retgrid=True,
):
"""
Rosenblatt-Parzen univariate kernel density estimator.
Parameters
----------
x : array_like
The variable for which the density estimate is desired.
kernel : str
The Kernel to be used. Choices are
- "biw" for biweight
- "cos" for cosine
- "epa" for Epanechnikov
- "gau" for Gaussian.
- "tri" for triangular
- "triw" for triweight
- "uni" for uniform
bw : str, float, callable
The bandwidth to use. Choices are:
- "scott" - 1.059 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "silverman" - .9 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "normal_reference" - C * A * nobs ** (-1/5.), where C is
calculated from the kernel. Equivalent (up to 2 dp) to the
"scott" bandwidth for gaussian kernels. See bandwidths.py
- If a float is given, its value is used as the bandwidth.
- If a callable is given, it's return value is used.
The callable should take exactly two parameters, i.e.,
fn(x, kern), and return a float, where:
* x - the clipped input data
* kern - the kernel instance used
weights : array or None
Optional weights. If the x value is clipped, then this weight is
also dropped.
gridsize : int
If gridsize is None, max(len(x), 50) is used.
adjust : float
An adjustment factor for the bw. Bandwidth becomes bw * adjust.
clip : tuple
Observations in x that are outside of the range given by clip are
dropped. The number of observations in x is then shortened.
cut : float
Defines the length of the grid past the lowest and highest values of x
so that the kernel goes to zero. The end points are
-/+ cut*bw*{min(x) or max(x)}
retgrid : bool
Whether or not to return the grid over which the density is estimated.
Returns
-------
density : ndarray
The densities estimated at the grid points.
grid : ndarray, optional
The grid points at which the density is estimated.
Notes
-----
Creates an intermediate (`gridsize` x `nobs`) array. Use FFT for a more
computationally efficient version.
"""
x = np.asarray(x)
if x.ndim == 1:
x = x[:, None]
clip_x = np.logical_and(x > clip[0], x < clip[1])
x = x[clip_x]
nobs = len(x) # after trim
if gridsize is None:
gridsize = max(nobs, 50) # do not need to resize if no FFT
# handle weights
if weights is None:
weights = np.ones(nobs)
q = nobs
else:
# ensure weights is a numpy array
weights = np.asarray(weights)
if len(weights) != len(clip_x):
msg = "The length of the weights must be the same as the given x."
raise ValueError(msg)
weights = weights[clip_x.squeeze()]
q = weights.sum()
# Get kernel object corresponding to selection
kern = kernel_switch[kernel]()
if callable(bw):
bw = float(bw(x, kern))
# user passed a callable custom bandwidth function
elif isinstance(bw, str):
bw = bandwidths.select_bandwidth(x, bw, kern)
# will cross-val fit this pattern?
else:
bw = float_like(bw, "bw")
bw *= adjust
a = np.min(x, axis=0) - cut * bw
b = np.max(x, axis=0) + cut * bw
grid = np.linspace(a, b, gridsize)
k = (
x.T - grid[:, None]
) / bw # uses broadcasting to make a gridsize x nobs
# set kernel bandwidth
kern.seth(bw)
# truncate to domain
if (
kern.domain is not None
): # will not work for piecewise kernels like parzen
z_lo, z_high = kern.domain
domain_mask = (k < z_lo) | (k > z_high)
k = kern(k) # estimate density
k[domain_mask] = 0
else:
k = kern(k) # estimate density
k[k < 0] = 0 # get rid of any negative values, do we need this?
dens = np.dot(k, weights) / (q * bw)
if retgrid:
return dens, grid, bw
else:
return dens, bw
def kdensityfft(
x,
kernel="gau",
bw="normal_reference",
weights=None,
gridsize=None,
adjust=1,
clip=(-np.inf, np.inf),
cut=3,
retgrid=True,
):
"""
Rosenblatt-Parzen univariate kernel density estimator
Parameters
----------
x : array_like
The variable for which the density estimate is desired.
kernel : str
ONLY GAUSSIAN IS CURRENTLY IMPLEMENTED.
"bi" for biweight
"cos" for cosine
"epa" for Epanechnikov, default
"epa2" for alternative Epanechnikov
"gau" for Gaussian.
"par" for Parzen
"rect" for rectangular
"tri" for triangular
bw : str, float, callable
The bandwidth to use. Choices are:
- "scott" - 1.059 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "silverman" - .9 * A * nobs ** (-1/5.), where A is
`min(std(x),IQR/1.34)`
- "normal_reference" - C * A * nobs ** (-1/5.), where C is
calculated from the kernel. Equivalent (up to 2 dp) to the
"scott" bandwidth for gaussian kernels. See bandwidths.py
- If a float is given, its value is used as the bandwidth.
- If a callable is given, it's return value is used.
The callable should take exactly two parameters, i.e.,
fn(x, kern), and return a float, where:
* x - the clipped input data
* kern - the kernel instance used
weights : array or None
WEIGHTS ARE NOT CURRENTLY IMPLEMENTED.
Optional weights. If the x value is clipped, then this weight is
also dropped.
gridsize : int
If gridsize is None, min(len(x), 512) is used. Note that the provided
number is rounded up to the next highest power of 2.
adjust : float
An adjustment factor for the bw. Bandwidth becomes bw * adjust.
clip : tuple
Observations in x that are outside of the range given by clip are
dropped. The number of observations in x is then shortened.
cut : float
Defines the length of the grid past the lowest and highest values of x
so that the kernel goes to zero. The end points are
-/+ cut*bw*{x.min() or x.max()}
retgrid : bool
Whether or not to return the grid over which the density is estimated.
Returns
-------
density : ndarray
The densities estimated at the grid points.
grid : ndarray, optional
The grid points at which the density is estimated.
Notes
-----
Only the default kernel is implemented. Weights are not implemented yet.
This follows Silverman (1982) with changes suggested by Jones and Lotwick
(1984). However, the discretization step is replaced by linear binning
of Fan and Marron (1994). This should be extended to accept the parts
that are dependent only on the data to speed things up for
cross-validation.
References
----------
Fan, J. and J.S. Marron. (1994) `Fast implementations of nonparametric
curve estimators`. Journal of Computational and Graphical Statistics.
3.1, 35-56.
Jones, M.C. and H.W. Lotwick. (1984) `Remark AS R50: A Remark on Algorithm
AS 176. Kernal Density Estimation Using the Fast Fourier Transform`.
Journal of the Royal Statistical Society. Series C. 33.1, 120-2.
Silverman, B.W. (1982) `Algorithm AS 176. Kernel density estimation using
the Fast Fourier Transform. Journal of the Royal Statistical Society.
Series C. 31.2, 93-9.
"""
x = np.asarray(x)
# will not work for two columns.
x = x[np.logical_and(x > clip[0], x < clip[1])]
# Get kernel object corresponding to selection
kern = kernel_switch[kernel]()
if callable(bw):
bw = float(bw(x, kern))
# user passed a callable custom bandwidth function
elif isinstance(bw, str):
# if bw is None, select optimal bandwidth for kernel
bw = bandwidths.select_bandwidth(x, bw, kern)
# will cross-val fit this pattern?
else:
bw = float_like(bw, "bw")
bw *= adjust
nobs = len(x) # after trim
# 1 Make grid and discretize the data
if gridsize is None:
gridsize = np.max((nobs, 512.0))
gridsize = 2 ** np.ceil(np.log2(gridsize)) # round to next power of 2
a = np.min(x) - cut * bw
b = np.max(x) + cut * bw
grid, delta = np.linspace(a, b, int(gridsize), retstep=True)
RANGE = b - a
# TODO: Fix this?
# This is the Silverman binning function, but I believe it's buggy (SS)
# weighting according to Silverman
# count = counts(x,grid)
# binned = np.zeros_like(grid) #xi_{k} in Silverman
# j = 0
# for k in range(int(gridsize-1)):
# if count[k]>0: # there are points of x in the grid here
# Xingrid = x[j:j+count[k]] # get all these points
# # get weights at grid[k],grid[k+1]
# binned[k] += np.sum(grid[k+1]-Xingrid)
# binned[k+1] += np.sum(Xingrid-grid[k])
# j += count[k]
# binned /= (nobs)*delta**2 # normalize binned to sum to 1/delta
# NOTE: THE ABOVE IS WRONG, JUST TRY WITH LINEAR BINNING
binned = fast_linbin(x, a, b, gridsize) / (delta * nobs)
# step 2 compute FFT of the weights, using Munro (1976) FFT convention
y = forrt(binned)
# step 3 and 4 for optimal bw compute zstar and the density estimate f
# do not have to redo the above if just changing bw, ie., for cross val
# NOTE: silverman_transform is the closed form solution of the FFT of the
# gaussian kernel. Not yet sure how to generalize it.
zstar = silverman_transform(bw, gridsize, RANGE) * y
# 3.49 in Silverman
# 3.50 w Gaussian kernel
f = revrt(zstar)
if retgrid:
return f, grid, bw
else:
return f, bw